Distributions: Generalized Functions with Applications in Sobolev Spaces 9783110269291, 9783110269277

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De Gruyter Textbook Bhattacharyya • Distributions

Pulin Kumar Bhattacharyya

Distributions Generalized Functions with Applications in Sobolev Spaces

De Gruyter

Mathematics Subject Classification 2010: 46FXX , 46F10, 46F12, 35E05, 46E35, 46XX, 46-01, 35-01, 35J40.

ISBN: 978-3-11-026927-7 e-ISBN: 978-3-11-026929-1 Library of Congress Cataloging-in-Publication Data Bhattacharyya, Pulin K. Distributions : generalized functions with applications in Sobolev spaces / by Pulin K.  Bhattacharyya. p. cm. – (De Gruyter textbook) Includes bibliographical references and index. ISBN 978-3-11-026927-7 (hardcover : alk. paper) – ISBN 978-3-11-026929-1 (e-book)  1. Theory of distributions (Functional analysis)–Textbooks.  2. Sobolev spaces–Textbooks.  I. Title. QA324.B46 2012 515′.782–dc23 2011042975

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen  Printed on acid-free paper Printed in Germany www.degruyter.com

To my daughter Marina .Tonushree/

Preface

The term distribution1 was introduced by the celebrated French mathematician Laurent Schwartz in his Theory of Distributions (Théorie des distributions), which was developed in the late 1940s to denote new mathematical objects such as the popularly (though incorrectly) named Dirac delta function ı.x/ and its derivatives ı .k/ .x/, and idealized concepts such as the density of mass/charge at a point, the magnitude of instantaneous force applied at a point, the excitation caused by an instantaneous source of heat placed at a point, etc. The definitions of these new mathematical objects, which do not have any point-values and cannot be represented by the usual functions having point-values, were intended to represent some kind of physical distribution or spread of mass, charge, force, etc. over an interval on R, an area in R2 , a volume in R3 , etc. Theory of Distributions provides rigorous mathematical foundations for these new mathematical objects and also generalizes in some sense the notion of functions from classical analysis which have point-values. Hence, (Schwartz) distributions are also called generalized functions—specifically by the Russian school of Gelfand–Schilov–Vilenkin–Graev (see [1]) and also by many other non-Russian mathematicians—though Courant [2] prefers to call distributions ideal functions. In our treatment, although distributions, generalized functions and ideal functions are all synonyms, we find the term distributions more appropriate and more exact from the physical point of view, and will therefore use it in the remainder of the present book (including the title). Owing to the nice properties of distributions, Theory of Distributions found early favour with physicists, and has had a profound influence on the development of topological vector spaces, nuclear spaces, etc., becoming an integral part of modern functional analysis. But its impact on mathematical physics is the most profound. Consequently, we have considered Sobolev spaces as the most important application in general, being essential tools for boundary value problems of elliptic partial differential equations. There are several very good books on functional analysis (for example, Rudin [3] and Yoshida [4]), on partial differential equations (for example, Hörmander [5]) and on mathematical physics (for example, Vladimirov [6]), which contain a chapter or two or more on distributions, but the prohibitively brief and concise treatment of the topics, possibly combined with the necessarily high mathematical level of their presentation, make them not easily understandable for applied scientists. The solitary exception to this is [7] by Laurent Schwartz himself (the 1966 translation of the original, Mathématiques pour les sciences physiques), which contains an excep1 Distributions must not be confused with probability or statistical distributions, since these are completely different objects (see also the last part of Section 1.11).

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tionally well-written chapter on distributions giving almost all the basic results. It does not, however, contain vector-valued distributions, and Sobolev spaces are not even mentioned, possibly due to historical reasons. Almost all good books on distributions/generalized functions were published over four decades ago, were written by mathematicians for mathematicians and were primarily addressed at pure mathematicians to present distributions as new mathematical objects. The best of these is probably the monograph [8], Théorie des distributions by Laurent Schwartz himself, which has only ever been published in French, the other one being the monumental work of Gelfand–Schilov–Vilenkin–Graev in five volumes, Generalized Functions [1], which contains diverse applications of distributions to different branches of higher mathematics. Two additional important books are Topological Vector Spaces and Distributions by J. Horvath [9] and Topological Vector Spaces, Distributions and Kernels by F. Trèves [10], which give further developments of the theory of distributions, dealing particularly with linear topological vector space aspects, and are therefore suitable for researchers in the theory of distributions. Linear Partial Differential Equations by L. Hörmander [5] gives almost all the basic results on distributions. This is probably the most elegant, concise presentation of the results of the theory of distributions, but also probably in the most difficult style for an applied scientist. Generalized Functions and Partial Differential Equations by A. Friedman [11] is specially oriented to the study of partial differential equations in the distributional sense and a highly specialized book. There are also many interesting books which discuss the theory of distributions but which are not available in English, and thus will not be discussed here. Although Sobolev Spaces by R. A. Adams [12] is possibly one of the best reference books on Sobolev spaces, specifically for imbedding results, the books of Lions [13], [14]; Lions and Magenes [15]; Neˇcas [16]; Grisvard [17], [18], [19]; etc. contain more interesting results on Sobolev spaces for application to boundary value problems of partial differential equations. The treatment of the topics in all of these books is far beyond the reach of the average reader belonging to the large community of applied mathematicians, physicists and engineers. But a book giving a rigorous treatment of distributions and their applications in a simple style and form such that the proofs and results are understandable to the applied community of readers is very much in demand. To our knowledge, such a book dedicated solely to distributions and their application primarily to Sobolev spaces is conspicuous by its absence. Hence, the rationale for writing the present book is to fill this gap, and the scope of the book has been increased by including some additional topics and innumerable examples of different applications in order to widen the readership circle. This book therefore differs from all the good books mentioned above in the style, form and content of the presentation of the theory of distributions, and is addressed in principle to the large community of applied mathematicians, engineers, physicists, etc. In general, it follows the principles of presentation of concepts with proper motivations, the gradual development of concepts with suitable examples

Preface

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and counterexamples and, finally, identifying and listing all the important properties and results so that an applied scientist or engineer can easily apply these results. 

For easy reading, Appendices A–D have been added.



Innumerable examples are given with all the intermediate steps and explanations which are not usually given in the advanced treatises.



Proofs also include all the intermediate steps and necessary explanations to make them easily understandable.



Practical applications are given, such as the physical interpretation of the duality principle, discussions on physical versus mathematical distributions and the application of convolution of distributions to the R-L-C circuit in electrical engineering and in the heat flow problem in a rod.



Distributional derivatives of discontinuous piecewise smooth functions of several variables have been dealt with in all details together with their application in the construction of finite element spaces, a new concept which will be extremely useful in understanding the mathematical foundation of finite element methods for boundary value problems.



Different methods of construction of elementary solutions of linear differential operators with constant coefficients have been given with a lot of details, which will be useful in boundary integral methods and boundary element methods.



Convolution matrices, determinants and the convolution system of equations etc. in Schwartz’s convolution algebra A [7] have been dealt with in detail.



Unusually for a book on distributions, systematic treatment has been given to Fourier transforms of tempered distributions and their applications to Sobolev spaces H s .Rn / of arbitrary order s 2 R; the nice properties of Sobolev spaces on Rn and also the problems of extension of these properties to domains  ¤ Rn ; compactness results in Sobolev spaces; Sobolev’s imbedding results; Sobolev spaces on manifolds , which are boundaries of a domain  in Rn ; trace theorems for Sobolev spaces on Rn , RnC , C m -regular domains  and polygonal domains in R2 ; etc.



Almost all the basic results of the theory of distributions are contained in this book. It can therefore be read as an introduction to advanced treatises on distributions such as, for example, Théorie des distributions by Laurent Schwartz.



Finally, the present book is written by an applied scientist, meant for the applied community and will serve as a reference-cum-text book of that same applied community.

The present book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists, etc. who were interested in studying the problems of mathematical physics in general and their approximate

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solutions on computer in particular. Although this book contains almost all the topics which will be essential for the study of Sobolev spaces and their application to elliptic boundary value problems and their finite element approximations, many additional topics of interest have been included, along with many interesting examples, for specific applied disciplines and engineering: elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations, etc. Moreover, the topics have been presented in such a manner that the reader may concentrate on topics of his or her interest, omitting others. While teaching engineers and others, the author found that even mathematically alert students of applied disciplines found extreme difficulty with 1. brief presentations, without sufficient explanation and motivation; 2. omission of the intermediate steps of involved computations in some problems and of justifications, however trivial these might be; 3. mathematical notations, which can instil fear or distaste when not judiciously chosen. The author has therefore addressed all these problems with sufficient care and due respect so that readers from applied disciplines and engineering should find that the theory of distributions and their applications are within their reach and understandable for subsequent successful and active application in the study of boundary value problems and their approximate solutions on computers. The book can be used either as a reference book or as a text book for different courses as shown separately later. Of all the books on distributions in English, French, German and Russian to which I have had access, the monographs Théorie des distributions and Mathematics for Physical Sciences helped me most in understanding various aspects of the theory of distributions, and their profound influence on me is reflected throughout the present book. For this I express my deepest sense of gratitude and indebtedness to the celebrated author of these monographs, Professor Laurent Schwartz (1915–2002). With very respectful sentiments of gratitude and indebtedness and très bons souvenirs de l’époque de sa grandeur, I recall the distinguished French mathematician Professor Jacques-Louis Lions (1928–2001), who kindly gave me all help, encouragement and opportunity to do research in shell analysis during 1976–77 in the wonderful research environment created by him in the Institut National de Recherche en Informatique et en Automatique (INRIA), France (called IRIA, France at that time). During this stay of mine at INRIA I was exposed to the great achievements of the new French school of research in applied mathematics developed by Professor Lions, which, in fact, inspired me and influenced my research and academic life in a definitive manner. For this I owe a lot to Professor Jacques-Louis Lions and express my grateful indebtedness to him. In addition to my feelings of gratitude and indebtedness to Prof. L. Schwartz mentioned above, I would like to acknowledge further that almost all the basic results,

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concepts, theorems, etc. on distributions presented in this book belong uniquely to Professor L. Schwartz, although his name is not specifically mentioned in each case in the book. In spite of my best efforts to make the manuscript free from mistakes, some may still remain, having defied all rigid checks and correction operations. For this, sincere regret and apology are expressed by me. I further request sympathetic readers to send their criticisms of the book and suggestions for improvement to me at [email protected] or [email protected], which will be thankfully acknowledged in future editions. Finally, all my efforts will only be fruitful if the readers are benefited and find the book readable and interesting. Bon courage to all readers! New Delhi, September 2011

P. K. Bhattacharyya

Contents

Preface How to use this book in courses Acknowledgment Notation 1

vii xxi xxv xxvii

Schwartz distributions 1.1 Introduction: Dirac’s delta function ı.x/ and its properties . . . . . . 1.2 Test space D./ of Schwartz . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Support of a continuous function . . . . . . . . . . . . . . . . 1.2.2 Space D./ . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Space D m ./ . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Space DK ./ . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Properties of D./ . . . . . . . . . . . . . . . . . . . . . . . 1.3 Space D 0 ./ of (Schwartz) distributions . . . . . . . . . . . . . . . . 1.3.1 Algebraic dual space D ? ./ . . . . . . . . . . . . . . . . . . 1.3.2 Distributions and the space D 0 ./ of distributions on  . . . 1.3.3 Characterization, order and extension of a distribution . . . . 1.3.4 Examples of distributions . . . . . . . . . . . . . . . . . . . 1.3.5 Distribution defined on test space D./ of complex-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Some more examples of interesting distributions . . . . . . . . . . . . 1.5 Multiplication of distributions by C 1 -functions . . . . . . . . . . . . 1.6 Problem of division of distributions . . . . . . . . . . . . . . . . . . 1.7 Even, odd and positive distributions . . . . . . . . . . . . . . . . . . 1.8 Convergence of sequences of distributions in D 0 ./ . . . . . . . . . 1.9 Convergence of series of distributions in D 0 ./ . . . . . . . . . . . . 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions . . . . . . . . 1.10.1 Periodic distributions . . . . . . . . . . . . . . . . . . . . . . 1.11 Physical distributions versus mathematical distributions . . . . . . . . 1.11.1 Physical interpretation of mathematical distributions . . . . . 1.11.2 Load intensity . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.3 Electrical charge distribution . . . . . . . . . . . . . . . . . . 1.11.4 Simple layer and double layer distributions . . . . . . . . . . 1.11.5 Relation with probability distribution [7] . . . . . . . . . . .

1 1 6 6 9 13 13 14 25 25 26 27 29 40 41 51 54 57 59 67 68 75 84 84 85 88 90 94

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Differentiation of distributions and application of distributional derivatives 96 2.1 Introduction: an integral definition of derivatives of C 1 -functions . . . 96 2.2 Derivatives of distributions . . . . . . . . . . . . . . . . . . . . . . . 100 2.2.1 Higher-order derivatives of distributions T . . . . . . . . . . 101 2.3 Derivatives of functions in the sense of distribution . . . . . . . . . . 102 2.4 Conditions under which the two notions of derivatives of functions coincide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2.5 Derivative of product ˛T with T 2 D 0 ./ and ˛ 2 C 1 ./ . . . . . 121 2.6 Problem of division of distribution revisited . . . . . . . . . . . . . . 125 2.7 Primitives of a distribution and differential equations . . . . . . . . . 131 2.8 Properties of distributions whose distributional derivatives are known 141 2.9 Continuity of differential operator @˛ W D 0 ./ ! D 0 ./ . . . . . . 142 2.10 Delta-convergent sequences of functions in D 0 .Rn / . . . . . . . . . . 149 2.11 Term-by-term differentiation of series of distributions . . . . . . . . . 154 2.12 Convergence of sequences of C k ./ (resp. C k; .// in D 0 ./ . . . 173 2.13 Convergence of sequences of Lp ./, 1  p  1, in D 0 ./ . . . . . 173 2.14 Transpose (or formal adjoint) of a linear partial differential operator . 175 2.15 Applications: Sobolev spaces H m ./; W m;p ./ . . . . . . . . . . . 177 2.15.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 177 2.15.2 Space H m ./ . . . . . . . . . . . . . . . . . . . . . . . . . 178 2.15.3 Examples of functions belonging to or not belonging to H m./ 182 2.15.4 Separability of H m ./ . . . . . . . . . . . . . . . . . . . . . 184 2.15.5 Generalized Poincaré inequality in H m ./ . . . . . . . . . . 186 2.15.6 Space H0m ./ . . . . . . . . . . . . . . . . . . . . . . . . . 187 2.15.7 Space H m ./ . . . . . . . . . . . . . . . . . . . . . . . . . 191 2.15.8 Quotient space H m ./=M . . . . . . . . . . . . . . . . . . 191 2.15.9 Quotient space H m ./=Pm1 . . . . . . . . . . . . . . . . . 193 2.15.10 Other equivalent norms in H m ./ . . . . . . . . . . . . . . . 194 2.15.11 Density results . . . . . . . . . . . . . . . . . . . . . . . . . 195 2.15.12 Algebraic inclusions () and imbedding (,!) results . . . . . 195 2.15.13 Space W m;p ./ with m 2 N, 1  p  1 . . . . . . . . . . 196 m;p 2.15.14 Space W0 ./, 1  p < 1 . . . . . . . . . . . . . . . . . 200 2.15.15 Space W m;q ./ . . . . . . . . . . . . . . . . . . . . . . . . 203 2.15.16 Quotient space W m;p ./=M for m 2 N; 1  p < 1 . . . . 203 2.15.17 Density results . . . . . . . . . . . . . . . . . . . . . . . . . 207 2.15.18 A non-density result . . . . . . . . . . . . . . . . . . . . . . 208 2.15.19 Algebraic inclusion  and imbedding (,!) results . . . . . . 209 2.15.20 Space W s;p ./ for arbitrary s 2 R . . . . . . . . . . . . . . 209

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3

4

5

6

Derivatives of piecewise smooth functions, Green’s formula, elementary solutions, applications to Sobolev spaces 3.1 Distributional derivatives of piecewise smooth functions 3.1.1 Case of single variable (n D 1) . . . . . . . . . . 3.1.2 Case of two variables (n D 2) . . . . . . . . . . 3.1.3 Case of three variables (n D 3) . . . . . . . . . . 3.2 Unbounded domain   Rn , Green’s formula . . . . . . 3.3 Elementary solutions . . . . . . . . . . . . . . . . . . . 3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

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Additional properties of D 0 ./ 4.1 Reflexivity of D./ and density of D./ in D 0 ./ . . . . . . . 4.2 Continuous imbedding of dual spaces of Banach spaces in D 0 ./ 4.3 Applications: Sobolev spaces H m ./; W m;q ./ . . . . . . . . 4.3.1 Space W m;q ./, 1 < q  1, m 2 N . . . . . . . . . . Local properties, restrictions, unification principle, space E 0 .Rn / of distributions with compact support 5.1 Null distribution in an open set . . . . . . . . . . . . . . . . . 5.2 Equality of distributions in an open set . . . . . . . . . . . . . 5.3 Restriction of a distribution to an open set . . . . . . . . . . . 5.4 Unification principle . . . . . . . . . . . . . . . . . . . . . . 5.5 Support of a distribution . . . . . . . . . . . . . . . . . . . . 5.6 Distributions with compact support . . . . . . . . . . . . . . . 5.7 Space E 0 .Rn / of distributions with compact support . . . . . . 5.7.1 Space E.Rn / . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Space E 0 .Rn / . . . . . . . . . . . . . . . . . . . . . . 5.8 Definition of hT; i for  2 C 1 .Rn / and T 2 D 0 .Rn / with non-compact support . . . . . . . . . . . . . . . . . . . . . .

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211 211 211 215 230 235 238 257

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263 263 265 269 273

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280 280 280 280 283 285 286 287 287 288

. . . . 296

Convolution of distributions 6.1 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Convolution of functions . . . . . . . . . . . . . . . . . . . . . . . 6.3 Convolution of two distributions . . . . . . . . . . . . . . . . . . . 6.4 Regularization of distributions by convolution . . . . . . . . . . . . 6.5 Approximation of distributions by C 1 -functions . . . . . . . . . . 6.6 Convolution of several distributions . . . . . . . . . . . . . . . . . 6.7 Derivatives of convolutions, convolution of distributions on a circle  and their Fourier series representations on  . . . . . . . . . . . . . 6.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Convolution equations (see also Section 8.7, Chapter 8) . . . . . . .

. . . . . .

298 298 303 315 327 329 331

. 333 . 349 . 364

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6.10 Application of convolutions in electrical circuit analysis and heat flow problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 6.10.1 Electric circuit analysis problem [7] . . . . . . . . . . . . . . 375 6.10.2 Excitations and responses defined by several functions or distributions [7] . . . . . . . . . . . . . . . . . . . . . . . . . 380 7

8

Fourier transforms of functions of L1 .Rn / and S.Rn / 7.1 Fourier transforms of integrable functions in L1 .Rn / . . . . . . . 7.2 Space S.Rn / of infinitely differentiable functions with rapid decay at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Space S.Rn / . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Continuity of linear mapping from S.Rn / into S.Rn / . . . . . . . 7.4 Imbedding results . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Density results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Fourier transform of functions of S.Rn / . . . . . . . . . . . . . . 7.7 Fourier inversion theorem in S.Rn / . . . . . . . . . . . . . . . .

383 . . 383 . . . . . . .

. . . . . . .

Fourier transforms of distributions and Sobolev spaces of arbitrary order H S .Rn / 8.1 Motivation for a possible definition of the Fourier transform of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Space S 0 .Rn / of tempered distributions . . . . . . . . . . . . . . . . 8.2.1 Tempered distributions . . . . . . . . . . . . . . . . . . . . . 8.2.2 Space S 0 .Rn / . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Examples of tempered distributions of S 0 .Rn / . . . . . . . . 8.2.4 Convergence of sequences in S 0 .Rn / . . . . . . . . . . . . . 8.2.5 Derivatives of tempered distributions . . . . . . . . . . . . . 8.3 Fourier transform of tempered distributions . . . . . . . . . . . . . . 8.3.1 Fourier transforms of Dirac distributions and their derivatives 8.3.2 Inversion theorem for Fourier transforms on S 0 .Rn / . . . . . 8.3.3 Fourier transform of even and odd tempered distributions . . . 8.4 Fourier transform of distributions with compact support . . . . . . . . 8.5 Fourier transform of convolution of distributions . . . . . . . . . . . . 8.5.1 Fourier transforms of convolutions . . . . . . . . . . . . . . . 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions . . . . . . . . . . . . . . . . . . 8.7 Fourier transform methods for differential equations and elementary solutions in S 0 .Rn / . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Laplace transform of distributions on R . . . . . . . . . . . . . . . . 8.8.1 Space D 0C . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Distribution T 1 2 D 0C (see also convolution algebra A D D 0C (6.9.15b)) . . . . . . . . . . . . . . . . . . . . . .

405 407 412 413 415 417 418

423 423 424 424 426 426 429 432 435 438 440 441 445 450 451 458 476 492 492 496

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8.9

8.10

8.11 8.12 8.13

8.8.3 Inverse L1 of Laplace transform L . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Sobolev spaces H s .Rn / . . . . . . . . . . . . . . . . . . . . 8.9.2 Imbedding result . . . . . . . . . . . . . . . . . . . . . . . . 8.9.3 Sobolev spaces H m .Rn / of integral order m on Rn . . . . . . 8.9.4 Sobolev’s Imbedding Theorem (see also imbedding results in Section 8.12) . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.5 Imbedding result: S.Rn / ,! H S .Rn / . . . . . . . . . . . . . 8.9.6 Density results H S .Rn / . . . . . . . . . . . . . . . . . . . . 8.9.7 Dual space .H s .Rn //0 . . . . . . . . . . . . . . . . . . . . . 8.9.8 Trace properties of elements of H s .Rn / . . . . . . . . . . . . Sobolev spaces on  ¤ Rn revisited . . . . . . . . . . . . . . . . . . 8.10.1 Space H s ./ with s 2 R,    Rn . . . . . . . . . . . . . . 8.10.2 m-extension property of  . . . . . . . . . . . . . . . . . . . 8.10.3 m-extension property of RnC . . . . . . . . . . . . . . . . . . 8.10.4 m-extension property of C m -regular domains  . . . . . . . 8.10.5 Space H s ./ with s 2 RC ,   Rn . . . . . . . . . . . . . 8.10.6 Density results in H s ./ . . . . . . . . . . . . . . . . . . . . 8.10.7 Dual space H s ./ . . . . . . . . . . . . . . . . . . . . . . 8.10.8 Space H0s ./ with s > 0 . . . . . . . . . . . . . . . . . . . . 8.10.9 Space H s ./ with s > 0 . . . . . . . . . . . . . . . . . . . 8.10.10 Space W s;p ./ for real s > 0 and 1  p < 1 . . . . . . . . s 8.10.11 Space H00 ./ with s > 0 . . . . . . . . . . . . . . . . . . . s 8.10.12 Dual space .H00 .//0 for s > 0 . . . . . . . . . . . . . . . . s;p 8.10.13 Space W00 ./ for s > 0, 1 < p < 1 . . . . . . . . . . . . 8.10.14 Restrictions of distributions in Sobolev spaces . . . . . . . . . 8.10.15 Differentiation of distributions in H s ./ with s 2 R . . . . . 8.10.16 Differentiation of distributions u 2 H s ./ with s > 0 . . . . Compactness results in Sobolev spaces . . . . . . . . . . . . . . . . . s ./ . 8.11.1 Compact imbedding results in H s ./, H0s ./ and H00 Sobolev’s imbedding results . . . . . . . . . . . . . . . . . . . . . . 8.12.1 Compact imbedding results . . . . . . . . . . . . . . . . . . Sobolev spaces H s ./, W s;p ./ on a manifold boundary  . . . . . 8.13.1 Surface integrals on boundary  of bounded   Rn . . . . . 8.13.2 Alternative definition of H s ./ with  2 C m -class (resp. C 1 -class) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.13.3 Space H s ./ (s > 0) with  in C m -class (resp. C 1 -class) . 8.13.4 Sobolev spaces on boundary curves  in R2 . . . . . . . . . . s 8.13.5 Spaces H0s .i /; H00 .i / for polygonal sides i 2 C 1 -class, 1i N . . . . . . . . . . . . . . . . . . . . . . . . . . .

497 502 502 503 507 512 521 522 523 526 546 546 550 558 569 573 578 579 579 580 580 585 591 591 593 598 601 605 616 617 632 634 634 637 638 641 651

xviii

9

Contents

8.14 Trace results in Sobolev spaces on    Rn . . . . . . . . . . . . . . 8.14.1 Trace results in H m .RnC / . . . . . . . . . . . . . . . . . . . 8.14.2 Trace results in H m ./ with bounded domain  ¨ Rn . . . 8.14.3 Trace results in W s;p -spaces . . . . . . . . . . . . . . . . . . 8.14.4 Trace results for polygonal domains   R2 . . . . . . . . . 8.14.5 Trace results for bounded domains with curvilinear polygonal boundary  in R2 . . . . . . . . . . . . . . . . . . . . . . . 8.14.6 Traces of normal components in Lp .divI / . . . . . . . . . . 8.14.7 Trace theorems based on Green’s formula . . . . . . . . . . . 8.14.8 Traces on 0   . . . . . . . . . . . . . . . . . . . . . . .

685 686 691 710

Vector-valued distributions 9.1 Motivation . . . . . . . . . . . . . . . . . 9.2 Vector-valued functions . . . . . . . . . . 9.3 Spaces of vector-valued functions . . . . 9.4 Vector-valued distributions . . . . . . . . 9.5 Derivatives of vector-valued distributions 9.6 Applications . . . . . . . . . . . . . . . . 9.6.1 Space E.0; T I V; W / . . . . . . . 9.6.2 Hilbert space W1 .0; T I V / . . . . 9.6.3 Hilbert space W2 .0; T I V / . . . . 9.6.4 Green’s formula . . . . . . . . .

712 712 712 715 718 723 724 725 725 728 729

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A Functional analysis (basic results) A.0 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . A.0.1 An important result on logical implication (H)) and non-implication (H)) 6 . . . . . . . . . . . . . . . . . . . . . A.0.2 Supremum (l.u.b.) and infimum (g.l.b.) . . . . . . . . . . . . A.0.3 Metric spaces and important results therein . . . . . . . . . . A.0.4 Important subsets of a metric space X  .X; d / . . . . . . . A.0.5 Compact sets in Rn with the usual metric d2 . . . . . . . . . A.0.6 Elementary properties of functions of real variables . . . . . . A.0.7 Limit of a function at a cluster point x0 2 Rn . . . . . . . . . A.0.8 Limit superior and limit inferior of a sequence in R . . . . . . A.0.9 Pointwise and uniform convergence of sequences of functions A.0.10 Continuity and uniform continuity of f 2 F ./ . . . . . . . A.1 Important properties of continuous functions . . . . . . . . . . . . . . A.1.1 Some remarkable properties on compact sets in Rn . . . . . . A.1.2 C01 ./-partition of unity on compact set K    Rn . . A.1.3 Continuous extension theorems . . . . . . . . . . . . . . . . A.2 Finite and infinite dimensional linear spaces . . . . . . . . . . . . . . A.2.1 Linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . .

651 652 654 670 672

731 731 731 732 732 735 737 738 738 739 740 740 741 741 741 741 743 743

xix

Contents

A.3

A.4 A.5

A.6 A.7

A.8

A.9 A.10

A.11 A.12 A.13 A.14 A.15 A.16

A.2.2 Linear functionals . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . Normed linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Semi-norm and norm . . . . . . . . . . . . . . . . . . . . . . A.3.2 Closed subspace, dense subspace, Banach space and its separability . . . . . . . . . . . . . . . . . . . . . . . . . . . Banach spaces of continuous functions . . . . . . . . . . . . . . . . . A.4.1 Banach spaces C 0 ./, C k ./ . . . . . . . . . . . . . . . . . Banach spaces C 0; ./, 0 <  < 1, of Hölder continuous functions . A.5.1 Hölder continuity and Lipschitz continuity . . . . . . . . . . A.5.2 Hölder space C 0; ./ . . . . . . . . . . . . . . . . . . . . . A.5.3 Space C k; ./, 0 <   1 . . . . . . . . . . . . . . . . . . . Quotient space V =M . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous linear functionals on normed linear spaces . . . . . . . . A.7.1 Space V 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7.2 Hahn–Banach extension of linear functionals in analytic form A.7.3 Consequences of the Hahn–Banach theorem in normed linear spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous linear operators on normed linear spaces . . . . . . . . . A.8.1 Space L.V I W / . . . . . . . . . . . . . . . . . . . . . . . . . A.8.2 Continuous extension of continuous linear operators by density . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.8.3 Isomorphisms and isometric isomorphisms . . . . . . . . . . A.8.4 Graph of an operator A 2 L.V I W / and graph norm . . . . . Reflexivity of Banach spaces . . . . . . . . . . . . . . . . . . . . . . Strong, weak and weak-* convergence in Banach space V . . . . . . . A.10.1 Strong convergence ! . . . . . . . . . . . . . . . . . . . . . A.10.2 Weak convergence * . . . . . . . . . . . . . . . . . . . . . A.10.3 Weak-* convergence * in Banach space V 0 . . . . . . . . . Compact linear operators in Banach spaces . . . . . . . . . . . . . . Hilbert space V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual space V 0 of a Hilbert space V , reflexivity of V . . . . . . . . . . Strong, weak and weak-* convergences in a Hilbert space . . . . . . . Self-adjoint and unitary operators in Hilbert space V . . . . . . . . . Compact linear operators in Hilbert spaces . . . . . . . . . . . . . . .

B Lp -spaces B.1 Lebesgue measure  on Rn . . . . . . . . . . . . . . . . B.1.1 Lebesgue-measurable sets in Rn . . . . . . . . . B.1.2 Sets with zero (Lebesgue) measure in Rn . . . . B.1.3 Property P holds almost everywhere (a.e.) on 

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746 747 748 748 750 750 750 753 753 754 754 756 756 756 757 758 760 760 761 762 762 763 763 763 764 764 764 765 768 769 769 769 771 771 771 772 775

xx

Contents

B.2 Space M./ of Lebesgue-measurable functions on  . . . . . . . . . B.2.1 Measurable functions and space M./ . . . . . . . . . . . . B.2.2 Pointwise convergence a.e. on  . . . . . . . . . . . . . . . . B.3 Lebesgue integrals and their important properties . . . . . . . . . . . B.3.1 Lebesgue integral of a bounded function on bounded domain  . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.2 Important properties of Lebesgue integrals (Kolmogorov and Fomin [20]) . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.3 Some important approximation and density results in L1 ./ . B.4 Spaces Lp ./, 1  p  1 . . . . . . . . . . . . . . . . . . . . . . B.4.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . B.4.2 Dual space .Lp .//0 of Lp ./ for 1  p  1 . . . . . . . . B.4.3 Space L2 ./ . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.4 Some negative properties of L1 ./ . . . . . . . . . . . . . . B.4.5 Some nice properties of L1 ./ . . . . . . . . . . . . . . . . p B.4.6 Space Lloc ./ inclusion results . . . . . . . . . . . . . . . .

776 776 778 778 778 780 784 788 788 794 797 798 799 799

C Open cover and partition of unity 803 C.1 C01 ./-partition of unity theorem for compact sets . . . . . . . . . . 803 D Boundary geometry D.1 Boundary geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1.1 Locally one-sided and two-sided bounded domains  . . . . . D.1.2 Star-shaped domain  . . . . . . . . . . . . . . . . . . . . . D.1.3 Cone property and uniform cone property . . . . . . . . . . . D.1.4 Segment property . . . . . . . . . . . . . . . . . . . . . . . . D.2 Continuity and differential properties of a boundary . . . . . . . . . . D.2.1 Continuity and differential properties . . . . . . . . . . . . . r n D.2.2 Open cover ¹r ºN rD1 of , local coordinate systems ¹i ºiD1 N and mappings ¹r ºrD1 . . . . . . . . . . . . . . . . . . . . . D.2.3 Properties of the mappings r W Rn1 ! R, 1  r  N . . D.3 Alternative definition of locally one-sided domain . . . . . . . . . . . D.4 Alternative definition of continuity and differential properties of  as a manifold in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 Atlas/local charts of  . . . . . . . . . . . . . . . . . . . . . . . . .

808 808 808 808 809 811 812 812

Bibliography Index

819 823

813 814 816 817 818

How to use this book in courses

This book can be used either as a reference book or as a text book for many specialized advanced courses, as shown below.

Reference book As the book contains almost all the basic results on distributions (generalized functions) and Sobolev spaces, and also many other applications, it can be used as a reference book by applied mathematicians, functional analysts, physicists, engineers and also by Ph.D. scholars and postdoctoral fellows in computational mathematics, mechanics and engineering disciplines.

Text book It can also be used as a text book for self study or for different courses at the M.S./ M.Sc., M.Phil., M.Tech. and Ph.D. levels. For example:

Theory of Distributions or Generalized Functions Syllabus for a two-semester course Chapters 1–9 (omitting some sections if necessary, particularly those containing specialized applications).

Introduction to Theory of Distributions or Generalized Functions Syllabus for a one-semester course Chapter 1, Sections 1.1–1.3, 1.5, 1.8, 1.11; Chapter 2, Sections 2.1, 2.2, 2.5, 2.8, 2.9, 2.12; Chapter 3, Sections 3.1.1, 3.1.2, 3.3; Chapter 4, Sections 4.1, 4.2; Chapter 5, Sections 5.1–5.7; Chapter 6, Sections 6.3–6.7; Chapter 7, Sections 7.2, 7.6, 7.7; Chapter 8, Sections 8.1–8.7.

Sobolev Spaces with Distributions Syllabus for a two-semester course Chapter 1, Sections 1.2–1.4, 1.8; Chapter 2, Sections 2.1–2.3, 2.5, 2.8, 2.9, 2.12, 2.13; Chapter 3, Sections 3.1.1–3.1.3, 3.4; Chapter 4, Sections 4.2, 4.3; Chapter 5, Sec-

xxii

How to use this book in courses

tions 5.1–5.3, 5.5, 5.6; Chapter 6, Sections 6.2–6.4, 6.8; Chapter 7, Sections 7.2, 7.6, 7.7; Chapter 8, Sections 8.2–8.6, 8.9–8.14; Chapter 9, Sections 9.4–9.6; Appendix D: Boundary Geometry.

Sobolev Spaces Syllabus for a one-semester course Chapter 1, Sections 1.1–1.3, 1.5, 1.8; Chapter 2, Sections 2.1, 2.2, 2.5, 2.13; Chapter 3, Section 3.4; Chapter 4, Sections 4.2, 4.3; Chapter 5, Sections 5.2, 5.5; Chapter 6, Sections 6.3, 6.4, 6.8; Chapter 7, Sections 7.2, 7.6; Chapter 8, Sections 8.2, 8.3, 8.5, 8.6, 8.9–8.14; Appendix D: Boundary Geometry. Some sections and some special results may be omitted without disturbing the sequence of topics.

Fourier Series, Fourier and Laplace Transforms Syllabus for a half-semester course Chapter 1, Sections 1.2, 1.3, 1.8, 1.10; Chapter 2, Sections 2.1–2.3, 2.5, 2.9, 2.11; Chapter 5, Sections 5.5, 5.6; Chapter 6, Sections 6.2, 6.3, 6.7; Chapter 7, Sections 7.1– 7.3, 7.6, 7.7; Chapter 8, Sections 8.1–8.8.

Dirac Distributions and their Properties Syllabus for a half-semester course Chapter 1, Sections 1.1–1.3, 1.6–1.8, 1.10, 1.11; Chapter 2, Sections 2.1–2.3, 2.5, 2.6, 2.10, 2.11; Chapter 3, Sections 3.1–3.3; Chapter 5, Sections 5.5–5.7; Chapter 6, Sections 6.3, 6.4, 6.7; Chapter 7, Sections 7.2, 7.7; Chapter 8, Sections 8.2, 8.8.

Elementary Solutions for Boundary Integral Equations Syllabus for a half-semester course Chapter 1, Sections 1.1–1.3, 1.5, 1.8; Chapter 2, Sections 2.1–2.3, 2.5, 2.14; Chapter 3, Sections 3.2, 3.3; Chapter 5, Sections 5.5–5.7; Chapter 6, Sections 6.3, 6.4, 6.9; Chapter 7, Section 7.2; Chapter 8, Sections 8.2–8.7. Additional topics may be included or some sections may be deleted in any of these courses. Other courses are possible, for example: 

Distributions and Differential Equations;



Introduction to Vector-Valued Distributions for Evolution Equations of Parabolic and Hyperbolic Types.

How to use this book in courses

xxiii

A few chapters of this book have been class tested in a short course on distributions entitled Selected Topics for M.Tech. students and Ph.D. research scholars in the Electrical Engineering Dept., I.I.T., Delhi, in 2006–2008. The responses from the participants of the course were quite encouraging for the author of the book.

Acknowledgment

The author expresses his heartiest thanks to Prof. Neela Nataraj of I.I.T., Mumbai, Prof. Kallol Ghosh of Jadavpur University, Calcutta and Prof. B. N. Mandal of I.S.I., Calcutta for their assistance in the preparation of the manuscript in LATEX. Dr. Subhashish Roy Chaudhury of Jadavpur University and Mr. Hariharan, Mr. Anoop Nair, Mr. Girish, Mr. Himanshu Tyagi, Tarun Vir Singh, Mr. Sarat and Mr. Srikamal of I.I.T., Delhi helped in typing the manuscript in LATEX, for which the author expresses grateful thanks to all of them. For almost all the figures, graphs and tables, which were primarily drawn in the U.S. by Mr. Ashish Das, a former student of I.I.T., Delhi, and his energetic wife, Anindita, the author expresses special thanks to the couple. Mr. Anoop Nair, Mr. Girish, Mr. Srikamal, Mr. Srihari, Mr. Sarat and Mr. Mohit Garg took the responsibility of preparing the final version of the manuscript in the VLSI Laboratory, I.I.T., Delhi, for which the author thanks them again, along with their supervisors, Prof. G. S. Visweswaran and Prof. Jayadeva, and Prof. B. Bhaumik for all their help. The author thanks Prof. S. C. Dutta Roy for his written comments on the application of convolution to R-L-C circuit analysis in Chapter 6. The author thanks his former students Prof. S. Balasundaram and Dr. S. Gopalsamy for taking an interest in the publication of the book. In particular, the author is grateful to Dr. Gopalsamy for taking the trouble to read and correct a few chapters. With feelings of gratitude, the author recalls the encouragement of many colleagues from I.I.T., Delhi; J.N.U., New Delhi; Jadavpur University, Calcutta, where the author taught in recent years. In particular, Prof. Suresh Chandra, Prof. B. R. Handa, Dr. W. Shukla, Mr. A. Nagabhusanam of I.I.T., Delhi, Prof. Karmeshu of J.N.U., New Delhi and Prof. A. K. Pani of I.I.T., Mumbai are thanked for their interest in the publication of the book. The author expresses grateful thanks to Prof. Olivier Pironneau of Pierre and Marie Curie University, Paris, Prof. Michel Bernadou of Leonard de Vinci University, La Defense, France and Prof. Maurice A. Jaswon of City University, London for their active interest in the book. The author thanks the Dept. of Electrical Engineering, I.I.T., Delhi, for providing the facilities for putting some chapters of the present book to class test at a highest level departmental course, and also thanks Dept. of Mathematics, I.I.T., Delhi, for giving the opportunity to start the teaching of distributions about three decades ago. Prof. Balasundaram took up the extensively laborious job of preparing the errata of all Chapters and Appendices of the book in typed form. Without his sincere and encouragement I could not have completed the whole task of correction of the galley-proof by this time. For all these, I express my most grateful thanks to Prof. S. Balasundaram.

xxvi

Acknowledgment

Finally, the author thanks De Gruyter for accepting my book for publication in their text book series and also their editorial and publishing division and in particular, Madame Friederike Dittberner, Madame Anja Möbius, Madame Ulrike Swientek and Mr. Christoph von Friedeburg for their excellent professional cooperation and gracious help on all occasions.

Notation

This section summarizes the notation used within this book. Where page numbers are given, these are either the page of the first occurrence of the notation or the page of its definition in Appendices A–D. Logical symbols 9x 8x

there exists x for every x

H)

Logical implication: .P / H) .Q/

H) 6

Logical non-implication: .P /

implies

does not imply

Set notations a2A ; AB A  B A{  {A B nA A[B A\B AB An

H) 6

.Q/

a is an element of the set A empty set A is a subset of the set B A is a compact subset of B complement of the set A complement of A in B the union of the sets A and B the intersection of the sets A and B Cartesian product of the sets A and B A    A „  A ƒ‚ … n times

A˙B

(p. 303)

Number systems N N0 Z Q R RC R RnC , Rn C F

¹1; 2; 3; : : : ; n; : : : º; the set of all natural numbers N [ ¹0º D ¹0; 1; 2; : : : º ¹0; ˙1; ˙2 : : : ; ˙n; : : : º the set of all rational numbers the set of all real numbers  1; 1Œ 0; 1Œ 1; 0Œ (p. 553) the set of all complex numbers (p. 5) number field R or C

xxviii

Notation

Rn

R    R … „  R ƒ‚

Cn

C  C ƒ‚    C … „

n times

n times

Multi-index notations ˛ j˛| x  x˛ ˛ f .x/ @˛  D ˛ f @˛i i f  Di˛i f ˛ˇ ˛Š ˛ˇ

.˛1 ; ˛2 ; : : : ; ˛n /, ˛i 2 N0 (p. 5) ˛1 C ˛2 C    C ˛n , ˛i 2 N0 (p. 5) .x1 ; x2 ; : : : ; xn / 2 Rn (p. 5) .1 ; 2 ; : : : ; n / 2 Rn x1˛1 x2˛2    xn˛n 1˛1 2˛2    n˛n f .x1 ; x2 ; : : : ; xn / @j˛j f ˛ ˛ ˛ @x1 1 @x2 2 :::@xn n @˛i f ˛ (p. 5) @xi i

˛  ˇ ” ˛i  ˇi for 1  i  n (p. 5) ˛1 Š ˛2 Š    ˛n Š .˛1  ˇ1 ; ˛2  ˇ2 ; : : : ; ˛n  ˇn / with ˛i  ˇi  0 (p. 5)

Notations used for properties in Rn x .x1 ; : : : ; xn / 2 Rn , an ordered n-tuple of real numbers xi , 1  i  n xCy .x1 C y1 ; : : : ; xn C yn / 2 Rn ˛x .˛x1 ; : : : ; ˛xn / 2 Rn 0 .0; 0; : : : ; 0/ 2 Rn  open subset of Rn  closure of  in Rn n d2 .x; y/ D d.x; y/ the Pnusual (Euclidean) metric in R d1 .x; y/ iD1 jxi  yi j d1 .x; y/ max1in ¹jxi yi jº: other equivalent metrics in Rn kxk D kxk2 the usual (Euclidean) norm in Rn with kxk2 D x12 C x22 C    C xn2 P n kxk1 iD1 jxi j kxk1 max1in ¹jxi jº: other equivalent norms in Rn B.0I "/ open ball, ¹xI x 2 Rn ; kxk < "; " > 0º B.0I "/ closed ball: closure of B.0I "/ in Rn (p. 9) n S.0I "/ sphere, ¹xI x 2 Rn ; kxk D "º in PR n n hx; yiRn inner product of x; y 2 R D iD1 xi yi

xxix

Notation

1=2

kxk x?y

hx; xiRn x ? y ” x 2 Rn is orthogonal to y 2 Rn

Mappings f W X ! Y f W   Rn ! R f W   Rn ! C J W V ! R A W V ! W L W V ! R ,!W X ! Y ,!,!W X ! Y

mapping f from set X into set Y real-valued functions f from  into R complex-valued function f from  into C functional J from vector space V into R operator A from vector space V into vector space W linear functional from V into R continuous imbedding operator compact imbedding operator

Notations for usual derivatives in the point-wise sense Œ dH .x/, Œ dH .x/, H 0 .x/ (p. 107) dx dx @f @f Œ @x .x/, Œ @x .x/ partial derivative of f with respect to xi at the point i i x in the usual point-wise sense (pp. 119, 212) 2f @f J0 ; Jk ; Jkl jump of f; Œ @x .x/,Œ @x@ @x .x/ across 0 (p. 213) k k l Œ f .x/ Laplacian of f in the usual point-wise sense (p. 223) Notations for distributional derivatives @T @j˛j T , @˛ T ˛1 ˛n (pp. 100–101) @x @x1 :::@xn @j˛j f ˛1 ˛ @x1 :::@xn n

i

@˛ f dH dx Tf

 f

Multiple integrals R f .x/d x 

(p. 102)

(p. 105) Laplacian of f in the distributional sense (p. 223) R



R

f .x1 ; : : : ; xn / dx1 : : : dxn with d x D dx1 dx2 : : : dxn (p. 21)

Linear spaces used for distributions on   Rn  closure of  in Rn , for  with boundary :   [ F ./ space of real-valued (resp. complex-valued) functions (p. 738) C k ./; C 1 ./ (p. 7) C00 ./ C0 ./ (p. 7) (p. 13) C0m ./ 1 C0 ./ (p. 7)

xxx

C m ./ C k; ./ Lp ./; 1  p  1 L1 ./ L2 ./ L1loc ./ D./ D m ./ DK ./ D  ./ D 0 ./ D 00 ./ E.Rn / E 0 .Rn /

Notation

(p. 20) (p. 754) (p. 788) (p. 780) (p. 797) (p. 16) C01 ./ (p. 9) (p. 13) (p. 13) algebraic dual space of D./ (p. 25) space of distributions on  D algebraic and topological dual of D./ (p. 26) .D 0 .//0 D the second dual space of D./ (p. 263) C 1 .Rn / (p. 287) dual space of E.Rn / (p. 288)

Linear spaces used for distributions on circle  D./ D C 1 ./ test space of C 1 -functions on  (p. 77) 0 D ./ space of distributions on  D dual space of D./ (p. 78) DT .R/ space of periodic functions on R with period T > 0 (p. 78) DT0 .R/ dual space of DT .R/ (p. 78) Linear spaces used in tempered distributions on Rn S.Rn / (p. 407) (p. 426) S 0 .Rn /

M .Rn / (p. 456)

C0 .Rn / (p. 457) S./ S.Rn1 / (p. 527) 0C D 0C .R/ (p. 492) D Linear spaces used in vector-valued functions/distributions 0; T Œ time interval with T > 0 C 0 .  Œ0; T / (p. 712) C 0 .Œ0; T I V / (p. 715) C k .Œ0; T I V / (p. 715) C k .0; T ŒI V / C 1 .0; T ŒI V / D.0; T ŒI V / C01 .0; T ŒI V / (p. 715)

xxxi

Notation

L2 .  0; T Œ// Lp .0; T I V / D 0 .0; T ŒI V / L1loc .0; T I V / L2 .0; T I V / E.0; T I V; W / W1 .0; T I V / W2 .0; T I V /

(0 < T < C1) (p. 714) Lp .0; T ŒI V / (p. 716) (p. 718) (p. 720) (p. 722) (p. 725) (p. 725) (p. 728)

Sobolev spaces with Hilbert space structure on   Rn or on Rn H m ./ (p. 178) H 1 ./ (p. 180) H 2 ./ (p. 181) m H0 ./ (p. 187) H01 ./; H02 ./ (p. 187) (p. 191) H m ./ (p. 546) H s ./ s H ./ (p. 574) H  ./ (p. 574) (p. 579) H0s ./ H s ./ (p. 579) s H00 ./ (p. 585) X s ./ (p. 587) s .//0 (p. 591) .H00 s H .Rn / (p. 502) H m .Rn / (p. 507) H m .Rm / (p. 507) H  .Rn /; H s .Rn / (s D Œs C , 0 <  < 1) (p. 509) s 12 n1 .R / (p. 528) H sj  12 n1 .R / (p. 530) H Sobolev spaces with Banach space structure on   Rn W m;p ./; 1  p < 1 (p. 196) W m;1 ./ (p. 196) m;p W0 ./ (p. 200) (p. 203) W m;q ./ W s;p ./ (p. 580) W s;q ./ (p. 583) s;p N (p. 584) W ./ s;p W00 ./ (p. 591)

xxxii

p

Xs ./ s;p .W00 .//0

Notation

(p. 592) (p. 593)

Sobolev spaces with Hilbert space structure on boundary  and 0   H s ./ (p. 637) (p. 638) L2 ./ H s ./ (p. 641) Lp ./ (p. 635) H s .0 / (p. 642) H0s .0 / (p. 643) 1=2 H00 .0 / (p. 643) 3=2 H00 .0 / (p. 643) H0s .i / (p. 651) s (p. 651) H .i / s .H00 .i //0 (p. 651) Sobolev spaces with Banach space structure on boundary  and 0   (p. 636) W s;p ./ W s;q ./ (p. 643) W s;p .0 / (p. 642) s;p W00 .0 / (p. 642) s;p W0 .0 / (p. 642) s;p (p. 643) .W0 .0 //0 Notations for duality pairing duality pairing between V and V 0 (p. 715) h  ;  iV V 0 h  ;  iD 0 ./D./ duality pairing between D 0 ./ and D./ (p. 26) duality pairing between D 0 .Rn / and D.Rn / (p. 34) h  ;  iD 0 .Rn /D.Rn / T ./, hT; i, .T; / value of distribution T 2 D 0 ./ at a test function  (p. 26) h  ;  iS 0 .Rn /S.Rn / duality pairing between S 0 .Rn / and S.Rn / (p. 436) Notations for inner products h  ;  iV in V hh  ;  iiW in W h  ;  iV (p. 765) h  ;  i0; (p. 210) h  ;  iL2 ./ h  ;  iL2 .T / h  ;  iT (p. 158) m h  ;  iH ./ h  ;  im; (p. 179) h  ;  iH s ./ h  ;  is; (p. 575)

xxxiii

Notation

h  ;  iH s .Rn / s h  ;  iH00 ./ h  ;  iH 1 ./ Π ;  0; hh  ;  ii0; h  ;  iH s ./ h  ;  iH.divI/ h  ;  iH.4I/ hΠ ; Π iH m ./=M

h  ;  is;Rn (p. 502) h  ;  i00;s; (p. 585) h  ;  i1; (p. 181) (p. 703) hh.  /; .  /ii0; (p. 703) h  ;  is; (p. 650) (p. 687) h  ;  i0;4; (p. 693) (p. 192)

Notations for semi-norms j  jV in V j  jH m ./ j  jm; (p. 179) j  jm;p; (p. 196) j  jW m;p ./ jŒ  jW m;p ./=Pm1 (p. 206) j  jC k ./ (p. 753) j  jC k; ./ ,  C k; ./ (p. 754) p.  / semi-norm (p. 748) semi-norm in S.Rn / (p. 408) q˛;ˇ .  / semi-norm in S.Rn / (p. 408) ql;m .  /  q˛;ˇ .  / semi-norm in S.Rn / (p. 408)  ql;m .  / semi-norm in S.Rn / (p. 408) p˛ semi-norm (p. 14) pK;m.K/ , pQK;m.K/ semi-norm (p. 27) Notations for norms k  kV in V, jjj  jjjW in W (p. 752) k  kC k ./ , jjj  jjjC k ./ (p. 754) k  kC 0; ./ k  kLp ./ (p. 749) k  kL2 ./ (p. 766) k  kL2 .T / k  kT (p. 159) k  kL.V IW / (p. 760) (p. 760) k  kL.V IR/ k  kH m ./ k  km; (p. 179) k  kH 1 ./ k  k1; (p. 181) m k  kH ./ k  km; (p. 269) k  kH s ./ k  ks; (p. 575) s k  k00;s; (p. 585) k  kH00 k  ks;.Rn / (p. 528) k  kH s .Rn / k  kH s ./ (p. 546) (p. 587) k  kX s ./

xxxiv

k  km;p;  m;2; jjjujjjm;2; k  kLp .divI/ k  kH.4I/ k  kH 2 .ƒ;/ jjj  jjj0; k  kE.0;T IV;W / k  kW1 .0;T IV / k  kW2 .0;T IV / k  kV k  k1 kŒ  kH m ./=M jjjŒ  jjjH m ./=Pm1 kŒ  kW m;p ./=M

Notation

(p. 196) (p. 200) (p. 200) (p. 686) (p. 693) k:k2;ƒ; (p. 702) (p. 703) (p. 725) (p. 726) (p. 728) (p. 748) (p. 749) (p. 191) (p. 193) (p. 203)

Notations for tensor product f ˝g tensor product of functions f and g (p. 298) (p. 299) 1x ˝ g.y/; f .x/ ˝ 1y Tx ˝ Sy Tx  T .x/; Sy  S.y/ (p. 301) hTx ˝ Sy ; .x/  .y/i (p. 301) hTx ˝ Sy ; .x; y/i (p. 301) (p. 301) supp.Tx ˝ Ty / Notations for convolutions f g convolution of functions f and g (p. 304) T  convolution of distribution T and test function  (p. 315)

" f regularization of f by convolution with " (p. 308) ı f convolution of Dirac distribution ı and f (p. 322) T S convolution of two distributions T and S (p. 317) ı T convolution of Dirac distribution ı and T (p. 322) supp.T S / support of T S (p. 321) S1 .s/ S2 .s/ convolution of distributions S1 ; S2 on circle  (p. 336) TL (p. 316) a f; a T (pp. 323, 72) Convolution algebra A A ŒA

convolution algebra (p. 367) .Aij /1i;j n with Aij 2 A

xxxv

Notation

ŒA ŒB  .A/ ŒE D ŒA1 

convolution matrix product in A (p. 371) convolution determinant of ŒA (p. 372) convolution inverse of ŒA (p. 373)

Fourier transform F and co-transform F (p. 383) F ;F O f F f (p. 383) fL; .FLf / (p. 389) O T F T : Fourier transform of tempered distributions T 2 S 0 .Rn / (p. 435) Notations for notions of convergences ! strong convergence (p. 763) * weak convergence (p. 764) * weak- convergence (p. 764) Laplace transform L L1

(p. 492) inverse Laplace transform (p. 497)

Notations for trace operators ‚; ‚j ; ‚j and their right-hand inverses trace operator (p. 528) 0 j trace operator (p. 527)  trace operator (p. 708) 1 j W H s .Rn / 7! H sj  2 .Rn1 / (p. 530) j Q 1   W H s .Rn / ! jmD0 H sj  2 .Rn1 / (p. 545) (p. 663) j  trace operator (p. 666)  j j W H m .RnC / ! H mj 1=2 .Rn1 / (p. 654) Q mj 1=2 .Rn1 /   W H m .RnC / ! jm1  D0 H (p. 654) j j W H m ./ ! H mj 1=2 ./ (p. 666) Q mj 1=2 ./ (p. 666)   W H m ./ ! jm1  D0 H Ker.  / (p. 670) H0m ./ j W W s;p .Rn / ! W sp1=p;p .Rn1 / (p. 670) j Q   W W s;p .Rn / ! jkD0 W sj 1=p;p .Rn1 / (p. 671) Q lj lj W H s ./ ! lkD0 H sk1=2 .j / (p. 672)   W v 2 Lp .divI / 7! v 2 W 1=p;p ./ (p. 687)

xxxvi



j 



Notation

 W H 2 .ƒ; / ! H 3=2 ./  H 1=2 ./ (p. 709) right-hand inverse j W H sj 1=2 .Rn1 / ! H s .Rn / (p. 544) Q mj 1=2 .Rn1 / ! H m .Rn /  W jm1 D0 H C (p. 654) Q

W jm1 H mj 1=2 ./ ! H m ./ (p. 666) QkD0 sj 1=p;p n1

W j D0 W .R / ! W s;p .Rn / (p. 671)

W W 1=p;p ./ ! Lp .divI / (p. 687)

General notations used in the book an open subset of Rn   Rn n R boundary of   Rn with  D  [  j  jV semi-norm in V , p.  / in V (p. 748) k  kV norm in V (p. 748) jjj  jjjV (p. 703) norm in H s ./ (p. 575) k  ks; s k  k00;s; norm in H00 ./ (p. 585) k  ks;p; norm in W s;p ./ (p. 581) s;p k  k00;s;p; norm in W00 ./ (p. 591) d.  ;  / metric/distance function in normed linear space X : d.x; y/ D kx  ykX (p. 732) h  ;  iV inner product in V (p. 765) hh  ;  iiV (p. 703) h  ;  iV 0 V duality pairing between V 0 and V (p. 715) ıij Kronecker delta (p. 2) ı D ı0 D ı.x/ Dirac delta function with concentration at 0 2 R (p. 1) ıa D ı.x  a/ Dirac delta function with concentration at a 2 R (p. 1) ıa D ı.x  a/ Dirac delta function with concentration at a 2 Rn (p. 5) ıS Dirac delta function with concentration on surface S  Rn (p. 36) ı Dirac delta function with concentration on   Rn (p. 90) supp./ support of a continuous function  (p. 6) supp./   supp./ compact in  (p. 6) H.x/ Heaviside function on R (p. 3) ln.x ˙ i 0/ (p. 41)

xxxvii

Notation

  x  ; xC ; x Pf

c.p.v. JF .x/ FT

"

m .x/ eT .x/ ˆ ˆ.s/ @u , @u @nA @n  A

T , S , etc. T , S , etc. T Tf T 2 E 0 .Rn / T ./ hT; iD 0 ./D./ .T; / b b T; b ı T #0 X ,! Y X ,!,! Y

(p. 49) Finite part; Pf. x1k /  Pseudo-function (finite part of) Pf. x1k / (p. 42) Cauchy principal value x1 (p. 39) Jacobian of F W Rn ! Rn at x (p. 68) image of T under F (p. 69) regularizing function (p. 308) regularizing sequence (p. 307) test function in D./ (p. 5) periodic function in DT .R/ with period T > 0 (p. 78) test function in D./ (p. 77) conormal derivatives of u with respect to A; A (p. 689) generic distributions on  (p. 25) generic tempered distributions (p. 424) period of periodic distributions S on R (p. 77) (regular) distribution in D 0 ./ defined by f 2 L1loc ./ (p. 29) distribution T 2 D 0 .Rn / with compact support in Rn (p. 288) duality pairing between T 2 D 0 ./ and  2 D./ (p. 26) T ./ (p. 26) T ./ (p. 26) translation operator (p. 72) (p. 72) restriction to 0 of T (p. 280) continuous imbedding of X into Y , compact imbedding from X into Y (p. 761)

Notations used in the analysis of R-L-C circuit ı.t / unit excitation impulse (p. 377) E.t / impulse response (p. 377) R resistance (p. 377) L inductance (p. 377) C capacitance (p. 377) Z.t / impedance (p. 378) A.t / admittance (p. 378)

xxxviii

Notations used for traces in H 2 .ƒ; / for plate bending problems ƒ plate bending operator (p. 699) ˆ tensor-valued functions (p. 700) ˆ D .ij / bending moment tensor field (p. 700) normal moment (p. 700) Mn Mnt twisting moment (p. 700) Qn vertical shear (p. 700) Kn Kirchhoff force (p. 700)

Notation

Chapter 1

Schwartz distributions

1.1

Introduction: Dirac’s delta function ı.x/ and its properties

Although the theory of distributions of Laurent Schwartz [8] has diverse applications in various branches of mathematics, we will be primarily concerned with its application in the theory of Sobolev spaces and elliptic partial differential equations. In fact, distributions are also called generalized functions [1], since the theory of distributions in some senses generalizes the notion of function in classical analysis by including not only the usual functions f with point-values (i.e. functions f having point-values f .x/ at the points x) but also new mathematical objects. These include the Dirac distribution ı (popularly, but incorrectly, called the delta function ı.x/) and its derivatives ı .k/ , and also idealized concepts such as the density of a material point, the density of a point charge or dipole moment, the spatial volume density of simple or double layers in electrostatics, the magnitude of an instantaneous force applied at a point, etc. which frequently arise in physics and mechanics. But according to Courant and Hilbert [2, p. 766], ‘the term “Ideal functions” seems much indicative of the true role of this concept : : : This role is indeed that of functions, almost as the role of real numbers is that of ordinary numbers’. It is further stated in [2, p. 767], ‘“Distributions” are most appropriately introduced as ideal elements in function spaces’. Moreover, the theory of distributions provides rigorous mathematical foundations for all the new mathematical objects mentioned above. Indeed, in the late 1920s, Dirac1 introduced the so-called delta function ı.x/ which violated existing mathematical principles, having the following properties [21], [22], [23], [24]: 

ı.x/ is defined, and not only continuous but equal to zero, for all x ¤ 0, i.e. ´ 0 8x ¤ 0 ı.x/ D 1 for x D 0

such that ı.x/ D ı.x/ 8x 2 R, with Z

Z

1

1

ı.x/dx D 1 1 Paul

ı.x/dx D 1:

(1.1.1)

1

Dirac was a Nobel Laureate in Physics and one of the founders of quantum mechanics.

2

Chapter 1 Schwartz distributions

Then, by shifting the origin through a 2 R, ´ 0 8x ¤ a ı.x  a/ D ı.a  x/ D 1 for x D a with

R1

1 ı.x

 a/dx D

R1

1 ı.a

 x/dx D 1.

The ‘most important property’ of ı.x/ [21, p. 58] is given by: 8 functions f continuous on R  1; 1Œ, Z 1 f .x/ı.x  a/dx D f .a/I 2 (1.1.2)



Z

1 1

Z

1

f .x/ı.x  a/dx D 1 Z 1

f .a/ı.x  a/dx D f .a/I

(1.1.3)

1 Z 1

f .x/ı.a  x/dx D 1

f .a/ı.a  x/dx D f .a/: 1

Moreover, for f .x/ D 1 8x and for a D 0, ı.x  a/ D ı.x/ and (1.1.2) reduces to (1.1.1). ı is not only continuous but also infinitely differentiable, such that, for every kR1 times differentiable function f on R, 1 f .t /ı .k/ .x  t /dt D f .k/ .x/ 8k 2 N,



2 An intuitive (though incorrect) derivation of formula (1.1.2): Consider the system .f /n i iDn with fi 2 R and the (Kronecker) Pn delta (ı D .ıij /ni;j n ; ıij D 0 for i ¤ j and ıij D 1 for i D j /. For fixed j , n  j  n, iDn fi ıij D fj . Consider a real-valued function f defined on discrete numbers i D n; .n  1/; : : : ; 0; 1; : : : ; n with f .i / D fi 2 R 8i D n; .n  1/; : : : 0; 1; : : : ; n, and set ı.i  j / D ıij for n  i; j  n. Then n X

fi ıij D

iDn

n X

f .i /ı.i  j / D f .j /

8 fixed j; n  j  n;

( )

iDn

which holds for discrete variables i and j . Now we may intuitively try to extend ( ), which is a correct definition, to the continuous case of a real variable x 2 R D 1; 1Œ, and for any fixed real number a 2 R, by replacing:  ‘i 2 ¹0; ˙1; : : : ; ˙nº’ by ‘x 2 1; 1Œ’; ‘j D 0; ˙1; : : : ; ˙n’ by ‘a 2 1; 1Œ’; 

‘f .i /’ by ‘f .x/’ 2 R, i.e. f is defined for all x 2 R and may be continuous at a;



‘ı.i  j /’ by ‘ı.x  a/’, with ı.x  a/ D 0 8x ¤ a, a 2 R, and ı.x  a/ D 1 for x D a; R1 P ‘ nn .  /’ by ‘ 1 .  /dx’; and Z 1 n X ‘ f .i /ı.i  j / D f .j /’ by ‘ f .x/ı.x  a/dx D f .a/ ’. ( )





n

1

The intuitive definition ( ) is mathematically incorrect (although ( ) is correct), R 1 since the value of the integrand f .x/ı.x  a/ D 0 8x ¤ a, and hence the value of the integral 1 f .x/ı.x  a/dx D 0, not f .a/ 8 fixed a 2 R as claimed in ( ) (see Proposition 1.3.2). The name ‘delta function’ probably follows from this intuitive approach. See Section 1.11 for more interesting details.

Section 1.1 Introduction: Dirac’s delta function ı.x/ and its properties

3

8x 2 R, Z

1

H)

f .t /ı .k/ .a  t /dt D f .k/ .a/

8a 2 R; 8k 2 N;

(1.1.4)

1

where .  /.k/ D

dk.  / . dx k

Another interesting property listed by Dirac [21, p. 61] is xı.x/ D 0. Then, for B constants A and B, A D B H) A x D x C C ı.x/ with unknown constant C , i.e. A B x ¤ x in general. ² 1 for x > 0  For the Heaviside function H.x/ D , 0 for x < 0 

dH .x/ D ı.x/: dx

(1.1.5)

Finally, Dirac listed the ‘remarkable formula’ used in the quantum theory of collision processes:



d ln x D 1=x  {ı.x/ dx

({ D

p 1):

(1.1.6)

The properties of the so-called delta function ı.x/ in (1.1.1) are contradictory, and hence such a function can not exist (see also Proposition 1.3.2 and equations (1.11.6) and (1.11.7)): 1. If ı.x/ D 0 8x ¤ 0 and ı.0/R D 1 (as defined in (1.1.1)), then its Lebesgue/ 1 Riemann integral is zero, i.e. 1 ı.x/dx D 0, but, according to (1.1.1), Z 1 ı.x/dx D 1: (1.1.7) 1

2. ı.x/ D 0 8x ¤ 0 and ı.0/ D 1 ´ 0 8x ¤ 0 8˛ > 0; ˛ı.x/ D 1 for x D 0 Z 1 ˛ı.x/dx D 1; its integral

H) H)

(1.1.8)

1

but, from (1.1.1), Z

Z

1

1

˛ı.x/dx D ˛

8˛ > 0; 1

ı.x/dx D ˛  1 D ˛: 1

In other words, (1.1.7), (1.1.8) and (1.1.9) are contradictory results!

(1.1.9)

4

Chapter 1 Schwartz distributions

Hence, mathematicians immediately pointed out that from the point of view of existing ‘rigorous’ mathematics, all this is nonsense, but Dirac believed ‘. . . advancement in physics is to be associated with a continuous modification and generalization of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation’ ([22] as quoted by [6]), and applied formulae (1.1.1)–(1.1.6) in quantum mechanics with great success to obtain physically meaningful and important results. Consequently, the physics community and the applied mathematicians accepted (1.1.1)–(1.1.6) as ‘correct’. It was perfectly clear to Dirac himself that ı.x/ was not a function in the classical sense of the term. In fact, he stated, ı.x/ is not a function of x according to the usual mathematical definition of a function, which requires a function to have a definite value for each point in its domain, but is something more general, which we may call an “improper function” to show up its difference from a function defined by the usual definition. Thus, ı.x/ is not a quantity which can be generally used in mathematical analysis like an ordinary function. [21, p. 58]. What was actually important for him was to consider ı.x/ and ı .k/ .x/ as ‘symbolic devices’, i.e. how ı.x/ and ı .k/ .x/ act as operators on continuous functions and ktimes differentiable functions f according to the formulae (1.1.1) and (1.1.4) respectively. In fact, it took almost three decades to discover the mathematical foundations of a correct formulation of the definition and properties of Dirac’s delta function, and it turned out that Dirac’s brilliant intuitive results had been right in all cases (1.1.1)– (1.1.6), if: 1. the defining integrals in (1.1.1) and (1.1.2) are considered meaningless and dispensed with; .k/

2. ı and ı .k/ (respectively ıa and ıa ) are considered to be continuous linear functionals (i.e. not functions with point-values) on a suitable test space of functions. (Then, the defining meaningless integrals in (1.1.1)–(1.1.2) can be replaced by the ‘duality pairings’ between the test spaces and their duals). 3. the usual point-wise concept of the derivative of a function is replaced by a new notion of derivatives, in which (1.1.5) and (1.1.6) are to be understood. All of these will be explained eventually in this chapter and the subsequent chapters. In order to obtain a very large space of continuous linear functionals, which were called distributions by Laurent Schwartz3 and contain, in particular, new mathemati3 The history of the discovery of the theory of distributions by Laurent Schwartz is similar to that of the ‘search for iron which led to the discovery of gold’! As in the discovery of X-rays by Röntgen in physics, formula (1.1.2) did not motivate Schwartz to the discovery of distributions. In fact, an altogether different problem of approximations associated with polyharmonic functions posed by Choquet and

5

Section 1.1 Introduction: Dirac’s delta function ı.x/ and its properties .k/

cal objects ı and its derivatives ı .k/ (resp ıa and ıa ), Schwartz introduced the smallest space of ‘sufficiently good’ functions  (i.e. the test space of functions) on which these continuous linear functionals or distributions will be defined, i.e. 8 test functions , ı./ D hı; i D .0/I

ı .k/ ./ D hı .k/ ; i D .1/k  .k/ .0/ .k/

(1.1.10)

.k/

(resp. ıa ./ D hıa ; i D .a/; ıa ./ D hıa ; i D .1/k  .k/ .a/), which will be the definitions instead of (1.1.1), (1.1.2) and (1.1.4) (see also (6.3.22)–(6.3.24) and Section 1.11). Hence, first of all, we will define this ‘smallest’ space of ‘sufficiently good’ functions, i.e. the common set of functions on which all continuous linear functionals are defined. Multi-index notations here:

We first define the multi-index and other notations to be used

N D ¹1; 2; : : : ; n; : : : º; N0 D N [ ¹0º D ¹0; 1; 2; : : : ; n; : : : º; Z D ¹0; ˙1; ˙2; : : : ; ˙n; : : : º; R D the set of all real numbers; C D the set of all complex numbers; 

˛ D .˛1 ; ˛2 ; : : : ; ˛n / with ˛i 2 N0 , i.e. ˛i  0 for 1  i  n; j˛j D ˛1 C    C ˛n  0, i.e. j˛j 2 N0 ; j˛j D 0 ” ˛i D 0 for 1  i  n;





Rn D R    R „  R ƒ‚ … D the linear space of all ordered n-tuples of real numbers; n times

x D .x1 ; x2 ; : : : ; xn / 2 Rn I



x˛ D x1˛1 x2˛2    xn˛n I

(1.1.11)

For derivatives with respect to variables x1 ; x2 ; : : : ; xn , n  2, f .x/ D f .x1 ; x2 ; : : : ; xn /, we use the notations:      @ @f @f @˛ i f @ – D @i .f /I D D @˛i i f I     @xi @xi @xi @xi @xi˛i „ ƒ‚ …



˛i times

–

@j˛j f @x1˛1 @x2˛2 : : : @xn˛n

D @˛1 1 : : : @˛nn f D @.˛1 ;:::;˛n / f D @˛ f I

(1.1.12)

– @˛ f  f for ˛ D 0, i.e. ˛1 D ˛2 D    D ˛n D 0; – for n D 1 and f .x/, we use the usual notations: 

df d ˛ f , dx dx ˛

8˛ 2 N;

other equivalent notations for derivatives of f (n  2, f .x1 ; x2 ; : : : ; xn /) are:

Deny was solved by Schwartz for higher dimensions n > 2, leading him to the theory of distributions for which he was awarded the Fields Medal (the highest honour for original mathematical creativity and discovery, mathematics’s Nobel prize).

6

Chapter 1 Schwartz distributions

–

Di f  @i f I Di˛i f D @˛i i f I D ˛ f D @˛ f I etc. 8˛ D .˛1 ; : : : ; ˛n /I (1.1.13)

– for ˛ D 0, i.e. ˛1 D ˛2 D    D ˛n D 0, D ˛ f  f ; – for n D 1 and f .x/, Df D

df , dx

D˛ f D

d˛f dx ˛

8˛ 2 N; for ˛ D 0, D ˛ f  f ;

– ˛ D .˛1 ; : : : ; ˛n /, ˇ D .ˇ1 ; : : : ; ˇn /; ˛  ˇ ” ˛i  ˇi 81  i  n; – ˛Š D ˛1 Š ˛2 Š    ˛n Š.

1.2

Test space D./ of Schwartz

1.2.1 Support of a continuous function Let   Rn be an open subset of Rn and C 0 ./ be the linear space of functions  continuous on . Then the support of , denoted by supp./, is the closure of the set ¹x W x 2 ; .x/ ¤ 0º in Rn , i.e. supp./ D ¹x W x 2 ; .x/ ¤ 0º

in Rn .

(1.2.1)

For example, 1. for  D 0; Œ  R, 1 .x/ D sin x 8x 2 0; Œ H) supp.1 / D 0; Œ D Œ0;  in R H) supp.1 / D  with   supp.1 /; 2. for  D R, 2 .x/ D e x 8x 2 R H) supp.2 / D R  R (R is also closed!); 3. for  D ; Œ, 3 .x/ D cos x 8 jxj < =2 and 3 .x/ D 0 otherwise on  H) supp.3 / D =2; =2Œ D Œ=2; =2    RI 4. for  D B.0I 2/q D ¹.x1 ; x2 / W .x1 ; x2 / 2 R2 ; .x12 C x22 /1=2 < 2º,

4 .x1 ; x2 / D C 1  x12  x22 8.x1 ; x2 / 2 B.0I 1/ and 4 .x1 ; x2 / D 0 otherwise H) supp.4 /  B.0I 2/.

Hence, according to our definition (1.2.1), supp./ may be a proper subset of , or  may be a proper subset of supp./, or supp./ D  D Rn , Rn being both open and closed. We note the following:  

supp./ is a closed set. For  D Rn , let supp./ ¤ Rn . Then, supp./ is the smallest closed subset of Rn outside which  vanishes. Conversely, Rn n supp./ D the complement of supp./ in Rn is the largest open subset (1.2.1a) of Rn in which  vanishes.

7

Section 1.2 Test space D./ of Schwartz

Let supp./    Rn , and supp./ be a bounded subset of . Then supp./ is a compact (i.e. closed and bounded) subset of , which we denote by the notation:



supp./   (i.e.  ” compact subset):

(1.2.2)

Let C k ./; C0k ./; C 1 ./, 8k 2 N0 , be the spaces defined by: C k ./ D ¹ W @˛  2 C 0 ./ 8j˛j  kºI C0k ./ D ¹ W  2 C k ./; supp./  º D C k ./ \ C0 ./:

(1.2.3)

For k D 0, C0 ./  C00 ./, C00 ./ D ¹ W  2 C 0 ./; supp./  º:

(1.2.4)

For example, 3 in (3) above belongs to C0 .; Œ/, but 30 … C01 .; Œ/, since  0 … C 1 .; Œ/ ( 0 .x/ D  sin x on =2; =2Œ and  0 .x/ D 0 otherwise H)  0 . 2 C / D 0,  0 . 2  / D 1). C 1 ./ D ¹ W  2 C k ./ 8k 2 N0 º D

1 \

C k ./

kD0

D the set of all continuous functions whose partial derivatives of all orders with respect to all the variables are continuous in . Then, C01 ./ D ¹ W  2 C 1 ./; supp./  º D C 1 ./ \ C0 ./:

(1.2.5)

As an example of a function  2 C01 ./, we consider: Example 1.2.1. Let  D R D 1; 1Œ, and let  be a function constructed from Cauchy’s infinitely differentiable function, which assumes the value exp.1=t 2 / for t > 0 and the value zero for t  0, by: ´

.

e .x/ D 0

1 / 1x 2

for jxj < 1 for jxj  1

Then,  2 C01 .R/ with supp./ D Œ1; 1  R (Figure 1.1).

(1.2.6)

8

Chapter 1 Schwartz distributions

(x) 1/e

-1

0

x

1

Figure 1.1 Graph of .x/ with supp./ D Œ1; 1

1 Proof. #1;1Œ .x/ D exp. .1x 2 / / is continuous and infinitely differentiable for jxj < 1, and  D 0 is continuous and infinitely differentiable for jxj > 1 H)  2 C 1 .R n ¹1; 1º/; i.e.  is continuous and infinitely differentiable on R n ¹1; C1º. It remains to show that  is, in fact, continuous and infinitely differentiable at ˙1 also – then  will belong to C01 .R/, with supp./ D Œ1; 1. Since  is a symmetric function in x, it is sufficient to show the continuity and infinite differentiability of  at x D C1 only. 1

Continuity of  at x D C1: limx!1 .x/ D limx!1 .e .1x2 / / D 0 and lim .x/ D lim 0 D 0

x!1C

x!1C

H) H)

lim .x/ D lim .x/ D .1/ D 0

x!1

x!1C

 is continuous at x D 1:

Infinite differentiability of  at x D C1: Set u.x/ D 1  x 2 so that .x/ D e 1=u d for jxj < 1. Then, du D 2x, dx . u1 / D 2xu2 ,  0 .x/ D 2xu2 e 1=u H) dx 8jxj < 1,  0 .x/ D p1 .x/u2 e 1=u , where p1 .x/ D 2x is a polynomial in x with the subscript ‘1’ in p1 .x/ corresponding to the order of the derivative in  0 .x/. Then, 8jxj < 1,  00 .x/ D Œ.p1 .x//0 u2 C .4x/p1 .x/u C .p1 .x/2 u4 e 1=u D p2 .x/u4 e 1=u , where p2 .x/ is a polynomial in x with the subscript ‘2’ corresponding to the order of  00 .x/. Thus, by induction, we get, 8k 2 N,  .k/ .x/ D pk .x/u2k e 1=u ; 8jxj < 1, where pk .x/ is a polynomial in x with the subscript k corresponding to the order of  .k/ .x/. Hence, 8k 2 N, limx!1  .k/ .x/ D limx!1 Œpk .x/u2k e 1=u  D pk .1/  0 D 0 (by L’Hospital’s rule, e 1=u  u2k ! 0 as x ! 1 ). Again, 8jxj > 1, .x/ D 0 H) 8jxj > 1, 8k 2 N,  .k/ .x/ D 0 H) limx!1C  .k/ .x/ D 0 8k 2 N. Thus,  .k/ .x/ is defined for 0 < jx  1j < r with r > 0, and limx!1  .k/ .x/ D limx!1  .k/ .x/ D limx!1C  .k/ .x/ D 0 8k 2 N. D limx!1  0 .1 C 1 .x  1// D Hence, for k D 1,  0 .1/ D limx!1 .x/.1/ x1 0 0 ( 1 2 0; 1Œ). Thus,  .1/ is well defined and  0 .1/ D 0. Similarly,  00 .1/ D 0 0 .1/ D limx!1  00 .1 C 2 .x  1// D 0. 2 2 0; 1Œ/. Thus, in general, limx!1  .x/ x1

9

Section 1.2 Test space D./ of Schwartz

if it is known that for some k 2 N;  .k/ .1/ D 0, then  .k/ .x/   .k/ .1/ x!1 x1

 .kC1/ .1/ D lim

D lim  .kC1/ .1 C k .x  1// D 0 .0 < k < 1/: x!1

Hence, we have .1/ D  0 .1/ D    D  .k/ .1/ D    D 0. As explained earlier, by virtue of the symmetry of , .1/ D  0 .1/ D    D  .k/ .1/ D    D 0, i.e.  2 C01 .R/ with supp./ D Œ1; 1  R, with  .k/ .x/ D 08k 2 N0 ; 8x outside Œ1; 1. In general, we can define, 8" > 0: 8 < . 2"2 2 / e " x " .x/ D :0

for jxj < " for jxj  ":

(1.2.6a)

Then, " 2 C01 .R/ with supp." / D Œ"; "  R. .k/ " .x/ and its derivatives " .x/8k 2 N vanish outside Œ"; ". Example 1.2.2. For n  2, let  D Rn ; kxk2 D x12 C x22 C    C xn2 , 8" > 0, " W Rn ! R be defined by: 8 "2 < . "2 kxk 2/ e for kxk < " " .x/ D (1.2.6b) :0 for kxk  ": Then, " belongs to C01 .Rn / with supp." / D B.0I "/ D ¹x W x 2 Rn ; kxk  "º  Rn . " .x/ D 0; @˛ " .x/ D 0 8j˛j 2 N and 8x outside B.0I "/.

1.2.2 Space D./ Definition 1.2.1. Let   Rn be an open subset of Rn . Then the space C01 ./ (1.2.5) is called the test space (or, equivalently, the space of test functions) and denoted by D./ (using the notation of Schwartz [8]); i.e. D./ D C01 ./ D ¹ W  2 C 1 ./; supp./  º if it is equipped with the following notion of convergence: A sequence .n /1 nD1 in D./ is said to converge to  2 D./ if and only if: I. 9 a compact subset K   such that supp.n  /  K 8n 2 N (i.e. the supports of all .n  / are contained in the same compact set K (8n 2 N/). II. @˛ n ! @˛  uniformly in  as n ! 1 8 fixed ˛ with j˛j 2 N0 (1.2.7) (i.e. the derivatives of any given order j˛j 2 N0 of n converge uniformly to the corresponding derivatives of  of the same order j˛j as n ! 1, and for ˛ D 0, with j˛j D 0; @˛ n D n ; @˛  D ; n !  uniformly as n ! 1 (see also Remark 1.2.1)).

10

Chapter 1 Schwartz distributions

Examples of convergence in D./ For the sake of simplicity, we will consider the case  D R. Let  2 D.R/ be defined by (1.2.6) with supp./ D Œ1; 1, which is used to define the sequences in the following examples. 1 Example 1.2.3. Let .n /1 nD1 be a sequence in D.R/ defined by: n .x/ D n .x/ 8x 2 R; 8n 2 N. Hence, supp.n / D supp./ D Œ1; 1  R8n 2 R, and we ˛ can choose K D Œ1; 1  R such that supp.n /  K8n 2 N and ddx˛n ! 0 (null function) uniformly in R as n ! 18˛  0 H) n ! 0 in D.R/ as n ! 1 (Figure 1.2).

1/e

1(x) 2(x)

1/2e

3(x)

-1

1/3e

0

1

x

Figure 1.2 Graphs of n .x/ D n1 .x/ with supp.n / D Œ1; 1; n D 1; 2; 3; : : :

Counterexample 1.2.4. 8n 2 N, let n be defined by: ´ 2 1 1 x exp . nn for jxj < n 2 x 2 / n n .x/ D . / D n n 0 for jxj  n Then .n /1 nD1 is a sequence in D.R/ with supp.n / D Œn; n  R8 fixed n 2 N. But supp.n / D Œn; n ! 1; 1Œ as n ! 1 H) there does not exist any compact subset K  R containing supp.n / for all n 2 N. Hence, .n /1 nD1 does not converge to the null function 0 in D.R/, although ˛ d n ! 0 uniformly in R as n ! 1 8˛  0 (Figure 1.3). dx ˛ Example 1.2.5. For fixed non-zero a 2 R and 8n 2 N, let n be defined by:   ´1 2 x 1 exp . aa for jxj < a 2 x 2 / n .x/ D  D n n a 0 for jxj  a

11

Section 1.2 Test space D./ of Schwartz

1/e 1(x)

2(x) 3(x)

-3

-2

-1

0

1

2

3

x

Figure 1.3 Graphs of n .x/ D n1 . xn / with supp.n / D Œn; n; n D 1; 2; 3; : : :

H) .n /1 nD1 is a sequence in D.R/ with supp.n / D Œa; a  R8n 2 N. Hence, we can choose K D Œa; a  R satisfying supp.n /  K 8n 2 N ˛ and ddx˛n ! 0 uniformly in R as n ! 1 8˛  0 H) n ! 0 in D.R/ as n ! 1. Counterexample 1.2.6. 8n 2 N, let n be defined by: ´ 1 1 exp . 1.xn/ for jx  nj < 1 1 2/ n .x/ D .x  n/ D n n 0 for jx  nj  1 H) n 2 D.R/ with supp.n / D ¹x W x 2 R; jx  nj  1º D Œn  1; n C 1 8 fixed n 2 N. But there does not exist any compact set K  R such that supp.n /  K 8n 2 N. Hence, .n /1 nD1 does not converge to the null function 0 in D.R/, although ˛ d n ! 0 uniformly in R as n ! 1 8˛  0 (Figure 1.4). dx ˛ Counterexample 1.2.7. 8n 2 N, let n be defined by: ´ 1 1 1 exp. 1.nx/ 1 2 / for jnxj < 1. i.e. for jxj < n / n D .nx/ D n n 0 for jxj  n1 H) n 2 D.R/ with supp.n / D B.0I n1 / 8n 2 N, i.e. n has support shrinking to 0 as n ! 1. Hence, condition I in Definition 1.2.1, supp.n /  Œ1; 1, holds k 8n 2 N, but for k  2, ddxkn does not converge to 0 on any neighbourhood of 0 as n ! 1, i.e. condition II in (1.2.7) does not hold.

12

Chapter 1 Schwartz distributions

1/e 1

(x), supp( 1) = [0,2]

1/2e

2

(x), supp( 2) = [1,3]

1/3e 3

-1

0

1

2

3

(x), supp( 3) = [2,4]

4

x

Figure 1.4 Graphs of n .x/ D n1 .x  n/ with supp.n / D Œn  1; n C 1; n D 1; 2; 3; : : :

Remark 1.2.1. 

Since the uniform convergence of derivatives of all orders are involved in this notion of convergence, it is a notion of convergence of infinite order.



The derivatives of each order taken separately converge uniformly, i.e. derivatives of all orders need not simultaneously converge uniformly.



The elements  2 D./ are called test functions.







 D 0 2 D./, since the null function 0 2 C 1 ./, supp.0/ D ;, the empty set, which is a compact subset of .  2 D./ H) 9 a compact subset K   such that .x/ D 0; @˛ .x/ D 0 8x 2  n K (i.e. x lying outside K) 8j˛j 2 N.  is a bounded domain with boundary ;  2 D./. Hence: I. .x/ D 0, @˛ .x/ D 0 8x 2 , 8j˛j 2 N. II. For sufficiently small " > 0, 9 an "-boundary layer of  in  such that .x/ D 0, @˛ .x/ D 0 8j˛j 2 N, 8x belonging to the "-boundary layer of  in . For sufficiently small " > 0, define " D ¹x W x 2 ; d.x; / D inf d.x; y/ > "º  : y2

Then  n " is called an "-boundary layer of  in  (Figure 1.5). For more details, see Theorems A.0.5.1 and A.0.5.3, Appendix A.

(1.2.8)

13

Section 1.2 Test space D./ of Schwartz

Figure 1.5 "-boundary layer of  in 

1.2.3 Space D m ./ Definition 1.2.2. 8m 2 N0 , the space C0m ./ defined by (1.2.3), C0m ./ D ¹ W @˛  2 C 0 ./8j˛j  m; supp./  º; will be denoted by D m ./  C0m ./ if it is equipped with the following notion of m m convergence: a sequence .n /1 nD1 in D ./ converges to  2 D ./ if and only if: I. 9 a compact subset K   of  such that supp.n  /  K

8n 2 NI

(1.2.9)

II. @˛ n ! @˛  uniformly in  as n ! 1 8j˛j  m. Convergence of a sequence in D./ H) its convergence in D m ./ For example, the sequence .n / in Examples 1.2.3 and 1.2.5, which converges in D.R/, also converges in D m .R/, whereas the sequences in Examples 1.2.4 and 1.2.6 do not converge in both D.R/ and D m .R/, since condition I for convergence is the same in D.R/ and D m .R/ 8m 2 N0 . D.R/ ,! D m .R/ m

D ./  D./

8m 2 N0 with continuous imbedding

(1.2.10)

”

(1.2.11)

m D 1:

1.2.4 Space DK ./ Definition 1.2.3. 8 fixed compact subsets K   of   Rn ; DK ./ is defined by the set: DK ./ D ¹ W  2 D./; supp./  Kº: S Then, D./ D K DK ./.

(1.2.12)

14

Chapter 1 Schwartz distributions 

8 multi-index ˛ with j˛j 2 N0 , semi-norms p˛ on DK ./ are defined by: p˛ ./ D max j@˛ .x/j x2K

8 2 DK ./:

(1.2.13)



The family ¹p˛ º of all these semi-norms defines a topology on DK ./.



D./ is the strict inductive limit of the spaces DK ./ (see [8], [9], [10], [11]).



The topology of D./ is defined with the help of the topology of DK ./ (see [8], [9], [25]).

Now we will collect the important properties of D./.

1.2.5 Properties of D./ Property 1: D./ is a linear space: 1 ; 2 2 D./, ˛1 ; ˛2 2 R H) ˛1 1 C ˛2 2 2 D./,  D 0 2 D./. Property 2: D./ is not normable. D./ is a locally convex topological vector space. (1.2.14) Property 3: Topology on D./: For this, it is necessary to introduce the notion of inductive limit topology, for which we refer to Schwartz [8, p. 66], Horvath [9, p. 132], and Trèves [25]. But in our treatment and for the applications in this book, it is sufficient to know the concept of convergence in D./ from (1.2.7), which is, in fact, called the pseudo-topology of Schwartz in D./ by Lions [13], Lions–Magenes [15], Neˇcas [16], etc.4 Property 4: 1 ; 2 2 D./

H)

1 2 2 D./

(1.2.15)

with supp.1 2 / D supp.1 / \ supp.2 /I  2 D./;

2C

1

./

H)

 2 D./

(1.2.16)

with supp. / D supp. / \ supp./: Property 5:

 2 D./

H)

@˛  2 D./

8j˛j 2 N:

(1.2.17)

Property 6: n !  in D./ in the sense of (1.2.7) H) @˛ n ! @˛  in D./ 8j˛j 2 N. (1.2.18) 4 Moreover, Trèves also writes in [10], ‘In teaching it [sophisticated functional analysis of topological vector spaces] can easily be, and most of the time is, bypassed. Most of the basic tenets of the theory can be stated and proved using solely [sequences] of test functions or distributions. The great success and usefulness of distribution theory lies in its simplicity and in the easy, automatic nature of operations. Many accused it of being shallow : : : With the easy part taken care of, analysts could push further and take care of the finer and more difficult points.’

15

Section 1.2 Test space D./ of Schwartz

Property 7: For sufficiently small d > 0, the d -neighbourhood of a compact set K   is the set Kd of points in  whose distance from K is  d , i.e. Kd D ¹x W x 2 ; d.xI K/ D inf d.x; y/  d º: y2K

(1.2.19)

Then, Kd   is a compact set containing K  Kd (Figure 1.6). For more details on the properties of compact sets in Rn , see Section A.0.5, Appendix A. K d with K Kd by the boundary

K

for d

0 enclosed

d

d : Boundary of K d K d : d-neighbourhood of compact set K

Figure 1.6 d -neighbourhood of Kd in  with K  Kd   for d > 0

Theorem 1.2.1 (Approximation Theorem). Let f 2 C0 ./  C00 ./ (1.2.4) be a continuous function with compact support K D supp.f /  , and Kd   be a d -neighbourhood of K with d > 0 as defined by (1.2.19). Then, for any " > 0; 9 a function  2 D./ with supp./  Kd such that supx2 jf .x/  .x/j  ", i.e. 8f 2 C0 ./; 9 2 D./ arbitrarily close to f . (1.2.20) In other words, D./ is dense in C0 ./. Proof. Let f 2 C0 ./ with compact supp.f / D K  , and let fQ be its null extension to Rn , i.e. fQ.x/ D f .x/ 8x 2  and fQ.x/ D 0 8x 2 Rn n , supp.fQ/ D supp.f / D K, fQ 2 C0 .Rn /. Using regularizing functions ı (see Definition 6.2.1) with the properties (6.2.17)– 2 / for kxk < ı and (6.2.19): 8ı > 0; ı .x/ D k1 . ıx /, where . ıx / D exp. ı 2ı kxk2 x N ı/ D ¹x W kxk  ıº;

. ı / D 0 for kxk  ı, 0  ı .x/  1, supp. ı / D B.0I R R R x n Q Rn ı .x/d x DR 1, ı 2 D.R /, k D Rn . ı /d x, define ı .x/ RD Rn f .x  / ı ./d  D Rn fQ./ ı .x  /d  (by change of variables) D K f ./ ı .x  /d  8x 2 Rn , since supp.fQ/ D supp.f / D K. Then ı 2 C01 .Rn / with supp. ı /  Kd , if we choose 0 < ı  d (see the proof N ı/; ı of Lemma 6.2.1 for all the details); since ı 2 C01 .Rn / with supp. ı / D B.0; can be differentiated indefinitely withR respect to variables x1 ; x2 ; : : : ; xn , which can be carried out under the integral sign , i.e. R ˛ 1 n  8j˛j 2 N, @˛ ı .x/ D K f ./@x Œ ı .x  /d  H) ı 2 C .R /; R  .x  / D 0 for kx  k  ı for all  2 K H) ı ı .x/ D K f ./ ı .x  /d  D 0 for x … Kı with Kı D ¹x W d.xI K/ D inf2K kx  k  ıº. But

16

Chapter 1 Schwartz distributions

x … Kı H) for all  2 K, kx  k  inf2K kx  k > ı. Hence supp. ı / will be contained in Kd  , i.e. supp. ı /  Kd  , if we choose ı  d. (1.2.20a) R Since Rn ı ./d  D 1, Z Z fQ.x/  ı .x/ D fQ.x/ fQ.x  / ı ./d 

ı ./d   Rn Rn Z Z Q Q D Œf .x/  f .x  / ı ./d  D ŒfQ.x/  fQ.x  / ı ./d  Rn

kkı

(1.2.20b) (since ı ./ D 0 for kk  ı). But fQ is continuous in Rn and supp.fQ/ D supp.f / D K is compact. Hence, fQ is uniformly continuous and consequently, 8" > 0; 9 > 0 such that jfQ.x/ fQ.x/j  " 8kk  . Then jfQ.x/  fQ.x  /j  " 8kk  ı   (if we choose ı  ) H) from (1.2.20b), Z Z jfQ.x/  ı .x/j  jfQ.x/  fQ.x  /j ı ./d   "

ı ./d  D "  1 D " Rn

kkı

(1.2.20c) for 0 < ı  . Define  D ı # with ı 2 C01 .Rn /, supp. ı /  Kd  , 0 < ı  d and ı   (by virtue of (1.2.20a) and (1.2.20c)). Then,  2 C01 ./ with supp./ D supp. ı /  Kd   and, 8x 2 , jf .x/  .x/j D jfQ.x/  ı .x/j  " for 0 < ı  d and ı  . Thus, for 0 < ı  d and ı  , 9 2 D./ with supp./  Kd such that supx2 jf .x/  .x/j  ". Theorem 1.2.2. Let   Rn be an open subset of Rn and L1loc ./ be the space of locally integrable functions on , i.e. Z 1 Lloc ./ D ¹f W 8K   jf .x/j d x < C1; KV D i nt .K/º: (1.2.21) KV

R

If u 2 L1loc ./ and  u d x D 08 2 C0 ./, then u.x/ D 0 almost everywhere (a.e. – see Definition B.1.3.1 in Appendix B) on . (1.2.21a) Proof. Following the interesting proof in [26, pp. 61–63], we give the proof in two steps (see also [12, p. 60]). Step 1. Let  be a bounded open subset of Rn and u 2 L1 ./. In Appendix B, the density of C0 ./ in L1 ./ is given by Theorem B.3.3.4. Hence, for any given " > 0; 9 2 C0 ./ such that ku  kL1 ./  ". (1.2.21b)

17

Section 1.2 Test space D./ of Schwartz

Then, using (1.2.21a) and (1.2.21b) and Hölder’s inequality for .u  / 2 L1 ./ and  2 C0 ./  L1 ./, we get ˇ ˇZ ˇ ˇZ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ d x  ud xˇ D ˇ .  u/d xˇˇ  k  ukL1 ./  kkL1 ./ ˇ    ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ H) ˇ dx  0ˇ D ˇ d xˇˇ  "kkL1 ./ 8 2 C0 ./: 



(1.2.21c) Now we will construct a function 0 2 C0 ./ with 1  0 .x/  1 8x 2  and some additional properties to be explained now. For fixed " > 0 satisfying (1.2.21b), define subsets A and B of  by: A D ¹x W x 2 ; .x/  "º;

B D ¹x W x 2 ; .x/  "º:

(1.2.21d)

Then A and B are closed subsets of a bounded set   Rn (see the proof of Proposition A.1.3.2 in Appendix A) H) A and B are compact subsets of , i.e. A  ; B  . Again, x 2 A H) .x/  " H) x … B and vice versa. Hence, A and B are disjoint, closed sets, i.e. A \ B D ;. Moreover, A; B  K D supp. /  . Using Proposition A.1.3.2 in Appendix A, we can define 0 2 C0 ./ such that 0 .x/ D C1 for x 2 A, 0 .x/ D 1 for x 2 B and 1  0 .x/  1 (1.2.21e) 8x 2  with k0 kL1 ./ D supx2 j0 .x/j D 1. Set E D A[B. Then, from (1.2.21d) and (1.2.21e), for fixed " > 0, with (1.2.21b), j .x/j  "8x 2 E, j .x/j < "8x 2  n E, j .x/j D .x/0 .x/ 8x 2 E and j .x/0 .x/j  j .x/j 8x 2  n E. Z Z Z Z j .x/jd x D .x/0 .x/d x D .x/0 .x/d x  .x/0 .x/d x E

E

ˇZ ˇ  ˇˇ  ˇZ ˇ  ˇˇ 



ˇ ˇZ ˇ ˇ .x/0 .x/d xˇˇ C ˇˇ ˇ Z ˇ .x/0 .x/d xˇˇ C

nE

nE

nE

ˇ ˇ .x/0 .x/d xˇˇ

ˇ ˇ ˇ ˇ ˇ .x/0 .x/ˇd x: ˇ ˇ

(1.2.21f)

R But j  .x/0 .x/d xj R  "k0 kL1 ./ D R " (using first (1.2.21c) with R  D 0 and then (1.2.21e)). Then E j .x/jd x  "C nE j .x/0 .x/jd x  "C nE j .x/jd x (using (1.2.21e) and (1.2.21f)). Hence, Z Z Z j .x/jd x D j .x/jd x C j .x/jd x 

E



nE

 Z j .x/jd x C

Z

 "C nE

Z

j .x/jd x nE

Z

j .x/jd x  " C 2"

D"C2 nE

d x  "Œ1 C 2./; nE

18

Chapter 1 Schwartz distributions

since j .x/j < "8x 2  n E and vol. n E/  vol./ D ./ ( being the n-dimensional Lebesgue volume measure) H) k kL1 ./  "Œ1 C 2./. Finally, kukL1 ./ D k.u 

/C

kL1 ./  ku 

kL1 ./ C k kL1 ./

 " C "Œ1 C 2./ D "Œ2 C 2./ 8 fixed " > 0 satisfying (1.2.21b) H) kukL1 ./  lim"!0C Œ".2 C 2./ D 0, i.e. kukL1 ./ D 0 H) u.x/ D 0 almost everywhere on . Step 2. Now we will consider the general case of   Rn and u 2 L1loc ./ (i.e. u maySor may not belong to L1 .//. Let   Rn be an open subset of Rn . Set  D m m with m 2 N, where m is a bounded subset of  and m is a compact subset of  8m 2 N, i.e. m  m   for m 2 N. In fact, for m 2 N; m can be constructed, for example, as follows: let c be the complement of  in Rn , i.e. c D ¹x W x 2 Rn ; x … º. 1 Then, for m 2 N, define m D ¹x W x 2 ; d.xI c / > mC1 and kxkRn < mº. To fix our ideas, consider the simple, i.e. one-dimensional, case. For  D 0; 1Œ, c D  1; 0, and 1 D  12 ; 1Œ, 2 D  13 , 2Œ; : : : ; m D 1 1 ; mŒ; : : : ; 1 D Œ 12 ; 1, 2 D Œ 13 ; 2; : : : ; m D Œ mC1 ; m; : : : being the corre mC1 S sponding compact subsets of 0; 1Œ. Then, 0; 1Œ D m2N m . Let u 2 L1loc ./. Then, u#m 2 L1 .m / 8m 2 N. Define C0 .m / D ¹ W  2 C0 ./; supp./  m º. Hence, 8m 2 N,  2 C0 .m / H)  2 C0 ./ and, 8 fixed m, Z Z .u#m /dx D udx D 0 8 2 C0 .m /: (1.2.21g) m



Since, 8 fixed m; m is a bounded domain, u#m 2 L1 .m / satisfying (1.2.21g), from Step 1, u#m D 0 almost everywhere on m . For fixed m, define Em D ¹x W x 2 m ; u#m .x/ ¤ 0º. Then u#m D 0 a.e. on m H) S the n-dimensional (Lebesgue) volume measure .Em / D 0 8m 2 N. Set E0 DP m Em . Then E0 D ¹x W x 2 ; u.x/ ¤ 0º with .E0 / D 0, since 0  .E0 /  m .Em / D 0. Hence, u.x/ D 0 for x 2  n E0 with .E0 / D 0 H) u.x/ D 0 for almost all x 2 . CorollaryR 1.2.1. Let   Rn be an open subset of Rn and u 2 Lp ./, 1  p  1, such that  u.x/d x D 0 8 2 C0 ./. Then u.x/ D 0 almost everywhere on . Proof. Let u 2 Lp ./; 1  p  1. Then, 8 bounded 0   with 0  , u#0 2 Lp .0 / ,! L1 .0 /, 1  p  1, by the imbedding result for Lp -spaces 1 Ron every bounded domain 0 ((B.4.1.12) in Appendix B). Hence, u 2 Lloc ./ and  ud x D 08 2 C0 ./. Then, by Theorem 1.2.2, u.x/ D 0 almost everywhere on .

19

Section 1.2 Test space D./ of Schwartz

Theorem 1.2.3. Let   Rn be an open (bounded or unbounded) subset of Rn . Then C0 ./ is a dense subspace of Lp ./ for 1  p < 1. Proof. For p D 1, see the proof of Theorem B.3.3.4 in Appendix B. Hence, it p remains to prove the result 7! R for 1 < p < 1. The mapping v 2 L ./ hu; vi.Lp .//0 Lp ./ D  uvd x is a bounded, linear functional on Lp ./ for p u 2 .Lp .//0  Lq ./ with 1 < q D p1 < 1; 1 < p < 1, since, by virtue of R Hölder’s inequality, jhu; vij D j  uvd xj  kukLq /./ kvkLp ./ 8v 2 Lp ./; 1 < p < 1. As a consequence of the Hahn–Banach theorem, C0 ./  Lp ./; 1 < p < 1, will be a dense subspace of Lp ./ if the vanishing of this bounded, linear functional on C0 ./R  Lp ./; 1 < p < 1, implies that it is a null functional on Lp ./, q p 0 i.e. if hu; i D  ud x D 0 8 2 C0 ./, then u D R 0 in L ./  .L .// . Hence, we assume that hu; i.Lp ./0 Lp ./ D  ud x D 0 8 2 C0 ./  p L ./, 1 < p < 1. We are to prove that u D 0 in .Lp .//0 D Lq ./ with 1 < p p q D p1 < 1. In fact, u 2 Lq ./  .Lp .//0 with q D p1 ; 1 < p < 1. Then, q 1 8 bounded 0   with 0  , u#0 2 L .0 / ,! L .R0 / by the imbedding result for bounded domain 0 H) u 2 L1loc ./ and hu; i D  ud x D 0 8 2 C0 ./ H) u D 0 a.e. on  by Theorem 1.2.2 H) u D 0 in Lq ./  .Lp .//0 , 1 < p < 1. Hence, C0 ./ is a dense subspace of Lp ./; 1 < p < 1, and the result follows. Remark 1.2.1A. For p D 1, C0 ./ is not a dense subspace of L1 ./. Property 8: Theorem 1.2.3A. Let   Rn be an open subset of Rn and L1loc ./ be the space of locally integrable functions on . R 1 If u 2 Lloc ./ and  u.x/.x/ D 0 8 2 D./, then u.x/ D 0 almost everywhere on . (1.2.22) Proof. Let u 2 L1loc ./ satisfy (1.2.22). Let 2 C0 ./R be any continuous function with compact supp. / D K  . Then we show that  u.x/ .x/d x D 0. By the Approximation Theorem 1.2.1 in Property 7, for any " > 0 and d > 0; 9 2 D./ with supp./  Kd ; supp. / D K  Kd  , such that supx2 j .x/  .x/j  ". Since .x/  .x/ D 0 8x 2  n Kd , ˇ ˇZ Z Z ˇ ˇ ˇ ˇ u.x/. .x/  .x//d xˇ  sup j .x/  .x/j ju.x/jd x  " ju.x/jd x: ˇ 

x2

Kd

Kd

(1.2.23) R

R

R

But  u.x/.x/d x D 0 H) j  u.x/ .x/d xj  " Kd ju.x/jd x ! 0 as " ! 0 R with fixed u and d > 0. Hence,  u.x/ .x/d x D 0 8 2 C0 ./. Now, applying Theorem 1.2.2, we get the result (1.2.22).

20

Chapter 1 Schwartz distributions

Property 9: Corollary 1.2.2. For 1  p  1; u 2 Lp ./, Z u.x/.x/dx D 0 8 2 D./ H) u D 0 in Lp ./ (i.e. u.x/ D 0 a.e. on ): 

(1.2.24) Proof. From the proof of Corollary 1.2.1, Lp ./  L1loc ./ H) the result by Theorem 1.2.3A. Property 10: Density Results 

D./ is dense in D m ./ (see Definition 1.2.2) 8m 2 N0 with D./ ,! (1.2.25) D m ./, the imbedding being a dense, continuous one.

Theorem 1.2.3B. D./ is dense in Lp ./ for 1  p < 1. In particular, D./ is dense in L2 ./.

(1.2.26) (1.2.27)

Proof. The proof is given in Chapter 6 using convolutions. See the proof of Theorem 6.8.3. 

D./ is not dense in L1 ./, nor in C 1 ./, nor in C m ./ 8m 2 N0 .

Here, we show that D./ is not dense in C m ./ 8m 2 N0 , where C m ./ D ¹ W is bounded and uniformly continuous in  8j˛j  mº is a Banach space for the norm k  kC m ./ or jjj  jjjC m ./ defined by: @˛ 

kkC m ./ D kkm;1; D jjjjjjC m ./ D

m X

sup j@˛ .x/j;

or

j˛jD0 x2

max sup j@˛ .x/j;

0j˛jm x2

(1.2.28)

both the norms in (1.2.28) being equivalent (see Definition A.4.1.3 and (A.4.1.7) (equivalent norms) in Appendix A). Proof. Let .n /1 nD1 be a sequence in D./ with supp.n / D Kn   8n 2 N. Let u 2 C m ./ such that ku  n kC m ./ D

D

m X

sup j@˛ .u  n /.x/j

j˛jD0 x2 m X j˛jD0

² max

sup x2Kn

³ j@˛ u.x/j; sup j@˛ .u  n /.x/j : (1.2.29) x2Kn

21

Section 1.2 Test space D./ of Schwartz

For ku  n kC m ./ < " with " > 0, it is necessary that m X

sup

j@˛ u.x/j < ";

(1.2.30)

j˛jD0 x2Kn

which must hold 8" > 0 and 8n > n0 ."/; n0 ."/ 2 N. But this will be possible only if u 2 C m ./ satisfies some additional conditions. In other words, 8u 2 C m ./, (1.2.30) will not hold and, consequently, there does not exist a sequence .n / in D./ such that n ! u in C m ./ in general. Hence D./ is not dense in C m ./ 8m 2 N0 . Identification of additional conditions such that (1.2.30) holds 1. If  is a bounded domain with a sufficiently smooth boundary , then the condition (1.2.30) implies that u and all derivatives @˛ u of order j˛j  m must vanish on the boundary . (1.2.31) 2. If  D Rn , then (1.2.30) implies that u.x/ and @˛ u.x/ must vanish as kxk ! 1 8j˛j  m. (1.2.32) If u 2 C m ./ possess either additional property 1 or 2, then there will exist a sem quence .n /1 nD1 in D./ such that n ! u in C ./ as n ! 1 (see also [27, p. 70]). (1.2.33) Property 11: Poincaré–Friedrichs inequality Theorem 1.2.4. Let  2 D.Rn /. Then, the following properties hold: I. 8i D 1; 2; : : : ; n, Z

2

Z

j.x/j d x D 2 Rn

Rn

xi .x/

@ .x/d x: @xi

(1.2.34)

II. For bounded open subset   Rn ; 9 a constant C > 0 such that, 8 2 D.Rn / with supp./  , Z 

j.x/j2 d x  C

Z

jr j2 d x D C



i.e. 8 2 D./, (1.2.35) holds.

ˇ2 n Z ˇ X ˇ ˇ @ ˇ d x; ˇ .x/ ˇ ˇ @x i 

(1.2.35)

iD1

(1.2.36)

22

Chapter 1 Schwartz distributions

Proof. R R R R @ @ I. Rn xi .x/ @x .x/d x D R    R    . R xi .x/ @x d xi /dx1 : : : dxi1 dxiC1 i i n : : : dxn (since  2 D.R / H) supp./ is a compact subset of Rn and @ .x/ 2 L1 .Rn /, we can apply Fubini’s Theorem 7.1.2C on the interxi .x/ @x i change of the order of integration). Since the variables x1 ; : : : ; xi1 ; xiC1 ; : : : ; xn are treated as parameters in the integral involving the variables xi , we can apply an integration by parts to get Z Z @ @ xi .x/ d xi D .xi .x/.x//jxxii DC1  .xi .x//.x/dxi : D1 „ ƒ‚ … @x @x i i R R „ ƒ‚ … u dv

Since  2 D.Rn / H) .x/ D 0 for xi D ˙1, we have Z Z @ @ xi .x/ .x/d xi D  .xi .x//.x/d xi @xi R R @xi Z    @ D .x/.x/ dxi ..x//2 C xi @xi R R R @ H) R j.x/j2 dxi D 2 R xi .x/ @x .x/d xi . i Integrating with respect to the other variables x1 ; x2 ; : : : ; xi1 ; xiC1 ; : : : ; xn , we get Z Z Z   j.x1 ; x2 ; : : : xn /j2 dx1 dx2 : : : dxn R R R „ ƒ‚ … n times

Z

D 2

Z :::

„R H)

R

Rn

Z

j.x/j2 d x D 2 L2 ./,

::: R ƒ‚

n times

R

Rn

xi .x/ R

…

@ .x/d x1 d x2 : : : d xn @xi

@ xi .x/ @x .x/d x. i

D.Rn /

II. Since D./  8 2 with supp./  , we can apply the Cauchy–Schwarz inequality (see (B.4.1.7)–(B.4.1.8) with p D q D 2 in Appendix B): ˇ ˇ ˇ ˇ  Z Z ˇ ˇ ˇ ˇ @ @ 2 ˇ ˇ ˇ ˇ j.x/j d x D ˇ  2 xi .x/ .x/d xˇ D 2ˇ xi ; @xi @xi L2 ./ ˇ     ˇ Z  12  Z ˇ  12  @  ˇ @ ˇ2 2 2   ˇ ˇ D2 .xi / j.x/j d x  2kxi kL2 ./  ˇ ˇ dx @xi L2 ./   @xi ˇ Z  12  Z ˇ  12 ˇ @ ˇ2 2 ˇ ˇ  2C1 j.x/j d x ˇ ˇ dx ;   @xi

23

Section 1.2 Test space D./ of Schwartz

since  is bounded and x 2 , 9C1 ./ > 0 such that jxi j  C1 . Z H)

 12  Z j.x/j d x

Z

2

2

j.x/j d x  2C1 





ˇ ˇ  12 ˇ @ ˇ2 ˇ ˇ dx : ˇ @x ˇ i

For  D 0, the inequality is trivially satisfied. Otherwise, dividing both sides R 1 by .  j.x/j2 d x/ 2 , we get Z

ˇ ˇ  12 ˇ @ ˇ2 ˇ ˇ dx ˇ ˇ  @xi ˇ ˇ Z ˇ Z n Z ˇ X ˇ @ ˇ2 ˇ @ ˇ2 2 2 2 ˇ ˇ ˇ ˇ dx j.x/j d x  4C1 ˇ ˇ d x  4C1 ˇ ˇ   @xi  @xi

 12 Z j.x/j d x  2C1 2



H)

iD1

H) Property 12: result:

the inequality (1.2.35) holds with C D

4C12

> 0:

For the solution of many problems later, we will need the following

Proposition 1.2.1. Let  2 D.R/ such that supp./  ŒA; A  R with A > 0. For fixed n 2 N0 , let W R ! R be a function defined by: ´ Pn x k .k/ 1 .0/ for x ¤ 0 nC1 Œ.x/  kD0 kŠ  x .x/ D  .nC1/ .0/ (1.2.37) for x D 0: .nC1/Š Then, is continuous on R and 9C > 0, independent of x, such that

I.

sup j .x/j  C sup j .nC1/ .x/jI jxjA

II.

.x/ D .0/ C x 0 .0/ C

(1.2.38)

jxjA

x 2 00 x n .n/  .0/ C    C  .0/ C x nC1 .x/: 2Š nŠ (1.2.39)

Proof. I.  2 D.R/  C01 .R/, we can write Taylor’s formula for  with the remainder Rn in integral form: x n .n/ x nC1 x  .0/ C .x/ D .0/ C  0 .0/ C    C 1Š nŠ nŠ H) .x/ 

Pn

x k .k/ .0/ kD0 kŠ 

D

x nC1 nŠ

R1 0

Z

1

.1  t /n  .nC1/ .tx/dt

0

.1  t /n  .nC1/ .tx/dt .

24

Chapter 1 Schwartz distributions

R P k 1 1 1 n .nC1/ H) For x ¤ 0; .x/ D x nC1 Œ.x/  nkD0 xkŠ  k .0/ D nŠ 0 .1  t /  .tx/dt; which is continuous for all x ¤ 0. (1.2.40) Then, Z 1 1 lim .x/ D lim .1  t /n  .nC1/ .tx/dt x!0 x!0 nŠ 0 Z 1 1 D .1  t /n Œ lim  .nC1/ .tx/dt x!0 nŠ 0 Z 1 1 D .1  t /n  .nC1/ .0/dt nŠ 0 ˇ  .nC1/ .0/ .1  t /nC1 ˇˇ1  .nC1/ .0/ D  D .0/ D ˇ nŠ nC1 .n C 1/Š 0 (by definition (1.2.37)). H) is continuous at x D 0. Hence, together with (1.2.40), on R.

is continuous

– A  x ¤ 0  A: from (1.2.40), j .x/j 

1 sup j .nC1/ .tx/j nŠ jtxjA 0t 1

H)

j .x/j 

Z

1

.1  t /n dt

0

1 1 1 sup j .nC1/ .x/j  D sup j .nC1/ .x/j nŠ jxjA nC1 .n C 1/Š jxjA

for 0 < jxj  A. – x D 0: from (1.2.37), j .0/j D

j .nC1/ .0/j 1  sup j .nC1/ .x/j: .n C 1/Š .n C 1/Š jxjA

Hence, 8x 2 ŒA; A, j .x/j  C supjxjA j .nC1/ .x/j with C D 1 > 0 H) supjxjA j .x/j  C supjxjA j .nC1/ .x/j. .nC1/Š II. In particular, for n D 1, .x/ D .0/ C x .x/;

(1.2.41)

.x/ D .0/ C x 0 .0/ C x 2 .x/

(1.2.42)

for n D 2,

etc.

Section 1.3 Space D 0 ./ of (Schwartz) distributions

25

Remark 1.2.2 (Test space D./ of complex-valued functions). We have considered the test space D./ consisting of only real-valued test functions  2 C01 ./ since, except in Fourier transform and Fourier analysis, we do not require and will not use complex vector spaces (of complex-valued functions) in general. But all the important results and properties of D./ stated above for the real case can be extended to the complex case with minor, if any, modifications, i.e. to the test space D./ consisting of complex-valued test functions  2 C01 ./ with .x/ D

.x/ C i .x/;

supp. /  ;

.x/ D Re..x//;

supp. /  

and

.x/ D Im..x// 2

C01 ./;

2

8x 2 ;

C01 ./;

(1.2.43)

assuming that D./ is a complex-vector space, which is obviously closed under the addition operation and for multiplication by complex numbers ˛ 2 C. Convergence in a complex test space D./ is similar to that in (1.2.7): .n / !  in D./ if and only if: I. 9 a compact subset K   such that supp.n  /  K II.

@˛ n

!

@˛ 

8n 2 N;

uniformly in  as n ! 1; 8˛.

(1.2.44)

Thus, the extension of D./ to complex functions is a straightforward affair.

1.3

Space D 0 ./ of (Schwartz) distributions

1.3.1 Algebraic dual space D ? ./ Let D./ be the space of real-valued test functions  on  (see Definition 1.2.1), and T W  2 D./ 7! T ./ 2 R be a linear functional on D./, i.e. 8˛i 2 R; 8i 2 D./ .i D 1; 2/;

T .˛1 1 C ˛2 2 / D ˛1 T .1 / C ˛2 T .2 /; (1.3.1)

other notations equivalent to T ./ being hT; i or .T; / or ŒT; . For example, hT; ˛1 1 C ˛2 2 i D ˛1 hT; 1 i C ˛2 hT; 2 i

8˛i 2 R; 8i 2 D./; i D 1; 2:

As convenient, we will use interchangeably any one of the different notations indicating the same object T ./: T ./ D hT; i D .T; / D ŒT; 

8 2 D./:

(1.3.2)

Then D ? ./ D ¹T W T is a linear functional on D./º (1.3.3) is the linear space of all linear functionals T defined on D./, called the algebraic dual space of D./.

26

Chapter 1 Schwartz distributions

Continuity of linear functional T 2 D ? ./ on D./ Definition 1.3.1. A linear functional T 2 D ? ./ defined on D./ is said to be continuous on D./ if and only if n !  in D./ in the sense of (1.2.7)

H)

T .n / ! T ./ in R as n ! 1: (1.3.4)

1.3.2 Distributions and the space D 0 ./ of distributions on  Definition 1.3.2. A continuous linear functional T 2 D ? ./ is called a distribution on   Rn . Then T ./ 2 R is the value of the distribution T at  2 D./, other equivalent notations used for T ./ being those in (1.3.2), i.e. T ./ D hT; i D .T; / D ŒT;  8 2 D./. Space D 0 ./ of distributions on  The set of all distributions on , i.e. of all continuous linear functionals defined on D./, forms a vector space denoted by D 0 ./ if we define the sum T1 C T2 2 D 0 ./ of distributions T1 ; T2 2 D 0 ./, the product ˛T 2 D 0 ./ of ˛ 2 R and T 2 D 0 ./ and the null distribution 0 2 D 0 ./ by: Sum T1 C T2 2 D 0 ./: .T1 C T2 /./ D T1 ./ C T2 ./, 8 2 D./; Product ˛T 2 D 0 ./: .˛T /./ D ˛  T ./ 8 2 D./, 8˛ 2 R;

(1.3.5)

Null distribution 0 2 D 0 ./: T D 0 in D 0 ./ ” T ./ D 0 8 2 D./. Definition 1.3.3. The linear space D 0 ./ of all distributions T on  (i.e. continuous linear functionals on D./) is called the space of (Schwartz) distributions on , with T ./DhT; iD 0 ./D./ D.T; /D 0 ./D./ DŒT; D 0 ./D./

8 2 D./;

where h  ;  i or .  ;  / or Π ;   denotes the duality pairing between D 0 ./ and D./, i.e. any one of these notations can be used interchangeably to mean the same thing. Remark 1.3.1. D ? ./ in (1.3.3) is the set of all linear functionals on D./, which may be or may not be continuous on D./, whereas D 0 ./ is the set of all linear functionals of D ? ./, which are continuous on D./. Hence, D 0 ./  D ? ./ is a subspace of D ? ./. There are currently no known examples of linear functionals discontinuous on D./, and there is very little chance of ever encountering such a discontinuous linear functional in practical applications. But the existence of linear functionals discontinuous on D./ can be established mathematically with the help of the Axiom of Choice [7], [28].

Section 1.3 Space D 0 ./ of (Schwartz) distributions

27

Physical interpretation of duality The abstract concepts of duality and duality pairing can be interpreted physically with the help of problems from mechanics (or optimization). Consider a conservative mechanical system (for example, a deformable elastic body). We can associate with this system a vector space V of admissible displacement fields v. Then there exists another vector space F such that there is a duality between the vector space V of admissible displacements v and the vector space F , the elements of which are the admissible force fields f acting on the system. In fact, an element f 2 F of the dual F associates with each element v 2 V a scalar hf; vi 2 R called the work done by the force f 2 F for the displacement v 2 V . Thus, the rôle of f 2 F of the dual F is to associate a scalar hf; vi 2 R to each v 2V.

1.3.3 Characterization, order and extension of a distribution Characterization of a distribution T 2 D 0 ./ Proposition 1.3.1. A linear functional T on D./ is a distribution on  if and only if any one of the following two equivalent properties hold: I. n ! 0 in D./ in the sense of (1.2.7) H) T .n / D hT; n i ! 0 in R as n ! 1 (i.e. T is continuous on D./ and hence T 2 D 0 ./). II. 8 compact subsets K   of , 9 an integer m.K/  0 and a constant CK > 0 such that, 8 2 D./ with supp./  K (i.e. 8 2 DK .//, jT ./j D jhT; ij  CK

max

max j@˛ .x/j D CK .pK;m.K/ .// (1.3.6)

j˛jm.K/ x2K

P (or, equivalently, jT ./j  CQK j˛jm.K/ maxx2K j@˛ .x/j D CQK .pQK;m.K/ .//), where pK;m.K/ ./ D maxj˛jm.K/ maxx2K j@˛ .x/j (or, equivalently, P pQK;m.K/ ./ D j˛jm.K/ maxx2K j@˛ .x/j, j˛j D ˛1 C ˛2 C    C ˛n ). Proof. I. follows from Definition 1.3.2 of a distribution T 2 D 0 ./. II. Assume that T is a linear functional on D./, which satisfies (1.3.6). We are to show that T is a distribution on . Let .n / be a sequence in D./ such that n ! 0 in D./. Then 9 a compact subset K   such that supp.n /  K 8n 2 N, and @˛ n ! 0 uniformly 8j˛j as n ! 1 H) 8j˛j 2 N0 , maxx2 j@˛ n .x/j D maxx2K j@˛ n .x/j ! 0 as n ! 1. P Hence, jhT; n ij  CK j˛jm maxx2K j@˛ n .x/j ! 0 as n ! 1, i.e. II H) I. Conversely, let T 2 D 0 ./ be a distribution. Suppose that (1.3.6) does not hold for T , i.e. 9 a compact set K   such that (1.3.6) does not hold for all m and for all C > 0. Hence, for this compact set K, (1.3.6) also

28

Chapter 1 Schwartz distributions

does P not hold for C˛ D m D n, i.e. 9n 2 DK ./ such that jhT; n ij > n j˛jn maxx2K j@ n .x/j8n 2 N. Since hT; n i ¤ 0, we can define n n 1 n D jhT;n ij such that jhT; n ij D jhT; jhT;n ij ij D jhT;n ij jhT; n ij D 1 8n 2 N. Then, X n 1 D jhT; n ij > max j@˛ n .x/j jhT; n ij x2K j˛jn X max j@˛ n .x/j 8n 2 N Dn j˛jn

x2K

P

1 ˛ n .x/j < n 8n 2 N j˛jn maxx2K j@ H) maxx2K j@˛ n .x/j < n1 8n 2 N; 8j˛j  n; with supp. n /  supp.n /  K H) 8˛, the sequence .@˛ n / tends to 0 uniformly in  as n ! 1 H)

H)

n ! 0 in D./ as n ! 1. But jhT; n ij D 1 8n 2 N H) hT; n i does not tend to 0 as n ! 1. Hence, T is not a distribution, which contradicts the hypothesis that T 2 D 0 ./. Therefore, our original assumption that T 2 D 0 ./ does not satisfy (1.3.6) is wrong, i.e. T 2 D 0 ./ must satisfy (1.3.6). In other words, we have proved I H) II.

Order of a distribution T 2 D 0 ./ Definition 1.3.4. A distribution T 2 D 0 ./ is said to be of order m0  0 if and only if, for any compact subset K   of , m0  0 is the smallest integer with m.K/  m0 for which T ./ D hT; i satisfies the inequality (1.3.6). That is, 8 2 D./ with supp./  K, jT ./j D jhT; ij  CK max max j@˛ .x/j D CK .pK;m0 .//: j˛jm0 x2K

Measures Definition 1.3.5. Distributions of order zero are called measures. Remark 1.3.2. The measures defined here are in bijective correspondence with measures defined on Rn , although we are not interested in pursuing this any further. Equality of distributions in D 0 ./ T1 ; T2 2 D 0 ./ are equal ” T1  T2 D 0 in D 0 ./ ” hT1 ; i D hT2 ; i 8 2 D./. (1.3.7) Without proof,5 we state the extension results for distributions of order m  0. 5 The proof is based on the density of D./ in D m ./8m

2 N0 (i.e. D./ ,! D m ./ with dense,

Section 1.3 Space D 0 ./ of (Schwartz) distributions

29

Extension of a distribution of order m  0 Theorem 1.3.1. A distribution T 2 D 0 ./ of order m  0 can be extended to a unique, continuous, linear functional T on D m ./, i.e. T 2 D 0m ./  .D m .//0 with hT ; i D hT; i 8 2 D./ and, 8K  , 9CK > 0 such that, 8 2 D m ./ with supp./  K, jhT ; ij  CK max max j@˛ .x/j D CK .pK;m .//: j˛jm x2K

(1.3.8)

Corollary 1.3.1. If T 2 D 0 ./ is of order 0, then T can be extended to a unique, continuous, linear functional T on D 0 ./  C00 ./  C0 ./ with hT ; i D hT; i 8 2 D./ and, 8K  , 9CK > 0 such that, 8 2 C0 ./ with supp./  K, jT ./j  CK max j.x/j:

(1.3.9)

x2K

We agree to denote the extended functional by the same notation T . Hence, from now on, T will also denote the extended continuous linear functional on D m ./.

1.3.4 Examples of distributions Example 1.3.1 (Usual functions). For open subsets   Rn , every locally integrable (summable) function f 2 L1loc ./ (see (1.2.21) for definition) defines a distribution Tf 2 D 0 ./ of order 0 by: Z Tf ./ D hT; i D

f .x/.x/d x

8 2 D./:

(1.3.10)



In fact, 1. 8 2 D./, supp./ D K   with .x/ D 0 8x lying outside K; R 2. f 2 L1loc ./ is integrable on this support K, i.e. K jf .x/jd x < C1; 3.  is continuous, and hence f  is also integrable on K, i.e. ˇZ ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ f .x/.x/d xˇ D ˇ f .x/.x/d xˇˇ  jf .x/k.x/jd x (1.3.11) ˇ  K K Z jf .x/jd x < C1  max j.x/j x2K

K

H) Tf ./ D hTf ; i 2 R 8 2 D./ H) Tf is a functional on D./. continuous imbedding (1.2.25)) and on the application of Corollary A.7.3.7 of the well-known Hahn– Banach Theorem A.7.2.1 (Section A.7, Appendix A) for the continuous, linear extension of continuous, linear functionals on D./ to D m ./ (see also [9], [25], for example).

30

Chapter 1 Schwartz distributions

Tf is a linear functional on D./

8˛1 ; ˛2 2 R, 81 ; 2 2 D./,

Z hTf ; ˛1 1 C ˛2 2 i D

f .x/.˛1 1 .x/ C ˛2 2 .x//d x Z Z D ˛1 f .x/1 .x/d x C ˛2 f .x/2 .x/d x 





D ˛1 hTf ; 1 i C ˛2 hTf ; 2 i: From (1.3.11), 8 compact subsets K   of ; 8 2 D./ with supp./  K, Z jTf ./j  max j.x/j x2K

jf .x/jd x D CK max j.x/j; x2K

K

Z

CK D

jf .x/jd x < C1: K

H) the inequality (1.3.6) holds with CK > 0 and m.K/ D 0 for all K   H) by Proposition 1.3.1, Tf is a distribution on , i.e. Tf 2 D 0 ./ and, by Definition 1.3.4, Tf 2 D 0 ./ is a distribution of order m0 D 0. In particular, for  D Rn , f 2 L1loc .Rn / defines a distribution Tf 2 D 0 .Rn / of order 0 by Z Tf ./ D hTf ; i D

f .x/.x/d x

8 2 D.Rn /:

(1.3.12)

Rn

Theorem 1.3.2. Let f; g 2 L1loc ./ be any two locally summable (integrable) func0 tions on   Rn , and Tf ; T R g 2 D ./ be the distributionsR defined by f and g respectively, i.e. hTf ; i D  f .x/.x/d x and hTg ; i D  g.x/.x/d x 8 2 D./. Then, Tf D Tg in D 0 ./

”

f D g in L1loc ./:

(1.3.13)

Proof. Let f D g Rin L1loc ./. Then fR.x/ D g.x/ a.e. on  H) hTf ; i D  f .x/.x/d x D  g.x/.x/d x D hTg ; i 8 2 D./ H) Tf D Tg 2 D 0 ./ (by (1.3.7)). Let TRf D Tg in D 0 ./. RThen, from (1.3.7), hTf ; i D hTg ; i 8 2 D./ H) R f .x/.x/d x D  g.x/.x/d x 8 2 D./ H)  h.x/.x/d x D 0 8 2 D./, where h.x/ D f .x/  g.x/ for almost all x 2 , and f  g D h 2 L1loc ./. Then, by Theorem 1.2.3A, h D 0 in L1loc ./, i.e. f D g in L1loc ./.

Section 1.3 Space D 0 ./ of (Schwartz) distributions

31

Important consequences of Theorem 1.3.2 

Distinct functions in L1loc ./ define distinct distributions in D 0 ./ with   Rn , i.e. f ¤ g in L1loc ./



H)

Tf ¤ Tg in D 0 ./:

(1.3.14)

As a consequence of (1.3.14), there will be no chance of confusion if we identify a locally summable function f 2 L1loc ./ with the distribution Tf 2 D 0 ./ defined by it. Hence, we will make this identification, and the same notation f will be used instead of Tf to denote the distribution it defines: f 2 L1loc ./ H) f 2 D 0 ./ H) L1loc ./  D 0 ./ such that Z hf; i D hTf ; i D f .x/.x/d x8 2 D./: (1.3.15) 

R



In particular, hT; i D  .x/d x 8 2 D./ defines a distribution T which R will be identified with f D 1, i.e. h1; i D  .x/d x 8 2 D./. (1.3.16)



All continuous functions (which include k-time differentiable or Hölder continuous or Lipschitz continuous functions) on  and Lp -functions (1  p  1) on  (see Appendix B for details) are locally summable on , and hence define distributions, i.e. D./, C 0 ./, C k ./, C k; ./, Lp ./, 1  p  1, are subspaces of L1loc ./ (for C k ./, C k; ./, 0    1, see Appendix A, Section A.4 for details, and for Lp ./, 1  p  1, see Appendix B, Section B.4). H)

D./; C 0 ./; C k ./; C k; ./; Lp ./  D 0 ./:

(1.3.17)

In other words, the usual functions on  are distributions. p For example, for n D 1;  D a; bŒ; f .x/ D sin x or cos x or e x or x m or 1= jxj etc. are all elements of L1loc .a; bŒ/ and, hence, elements of D 0 .a; bŒ/ in the following sense: 8 2 D.a; bŒ/, Z b Z b x hsin x; i D sin x.x/dxI he ; i D e x .x/dxI a a Z b Z b cos x.x/dxI hx m ; i D x m .x/dxI hcos x; i D a a   Z b 1 1 (1.3.18) p ; D p .x/dxI jxj jxj a etc., where a, b may have arbitrary values including 1 to C1 respectively, since sin x.x/, e x .x/, cos x.x/, p1jxj .x/, etc. vanish outside Œa0 ; b0   1; 1Œ for supp./  Œa0 ; b0 , and the corresponding integrals in (1.3.18) exist. Similarly, for n D 2,  D a; bŒ  c; d Œ  R2 ; f .x1 ; x2 / D sin x1 cos x2 or e x1 sin x2 or x1m e x2 etc. are all elements of L1loc .a; bŒ  c; d Œ/ and, hence, elements

32

Chapter 1 Schwartz distributions

of D 0 .a; bŒ  c; d Œ/ in the following sense: 8 2 D.a; bŒ  c; d Œ/, Z bZ d sin x1 cos x2 .x1 ; x2 /dx1 dx2 I hsin x1 cos x2 ; i D a

Z

he x1 sin x2 ; i D

c

b

a

hx1m e x2 ; i

Z

Z

d

e x1 sin x2 .x1 ; x2 /dx1 dx2 I

c

b

Z

d

D a

c

x1m e x2 .x1 ; x2 /dx1 dx2 I

(1.3.19)

where a; b (resp. c; d ) may have arbitrary values including 1 and C1 respectively. Similarly, the definitions as elements of D 0 ./ for other elementary functions f of several variables belonging to L1loc ./ can be written. 



f .x/ D x1 … L1loc .R/, i.e. is not locally integrable on R (see Example 1.3.7), and hence does not define a distribution on R. f .x/ D x1 for x > 0 (i.e. on RC D 0; 1Œ) is both continuous and locally summable on RC , but not summable on RC , i.e. x1 2 L1loc .RC / H) x1 defines a distribution on 0; 1Œ by   Z 1 1 1 ; D .x/dx 8 2 D.0; 1Œ/; (1.3.20) x x 0 since .0/ D 0 8 2 D.0; 1Œ/.



f .x/ D x1 for x < 0 (i.e. on R D 1; 0Œ) is both continuous and locally summable on R , i.e. x1 2 L1loc .R / H) x1 defines a distribution on 1; 0Œ by:  Z 0  1 1 ; D .x/dx 8 2 D.1; 0Œ/; (1.3.21) x 1 x since .0/ D 0 8 2 D.1; 0Œ/.



f .x/ D x1 for x 2 R n ¹0º is locally integrable on R n ¹0º D R [ RC and defines a distribution on R n ¹0º by:   Z 1 1 1 ; D .x/dx 8 2 D.R n ¹0º/: (1.3.22) x 1 x R C1

1 1 x .x/dx

8 2 D.R/, see Example 1.3.8.



For Cauchy principal value



ln jxj is locally integrable on R (see Example 1.4.1), and hence defines a distribution by Z 1 lnjxj.x/dx 8 2 D.1; 1Œ/: hln jxj; i D 1

Section 1.3 Space D 0 ./ of (Schwartz) distributions 

33

In fact, every test function  2 D./  L1loc ./ defines a distribution T D  2 D 0 ./ in the sense: Z h; i D

.x/ .x/d x

8

2 D./:

(1.3.23)



Hence, D./  D 0 ./. 

(1.3.24)

R But  u.x/.x/d x may not exist for u 2 M./ and  2 D./, M./ being the space of Lebesgue measurable functions on  (see Appendix B), since M./ is not a subspace of L1loc ./ (i.e. M./ 6 L1loc ./). (1.3.25)

Regular distributions on   Rn Definition 1.3.6. A distribution T 2 D 0 ./ is called a regular distribution if and only if T can be identified with a locally (integrable) summable function f 2 L1loc ./ such that (1.3.15) holds. In other words, for a regular distribution T 2 D 0 ./, 9 a unique f 2 L1loc ./ such that the integral representation of T in (1.3.15) holds, i.e. Z hT; i D hf; i D

f .x/.x/d x

8 2 D./:

(1.3.26)



Functions of D./, C 0 ./, C k ./, C k; ./, Lp ./, 1  p  1 (see Appendices A and B), which are subspaces of L1loc ./ (see (1.3.17)) define regular distributions of D 0 ./. (1.3.18)–(1.3.23) represent some examples of regular distributions.

Singular distributions on   Rn Definition 1.3.7. A distribution T 2 D 0 ./ is called a singular distribution if it is not a regular distribution. Hence, for singular distributions T 2 D 0 ./, there does not exist any locally (integrable) summable function f 2 L1loc ./ with which T can be identified. In other words, for a singular distribution T 2 D 0 ./, an integral representation of T with the help of a function f 2 L1loc ./ in the form (1.3.26) is not possible.

34

Chapter 1 Schwartz distributions

Examples of singular distributions Example 1.3.2. 

The Dirac distribution at the origin 0 2 Rn (i.e. mass/charge/force concentrated at the origin), denoted by ı, or equivalently by ı0 or by ı.x/, is defined by: hı; i D hı0 ; i D hı.x/; i D .0/

8 2 D.Rn /;

(1.3.27)

where T D ı D ı0 D ı.x/ 2 D 0 .Rn / (which is proved below). 

The Dirac distribution at the point a 2 Rn (i.e. mass/charge/force concentrated at the point a), denoted by ıa or ı.x  a/, is defined by: hıa ; i D hı.x  a/; i D .a/

8 2 D.Rn /;

(1.3.28)

where T D ıa D ı.x  a/ 2 D 0 .Rn /, from which, for a D 0 2 Rn , we get ı0 D ı D ı.x/ and (1.3.27). Hence it is sufficient to show that ıa is a distribution of D 0 .Rn /. 

ıa is a linear functional on D.Rn / 8 fixed a 2 Rn : using (1.3.28), 81 ; 2 2 D.Rn /, hıa ; 1 C 2 i D .1 C 2 /.a/ D 1 .a/ C 2 .a/ D hıa ; 1 i C hıa ; 2 iI hıa ; ˛i D .˛/.a/ D ˛.a/ D ˛hıa ; i



8˛ 2 R; 8 2 D.Rn /:

jhıa ; ij D j.a/j  maxx2K j.x/j 8 fixed compact subsets K  Rn , 8 2 D.Rn / with supp./  K, i.e. 8 2 DK .Rn / H) the inequality (1.3.6) holds with CK D 1, m.K/ D m0 D 0 8 compact subsets K  Rn H) by Proposition 1.3.1 and Definition 1.3.4, 8 fixed a 2 Rn , ıa defined by (1.3.28) is a distribution of order 0 in D 0 .Rn /. Hence, the Dirac distribution ıa 2 D 0 .Rn / defined by (1.3.28) is a measure by Definition 1.3.5, and is also called the Dirac measure.

Remark 1.3.3. 

Equation (1.3.28) is often written in the (incorrect) form: Z ı.x  a/.x/d x D .a/ 8 2 D.Rn /;

(1.3.29)

Rn

where the integral is meaningless, and hence must be understood as the duality pairing hıa ; iD 0 .Rn /D.Rn / between ıa 2 D 0 .Rn / and  2 D.Rn /. But we will try to avoid this, and write in the correct form (1.3.28) unless stated otherwise.

Section 1.3 Space D 0 ./ of (Schwartz) distributions

35



The Dirac distribution (measure) ıa is usually called the delta function ı.x  a/ (see (1.1.2)) with mass/charge/load etc. concentrated at the point a 2 Rn . Hence, the precise mathematical definition of the delta function ı.xa/ is given by (1.3.28) and does not suffer from the contradictory properties in (1.1.7)– (1.1.9), since ı.x  a/ is the distribution ıa and not a function, and the meaningless integral representation in (1.1.2) is replaced by the correct representation (1.3.28) with the duality pairing hıa ; iD 0 .Rn /D.Rn / (see also Section 1.11).



In fact, we will prove that (1.3.28) can not be written in the integral form (1.3.29) to justify that ıa 2 D 0 .Rn / is a singular distribution.

Proposition 1.3.2. There does not exist any locally integrable function f 2 L1loc .Rn / such that Z f .x/.x/d x D .a/ 8 2 D.Rn /: (1.3.30) hıa ; i D Rn

(In other words, the Dirac distribution ıa D ı.xa/ in (1.3.28) can not be represented by a locally integrable function f 2 L1loc .Rn / and is thus a singular distribution.) Proof. Without loss of generality, and for the sake of simplicity, we consider a D 0 2 Rn , i.e. ı D ı0 D ı.x/. 8" > 0, let " 2 D.Rn / be a test function defined by: 8 "2 <  "2 kxk 2 for x 2 B.0I "/ D ¹x W kxk D .x12 C    C xn2 /1=2 < "º e " .x/ D :0 for kxk  " N "/ D ¹x W kxk  "º; " .x/  e 1 8x 2 Rn . with supp." / D B.0I Then hı; " i D " .0/ D 1e 8" > 0 H) lim"!0 hı; " i D lim"!0 " .0/ D 1e . 1 n Suppose that the contrary ı can be represented R holds, i.e. 9f 2 Lloc .R / such that by f : hı; i D hf; i D Rn f .x/.x/d x D (0) 8 2 D.Rn /. Then we have hf; " i D hı; " i D " .0/ D

1 e

1 lim hf; " i D : "!0 e

H)

(1.3.31)

Since f 2 L1loc .Rn /; " 2 D.Rn / 8" > 0, we have Z Z f .x/" .x/d x D f .x/" .x/d x; hf; " i D Rn

B.0I"/

because .x/ D 0 for all x outside B.0I "/, ˇZ ˇ ˇ ˇ f .x/" .x/d xˇˇ  H) jhf; " ij D ˇˇ B.0I"/

1 D e since f 2 L1loc ./.

Z

Z sup " .x/ x2B.0I"/

jf .x/jd x < C1; B.0;"/

jf .x/jd x B.0I"/

36

Chapter 1 Schwartz distributions

But Lebesgue n-dimensional volume measure .B.0I "// ! 0 as " ! 0, Z H) jf .x/jd x ! 0 as " ! 0 B.0I"/

H)

1 e

Z

jf .x/jd x ! 0 as " ! 0: B.0I"/

Hence, lim"!0 hf; " i D 0, which contradicts (1.3.31), i.e. our assumption is wrong and the result follows. Thus, the Dirac distribution ıa can not be written in the form (1.3.30) and is a singular distribution. Remark 1.3.4. For more details, see Section 1.11. Example 1.3.3. For an open subset   Rn , functional T defined by: 8 fixed ˛ with j˛j D m and fixed point a 2 , T ./ D hT; i D @˛ .a/

8 2 D./

(1.3.32)

is a distribution of order m in D 0 ./. T defined by (1.3.32) is a linear functional: 81 ; 2 2 D./, hT; 1 C 2 i D @˛ .1 C 2 /.a/ D @˛ 1 .a/ C @˛ 2 .a/ D hT; 1 i C hT; 2 i hT; i D @˛ ./.a/ D @˛ .a/ D hT; i

8 2 R; 8 2 D./:

T satisfies the inequality (1.3.6) with CK D 1 and m.K/ D m for all compact subsets K  : 8 2 D./ with supp./  K, jT ./j D jhT; ij D j@˛ .a/j  max max j@˛ .x/j j˛jDm x2K

H) by Proposition 1.3.1 and Definition 1.3.4, T is a distribution of order m on , i.e. T 2 D 0 ./. Hence, for m  1, T is not a measure, although it is still a singular distribution. In particular, for  D Rn and m D 0, we get T D ıa 2 D 0 .Rn / defined by (1.3.28), i.e. m D 0; T D ıa is a measure. Example 1.3.4 (Dirac distribution ıS on a surface S, see Section 1.11 for interesting details). Let x1 D 0 be the equation of a hyperplane S in Rn . Then ıS 2 D 0 .Rn / is a Dirac distribution (measure) concentrated on S (i.e. with mass/charge/force etc. concentrated on the hyperplane S  Rn ): 8 2 D.Rn / with supp./  K  Rn , Z hıS ; i D .0; x2 ; : : : ; xn /dx2 dx3 : : : dxn : (1.3.33)



Rn1

Section 1.3 Space D 0 ./ of (Schwartz) distributions

37

Since ıS is a linear functional on D.Rn / and, 8 2 D.Rn / with supp./  K, ˇZ ˇ ˇ ˇ .0; x2 ; : : : ; xn /dx2 : : : dxn ˇˇ jhıS ; ij D ˇˇ Rn1 ˇ ˇZ ˇ ˇ ˇ .0; x2 ; : : : ; xn /dx2 : : : dxn ˇˇ  CK max j.x/j Dˇ x2K

K\Rn1

R

R

with CK D K\S dx2 : : : dxn  K dx1 dx2 : : : dxn D n-dimensional volume measure .K/ < C1 (supp./  K  Rn ) H) jhıS ; ij satisfies the inequality (1.3.6) with CK < C1 and m.K/ D m0 D 0 for all compact subsets K with supp./  K H) by Proposition 1.3.1 and Definition 1.3.4, ıS defined by (1.3.33) is a distribution of order 0 in D 0 .Rn / and, hence, a measure by Definition 1.3.5. Let f .x1 ; x2 ; : : : ; xn / D 0 be the equation of the smooth (i.e. infinitely differen@f tiable) hypersurface S  Rn such that @x ¤ 0 for 1  j  n (i.e. r f ¤ 0). Leray j form ! [1, Vol.1, p. 348] for the hypersurface S is the exterior differential form such that 

df ^ ! D dx1 ^ dx2 ^    ^ dxn D dx1 dx2 : : : dxn : Then on S with

@f @xj

¤ 0, 1  j  n, we have

dx1 ^ dx3 ^    ^ dxn dx2 ^ dx3 ^    ^ dxn D @f =@x1 @f =@x2 dx ^    ^ dx ^ dx 1 j 1 j C1 ^    ^ dxn D .1/j 1 .1  j  n/; @f =@xj

!D

and Dirac distribution ıS 2 D 0 .Rn / is defined by: for fixed j with 1  j  n, Z Z .x/ ! D .1/j 1 dx1 dx2 : : : dxj 1 dxj C1 : : : dxn : (1.3.34) hıS ; i D @f =@xj S S The integral in (1.3.34) is taken over the hypersurface S W f D 0, which is why ıS is said to be concentrated on hypersurface S W f .x/ D 0. n Since ıS is a linear functional on D.R R / and jhıS ; ij  CK maxx2K j.x/j satisfies the inequality (1.3.6) with CK D K\S ! < C1 and m.K/ D m0 D 0 for all compact subsets K of Rn , with supp./  K, ıS defined by (1.3.34) is also a distribution of order 0 by Proposition 1.3.1 and Definition 1.3.4, and, hence, a measure by Definition 1.3.5. For more details on (1.3.34), see [1, Vol.1, pp. 210–211, 348–349]). Dirac distributions ıS 2 D 0 .Rn / defined by (1.3.33) and (1.3.34) are singular distributions on Rn , since hıS ; i8 2 D.Rn / can not be represented by an n-dimensional volume integral over Rn .

38

Chapter 1 Schwartz distributions

Example 1.3.5 (Magnetic dipole moment layer on smooth surface S; see Section 1.11 for interesting details). Let S be a smooth, bounded surface in R3 , dS the surface area measure on S, @ the derivative of  in the direction of the unit normal nO to S and @n be a continuous function on S defining the surface moment density on S. Then, Z hT; i D

.x/ S

@ dS @n

8 2 D.R3 /

(1.3.35)

defines a distribution T on R3 , which is the normally oriented magnetic dipole distribution over S with surface moment density , since T is a linear functional on D.R3 / and ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ @ ˇ @ jhT; ij D ˇˇ .x/ .x/dS ˇˇ  max ˇˇ .x/ˇˇ j .x/jdS 8 2 D.R3 / @n @n x2K S S\K with supp./  K. p O R3 j  .x/j D jhr .x/; ni 3 max1i3 maxx2K j@i .x/j, Since j @ @n p R ˛ jhT; ij  maxj˛j1 maxx2K j@ .x/jCK with CK D 3 S\K j .x/jdS < C1 with supp./  K, and for all compact subsets K  R3 , H) by Proposition 1.3.1 and Definition 1.3.4, T is a distribution of order m0 D 1. Remark 1.3.5. T defined by (1.3.35) is a singular distribution (see Section 1.11 for explanations). This distribution must not be confused with the regular distribution defined by volume R density f , which is identified with the function f itself, i.e. hf; i D hT; i D R3 f .x/.x/ 8 2 D.R3 /. We will also meet with distributions of the following form in the definition of derivatives of distributions later. Example 1.3.6. For open subset   Rn of Rn , let f 2 L1loc ./ be a fixed locally integrable (summable) function on . Then the functional T defined by: 8 fixed ˛ with j˛j D m 2 N, 8 2 D./, Z

f .x/@˛ .x/d x D hf; @˛ i

T ./ D hT; i D

(1.3.36)



is also a distribution of order m in D 0 ./, since 8 2 D./ with supp./  K, ˇ ˇZ ˇ ˇ f .x/@˛ .x/d xˇˇ jT ./j D jhT; ij D ˇˇ  Z ˛ jf .x/jd x D CK max max j@˛ .x/j;  max max j@ .x/j j˛jDm x2K

K

j˛jDm x2K

Section 1.3 Space D 0 ./ of (Schwartz) distributions

39

R with CK D K jf .x/jd x < C1, m.K/ D m for all compact subsets K   of . In particular, for  D Rn and fixed f 2 L1loc .Rn /, T 2 D 0 .Rn / defined by: 8 fixed ˛ with j˛j D m, Z f .x/@˛ .x/d x 8 2 D.Rn / (1.3.37) T ./ D hT; i D 

is also a distribution of order m in D 0 .Rn /. Example 1.3.7. f .x/ D x1 is not locally summable on R. Hence, x1 does not define a distribution on R. R Ra 0 1 In fact, for a > 0; a jxj dx D 1, 0 x1 dx D 1, and hence, 8 compact intervals Ra 1 1 Œa; a with a > 0, a jxj dx D 1; jxj is not integrable in the neighbourhood of x D 0, and consequently x1 is not locally integrable (summable) on R D 1; 1Œ. R1 Hence, 1 x1 .x/dx is not defined 8 2 D.R/. But x1 is locally summable on R n ¹0º and defines a distribution on R n ¹0º (see (1.3.22)). Example 1.3.8 (Cauchy principal value). R1 1 R1  c:p:v: T on R by: hT; i D c:p:v: 1 x1 .x/ 1 x .x/dx defines a distribution Ra 1 R1 1 dx 8 2 D.R/, since c:p:v: 1 x .x/dx D c:p:v: a x .x/dx D R " Ra lim"!0C Œ a x1 .x/dx C " x1 .x/dx exists 8 compact intervals Œa; a with supp./  a; aŒ, a > 0. Z Z 1 1 1  .x/dx D lim .x/dx 8 2 D.R/ (1.3.38) hT; i D c:p:v: C x x "!0 jxj" 1 defines a distribution on R. For this we are to show that T is a continuous linear functional on D.R/. R By virtue of the properties of integral and limit, hT; ˛1 1 C ˛2 2 i D ˛1 lim"!0C jxj" x1 1 .x/ R dx C ˛2 lim"!0C jxj" x1 2 .x/dx D ˛1 hT; 1 i C ˛2 hT; 2 i 81 ; 2 2 D.R/. R1 1  T is continuous on D.R/ ” hT;  i D c:p:v: n 1 x n .x/dx ! 0 in R as n ! 0 in D.R/. Let a; aŒ be any bounded interval such that supp.n /  a; aŒ 8n 2 N; a > 0:

(1.3.39)

Hence, Z

1

c:p:v: 1

1 n .x/dx D c:p:v: x

Z

a

a a

Z

D c:p:v: a

1 n .x/dx (1.3.40) x Z a n .0/ n .x/  n .0/ dx C c:p:v: dx: x x a

40

Chapter 1 Schwartz distributions

But   Z a n .0/ 1 c:p:v: dx D n .0/ c:p:v: dx D n .0/  0 D 0; (1.3.41) x a a x R " Ra Ra since x1 is an odd function, . a x1 dx C " x1 dx/ D 0 8" > 0 and c:p:v: a x1 dx D R " 1 Ra 1 lim"!0C . a x dx C " x dx/dx D 0. R0 n .0/ n .0/ D n0 .0/, the integrals a n .x/ dx and Since limx!0 n .x/ x x R a n .x/n .0/ dx exist (are finite) and, consequently, 0 x Z

a

Z

a

c:p:v: a

n .x/  n .0/ dx D x

Z

a

a

n .x/  n .0/ dx: x

(1.3.42)

Moreover, by the Mean Value Theorem, jn .x/  n .0/j  jxj max jn0 .x/j x2Œa;a

H) H)

ˇ ˇ ˇ n .x/  n .0/ ˇ ˇ  max j 0 .x/j ˇ ˇ x2Œa;a n ˇ x ˇ ˇZ a Z ˇ n .x/  n .0/ ˇˇ 0 ˇ dx j .x/j  max ˇ x2Œa;a n ˇ x a

(1.3.43) a

a

Using (1.3.41)–(1.3.43) in (1.3.40), we get ˇ ˇ Z 1 ˇ n .x/ ˇˇ ˇ c:p:v: dx ˇ  2a max jn0 .x/j ! 0 ˇ x x2Œa;a 1

dx D 2a max jn0 .x/j x2Œa;a

in R as n ! 1;

(1.3.44)

since supp.n /  a; aŒ 8n 2 N, n ! 0 in D.R/ H) n0 .x/ ! 0 uniformly in R as n ! 1 H) maxx2Œa;a jn0 .x/j ! 0 as n ! 1. R1 Thus, hT; n i D c:p:v: 1 nx.x/ dx ! 0 as n ! 1 H) T is continuous on D.R/ H) T is a distribution, which we agree to denote by c:p:v: x1 such that 

 Z 1 1 1 c:p:v: ;  D c:p:v: .x/dx x 1 x

8 2 D.R/;

(1.3.45)

i.e. c:p:v: x1 2 D 0 .R/. From (1.3.44), c:p:v: x1 is a distribution of order 1.

1.3.5 Distribution defined on test space D./ of complex-valued functions Up to now we have considered distributions T defined on D./ consisting of realvalued test functions  such that T ./ 2 R. But in some situations we are to consider test space D./ consisting of complex-valued test functions  2 C01 ./ satisfying

Section 1.4 Some more examples of interesting distributions

41

(1.2.43) and (1.2.44); the complexification of D./ is a quite straightforward affair (see Remark 1.2.2), since almost all results stated for the real case in this section can be extended to the complex case with minor or no modifications at all. For example, D./ is now a complex vector space, which is obviously closed under multiplication by complex numbers ˛ 2 C. Hence, for the complex test space D./ consisting of complex-valued functions  of real variables x1 ; : : : ; xn in   Rn , satisfying (1.2.43) and (1.2.44), T is a distribution if and only if 

T W  2 D./ 7! T ./ D hT; i 2 C is linear on D./ W T .1 C 2 / D T .1 / C T .2 / T .˛/ D ˛T ./



(1.3.46)

81 ; 2 2 D./I

8˛ 2 C; 8 2 D./:

(1.3.47)

T is continuous on D./: n ! 0 in D./ in the sense of (1.2.44) (together with (1.2.43)) H) T .n / ! 0 in C as n ! 1. (1.3.48)

D 0 ./ D ¹T W T is a continuous linear functional on D./ satisfying (1.3.46)– (1.3.48)º is the space of distributions defined on complex test space D./. (1.3.49) All the basic results of this section for the real case will also hold for the complex case with the necessary modifications satisfying (1.2.43) and (1.2.44), (1.3.46)– (1.3.49) and without anypadditional theoretical complications, since the variables are real and the role of i D 1 is that of a parameter in all operations of differentiation and integration. Hence, the complexification of D 0 ./ is also a straightforward affair and consequently we will ignore the complexification aspects of later problems.

1.4

Some more examples of interesting distributions

Example 1.4.1. Function ln jxj defines a regular distribution on R. Solution. Function ln jxj .x ¤ 0/ is integrable in the of 0 (see ˇ neighbourhood ˇ Example 2.3.6ˇ for anˇ alternate proof) since 8" < 1, jxj" ˇln jxjˇ ! 0 as jxj ! 0 and consequently, ˇln jxjˇ  jxj1 " 8x ¤ 0 in some neighbourhood a; aŒ of 0 with a > 0, ˇ R Ra ˇ ˇln jxjˇdx  a 1 " dx < 1 for " < 1 H) ln jxj is locally integrable on R, i.e. a

a jxj

ln jxj 2 L1loc .R/ and defines a regular distribution on R by Z 1 .ln jxj/.x/dx 8 2 D.R/: hln jxj; i D

(1.4.1)

1

Example 1.4.2. Functions ln.x C i 0/ and ln.x  i 0/ define regular distributions on R. Solution. Functions ln.x ˙ i 0/ are defined by : ln.x C i 0/ D lim ln.x C iy/I y!0C

ln.x  i 0/ D lim ln.x  iy/: y!0C

(1.4.2)

42

Chapter 1 Schwartz distributions

In fact, for fixed y > 0 in the upper half-plane of z D x C iy,   q y 2 2 ln z D ln jzj C i arg z D ln x C y C i arctan is analytic: (1.4.3) x p But limy!0C ln x 2 C y 2 D limy!0C 12 ln.x 2 C y 2 / D 12 ln x 2 D ln jxj, and ´   y i lim i arctan D x 0 y!0C

for x < 0 for x > 0

³ D i H.x/;

(1.4.4)

where the Heaviside function H.x/ D 1 for x < 0, H.x/ D 0 for x > 0. Hence, ´ ln jxj C i  for x < 0 lim ln.x C iy/ D C ln jxj for x > 0 y!0 H)

ln.x C i 0/ D lim ln.x C iy/ D ln jxj C i H.x/: y!0C

Similarly, for fixed y > 0, ln.x  iy/ D

1 2

(1.4.5)

ln.x 2 C y 2 /  i arctan. yx /

´ ln jxj  i  H) ln.x  i 0/ D lim ln.x  iy/ D ln jxj y!0C

for x < 0 for x > 0

(1.4.6)

D ln jxj  i H.x/: Since ln jxj 2 L1loc .R/ (see Example 1.4.1) and H.x/ 2 L1loc .R/, ln.x ˙ i 0/ 2 L1loc .R/ and define regular distributions by: 8 2 D.R/, Z hln.x C i 0/; i D ln.x C i 0/.x/dx R 1

Z D

Z

1

.x/dxI

(1.4.7)

.x/dx:

(1.4.8)

1

Z hln.x  i 0/; i D

0

ln jxj.x/dx C i  ln.x  i 0/.x/dx R Z 1

D

Z

0

ln jxj.x/dx  i  1

1

Distributions defined by pseudo-functions (or finite part) Pf x1k ; Pf H.˙x/ xk From Examples 1.3.7 and 1.3.8, we know that x1 … L1loc .R/ and hence does not define R1 a distribution on R, but c:p:v: 1 .x/ x dx is well defined and defines a distribution

43

Section 1.4 Some more examples of interesting distributions

on R denoted by c:p:v: x1 . For k  2, 8 2 D.R/, lim"!0C but 8" > 0, if we can write Z .x/ dx D I."/ C F ."/; k jxj" x

R

.x/ jxj" x k dx

D ˙1,

(1.4.9)

where I."/ is called the infinite part of the divergent integral, since I."/ tends to ˙1 as " ! 0C , and F ."/ is called the finite part (partie finie (Pf) in French), since F ."/ tends to a finite limit F as " ! 0C . Then,  Z .x/ lim dx  I."/ D lim F ."/ D F 2 R; and (1.4.10) k "!0C "!0C jxj" x Z 1 .x/ dx 8 2 D.R/ (1.4.11) F D Pf k 1 x R1 is called the finite part .Pf/ of the divergent integral 1 .x/ dx, k  2. This concept xk of the separation of the finite part from a divergent integral is due to Hadamard [29, p. 38]. Then, pseudo-function Pf x1k .k  2/, i.e. (1.4.11) defines a distribution on R by: 8 2 D.R/ with supp./  ŒA; A, A > 0,    Z Z 1 .x/ .x/ 1 dx D lim dx  I."/ Pf k ;  D Pf k k x "!0C jxj" x 1 x   Z " Z A .x/ .x/ D lim dx C dx  I."/ ; (1.4.12) k xk "!0C A x " where I."/ ! ˙1 as " ! 0C is the infinite part of the divergent integral. The expression in the square bracket on the r.h.s. of (1.4.12), which is obtained by subtracting the infinite part I."/, has finite limit F 2 R as " ! 0C , i.e. we R 1are considering the finite part (partie finie) of the divergent integral denoted by Pf 1 .x/ dx. xk Determination of I."/ Let  2 D.R/ with supp./ D K  ŒA; A; A > 0. x 0 x k1 .k1/  .0/C  C .k1/Š  .0/C Then, by Proposition 1.2.1, we have .x/ D .0/C 1Š x k .x/, where 2 C 0 .R/

with sup j .x/j  C sup j .k/ .x/j; jxjA

C > 0:

(1.4.13)

jxj2K

Hence 8" > 0, for k  2, Z Z Z .x/ 1 dx 0 dx D .0/ dx C  .0/ C  k k k1 x x x "jxjA "jxjA "jxjA Z Z  .k1/ .0/ 1 dx C .x/dx: (1.4.14) C .k  1/Š "jxjA x "jxjA

44

Chapter 1 Schwartz distributions

But x1 is an odd function on "  jxj  A H) Using (1.4.14) and (1.4.15),

Z "jxjA

D

k1 X j D1

R

dx "jxjA x

D 0.

(1.4.15)

Z k1 X  .j 1/ .0/ Z .x/ 1 dx D dx C .j  1/Š "jxjA x k.j 1/ xk "jxjA j D1

 .j 1/ .0/ .j  1/Š



.x/dx

ˇ"  ˇA   ˇ ˇ x .kj C1/C1 x .kj C1/C1 ˇ C ˇ ˇ .k  j C 1/ C 1 A .k  j C 1/ C 1 ˇ"

Z

.x/dx

C "jxjA

 k1 X

1 .1/j k  1  .j 1/ .0/ D   .j  1/Š j  k "kj j D1



 k1 X

 Z 1 1  .1/j k  .j 1/ .0/ C   C .j  1/Š j  k Akj "jxjA j D1

.x/dx:

We set

k1 X

Cj  .j 1/ .0/ D

j D1

k1 X j D1

I."/ D

 1  .1/j k  .j 1/ .0/ .j  1/Š.j  k/Akj

k1 X

.1/j k  1  .j 1/ .0/  ; .j  k/.j  1/Š "kj j D1

and

(1.4.16)

where constants Cj do not depend on ", but depend on A, I."/ ! infinity .˙1/ as R R P .j 1/ .0/ C " ! 0C . Then "jxjA .x/ dx D I."/ C jk1 D1 Cj  "jxjA .x/dx xk

H) hPf

1 ; i D Pf xk

Z

1

.x/ dx D lim k "!0C 1 x Z k1 X .j 1/ D Cj  .0/ C j D1

Z

jxjA

"jxjA

.x/dx;

 .x/ dx  I."/ xk

45

Section 1.4 Some more examples of interesting distributions

the r.h.s. being a finite number (since H)

2 C 0 .R/),

ˇ ˇ k1 ˇ ˇ X ˇ Pf 1 ;  ˇ  jCj j sup j .j 1/ .x/j C . sup j .x/j/2A ˇ ˇ xk x2K

j D1



k1 X

jCj j sup j .j 1/ .x/j C C sup j .k/ .x/j .using(1.4.13)/ x2K

j D1

C

k X



jxjA

jxj2A

sup j .j / .x/j; C   max¹jC1 j; : : : ; jCk1 jI C º > 0

j D0 x2K

H) Pf x1k defines a distribution of order k. Thus,  Pf

   Z " Z 1 1 .x/ .x/ ;  D lim dx C dx  I."/ k xk xk "!0C 1 x "

(1.4.17)

with I."/ D

k1 X

.1/j k  1  .j 1/ .0/ kj " .j  1/Š.j  k/ j D1

(1.4.18)

defines a distribution of order k on R. Remark 1.4.1. Gelfand and Schilov [1] call this procedure regularization of divergent integrals of functions with algebraic singularities and give an exhaustive analysis of this involved problem. For all details, we refer to [1]. Example 1.4.3. 8 2 D.R/,   Z  Z 1 1 .x/ .x/ .0/ Pf 2 ;  D Pf dx D lim dx  2 2 2 x " "!0C jxj" x 1 x

(1.4.19)

defines a distribution on R. Proof. From (1.4.17) and (1.4.18), for k D 2, j D 1, I."/ D  H)

.1/1 .0/ "0Š.1/

D

2.0/ "

 Z 1 1 .x/ Pf 2 ;  D Pf dx 2 x 1 x  Z "  Z 1 .x/ .x/ .0/ dx C dx  2 D lim 2 x2 " "!0C 1 x "

8 2 D.R/ defines a distribution of order 2 on R.

46

Chapter 1 Schwartz distributions

Example 1.4.4. 8 2 D.R/,  Pf

  Z 1 .x/ H.x/ ;  D lim dx  I."/ xk xk "!0C "

(1.4.20)

with

I."/ D 

k1 X

 .j 1/ .0/  .k1/ .0/ 1  ln ";  kj .j  1/Š .k  1/Š .j  k/" j D1

(1.4.21)

defines a distribution of order k, H being the Heaviside function: H.x/ D 1 for x > 0, H.x/ D 0 for x < 0.

Proof. From (1.4.13) and (1.4.14), we get: 8 2 D.R/ with supp./ D K  ŒA; A; A > 0, Z

A "

Z Z A k1 X  .j 1/ .0/ Z A .x/ dx  .k1/ .0/ A dx C dx D C .j  1/Š " x k.j 1/ .k  1/Š " x xk " j D1

.x/dx

 k1 X

 X k  .j 1/ .0/ 1  .k1/ .0/ D ln " C  Cj  .j 1/ .0/ .j  1/Š .j  k/"kj .k  1/Š j D1 j D1 Z C

A

.x/dx; "

P  .j 1/ .0/  .k1/ .0/ where I."/ D  jk1 D1 .j 1/Š.j k/"kj  .k1/Š ln " is the infinite part; coefficients Cj do not depend on ", but depend on A. 

  Z A .x/ H.x/ dx  I."/ Pf k ;  D lim x xk "!0C " Z A k X D Cj  .j 1/ .0/ C lim j D1

; ij  CQ which is a finite number, and jhPf H.x/ xk CQ .A/ > 0,

"!0C

Pk

.x/dx;

"

j D0 supx2K

j .j / .x/j with CQ D

47

Section 1.4 Some more examples of interesting distributions

H) Pf H.x/ is a distribution of order k. In particular, for k D 2, j D 1, xk .0/  0 .0/ .0/  ln " D   0 .0/ ln "; (1.4.22) .1/0Š" 1Š "   Z 1  Z 1 .0/ .x/ .x/ H.x/ 0 C  .0/ ln " ; dx D lim dx  Pf 2 ;  D Pf x x2 x2 " "!0C 0 " (1.4.23)     Z " Z 0 .x/ .x/ H.x/ .0/ 0   .0/ ln " Pf ;  D Pf dx D lim dx  2 2 x2 " "!0C 1 x 1 x (1.4.24) I."/ D 

(with I."/ D

.0/ "

C  0 .0/ ln " in (1.4.24)) are distributions on R.

Example 1.4.5. 1: 2:

3:

1 1 D c:p:v: in D 0 .R/I (1.4.25) x x    Z 1 Z 1 H.x/ .x/ .x/ Pf ;  D Pf dx D lim dx C .0/ ln " I x x x "!0C 0 " (1.4.26)     Z " Z 0 H.x/ .x/ .x/ Pf ;  D Pf dx D lim dx  .0/ ln " : x "!0C 1 x 1 x (1.4.27) Pf

Proof. R1 R 1. hPf x1 ; i D Pf 1 .x/ x dx D lim"!0C jxj" D.R/ H) Pf x1 D c:p:v: x1 in D 0 .R/.

.x/ x dx

D hc:p:v: x1 ; i 8 2

2. For  2 D.R/ with supp./ D K  ŒA; A, A > 0, .x/ D .0/ C x .x/ with 2 C 0 .R/, supjxjA j .x/j  C supx2K j .1/ .x/j, Z "

1

.x/ dx D x

Z

A "

.x/ dx D .0/ x

Z

A "

1 dx C x

Z

A

.x/dx "

Z

A

D .0/Œln A  ln " C

.x/dx "

D .0/ ln " C.0/ ln A C „ ƒ‚ … I."/

Z

A

.x/dx "

48

Chapter 1 Schwartz distributions

RA RA H) lim"!0C Œ " .x/ .x/dx, which is a x dx  I."/ D .0/ ln A C 0 finite number, with I."/ D .0/ ln ".    Z 1 Z 1 .x/ .x/ H.x/ H) Pf ;  D Pf dx D lim dx C .0/ ln " x x x "!0C 0 " Z A D .0/ ln A C .x/dx D a finite number 0    Z 1 H.x/ .x/ H) Pf ;  D lim dx C .0/ ln " 8 2 D.R/ x x "!0C " defines a distribution on R, since ˇ ˇ ˇ ˇ ˇhPf H.x/ ; iˇ  j.0/j  j ln Aj C CA sup j .1/ .x/j ˇ ˇ x x2K

 C1

1 X

sup j .j / .x/j;

j D0 x2K

with C1 D C1 .A/ > 0. 3. For  2 D.R/ as defined in the proof of (2), we have, 8" > 0, Z " Z " Z " .x/ 1 dx D .0/ dx C .x/dx 1 x A x A Z " .x/dx D .0/Œln "  ln A C A

Z

"

.x/dx

D .0/ ln "  .0/ ln A C A

   Z " Z 0 .x/ .x/ H.x/ ;  D Pf dx D lim dx  .0/ ln " x "!0C 1 x 1 x Z 0 D .0/ ln A C .x/dx D a finite number A     Z " H.x/ .x/ H) Pf ;  D lim dx  .0/ ln " 8 2 D.R/ x "!0C 1 x defines a distribution on R, since ˇ ˇ ˇ ˇ ˇ Pf H.x/ ;  ˇ  j.0/j j ln Aj C CA sup j .1/ .x/j ˇ ˇ x 

H)

Pf

x2K

 C2

1 X

sup j .j / .x/j;

j D0 x2K

with C2 D C2 .A/ > 0. Remark 1.4.2. From (1.4.26) and (1.4.27), the result (1.4.25) follows.

49

Section 1.4 Some more examples of interesting distributions   Distributions defined by x  ; xC ; x

Function x  Let  D a C i b 2 C be a complex number with a; b 2 R. Then, for real x ¤ 0; x  is defined by: x  D e  ln x D x a e ib ln x

for x > 0

(1.4.28)

x  D Œ.1/.x/ D e i .x/ D jxja :e bCi.b ln jxjCa/

for x < 0 (1.4.29)

H) jx  j  jxja 8x ¤ 0, 8 D a C i b H) x  2 L1loc .R/ if jxja is locally integrable on R. But jxja is locally integrable on R for a D Re./ > 1: Z

A

jx  jdx 

A

Z

A

jxja dx < C1

8a D Re./ > 1; 8A > 0:

(1.4.30)

A

Distribution x  For  with Re./ > 1, x  2 L1loc .R/ and defines a regular distribution Tx  D x  2 D 0 .R/ by: 

Z

hTx  ; i D hx ; i D

1

x  .x/dx

8 2 D.R/:

For  D k, k 2 N, Pf x1k 2 D 0 .R/ is defined by (1.4.17).   Functions xC ; x

(1.4.32)

  are defined by: For  2 C with Re./ > 1, functions xC ; x

´

 xC

(1.4.31)

1

x D 0

´

for x > 0 I for x  0

 x

D

0 jxj

for x  0 : for x < 0

(1.4.33)

  For Re./ > 1, functions x  ; x  2 L1 .R/ are locally Distributions xC ; x C  loc integrable (see (1.4.30)) and hence define regular distributions on R by:

 hxC ; i D  hx ; i D

Z Z

1

x  .x/dxI

(1.4.34)

jxj .x/dx:

(1.4.35)

0 0 1

50

Chapter 1 Schwartz distributions

Alternative forms of (1.4.34) and (1.4.35) for Re./ > 2 and  ¤ 1 Since for R1 C1 1 2 C; 8 2 D.R/, Re./ > 1, 0 x  D xC1 j10 D C1  hxC ; i

Z

1

D Z

x  .x/dx

0 1

D

Z



1

x Œ.x/  .0/dx C .0/ Z

0 1

Z



1

0

1

x dx C 0

x  .x/dx

1

x  .x/dx C

x Œ.x/  .0/dx C

D

Z



1

.0/ : C1

(1.4.36)

L Since .x/ D .x/ 8x, 8 2 D.R/; for Re./ > 1, Z 1   L hx ; i D hxC ; i D jxj .x/dx Z

0

1

D

x  Œ.x/  .0/dx

0

Z

1

C

x  .x/dx C

1

.0/ C1

8 2 D.R/:

(1.4.37)

 and x  defined by (1.4.36) and (1.4.37) respectively hold for Distributions xC  Re./ > 2, and  ¤ 1. Consider the r.h.s. of (1.4.36). In fact,R by Proposi1 tion 1.2.1, .x/ D .0/Cx .x/ with 2 C 0 .R/ H) the first integral 0 x  Œ.x/ R 1 C1 .0/dx D 1 x .x/dx is well defined for Re. C 1/ > 1 H) Re./ > 2. R1   The second integral 1 x .x/dx is well defined 8, since x  1. 

.0/ The third expression .C1/ is defined for  ¤ 1. Hence, the expression on the r.h.s. of (1.4.36) is well defined for Re./ > 2 and  ¤ 1. Similarly, we can prove that the r.h.s. of (1.4.37) is well defined for Re./ > 2 and  ¤ 1. 

 Moreover, in a similar way we can show that for Re./ > n,  ¤ 1; 2; : : : ; .n  1/, Z 1  hxC ; i D jxj .x/dx

Z

0

1

D 0

  n X x j 1  .j 1/ .0/ dx x .x/  .j  1/Š 

j D1

Z

1

C 1

jxj .x/dx C

n X kD1

 .k1/ .0/ : . C k/.k  1/Š

(1.4.38)

Section 1.5 Multiplication of distributions by C 1 -functions

51

Remark 1.4.3. Formulae (1.4.36) and (1.4.38) are not contradictory ones. (1.4.38) is an analytic extension of (1.4.36) to the left of  D 1 with  ¤ 2; 3; : : : ; .n1/ (see [1] for all details).

1.5

Multiplication of distributions by C 1 -functions

For arbitrary distributions S 2 D 0 ./ and T 2 D 0 ./, the product T S or S T is not defined in general. (1.5.1) For example, for n D 1;  D R; f .x/ D p1jxj .x ¤ 0/ 2 L1loc .R/ and defines a regular distribution Tf D f on R since, 8 compact Œa; b  R containing 0, Z

b a

1 p dx D lim "1 !0C jxj

Z a

"1

1 p dx C lim "2 !0C jxj

Z

b "2

1 p dx < C1: jxj

1 But f 2 D f:f D jxj … L1loc .R/ (see Example 1.3.7) and hence f 2 does not define a distribution, i.e. Tf 2 is not a distribution. But in particular cases, the product S T of distributions S; T 2 D 0 ./, may have a meaning, when for an ‘irregular’ T 2 D 0 ./, S 2 D 0 ./ is ‘highly regular’. For example, let T 2 D 0 ./ be an arbitrary distribution and S 2 D 0 ./ be defined by an infinitely differentiable function ˛ 2 C 1 ./ (i.e. ˛ 2 CR1 ./ H) ˛ 2 L1loc ./ H) ˛ defines a regular distribution S 2 D 0 ./ by S./ D  ˛d x 8 2 D./). Then, ˛T is defined and a distribution. Hence, we have:

Definition 1.5.1. Let T 2 D 0 ./; ˛ 2 C 1 ./. Then ˛T 2 D 0 ./ and is defined by: h˛T; i D hT; ˛i

8 2 D./;

(1.5.2)

with ˛ 2 D./. In particular, f 2 L1loc ./; g 2 C 1 ./

H)

fg 2 L1loc ./

H)

Tfg 2 D 0 ./: (1.5.3)

Justification of Definition 1.5.1 ˛ 2 C01 ./ with supp.˛/ D supp.˛/ \ supp./  supp./  , n ! 0 in D./ H) ˛n ! 0 in D./, which is established by applying Leibniz’ theorem to successive differentiations of .˛/. ˛T is a continuous linear functional on D./ W n ! 0 in D./ H) ˛n ! 0 in D./ H) h˛T; n i D hT; ˛n i ! 0 in R, since T 2 D 0 ./.

52

Chapter 1 Schwartz distributions

Example 1.5.1. 8˛ 2 C 1 .Rn / and for Dirac distribution ı 2 D 0 .Rn /, the product ˛ı is defined by: ˛ı D ˛.0/ı: In fact, h˛ı; i D hı; ˛i D ˛.0/.0/ D ˛.0/hı; i D h˛.0/ı; i

8 2 D.Rn /

H) ˛ı D ˛.0/ı in D 0 .Rn /, i.e. any product in which ı occurs, is proportional to ı. In particular, for n D 1, ˛.x/ D x 8x 2 R, xı D 0 2 D 0 .R/. (1.5.4) n For ˛.x/ D p.x/ D a0 C a1 x C    C an x , p.x/ı D p.0/ı D a0 ı 2 D 0 .R/:

(1.5.5)

Example 1.5.2. 8˛ 2 C 1 .Rn / and for Dirac distribution ıa 2 D 0 .Rn / defined by hıa ; i D .a/ 8 2 D.Rn /, the product ˛ıa is given by ˛ıa D ˛.a/ıa . Indeed, h˛ıa ; i D hıa ; ˛i D ˛.a/.a/ D ˛.a/hıa ; i D h˛.a/ıa ; i

8 2 D.Rn / (1.5.6)

H) ˛ıa D ˛.a/ıa in D 0 .Rn /. In particular, for n D 1, ˛.x/ D x 8x 2 R, xıa D aıa

(1.5.7)

p.x/ıa D p.a/ıa I

(1.5.8)

e bx ıa D e ab ıa I

(1.5.9)

.x  a/ıa D 0:

(1.5.10)

Example 1.5.3. Show that the following equalities hold in D 0 .R/:   1 1: x c:p:v: D 1I x   H.x/ (1.5.11) D HI 2: x Pf x   1 1 3: x Pf D x.c:p:v: / D 1: x x R1 R1 R 1Solution 1.  2 D.R/ H) 1 .x/dx exists H) c:p:v: 1 .x/dx D 1 .x/dx.

Section 1.5 Multiplication of distributions by C 1 -functions

53

1. 8 2 D.R/,       Z 1 1 1 1 .x.x//dx x c:p:v: ;  D c:p:v: ; x D c:p:v: x x 1 x   Z 1 1 D D 1 in D 0 .R/: .x/dx D h1; i H) x c:p:v: x 1 2. 8 2 D.R/,       Z 1 Z 1 1 H.x/ H.x/ ; x D Pf .x/dx D .x/dx ;  D Pf x Pf x x x 0 0   Z 1 H.x/ D D H in D 0 .R/: H.x/.x/dx D hH; i H) x Pf x 1 3. The result follows from (1) and Example 1.4.5(1).   for Re./ > 2 and  ¤ 1. Example 1.5.4. Calculate x  xC and x  x Solution. Using (1.4.36), for Re./ > 2 and  ¤ 1,   hx  xC ; i D hxC ; xi Z 1 Z  D x Œx.x/  .x/.0/dx C

Z

0 1

D

x

C1

H)

1

Z

1

.x/ C

0

1

x

C1

Z

dx D

1

C1  D xC x  xC

x  x.x/dx C 1

.x/.0/ C1

x C1 .x/dx

8 2 D.R/;

0

in D 0 .R/;

since .x/.0/ D 0 (using (1.4.33)): (1.5.12)

   L D hx  ; x i L (by (1.4.37)) ; i D hx ; xi D hxC ; .x/i hx  x C Z 1 Z 1 C1 L x  x.x/dx D  x C1 .x/dx D hxC ; i D 0

H)

x

 x

0

D

C1 hx ; i

D

C1 x

C1 hx ; i

D

8 2 D.R/

0

in D .R/:

(1.5.13)

Multiplication of several distributions The product of several distributions is well defined if all of the distributions, except at most one of them, are C 1 -functions and the product is associative. Otherwise, the product of several distributions is not defined in general. In particular, even if the product of several distributions is defined, it may not be associative. ˛; ˇ 2 C 1 ./; T 2 D 0 ./

H)

˛.ˇT / D .˛ˇ/T 2 D 0 ./:

(1.5.14)

54

Chapter 1 Schwartz distributions

In fact, h˛.ˇT /; i D hˇT; ˛i D hT; ˇ˛i D hT; ˛ˇi D h˛T; ˇi D hˇ˛T; i D h.˛ˇ/T; i

8 2 D./:

(1.5.15)

Counterexample 1.5.5. Let c:p:v: x1 2 D 0 .R/ be the distribution defined in ExR1 R .x/ ample 1.3.8: hc:p:v: x1 ; i D c:p:v: 1 .x/ x dx D lim"!0C jxj" x dx 8 2 D.R/. Then the product of Dirac distribution ı and c:p:v: x1 can not be defined. Proof. Set T D ı; S D c:p:v: x1 . Suppose that T S; S T are defined with T S D S T . Then, for ˛ 2 C 1 .R/, we would have ˛.T S/ D .˛T /S D T .˛S /. Choose ˛.x/ D x 8x 2 R. Then, we would have x.ı c:p:v: x1 / D .xı/ c:p:v: x1 D 0  c:p:v: x1 D 0 2 D 0 .R/, x.ı c:p:v: x1 / D x.c:p:v: x1 ı/ D .x c:p:v: x1 /  ı D 1  ı D ı 2 D 0 .R/, (x c:p:v: x1 D 1 2 D 0 .R/ from Example 1.5.3(1)), which is impossible. Hence, the product of ı and c:p:v: x1 can not be defined. Remark 1.5.1. ı and c:p:v: x1 are two distributions, x is a C 1 -function and xı D 0 2 D 0 .R/, x.c:p:v: x1 / D 1 2 D 0 .R/ are well defined.

1.6

Problem of division of distributions

Case of single variable (n D 1) Let   R be an open subset of R, in which a given function f does not vanish, i.e. f .x/ ¤ 0 8x 2 . Let S 2 D 0 ./ be a given distribution and f 2 C 1 ./. Then, 9 one and only one distribution T 2 D 0 ./ such that f T D S in D 0 ./, since f1 2 C 1 ./ and, multiplying both sides by f1 , we get

1 .f f

T/ D

1 S f

” T D S f1 2 D 0 ./, i.e. for f 2 C 1 ./ with f .x/ ¤ 0

for any x 2 , the division of S by f is the multiplication of S by f1 2 C 1 ./. In particular, for S D 0 in D 0 ./ and f 2 C 1 ./ with f .x/ ¤ 0 8x 2 , f T D 0 in D 0 ./

H)

T D 0 in D 0 ./:

(1.6.1)

Example 1.6.1. For f .x/ D 1Cx 2 2 C 1 .R/, .1Cx 2 /T D 0 in D 0 .R/ H) T D 0 in D 0 .R/. But if f .x/ D 0 for some x 2 , the situation is completely different. For example, if for f 2 C 1 ./; f .x/ D 0 for some x 2   R, then f T D 0 in D 0 ./ does not imply T D 0 in D 0 ./. In fact, we have: Proposition 1.6.1. 8 real a 2 R, .x  a/T D 0 in D 0 .R/

”

T D C ıa ;

(1.6.2)

55

Section 1.6 Problem of division of distributions

where ıa is the Dirac distribution with mass/charge/force etc. concentrated at a, C 2 R being an arbitrary constant. In particular, for a D 0, xT D 0 in D 0 .R/

”

T D C ı0 D C ı:

(1.6.3)

Proof. T D C ıa H)

h.x  a/T; i D hT; .x  a/i D hC ıa ; .x  a/i D C hıa ; .x  a/i D C Œ.x  a/.x/xDa D C  Œ0  .a/ D 0 8 2 D.R/:

Converse: Now we show that .x  a/T D 0 H) T D C ıa . In fact, hT; .x  a/i D 0 8 2 D.R/ H) T vanishes on all D .x  a/ 2 D.R/, i.e. on all 2 D.R/ which vanish at x D a. Let 2 D.R/ be a given fixed function such that .a/ D 1. Then every 2 D.R/ is of the form: D  C with D .x  a/ 2 D.R/;  2 D.R/ such that  D .a/; .a/ D 0. Hence, T . / D T . C / D T . / C T . / D C .a/ C hT; .x  a/i, where  D .a/; C D T . /; hT; .x  a/i D h.x  a/T; i D 0 (by hypothesis) H) T . / D C .a/ D C hıa ; i D hC ıa ; i 8 2 D.R/ H) T D C ıa ; C being an arbitrary constant. Remark 1.6.1. Let T 2 D 0 .R  ¹aº/ be a distribution on the open set R  ¹aº; then 1 2 C 1 .R  ¹aº/ does not vanish on the open set R  ¹aº. Then .x  a/T D .xa/ S 2 D 0 .R  ¹aº/ has the solution T D

1 S 2 D 0 .R  ¹aº/: xa

(1.6.4)

Example 1.6.2. Find the general solution (i.e. all the solutions) of the following equations for T in 2 D 0 .R/: for a; b 2 R; a ¤ b, 1. .x  a/T D ıb ; 2. .x  a/.x  b/T D 0; 3. .1 C x 2 /.1  x 2 /T D 0; 4. p.x/T D 0, where p.x/ D .x  a1 /.x  a2 / : : : .x  an / with a1 < a2 <    < an . (1.6.5) Solution. 1. For .x  a/T D ıb , where ıb is the Dirac distribution with force/mass/ charge etc. concentrated at b 2 R; T D Th C Tp is the general solution of .x  a/T D ıb , where Tp is a particular solution of .x  a/T D ıb , i.e. .x  a/Tp D ıb , and Th is the general solution of the corresponding homogeneous equation .x  a/T D 0, which is obtained by putting the right-hand side equal to 0 2 D 0 .R/, since .xa/T D ıb H) .x  a/.Tp C Th / D .x  a/Tp C .x  a/Th D ıb C .x  a/Th D ıb H) .x  a/Th D 0 H) Th satisfies the corresponding homogeneous equation.

56

Chapter 1 Schwartz distributions

Then, by Proposition 1.6.1, Th D C1 ıa , C1 2 R being an arbitrary constant. ıb Construction of Tp : Tp D ba with a ¤ b. Indeed, h.x  a/Tp ; i D hTp ; .x  ıb 1 1 ; .x  a/i D ba hıb ; .x  a/i D ba Œ.b  a/.b/ D .b/ D a/i D h ba ıb hıb ; i 8 2 D.R/ H) .x  a/Tp D ıb with Tp D ba . Hence, the general solution is T D C1 ıa C

ıb ; ba

C1 2 R being an arbitrary constant :

(1.6.6)

Remark 1.6.2. Case a D b can not be discussed here, since derivatives of Dirac distributions will be involved, which will be introduced in the next chapter. 2. .x  a/.x  b/T D 0 2 D 0 .R/.b ¤ a/. Set S D .x  b/T 2 D 0 .R/. Then .x  a/S D 0 in D 0 .R/ H) S D C1 ıa by Proposition 1.6.1, C1 2 R being an arbitrary constant H) .x  b/T D C1 ıa in D 0 .R/. From the solution (1.6.6) of (1), ıa T D C1 ab C C2 ıb in D 0 .R/ is the general solution, which can be rewritten as T D d1 ıa C d2 ıb ;

d1 and d2 being arbitrary real constants.

(1.6.7)

3. .1 C x 2 /.1  x 2 /T D 0 in D 0 .R/. Since .1 C x 2 /1 2 C 1 .R/ does not vanish on R, .1 C x 2 /1 .1 C x 2 /.1  x 2 /T D 0 in D 0 .R/ H) .1  x 2 /T D 0 H) .x C 1/.x  1/T D 0. Then from the solution (1.6.7) of (2) with a D 1, b D 1, T D d1 ı1 C d2 ı1

is the required general solution;

(1.6.8)

where ı1 (resp. ı1 ) is the Dirac distribution with concentration at x D 1 (resp. 1), and d1 and d2 are arbitrary real constants. 4. p.x/T D 0 in D 0 .R/ H) .x  a1 /.x  a2 / : : : .x  an /T D 0, where a1 < a2 <    < an . Set S1 D .x  a2 /    .x  an /T in D 0 .R/. Then .x  a1 /S1 D 0 H) S1 D C1 ıa1 by Proposition 1.6.1, C1 being an arbitrary constant. Then .x a2 / : : : .x an /T D C1 ıa1 2 D 0 .R/. Set S2 D .x a3 / : : : .x an /T 2 ı

a1 (by solution (1.6.6) D 0 .R/. Then .x  a2 /S2 D C1 ıa1 H) S2 D C2 ıa2 C C1 a1 a 2

ı

a1 . of (1) with a2 D a, a1 D b) H) .x  a3 /    .x  an /T D C2 ıa2 C C1 a1 a 2

ı

ı

a2 a1 C C1 .a1 a2 /.a . Then .x  a4 / : : : .x  an /T D C3 ıa3 C C2 a2 a 3 1 a3 / Continuing in this way, finally we get

T D

n X kD1

dk ıak

with dk D

Ck ; 1  k  n  1; dn D Cn ; .ak  akC1 /    .ak  an / (1.6.9)

where ıak is the Dirac distribution with concentration at ak ; 1  k  n.

57

Section 1.7 Even, odd and positive distributions

1.7

Even, odd and positive distributions

Even and odd distributions L 8 2 D.R/, .x/ D .x/ 8x 2 R, L 2 D.R/. Definition 1.7.1. A distribution T 2 D 0 .R/ is called even if and only if L D T ./; T ./

i.e. T ..x// D T ..x//

8 2 D.R/I

(1.7.1)

odd if and only if L D T ./; T ./

i.e. T ..x// D T ..x//

8 2 D.R/:

(1.7.2)

Examples of even distributions are: 1. f is even in L1loc .R/ ” f .x/ D f .x/ for almost all x 2 R. Then Z Z L L Tf ./ D hf; i D f .x/.x/dx D f .x/.x/dx (by change of variable) R R Z f .x/.x/dx D Tf ./ 8 2 D.R/ (since f .x/ D f .x/ a.e. on R) D R

H) Tf defined by even f 2 L1loc .R/ is an even distribution. L D .1/2m hı; ./ L .2m/ i D ./ L .2m/ .0/ D ./.2m/ .0/ 2. hı .2m/ ; i

(1.7.3)

D hı;  .2m/ i D .1/2m hı .2m/ ; i D hı .2m/ ; i 8 2 D.R/ (For derivatives of ı, see (2.3.8) in Chapter 2) H) ı .2m/ (i.e. even-order derivatives of Dirac distribution ı) are even distributions for all m 2 N0 . (1.7.4)   , with x  ; x  defined by (1.4.35) and (1.4.36) respectively, is an 3. jxj D xC C x C  even distribution. (1.7.5)

Examples of odd distributions are: 4. f is odd in L1loc .R/ ” f .x/ D f .x/ for almost all x 2 R. Then Z Z L Tf ./ D f .x/.x/dx D f .x/.x/dx (by change of variables) R R Z D f .x/.x/dx D Tf ./ 8 2 D.R/ R

H) Tf defined by odd f 2 L1loc .R/ is an odd distribution.

(1.7.6)

58

Chapter 1 Schwartz distributions

L D .1/.2mC1/ hı; ./ L .2mC1/ i D    D .1/.2mC1/ hı .2mC1/ ; i 5. hı .2mC1/ ; i D hı .2mC1/ ; i

8 2 D.R/

(see (2.3.8) in Chapter 2)

H) ı .2mC1/ (i.e. odd-order derivatives of Dirac distribution ı) are odd distributions (1.7.7) for all m 2 N0 .   , with x  ; x  defined by (1.4.35) and (1.4.36) respectively, 6. jxj sgn x D xC  x C  is an odd distribution. (1.7.8)

Every distribution T 2 D 0 .R/ can be expressed as a sum of an even distribution TE 2 D 0 .R/ and an odd distribution T0 2 D 0 .R/, i.e. T 2 D 0 .R/

H)

T D T E C T0 ;

(1.7.9)

the decomposition being a unique one. L Proof. Define TE and T0 by: TE ./ D 12 ŒT ./ C T ./, 1 L T0 ./ D ŒT ./  T ./ 2

8 2 D.R/:

(1.7.10)

LL D 1 ŒT ./ L D 1 ŒT ./ L C T ./ L C T ./ D TE ./ H) TE is even. Then TE ./ 2 2 LL D 1 ŒT ./ L  T ./ L  T ./ D  1 ŒT ./  T ./ L D T0 ./ L D 1 ŒT ./ T0 ./ 2 2 2 H) T0 is odd. .TE C T0 /./ D TE ./ C T0 ./ D T ./

8 2 D.R/

H) TE C T0 D T in D 0 .R/. Uniqueness: Let TE0 ; T00 2 D 0 .R/ such that TE0 C T00 D TE C T0 D T . Then, T  TE0 D T00 is odd H)

L D T ./ L C T 0 ./ L hT  TE0 ; i D hT  TE0 ; i E

H)

L C T 0 ./ L T ./  TE0 ./ D T ./ E

H)

L D T ./ C T ./ L 2TE0 ./

H) H)

L D TE ./ L D 1 ŒT ./ C T ./ TE0 ./ D TE0 ./ 2 TE0 D TE in D 0 .R/:

Similarly, T00 D T0 in D 0 .R/.

8 2 D.R/

Section 1.8 Convergence of sequences of distributions in D 0 ./

59

Positive distributions Definition 1.7.2. A distribution T 2 D 0 ./ with   Rn is called positive, i.e. T  0, if and only if T ./ D hT; i  0 8 2 D.R/ with   0. (1.7.11) T1  T2 in D 0 .R/ ” T1  T2  0 ” T1 ./  T2 ./ 8 2 D.R/ with   0. (1.7.12)

1.8

Convergence of sequences of distributions in D 0 ./

0 n Definition 1.8.1. Let .Tn /1 nD1 be a sequence of distributions in D ./ with   R . Then, if 9T 2 D 0 ./ such that

hTn ; i ! hT; i in R .resp. C/ 8 2 D./

as n ! 1;

(1.8.1)

0 the sequence .Tn /1 nD1 is said to converge to T 2 D ./, and we write Tn ! T in 0 D ./ as n ! 1. Then T is called the limit of the sequence .Tn / in D 0 ./:

lim Tn D T

n!1

in D 0 ./:

(1.8.2)

0 Proposition 1.8.1. Let .Tn /1 nD1 be a sequence of distributions in D ./ such that limn!1 hTn ; i exists in R (resp. C) 8 2 D./. Then the sequence .Tn /1 nD1 has a limit in D 0 ./, i.e. 9 a unique T 2 D 0 ./ such that

hTn ; i ! hT; i in R .resp. C/ 8 2 D./

as n ! 1:

(1.8.3)

Remark 1.8.1. As a consequence of Proposition 1.8.1, for the convergence of a sequence .Tn / in D 0 ./ it is not necessary to know its limit explicitly, i.e. it is sufficient to show that limn!1 hTn ; i exists in R (resp. C) 8 2 D./. Examples of convergence of sequences of distributions Example 1.8.1. Tn D sin nx in D 0 ./ 8n 2 N H) limn!1 sin nx D 0 in D 0 .R/, i.e. Z 1 lim hTn ; i D lim hsin nx; i D lim sin nx.x/dx D 0 8 2 D.R/: n!1

n!1

n!1 1

(1.8.4) In fact, 8 2 D.R/; 9A > 0 such that supp./  ŒA; A. Then, integrating by R1 RA parts, we get 1 sin nx.x/dx D A cosnnx  0 .x/dx; since .˙A/ D 0. ˇ ˇZ A Z A ˇ 1 ˇ cos nx 0  .x/dx ˇˇ  max j 0 .x/j dx But ˇˇ n n x2ŒA;A A A D

2A max j 0 .x/j ! 0 as n ! 1 n x2ŒA;A

60

Chapter 1 Schwartz distributions

RA H) limn!1 A cosnnx  0 .x/dx D 0 8 2 D.R/ with supp./  ŒA; A. Hence, limn!1 hTn ; i D limn!1 hsin nx; i D 0 8 2 D.R/. Then, by Proposition 1.8.1, 9T D 0 2 D 0 .R/ such that limn!1 hsin nx; i D h0; i 8 2 D.R/ H) limn!1 sin nx D 0 in D 0 .R/. Example 1.8.2. Tn D sinxnx in D 0 .R/ 8n 2 N H) limn!1 sinxnx D ı in D 0 .R/, i.e., 8 2 D.R/,   Z 1 sin nx sin nx lim hTn ; i D lim ;  D lim .x/dx D .0/ D hı; i: n!1 n!1 n!1 x x 1 (1.8.5) Indeed, 8 2 D.R/; 9A > 0 such that supp./  ŒA; A, and Z

1 1

sin nx .x/dx D x

Z

A

A A

Z

sin nx .x/dx x

.x/  .0/ D dx C .0/ sin nx x A From Proposition 1.2.1, .x/ D .0/ C x .x/ with Z

A

H)

sin nx A

.x/  .0/ dx D x

Z

Z

A

A

sin nx dx: (1.8.6) x

2 C 0 .R/,

A

sin nx .x/dx ! 0 as n ! 1; A

since 2 C 0 .ŒA; A/  L1 .A; AŒ/ by the Riemann–Lebesgue theorem6 (see also (2.11.7i)), Z lim

A

n!1 A 6 Riemann–Lebesgue

I. lim!1 II. lim!1

f .x/ sin xdx D 0;

Rb

f .x/cosxdx D 0.

a

(1.8.7)

Theorem: For compact Œa; b  R, let f 2 L1 .a; bŒ/. Then,

Rb a

sin nx .x/dx D 0:

Proof. We give the proof of (I) and (II) for f 2 C 1 .Œa; b/  L1 .a; bŒ/. Hence, for f 2 C 1 .Œa; b/, R Rb Rb cos x b 1 b 0 1 a f .x/ sin xdx D f .x/  ja C  a f .x/ cos xdx H) j a f .x/ sin xdxj   Œjf .b/j C Rb Rb 0 jf .a/j C a jf .x/jdx ! 0 as  ! 1 H) lim!1 a f .x/ sin xdx D 0. Rb The linear functional l.f / D lim!1 a f .x/ sin xdx 8f 2 L1 .a; bŒ/ is continuous on R b L1 .a; bŒ/, i.e. jl.f /j  lim!1 a jf .x/jdx H) jl.f /j  kf kL1 .a;bŒ/ 8f 2 L1 .a; bŒ/. The continuous linear functional l vanishes on C 1 .Œa; b/, which is dense in L1 .a; bŒ/. Hence, the result follows as a consequence of Hahn–Banach Theorem.

Section 1.8 Convergence of sequences of distributions in D 0 ./ n

Putting x D

R1

sin 1 d 

d n ,

with dx D

we get

RA

A

sin nx x dx

61 D

R nA

nA

sin d

!

D  as n ! 1, i.e. Z

A

lim

n!1 A

sin nx dx D : x

(1.8.8)

Finally, from (1.8.6)–(1.8.8), limn!1 h sinxnx ; i D 0 C .0/ D hı; i 8 2 D.R/ H) limn!1 sinxnx D ı in D 0 .R/. Rn Example 1.8.3. Tn D Un with Un .x/ D n e ixy dy 8x 2 R, 8n 2 N Z n e ixy dy D 2ı in D 0 .R/ with hı; i D .0/ 8 2 D.R/; H) lim n!1 n

(1.8.9) i.e.

Z

n

lim hTn ; i D lim

n!1

In fact, H)

Rn

n e

ixy dy

Z

D

n

e

lim

n!1

ixy

lim

 dy;  D h2ı; i D 2.0/:

D



e ix n e ix n ix Z n

dy;  D lim

n!1

sin nx x

D2

  sin nx sin nx .x/dx D lim 2 ; 2 n!1 x x n

D 2.0/ D h2ı; i n

n!1 n

(1.8.10)

n

e ixy yDn ix jyDn

n

Z H)

e

n!1

ixy

e ixy dy D 2ı

(by (1.8.5))

8 2 D.R/:

in D 0 .R/:

Example 1.8.4. For  > 0, let T 2 D 0 .R/ be defined by: Z  2 cos x hT ; i D Œ.x/  .0/dx 8 2 D.R/: x  2

(1.8.11)

Find 1. lim!1 T ; 2. lim!0C T in D 0 .R/. Solution. 1. By Proposition 1.2.1, .x/ D .0/ C x .x/ with 2 C 0 .R/. Then R 2 .x/  .0/ D x .x/ 8x H) hT ; i D  cos x .x/dx, where is 2

 1   continuous on Œ  2 ; 2  and hence belongs to L . 2 ; 2 Œ/. Consequently, by the Riemann–Lebesgue theorem (see footnote p. 60), Z  2 lim cos x .x/dx D 0: (1.8.12) !1

 2

Hence, lim!1 hT ; i D 0 8 2 D.R/ H) lim!1 T D 0 in D 0 .R/.

62

Chapter 1 Schwartz distributions

R R 2 2.  ! 0C H) cos x ! 1 H) 2 cos x .x/dx !  .x/dx by 2 2 Lebesgue’s dominated convergence theorem (see Theorem B.3.2.2 in  Appendix B), since jcos x .x/j  j .x/j 2 L1 .  2 ; 2 Œ/. R 2 1 Hence, lim!0C hT ; i D  x Œ.x/  .0/dx 8 2 D.R/. 2

Let T 2 D 0 .R/ be defined by: Z  2 1 Œ.x/  .0/dx hT; i D x  2

8 2 D.R/:

(1.8.13)

Then lim!0C T D T in D 0 .R/ with T defined by (1.8.13). Example 1.8.5. 8" > 0, let f" be defined by f" .x/ D 2" jxj"1 8x 2 R. Then show that lim f" D ı

"!0C

in D 0 .R/:

(1.8.14)

Proof. Since jxj is locally integrable on R for  D "  1 > 1 8" > 0 (see (1.4.30)), jxj"1 is locally integrable on R and defines a distribution on R, and conR " 0 "1 sequently, f" 2 D .R/ 8" > 0 with hf" ; i D 2 R jxj .x/dx. For  2 D.R/ with supp./ D K  ŒA; A; A > 0, Z Z " 0 " A "1 hf" ; i D .x/"1 .x/dx C x .x/dx 2 A 2 0 Z Z " A "1 " A "1 D x .x/dx C x .x/dx 2 0 2 0 Z " A "1 D x Œ.x/ C .x/dx: 2 0 From Proposition 1.2.1, .˙x/ D .0/ ˙ x .˙x/ with supjxjA j .x/j  C supjxjA j 0 .x/j H) .x/ C .x/ D 2.0/ C xŒ .x/  .x/ Z A Z " " A " x "1 dx C x Œ .x/  .x/dx: (1.8.15) H) hf" ; i D 2.0/ 2 2 0 0 RA " " C Then, for 0 < " < 1, " 0 x "1 dx D " x" jA 0 D A , which tends to 1 as " ! 0 ,    Z A H) lim .0/ " x "1 dx D .0/  lim A" D .0/: (1.8.16) "!0C

0

Again, for 0 < " < 1, 2" j " 2

RA 0

RA

" 2

RA 0

x " jΠ.x/ .x/jdx  1C"

x " dx  C supjxjA j 0 .x/j  A1C"  " ! 0 as " ! 0C Z " A " lim x Π.x/  .x/dx D 0: (1.8.17) "!0C 2 0

 2  C supjxjA j 0 .x/j  H)

x " Π.x/ .x/dxj 

"!0C

0

Section 1.8 Convergence of sequences of distributions in D 0 ./

63

Then, from (1.8.15)–(1.8.17), lim"!0C hf" ; i D .0/ C 0 D .0/ D hı; i8 2 D.R/ H) lim"!0C f" D ı in D 0 .R/. Example 1.8.6. 1. Let .fn / be a sequence of functions on R defined by: ´ n2 for jxj < n1 8n 2 N: fn .x/ D 0 for jxj  n1

(1.8.18)

Show that (a) .fn .x// converges to 0 in R 8x ¤ 0; (b) .fn / does not converge in D 0 .R/. 2. Let .gn / be a sequence defined by: ´ gn .x/ D

n 2

0

for jxj < for jxj 

1 n 1 n

8n 2 N:

(1.8.19)

Show that (a) .gn / converges to ı in D 0 .R/; (b) .gn .x// converges to 0 for x ¤ 0. Proof. 1. (a) 8x ¤ 0, fn .x/ ! 0 in R as n ! 1. In fact, 8 fixed x0 ¤ 0, 9n0 2 N such that 8n  n0 , n10 < jx0 j H) fn .x0 / D 0 8n  n0 H) 8x0 ¤ 0, fn .x0 / ! 0 in R as n ! 1. R1 R1 (b) hfn ; i D n1 n2 .x/dx D n2 n1 .x/dx: n

n

9 2 D.R/ such that .x/ D 1 8jxj  1. Then, for such a  2 D.R/ with R1 .x/ D 18jxj  1, hfn ; i D n2 n1 1dx D n2  n2 D 2n ! 1 as n ! 1, i.e. n

.hfn ; i/ does not converge in R, and hence .fn / does not converge in D 0 .R/. R1 2. (a) hgn ; i D n1 n2 .x/dx D n2  n2 .n / D .n / with jn j  n1 . n

But n ! 0 as n ! 1, and n ! 0 H) .n / ! .0/, since  is continuous. Hence, lim hgn ; i D .0/ D hı; i 8 2 D.R/

n!1

H)

lim gn D ı in D 0 .R/:

n!1

(b) For x ¤ 0, gn .x/ ! 0 in R as n ! 1 (see proof of 1(a)).

64

Chapter 1 Schwartz distributions

Example 1.8.7. Let .ın /1 nD0 be a sequence of Dirac distributions ın with unit mass/ charge/force etc. concentrated at x D n 2 N0 , hın ; i D .n/. Show that for an 1 arbitrary system .an /1 nD0 of real numbers an , the sequence .an ın /nD0 of distributions 0 0 in D .R/ converges to 0 2 D .R/. Proof. Set Tn D an ın 2 D 0 .R/ 8n 2 N. Then .Tn / converges in D 0 .R/ iff .hTn ; i/ with hTn ; i D han ın ; i D an hın ; i D an .n/ converges in R 8 2 D.R/. 8 fixed  2 D.R/, supp./ is a compact subset of R H) 9n0 2 N such that supp./  Œn0 ; n0  H) 8n  n0 , .n/ D 0 H) hTn ; i D an .n/ D 0 8n  n0 H)

H)

lim hTn ; i D h0; i 8 2 D.R/

lim hTn ; i D 0 in R 8 2 D.R/

n!1

H)

n!1

Tn D an ın ! 0 2 D 0 .R/

for arbitrary choice of an 2 R. Example 1.8.8. Prove that limy!0C ln.x ˙ iy/ D ln.x ˙ i 0/ in D 0 .R/, where   q y 2 2 ln.x ˙ iy/ D ln x C y ˙ i arctan for y > 0; (1.8.20) x ln.x ˙ i 0/ D lim ln.x ˙ iy/ D ln jxj ˙ i H.x/ y!0C

´ ln jxj ln jxj ˙ i H.x/ D ln jxj ˙ i 

with

for x > 0 for x < 0:

(1.8.21)

Proof. 8 fixed y > 0, ln.x ˙ iy/ is locally integrable on R H) ln.x ˙ iy/ 2 D 0 .R/ 8 fixed y > 0, with Z hln.x ˙ iy/; i D ln.x ˙ iy/.x/dx 8 2 D.R/: (1.8.22) R

D 0 .R/

is defined by (1.4.7) and (1.4.8): 8 2 D.R/, Z 0 Z ln jxj.x/dx ˙ i  .x/dx: hln.x ˙ i 0/; i D

Again, ln.x ˙ i 0/ 2

(1.8.23)

1

R

We will show that limy!0C hln.x C iy/; i D hln.x C i 0/; i 8 2 D.R/ Z Z H) lim ln.x C iy/.x/dx D ln.x C i 0/.x/dx y!0C

R

Z D

R

lim ln.x C iy/.x/dx;

R y!0C

(1.8.24)

which we will prove using Lebesgue’s Theorem B.3.2.2 on dominated convergence in Appendix B.

Section 1.8 Convergence of sequences of distributions in D 0 ./

65

In fact, limy!0C Œln j.x C iy/j.x/ D ln jxj.x/ for almost all x 2 R. For jx C iyj > 1 and 0 < y < 1, q ˇ ˇ ˇ ln jx C iyjˇ D ln jx C iyj D ln x 2 C y 2 D 1 ln .x 2 C y 2 /  1 ln .x 2 C 1/; 2 2 and for jx C iyj < 1, and y > 0, 

 ˇ ˇ 1  ln D ˇ ln jxjˇ: jxj ˇ ˇ ˇ ˇ ˇ ˇ Hence, ˇ ln jx C iyj.x/ˇ  Œ 12 ln .x 2 C 1/ C ˇ ln jxj ˇ j.x/j for 0 < y < 1 and for 1 2 almost all x 2 R, where Œ 2 ln .x C 1/ C ˇ ln jxjˇj.x/j is integrable on R 8 2 D.R/ (see Example 1.4.1). Hence, by Lebesgue’s dominated convergence theorem, 8 2 D.R/, Z Z Z ln j.x C iy/j.x/dxD . lim ln j.x C iy/j/.x/dxD ln jxj.x/dx: lim ˇ ˇ ˇ ln jx C iyjˇ D ln

y!0C

1 jx C iyj





R y!0C

R

R

(1.8.25) For x > 0, limy!0C Œarctan yx  D 0, and for x < 0, limy!0C Œarctan yx  D , i.e. limy!0C Œarctan yx .x/ D H.x/.x/ for almost all x 2 R and 8 2 D.R/. Moreover, for y > 0, j arctan yx j   H) j arctan yx .x/j  j.x/j for almost all x 2 R, 8y > 0. Hence, again by Lebesgue’s dominated convergence theorem, 8 2 D.R/,     Z Z  y y lim .x/dx D .x/ dx lim arctan arctan x x y!0C R R y!0C Z D H.x/.x/dx: (1.8.26) R

Finally, from (1.8.24)–(1.8.26), we get, 8 2 D.R/, Z Z Z lim ln.x C iy/.x/dx D ln jxj.x/dx C i H.x/.x/dx y!0C

R

Z D

R

R

Œln jxj C i H.x/.x/dx D R

H)

Z

ln.x C i 0/.x/dx R

lim ln.x C iy/ D ln.x C i 0/ in D 0 .R/:

y!0C

Similarly, we can show that limy!0C ln.x  iy/ D ln.x  i 0/ in D 0 .R/. Example 1.8.9. Show that limy!C1 e ixy .c:p:v: x1 / D i ı in D 0 .R/.

66

Chapter 1 Schwartz distributions

Proof. 8 fixed y > 0, e ixy .c:p:v: x1 / 2 D 0 .R/ is defined by: 8 2 D.R/,       Z e ixy .x/ 1 1 ixy ixy dx: (1.8.27) c:p:v: e ;  D c:p:v: ; e  D lim x x x "!0C "jxj Thus, for  2 D.R/ with supp./ D K  ŒA; A, A > 0, 8 fixed y > 0, Z I."/ D "jxjA

e ixy .x/ dx D x

Z

"

A

Z

e ixy .x/ dx C x

A

D "

Z

A

e ixy .x/ dx x " Z A ixy e ixy .x/ e .x/ dx C dx: x x " (1.8.28) 2 C 0 .R/.

From Proposition 1.2.1, .˙x/ D .0/ ˙ x .˙x/ with H) For fixed y > 0, Z

A

.e ixy  e ixy /.0/ C xŒe ixy .x/ C e ixy .x/ dx x " Z A Z A sin xy D 2i .0/ dx C Œe ixy .x/ C e ixy .x/dx D I1 ."/ C I2 ."/ x " " „ ƒ‚ … „ ƒ‚ …

I."/ D

I1 ."/

I2 ."/

For fixed y > 0, Z

A

lim I1 ."/ D 2i .0/ lim

"!0C

"!0C

"

Z

yA

D 2i .0/ lim

"!0C

sin xy dx x

y"

sin s ds D 2i .0/ s

since sins s has a removable discontinuity at s D 0, i.e. Similarly, for fixed y > 0, Z lim I2 ."/ D

"!0C

A

sin s s

Z

yA 0

sin s ds; s

(1.8.29)

is continuous at s D 0.

Œe ixy .x/ C e ixy .x/dx;

(1.8.30)

0

since the integrand is continuous at x D 0. Then, from (1.8.29) and (1.8.30), R yA RA he ixy .c:p:v: x1 /; i D 2i .0/ 0 sins s ds C 0 Œe ixy .x/ C e ixy .x/dx. But R 1 sin s R yA sin s R A ixy  limy!1 0 .x/dx D 0 by the s ds D 0 s ds D 2 and limy!1 0 e Riemann–Lebesgue theorem. Hence, limy!1 he ixy .c:p:v: x1 /; i D 2i .0/: 2 C 0 D i .0/ D hi ı; i 8 2 D.R/ H) limy!1 e ixy .c:p:v: x1 / D i ı in D 0 .R/.

Section 1.9 Convergence of series of distributions in D 0 ./

1.9

67

Convergence of series of distributions in D 0 ./

P 0 Definition 1.9.1. Let 1 nD1 Tn be a series of distributions Tn 2 D ./ 8n 2 N. PN 0 ./, Let SN DP nD1 Tn 8N 2 N. Then, if the sequence .SN /1 N D1 converges in PD 1 1 0 the series nD1 Tn is said to converge in D ./ or, equivalently, the series nD1 Tn P converges in D 0 ./ if and only if the number series 1 hT ; i converges in R n nD1 (resp. C) 8 2 D./. P 0 Then the series 1 (1.9.1) nD1 Tn is called summable in D ./. 

P1 Tn converges in D 0 ./ ” 9 a unique T 2 D 0 ./ such that T D PnD1 1 nD1 Tn with

hT; i D

1 X

hTn ; i

8 2 D./:

(1.9.2)

nD1



P

Ti converges in D 0 ./ ” D./. i2I

P

i2I hTi ; i converges in R (resp. C) 8

2 (1.9.3)

For more interesting properties of series of distributions, see Chapter 2. P PnDC1 n n Example 1.9.1. For a > 0, let 1 nD0 a ın and nD1 a ın be two given series in D 0 .R/, ın being Dirac distributions (with unit mass/charge/force etc.) concentrated at x D ˙n 2 Z. Show that both the series of distributions converge in D 0 .R/. PN n Proof. Set SN D a > 0. Then .SN / is a sequence nD0 a ın 8N 2 N with PN P 0 n of distributions in D .R/ with hSN ; i D h nD0 an ın ; i D N nD0 a hın ; i D PN n nD0 a .n/ 8 2 D.R/, 8N 2 N. Now, 8 fixed  2 D.R/, 9n0 2 N such that  Œn0 ; n0  H) 8n > n0 , .n/ D 0 H) limN !1 hSN ; i D Pn0supp./ n .n/ 2 R H) .S / converges to a distribution S 2 D 0 .R/ as N ! 1 a N nD0 P P1 n ı converges with S D n 8a > 0, and 1 a n nD0 nD0 a ın 8a > 0. Similarly, PnDCN n a ın and proceeding in the same way, we can show that setting TN D nDN PnDn n 0 limN !1 hTN ; i D nDn0 a .n/ 2 R 8 fixed  2 D.R/ with supp./  0 Œn0 ; n0  H) .TN / converges to a distribution P PnDC1 nT 2 D .R/ 8a > 0 as N ! 1 H) nDC1 n nD1 a ın converges with T D nD1 a ın 8a > 0.

Remark 1.9.1. More interesting problems will be discussed in Chapter 2.

68

1.10

Chapter 1 Schwartz distributions

Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions

Let F W Rn ! Rn be an invertible, infinitely differentiable mapping from Rn onto itself defined by: 8x 2 Rn ;

 D F(x) 2 Rn ;

(1.10.1)

which defines the bijective correspondence between the variables x1 ; x2 ; : : : ; xn and the variables 1 ; 2 ; : : : ; n , i.e. i D i .x1 ; x2 ; : : : ; xn / D fi .x1 ; x2 ; : : : ; xn /, 1  i  n, with x D .x1 ; x2 ; : : : ; xn /;

 D .1 ; 2 ; : : : ; n /;

F D .f1 ; f2 ; : : : ; fn /; fi 2 C 1 .Rn /

8i: (1.10.2)

Let F1 W Rn ! Rn also be an infinitely differentiable mapping from Rn onto itself defined by: 8 2 Rn , x D F1 ./ 2 Rn , i.e. xi D xi .1 ; : : : ; n / D gi .1 ; : : : ; n /; 1  i  n, F1 D .g1 ; g2 ; : : : gn /, gi 2 C 1 .Rn / 8i . In other words, F is a C 1 -diffeomorphism from Rn onto itself and       @i @fi JF .x/ D det .x/ .x/ D det (1.10.3) @xj @xj 1i;j n 1i;j n is the Jacobian of F such that JF .x/ ¤ 0, and the Jacobian of F1 is    @xi 1 JF1 ./ D det ./ ¤ 0: D @j J F 1i;j n

(1.10.4)

Let f .x/ be a function locally integrable on Rn in variables x1 ; x2 ; : : : ; xn defining the corresponding (regular) distribution in D 0 .Rn / by: Z f .x/.x/d x 8 2 D.Rn /: (1.10.5) hf .x/; .x/i D Rn

Now, under change of variables,  D F.x/, x D F1 ./ defined in (1.10.1)–(1.10.4), we have Z Z f .x/.x/d x D f .F1 .//.F1 .//jJF1 ./jd ; (1.10.6) Rn

Rn

which we can rewrite in distribution notation as follows: hf .x/; .x/i D hg./; ./i

8 2 D.Rn /;

(1.10.7)

Section 1.10 Images of distributions due to change of variables

where

69

./ D  ı F1 ./jJF1 ./ j. Then, (1.10.7) defines the function g: g./ D .Ff /./ D f .F1 .//;

(1.10.8)

which corresponds to the function f .x/ in new variables 1 ; 2 ; : : : ; n , i.e. g is the image of f under F. Now, we will extend this definition (1.10.7)–(1.10.8) to the case of an arbitrary distribution T 2 D 0 .Rn /. Let S; T 2 D 0 .Rn / be two distributions on Rn in variables 1 ; 2 ; : : : ; n and x1 ; x2 ; : : : ; xn respectively. By abuse of notations, we will write T .x/ (resp. S./) to denote that T (resp. S ) is associated with the variables x1 ; : : : ; xn (resp. 1 ; : : : ; n ). T .x/ (resp. S./) must not be understood as the value of T (resp. S) at the point x (resp. ), since distributions cannot have point-values. An alternative notation could have been Tx (resp. S ) to indicate the corresponding variables. (1.10.9) Definition 1.10.1. Let T .x/ 2 D 0 .Rn / be a distribution on Rn in variables x1 ; x2 ; : : : ; xn , and F be the invertible C 1 -diffeomorphism on Rn defined in (1.10.1)– (1.10.4). Then the corresponding distribution S./ D .FT /./ in the new variables 1 ; : : : ; n is defined by the continuous linear functional on D.Rn /: 8 ;  2 D.Rn /, hS./; ./i D h.FT /./; ./i D hT .F1 .//; ./i D hT .x/; .F (x)/jJF .x/ji D hT .x/; .x/i

(1.10.10)

with .FT /./ D .T ı F1 /./ D T .F1 .//; .x/ D

.F.x//jJF .x/j;

./ D .F

(1.10.11) 1

.//jJF1 ./j:

(1.10.12)

Hence, S D FT is called the image of T under F. The mapping  2 D.Rn / 7!  ı F1 jJF1 j D 2 D.Rn / (resp. 2 D.Rn / 7! ı FjJF j D  2 D.Rn /) is linear and continuous from D.Rn / into D.Rn /. Hence, FT D T ı F1 defined by (1.10.11) is linear and continuous on D.Rn / and, consequently, a distribution on Rn . Example 1.10.1. Under the change of variable  D F.x/ in (1.10.1)–(1.10.4), Dirac distribution ı D ı0 D ı.x/ with mass/charge/force etc. concentrated at 0 is transformed into Fı D jJF .0/jıF.0/ ;

(1.10.13)

where ıF.0/ D ı.  F(0)/ is the Dirac distribution with mass/charge/force etc. concentrated at F.0/, jJF .0/j D jJF .x/jxD0 D the absolute value of the Jacobian of F at x D 0.

70

Chapter 1 Schwartz distributions

In fact, hFı; i D hı.F1 ./; ./i D hı.x/; .F.x//jJF .x/ji D D hıF.0/ ; ./ijJF .0/j D hjJF .0/jıF.0/ ; i H)

8

.F.0//jJF .0/j 2 D.Rn /

Fı D jJF .0/jıF.0/ :

Change of variables defined by F.x; y/ D x 2 C y 2 , where F is not invertible Let F W R2 ! R be defined by F .x; y/ D x 2 C y 2 D  2 R

8.x; y/ 2 R2 :

(1.10.14)

8T 2 D 0 .R2 /; let F T 2 D 0 .R/ be the image of T under F defined by hF T; i D hT;  ı F i D hT; .F .x; y//i

8 2 D.R/:

(1.10.15)

Example 1.10.2. Under the change of variables defined by (1.10.14), let F T 2 D 0 .R/ be the image of T 2 D 0 .R2 / defined by (1.10.15). Then find 1. F ı.a;b/ ; 2. FH ; 3. F 1; where ı.a;b/ 2 D 0 .R2 / is the Dirac distribution with mass/charge/force etc. concentrated at .a; b/ 2 R2 : hı.a;b/ ; .x; y/i D

.a; b/ 8

2 D.R2 /I

(1.10.16)

H D H.x; y/ D 1 for x > 0 and y > 0 and H.x; y/ D 0 otherwise in R2 ; 1 D 1.x; y/ 8.x; y/ 2 R2 . (1.10.17) Solution. 1. From (1.10.15), hF ı.a;b/ ; ./i D hı.a;b/ ;  ı F .x; y/i D hı.a;b/ ; .F .x; y//i D hı.a;b/ ; .x 2 C y 2 /i D .a2 C b 2 / D hıa2 Cb 2 ; ./i H)

F ı.a;b/ D ıa2 Cb 2 in D 0 .R/;

8 2 D.R/:

2. hFH; i D hH.x; y/;  ı F .x; y/i D hH.x; y/; .F .x; y//i Z 1Z 1 Z 1Z  2 D .x 2 C y 2 /dxdy D .r 2 /rdrd

0

0

0

0

(1.10.18)

71

Section 1.10 Images of distributions due to change of variables

with change of variables: y D r sin ; J D r; x 2 C y 2 D r 2 ; (1.10.19) Z 1 Z  Z 2 dt  1 hFH; i D .t / d D .t /dt .setting t D r 2 / 2 0 4 0 0 Z   1 H.x/.x/dx D h H.x/; i 8 2 D.R/ D 4 1 4

x D r cos ; H)

 4 H.x/

in D 0 .R/.H.x/ is the Heaviside function). Z 1Z 1 3. hF 1; i D h1;  ı F .x; y/i D 1  .x 2 C y 2 /dxdy H) FH D

Z

1 Z 2

D

1

1

2

Z

1

.r /rdrd D 2 0

0

0

dt .t / D 2

Z

1

H.x/.x/dx 1

H) hF 1; i D hH.x/; i 8 2 D.R/ H) F 1 D H.x/ in D 0 .R/. (1.10.20) Invertible affine transformation of variables In particular, let F W Rn ! Rn be an invertible affine mapping satisfying (1.10.1)– (1.10.4) such that F.x/ D Ax C b D 

8x 2 Rn ;

(1.10.21)

where A D .aij /1i;j n is a non-singular square matrix of order n, and b 2 Rn is a fixed vector. Then, F1 ./ D A1 .  b/ D x 8 2 Rn ; JF D jAj ¤ 0;

JF1 D

1 ; jA1 j

(1.10.22)

and F is a C 1 -diffeomorphism from Rn onto itself. Hence, for T .x/ 2 D 0 .Rn /, the image S D FT of T under F is defined by: S./ D FT ./ D T .F1 .// D T .A1 .  b// such that, 8; 2 D.Rn / related by (1.10.12), hS./; ./i D hFT ./; ./i D hT .A1 .  b//; ./i D hT .x/; .Ax C b/j det.A/ji D hT .x/; .x/i

(1.10.23)

with .x/ D

ı F.x/jJF j D

.Ax C b/j det.A/j

./ D  ı F1 ./jJF1 j D .A1 .  b// 

1 : j det.A/j

(1.10.24)

72

Chapter 1 Schwartz distributions

Example 1.10.3. Let F.x/ D Ax D y 2 Rn with b D 0, x 2 Rn , A D .aij /1i;j n , det.A/ ¤ 0 and f 2 L1 .Rn /. Then the image Ff D Af of f under F is defined by: 8 2 D.Rn /, hAf .y/; .y/i D hATf .y/; .y/i D hTf .A1 .y/; .y/i Z Z D f .A1 .y// .y/d y D f .x/ .Ax/j det.A/jd x Rn

Rn

D hf .x/; .x/i

with .x/ D

.Ax/j det.A/j:

(1.10.25)

Important examples are translation, rotation, reflection at the origin, etc., which can be retrieved as particular cases of the affine transformation in (1.10.21)–(1.10.24), as shown below. Translation F.x/ D x C b D  with A D I (identity matrix), b ¤ 0, F1 ./ D   b D x with A1 D I , jJF j D jI j D jJF1 j D 1 in (1.10.21)–(1.10.24). Then S D FT D T with S./ D .FT /./ D T .F1 .// D T .  b/ such that 8; 2 D.Rn / satisfying (1.10.24): hFT; i D h.FT /./; ./i D hT .  b/; ./i D jJF jhT .x/; .x C b/i D hT .x/; .x C b/i D hT .x/; .x/i: (1.10.26) Example 1.10.4. 8b 2 Rn , 8T 2 D 0 .Rn /; b T D FT is called the translated distribution of T by the vector b, and is defined by: .b T /./ D T .  b/ 8 2 Rn such that hb T; i D hT .x/; .x C b/i

8

2 D.Rn /:

(1.10.27)

Example 1.10.5. For Dirac distribution ı D ı.x/ (with mass/charge/force etc.) concentrated at x D 0 2 Rn ; b ı D ıb D ı.  b/, which is the Dirac distribution with (mass/charge/force etc.) concentrated at  D b. In fact, from (1.10.27), hb ı; i D hı.x/; .x C b/i D

.b/ D hıb ; i

8

2 D.Rn /

H) b ı D ıb in D 0 .Rn /. Remark 1.10.1. Translation is often written as .b T /.x/ D T .x  b/ instead of introducing another variable . Example 1.10.6. For T D Tf D f 2 L1loc .Rn / and the translation defined by (1.10.26), Z Z f .  b/ ./d  D f .x/ .x C b/d x 8 2 D.Rn /: (1.10.28) Rn

Rn

73

Section 1.10 Images of distributions due to change of variables

Rotation F.x/ D Ax D , F1 ./ D A1  D At  D x with b D 0, where A is an orthogonal matrix with A1 D At I

jJF j D jAj D jA1 j D 1I

(1.10.29)

(using (1.10.24)) .x/ D

./ D .At /

.Ax/;

2 D.Rn /:

8;

(1.10.30)

Then the image FT of T 2 D 0 .Rn / under rotation is defined by: .FT /./ D T .At / such that 8; 2 D.Rn / satisfying (1.10.30), hFT; i D hT .At /; ./i D hT .x/; .Ax/ij det.A/j D hT .x/; .x/i: (1.10.31) Example 1.10.7. For T D Tf D f 2 L1loc .Rn / and the rotation defined by (1.10.30), Z Z Z f .At / ./d  D f .x/ .Ax/d x D f .x/.x/d x: (1.10.32) Rn

Rn

Rn

Example 1.10.8. For Dirac distribution ı D ı0 D ı.x/ with mass/charge/force etc. concentrated at x D 0; Fı D ı under rotation in Rn (see invariance later). 8 2 D.Rn /, hFı; i D hı.At /; ./i D hı.x/; .Ax/i D D

.0/ D hı./; ./i D hı; i

.A0/

”

(by definition of ı)

Fı D ı:

F.x/ D x D , F1 ./ D  D x with b D 0,

Reflection at the origin

A D A1 D I;

jJF j D jJF1 j D j det.A/j D 1;

(1.10.33)

(from (1.10.24)) .x/ D

.x/;

./ D ./ 8;

2 D.Rn /:

(1.10.34)

Then the image FT of T 2 D 0 .Rn / under reflection at the origin is defined by .FT /./ D T .F1 ./ D T ./ such that 8; 2 D.Rn / satisfying (1.10.34), hFT; i D hT ./; ./i D hT .x/; .x/ij det.A/j D hT .x/; .x/i D hT .x/; .x/i:

(1.10.35)

Example 1.10.9. For Dirac distribution ı D ı0 D ı.x/, .Fı/./ D ı.x/ D ı.x/. In fact hFı; i D hı./; ./i D hı.x/; .x/ij det.A/j D hı.x/; .x/i D

.0/ D hı; ./i D hı; i

8

2 D.Rn /

” Fı D ı, i.e. ı./ D ı./, which can be written as ı.x/ D ı.x/ (see (1.1.1)).

74

Chapter 1 Schwartz distributions

Example 1.10.10. For T D Tf D Rf 2 L1loc .Rn / and reflection at the origin, R n R8, 2 D.R / satisfying (1.10.34), Rn f ./ ./d  D Rn f .x/ .x/d x D Rn f .x/.x/d x. Homothetic transformation F1 ./ D

F.x/ D ˛x D  with ˛ > 0; b D 0; 1  D xI ˛

A D ˛I;

JF D jAj D ˛ n ;

A1 D

jA1 j D

1 I; ˛

1 : ˛n

(1.10.36)

From (1.10.24), .x/ D

.˛x/  jAj D ˛ n .˛x/I

 ./ D 

 1 1   n: ˛ ˛

(1.10.37)

Then the image FT of T under homothetic transformation with ˛ > 0 is given by: .FT /./ D T . ˛1 / such that 8, 2 D.Rn / satisfying (1.10.37),     1 hFT; i D T  ; ./ D hT .x/; .F.x//ijJF .x/j D ˛ n hT .x/; .˛x/i ˛ D hT .x/; .x/i D hT; i:

(1.10.38)

Homogeneous distributions A distribution T 2 D 0 .Rn / is called homogeneous of degree d 2 R if and only if, 8 > 0, T .x/ D d T .x/ in D 0 .Rn / such that hT .x/; .x/i D d hT .x/; .x/i

8 2 D.Rn /:

(1.10.39)

Example 1.10.11. For homogeneous T of degree d defined by: T D Tf D f 2 R R L1loc .Rn /, (1.10.39) implies that Rn f .x/.x/d x D d Rn f .x/.x/d x 8 2 R D.Rn / H) Rn Œf .x/  d f .x/.x/d x D 0 8 2 D.Rn / H) f .x/  d f .x/ D 0 a.e. in Rn (by Theorem 1.2.3A) H) f .x/ D d f .x/ a.e. in Rn H) f is a homogeneous function of degree d in L1loc .Rn /. Equation (1.10.39) can be rewritten as follows: Using (1.10.23) and (1.10.24), F.x/ D x D ; JF D jAj D n ;

F1 ./ D

jAj1 D n ;

1  D x;  .x/ D

 > 0;

1 A1 D I;    1  n : ./ D   (1.10.40)

A D I;

.x/  n ;

75

Section 1.10 Images of distributions due to change of variables

Then .FT /./DT .F1 .//DT . 1 / such that, 8;

2 D.Rn / satisfying (1.10.40),

    1 hFT ./; ./i D T  ; ./ D hT .x/; .x/in   d  1 H) T ./; ./ D d hT ./; ./i D n hT .x/; .x/i:  Alternative definition of homogeneous distributions From the previous steps we have: hT . 1 /; ./i D d hT ./; ./i D n hT .x/;  .x/i with  .x/ D .x/ H) hT;  i D .d Cn/ hT; i 8 2 D.Rn /. Hence, an alternative equivalent definition of homogeneous distributions is as follows: A distribution T 2 D 0 .Rn / is called homogeneous of degree ‘d ’ 2 R if and only if hT;  i D .d Cn/ hT; i

8 2 D.Rn /:

(1.10.41)

Invariance A distribution T 2 D 0 .Rn / is called invariant under transformation of variables F.x/ D  defined in (1.10.21)–(1.10.24) if and only if FT D T with .FT /./ D T .F1 .//, i.e. hFT; i D hT .F1 .//; ./i D hT; i

8

2 D.Rn /:

(1.10.42)

Example 1.10.12. Dirac distribution ı D ı0 D ı.x/ is invariant under rotation in Rn , i.e. Fı D ı (see Example 1.10.8). Spherically symmetric distribution A distribution T 2 D 0 .Rn / is called spherically symmetric if and only if it is invariant with respect to all rotations. (1.10.43) Important examples of spherically symmetric distributions are Dirac ı distributions and distributions defined by functions f D f .r/ 2

L1loc .Rn /

2

with r D

n X

xi2 :

(1.10.44)

iD1

1.10.1 Periodic distributions Periodic functions on R A function fQ W R ! C is called periodic with period p 2 R if and only if fQ.xCp/ D fQ.x/8x 2 R, which can be rewritten as p fQ D fQ, where .p fQ/.x/ D fQ.x  .p// D fQ.x C p/8x 2 R. We will use fQ rather than f to denote that it is a periodic function.

76

Chapter 1 Schwartz distributions

8a 2 R, a; a C pΠis called a period interval of periodic fQ with period p. fQ 2 C k .R/ is periodic on R H) fQ.l/ is also periodic on R for l  k. fQ 2 L1loc .R/ is periodic on R with period p Z

aCp

H)

fQ.x/dx D

a

In fact, for b ¤ a, Z Z bCp fQ.x/dx D b

Z

bCp

fQ.x/dx:

aCp

fQ.x/dx C

a

Z

bCp

aCp

„

fQ.x/dx  ƒ‚

a

a

Z

b a

I

By change of variable, x D  C p, dx D d, Z b Z b Z Z bCp Q Q Q f .x/dx D f . C p/d D f ./d D aCp

(1.10.45a)

b

b

fQ.x/dx

fQ.x/dx : …

H)

I D 0:

a

Periodic functions on Rn A function fQ W Rn ! C is called periodic with period p 2 Rn if and only if fQ.xCp/ D fQ.x/ 8x 2 Rn , which can be written as p fQ D fQ, where .p fQ/.x/ D fQ.x C p/ 8x 2 Rn . Periodic distributions Let fQ 2 L1loc .Rn / be a periodic function with period p 2 Rn . Then fQ 2 D 0 .Rn / and p fQ 2 D 0 .Rn / is a regular distribution on Rn defined by: h.p fQ/.x/; .x/iD 0 .Rn /D.Rn / D hfQ.x  p/; .x/iD 0 .Rn /D.Rn / Z Z Q f .x  p/.x/d x D fQ. /. C p/d D Rn

Rn

D hfQ.x/; .x C p/iD 0 .Rn /D.Rn /

8 2 D.Rn /:

For periodic fQ 2 L1loc .Rn / with period p, we have p fQ D fQ and h.p fQ/.x/; .x/iD 0 .Rn /D.Rn / D hfQ.x/; .x C p/iD 0 .Rn /D.Rn / 8 2 D.Rn / or, equivalently, h.p fQ/.x/; .x/iD 0 .Rn /D.Rn / D hfQ.x/; .x/i8 2 D.Rn /, i.e. p fQ D fQ 2 D 0 .Rn /, any one of which may be used to define periodic distributions on Rn with period p. Hence: Definition 1.10.1A. A distribution T 2 D 0 .Rn / is called periodic with period p 2 Rn if and only if p T D T 2 D 0 .Rn / or, equivalently, hT .x/; .x C p/i D hT .x/; .x/i, or h.p T /.x/; .x/i D hT .x/; .p /.x/i D hT .x/; .x/i

8 2 D.Rn /: (1.10.45b)

77

Section 1.10 Images of distributions due to change of variables

(By abuse of notation, we have written T .x/ to show that the variable involved is x, although T .x/ is not the value of T at x as distributions T cannot have point values.) In other words, a distribution T 2 D 0 .Rn / invariant under translations through the vector p D b is called periodic with periodic p. For n D 1, T 2 D 0 .R/ is periodic with period p ” hp T; iDhT; p iDhT; .x C p/iDhT; i8 2 D.R/ .see (1.10.27)/: For example, for p D 1, T 2 D 0 .R/ is periodic with period 1 if and only if h1 T; i D hT; 1 i D hT; .x C 1/i D hT; i8 2 D.R/:

(1.10.45c)

Constant distribution A distribution T 2 D 0 .Rn / is constant if and only if T has every p 2 Rn as a period. Alternatively, a distribution T invariant with respect to all translations is called constant. (1.10.46) Other equivalent notations for period are T (time period), L (length period), etc. Henceforth, we will use T instead of p 2 R as the period of functions and distributions on R, and then T will not be a distribution, for which we will use S or any other convenient notation. Since periodic functions and distributions on R are very important in applications and can be given Fourier series representations, we will give further interesting details here. Alternative definition of periodic distributions on R First of all, we define a completely new test space D./ on circle , and its dual D 0 ./ as the space of distributions on , which have a one-to-one correspondence with the space of periodic distributions on R with period T D circumference of ; this will be essential later for Fourier series of periodic distributions. Let  D .0I r/ be a circle in the xy-plane with circumference 2 r and centre at the origin 0. Let A D .r; 0/ be the point of intersection of  and the x-axis, from which the arc length coordinate s of any point P 2  is measured in the anti-clockwise direction as positive sense of orientation of . For r D 1, s D angle in radians. For functions ˆ W  ! C on circle  with ˆ D ˆ.s/ (we have used capital ˆ R k instead of  2 D.R/), we can define derivatives: ddsˆ k , integrals:  ˆ.s/ds, etc. in the usual way. Then C 1 ./ denotes the linear space of all ˆ D ˆ.s/ on  such that ˆ 2 C k ./ 8k 2 N. Definition 1.10.2. The test space D./ D ¹ˆ W ˆ W  ! C; ˆ 2 C 1 ./º

(1.10.47)

is the linear space of all infinitely differentiable complex-valued functions ˆ D ˆ.s/ defined on .

78

Chapter 1 Schwartz distributions

Convergence in D. / A sequence .ˆn /1 nD1 in D./ is said to converge to ˆ 2 d ˛ ˆk d˛ˆ D./ if and only if ds ˛ ! ds ˛ uniformly on  8˛ 2 N0 as n ! 1. Continuity on D. / A linear functional S on D./ is called continuous on D./ if and only if ˆk ! ˆ in D./ H) hS; ˆk i ! hS; ˆi in C as k ! 1. Since  is bounded, D./ is less complicated than D.R/. For example, ˆ D 1 belongs to D./, but it does not belong to D.R/. Definition 1.10.3. D 0 ./ is the linear space of all continuous linear functionals on D./ and is called the space of distributions on . Then S 2 D 0 ./ H) hS; ˆiD 0 ./D./ is the duality pairing between D 0 ./ and D./. We will show that the distributions on  are intimately related to periodic distributions on R with period T D 2 r. For this we need Q Wˆ Q W R ! C; ˆ.x Q C T / D ˆ.x/ Q Q 2 C 1 .R/º DT .R/ D ¹ˆ 8x 2 R; ˆ D CT1 .R/;

(1.10.48)

which is the linear space of infinitely differentiable, complex-valued periodic funcQ D ˆ.x/ Q tions ˆ on R with period T D 2 r, and DT0 .R/ is the linear space of continuous, linear functionals on DT .R/, which can be identified with the space of periodic distributions on R with period T . (1.10.48a) Isomorphism between two sets D. / and DT .R/ To each ˆ 2 D./ with Q 2 DT .R/ such that ˆ D ˆ.s/, we associate a unique periodic ˆ Q ˆ.x/ D ˆ.P /;

(1.10.49)

where P 2  has arc length coordinate s D x. Q 2 DT .R/ we associate a unique ˆ 2 D./ by Conversely, to each periodic ˆ Q ˆ.P / D ˆ.x/;

(1.10.50)

where x is one of the arc length distances of P from A.s D 0/, any two of which Q 2 DT .R/ is differ by kT with k 2 Z. Then the correspondence ˆ 2 D./ ! ˆ one-to-one from D./ onto DT .R/, i.e. defines an isomorphism between these two sets. DT .R/ is a subspace of C 1 .R/ (i.e. of E.R/; see Definition 5.7.1, Chapter 5). Q 2 DT .R/ H) ˆ Q does not belong to D.R/. 2 D.R/ H) does not belong ˆ to DT .R/.

Section 1.10 Images of distributions due to change of variables

79

Q 2 DT .R/ in terms of 2 D.R/ Construction of test functions ˆ 2 D. / and ˆ Let  2 D.R/. Then  does not belong to PDT .R/. But if we consider P1 the sum of all translated or shifted functions lT , i.e. 1 . /.x/ D lT lD1 lD1 .xClT P/, which will always converge, since supp./  R will imply a finite summation . Let this sum be denoted by Q ˆ.x/ D

1 X

.x C lT /

(1.10.51)

lD1

P1 Q Q Q such that ˆ.x C T/ D lD1 .x C .l C 1/T / D ˆ.x/ 8x 2 R, i.e. ˆ is a 1 1 Q periodic function on R with period T , and ˆ 2 C .R/ (since  2 C .R/). Hence, Q 2 DT .R/. To ˆ Q 2 DT .R/ we associate the unique function ˆ D ˆ.s/ on  by ˆ Q ˆ.P / D ˆ.x/

8x 2 R;

(1.10.52)

where x is one of the arc length coordinates of P (see (1.10.50)) such that 1 X

Q Q C kT / D ˆ.s/ D ˆ.x/ D ˆ.x

.x C lT /;

(1.10.53)

lD1

with k 2 Z and ˆ 2

C 1 ./,

i.e. ˆ 2 D./.

Example 1.10.13. For " > 0, let " 2 D.R/ be defined as in (1.2.6a): "2 " .x/ D exp. "2 jxj 2 / for jxj < " and D 0 for jxj  ". Choosing " < T =2, P Q " .x/ D 1 Q we get ˆ" 2 DT .R/ defined by: ˆ lD1 " .x C lT /. See Figure 1.7 for an illustration.

0

Q " 2 DT .R/ Figure 1.7 Periodic test function ˆ

One-to-one correspondence between distributions on  and periodic distributions on R with period T Let f 2 L1loc ./ be a locally summable function on  defining a regular distribution on  by: 8ˆ 2 D./, Z hf; ˆiD 0 ./D./ D f .s/ˆ.s/ds; (1.10.54) 

80

Chapter 1 Schwartz distributions

Q on R with period T , and defined by where ˆ is associated with the periodic ˆ 1 Q (1.10.52). Let f 2 Lloc .R/ be the associated (with f 2 L1loc ./) periodic locally summable function such that fQ.x/ D f .P / a.e., where P 2  has the arc length coordinate s D x, fQ.x C kT / D fQ.x/ a.e. 8k 2 Z. Then fQ 2 D 0 .R/ defines a regular distribution in D 0 .R/ by: Z Q hf .x/; .x/iD 0 .R/D.R/ D fQ.x/.x/dx 8 2 D.R/: (1.10.55) R

Now we will show that the integrals (1.10.54) and (1.10.55) are equal. Z

1

fQ.x/.x/dx D

1

1 Z X lD1

D

Z

T

D 0

Z 0

T

fQ. C lT /. C lT /d

0

(by change of variables: x D  C lT )  X  1 fQ.x/ .x C lT / dx lD1

T

D

fQ.x/.x/dx

lT

1 Z X lD1

.lC1/T

Q fQ.x/ˆ.x/dx D

Z f .s/ˆ.s/ds 

Q is defined by (1.10.51) and f .s/ D fQ.x/, ˆ.s/ D ˆ.x/ Q (fQ is periodic, ˆ (1.10.53), ds D dx). We write Z

T 0

Q Q 0 fQ.x/ˆ.x/dx D hfQ.x/; ˆ.x/i DT .R/DT .R/

Q 2 DT .R/: with ˆ

(1.10.56)

Hence, Q D 0 .R/D .R/ hfQ; iD 0 .R/D.R/ D hf; ˆiD 0 ./D./ D hfQ; ˆi T T

8 2 D.R/: (1.10.57)

(1.10.56) shows the relation between the periodic distribution on R defined by periodic function fQ 2 L1loc .R/ with period T and the distribution on  defined by the associated function f 2 L1loc ./ on . Finally, relation (1.10.56) suggests an alternative consistent definition of periodic distributions SQ on R with period T , which are associated in one-to-one correspondence with the distributions S 2 D 0 ./ on , by: Q 0 hSQ .x/; .x/iD 0 .R/D.R/ D hS.s/; ˆ.s/iD 0 ./D./ D hSQ .x/; ˆ.x/i DT .R/DT .R/ (1.10.58)

81

Section 1.10 Images of distributions due to change of variables

Q related by (1.10.51), and with ˆ Q and ˆ 8 2 D.R/ and ˆ 2 D./ with  and ˆ related by (1.10.53), since Q .x C T /iD 0 .R/D.R/ hSQ ; .x/iD 0 .R/D.R/ D hS; Q Q 0 D hS.x/; ˆ.x/i DT .R/DT .R/ D hS.s/; ˆ.s/iD 0 ./D./ Q and consequently ˆ, do not (if .x/ is replaced by the shifted value .x C T /, ˆ, change). Example 1.10.14. Let S D ı0 D ı.s/ 2 D 0 ./ be the Dirac distribution corresponding to unit (C1) mass/charge/force etc. concentrated at s D 0. Then the corresponding periodic distribution SQ D ıQ on R with period T , i.e. periodic Dirac distributions ıQ of unit (C1) mass/charge/force etc. concentrated at points x D lT 2 R for l 2 Z, is defined by: 8 2 D.R/, Q hı.x/; .x/iD 0 .R/D.R/ D hı.s/; ˆ.s/iD 0 ./D./ 1 X

Q D ˆ.0/ D ˆ.0/ D

.0 C lT /

lD1

D

1 X

.lT /

X . is a finite summation/

lD1

D

1 X

l

hılT ; .x/iD 0 .R/D.R/

lD1

D

 X 1

 ılT ; .x/

;

D 0 .R/D.R/

lD1

since hılT ; i D .lT / with ılT D ı.x  lT /

H)

ıQ D

1 X

ılT D

lD1

1 X

ı.x  lT /;

(1.10.59)

lD1

where ıQ 2 D 0 .R/ is a periodic Dirac distribution on R with period T , ılT is a Dirac distribution in D 0 .R/ with concentration at x D lT .

D ı.xlT /

Alternatively, Q Q ˆi Q Q D ˆ.0/ Q hı.x/; ˆ.x/i D hı; D  D

 X 1

 ılT ; 

lD1

But Q i D hı; Q ˆi Q D hı;

 X 1 lD1

 ılT ;  8 2 D.R/

”

: D 0 .R/D.R/

ıQ D

1 X lD1

ılT in D 0 .R/.

82

Chapter 1 Schwartz distributions

Remark 1.10.2. From (1.10.57) and (1.10.58), we find that periodic distributions SQ 2 D 0 .R/ on R with period T also satisfy the following relation: 8 2 D.R/ with Q ˆ.x/ D

1 X

.x C lT / 2 DT .R/;

lD1

Q Q 0 ˆ.x/i hSQ .x/; .x/iD 0 .R/D.R/ D hS.x/; DT .R/DT .R/ :

(1.10.60)

Denoting the set of all periodic distributions SQ on R with period T by DT0 .R/, we write formally: Q Q 0 hSQ .x/; .x/i D hSQ .x/; .x/i DT .R/DT .R/ :

(1.10.61)

For more details, see Remark 6.7.4 in Section 6.7, Chapter 6. 0 < t < 1=n and Example 1.10.15. Let .pn /1 nD1 be defined by: pn .t / D n for P pn .t / D 0 for t < 0 and t > 1=n. Let PQn be defined by: PQn .t / D 1 lD1 pn .t ClT / 8t 2 R, T > 0 being a number with n > 2=T . Show that 1. pn ! ı in D 0 .R/; 2. PQn defines a periodicPdistribution on R with period T 8n 2 N such that 0 Q limn!1 PQn D ıQ D 1 lD1 ılT in D .R/, where ı is a periodic Dirac distribution on R, ılT D ı.x  lT / is Dirac distribution of unit (C1) mass/charge/ force etc. concentrated at the points x D lT with l 2 Z.

0

Figure 1.8 Periodic rectangular pulse function PQn .

Solution. 1.

Z

lim hpn ; i D lim

n!1

n!1 0

1 n

n.t /dt D lim

n!1

 Z n



1 n

Œ.0/ C ..t /  .0//dt

0

  Z 1 n 1 0 D lim n  .0/  C n t   . /dt D .0/; n!1 n 0

83

Section 1.10 Images of distributions due to change of variables

since ˇ Z ˇ ˇn ˇ

0

1 n

ˇ Z ˇ 0 ˇ t   . /dt ˇ  max j .t /jn t2R 0

1 n

t dt

0

D max j 0 .t /j  n  t2R

1 !0 2:n2

as n ! 1

H) limn!1 hpn ; i D .0/ D hı; i 8 2 D.R/ H) limn!1 pn D ı 2 D 0 .R/. P 2. Since supp.pn ) D Œ0; n1 , 1 pn .t C lT / contains a finite number of terms PlD1 1 Q 8t 2 R. PQn .t C T / D lD1 pn .t C .l C 1/T / D Pn .t / 8t 2 R H) Q Pn is a periodic function on R with period T , i.e. a periodic extension to R of rectangular pulse function pn (see Figure 1.8). Hence, PQn 2 L1loc .R/ is a periodic function on R with period T and defines a periodic distribution in D 0 .R/ with period T by: hPQn ; .t /iD 0 .R/D.R/ D

Z

1 1

1 X

D

lD1

lD1

lD1

T

.lC1/T

PQn .t /.t /dt

lT

PQn . C lT /. C lT /d 

0 T

PQn . /. C lT /d 

0

1 Z X

D H)

lD1

Z

1 Z X

D

1 Z X

PQn .t /.t /dt D

1 n

n.t C lT /dt

8 2 D.R/

0

lim hPQn .t /; .t /iD 0 .R/D.R/

n!1

D lim

 X 1

n!1

D

1 X lD1

Z n

.t C lT /dt

n!1

.summation is finite/

0

lD1

lim



1 n

 Z n 0

Z

1 n

.lT /dt C n



1 n

..t C lT /  .lT //dt 0

.interchange of sum and limit is admissible/ D

1 X lD1

  Z 1 1 X n 1 0  lim .lT /  n  C n t  . /dt D .lT /; n!1 n 0 lD1

84

Chapter 1 Schwartz distributions

since   2 lT; t C lT Œ, jn n!1 Z H)

lim n

n!1

H)

1 n

R

1 n

0

t  0 .  /dt j  max t2R j 0 .t /j  n 

1 2n2

! 0 as

t  0 .  /dt D 0

0

lim hPQn ; .t /i D

n!1

1 X

.lT / D

lD1

D

1 X

hılT ; .t /i

lD1



 X 1

ılT ; 

8 2 D.R/

ŒılT D ı.x  lT /

lD1

”

lim PQn D

n!1

1 X

ılT D ıQ

in D 0 .R/:

lD1

(For convergence of series of distributions, see Definition 1.9.1 and Example 1.9.1.) P1 0 Q lD1 ılT D ı 2 D .R/ is a periodic distribution on R with periodic T : from (1.10.45b), Q iD 0 .R/D.R/ D hı; Q T i D hı; Q .x C T /i D hT ı;

 X 1

 ılT ; .x C T /

lD1

D

1 X

hılT ; .x C T /i D

lD1

D

1 X lD1

1 X

.lT C T / D

.lT / D

hılT ; .t /i D

lD1

.l 0 T / .l 0 D l C 1/

l 0 D1

lD1 1 X

1 X

 X 1

 ılT ; .t /

lD1

Q .t /i 8 2 D.R/ D hı; P1 P1 H) T ıQ D ıQ in D 0 .R/ H) ıQ D lD1 ılT D lD1 ı.x  lT / is a periodic distribution on R with period T .

1.11

Physical distributions versus mathematical distributions

1.11.1 Physical interpretation of mathematical distributions A mathematical distribution T 2 D 0 ./ on   Rn can be interpreted as the physical distribution or spread of mass, force, electric charges, magnetic dipole moments, etc. in . For example, a mathematical distribution T D Tf 2 D 0 ./ corresponding to

Section 1.11 Physical distributions versus mathematical distributions

85

a locally integrable function f on  can be interpreted as the physical distribution of mass/force/charge etc. with density f over a volume . Again, Dirac distribution ı (resp. ıa / can be interpreted physically as the mass/force/charge etc. with intensity C1 concentrated at x D 0 2  (resp. x D a 2 ). Hence, regular distributions T D Tf 2 D 0 ./ correspond to physical distributions of mass/force/charge etc. with density f 2 L1loc ./, whereas the singular Dirac distribution T D ı (resp. ıa / corresponds physically to point concentration of mass/force/charge etc. at 0 2  (resp. a 2 ) with intensity C1. In physical applications, we will meet with expression hTf ; i or hıa ; i, where  … D./ (see 5.6 in Chapter 5 for details). Tf (resp. ıa / can be given extension to a space of functions  larger than D./ (see, for example, Theorem 1.3.1 for a unique, R continuous extension of T ). T D Tf defined by hTf ; i D  f d x can be extended to a larger space of functions  such that f  is integrable on , whereas T D ıa with hıa ; i D .a/ can be extended to an extremely large space of functions  continuous at a 2  (see also Theorem 1.3.1 for a unique extension). From these examples, we find that different distributions Tf and ıa have been extended to different spaces of functions  which cannot be interchanged, i.e. hıa ; i (resp. hTf ; i) is not defined in general on the space of functions  such that f  is integrable on  (resp. such that  is continuous at a 2 ). Then the question arises: What is the role of D./? D./ is the common set of functions  on which all distributions on  are defined. From these discussions, we find that 8T 2 D 0 ./; hT; 1i, if it is defined, gives the total value. For example, 

for T D Tf D f 2 L1 ./, Z hT; 1i D hTf ; 1i D



f .x/d x;

(1.11.1)



for T D ıa 2 D 0 ./, hT; 1i D hı; 1i D C1;

(1.11.2)

gives the total value of mass/force/charge etc. (distributed) in  (see also Section 5.6 for justifications).

1.11.2 Load intensity The notion of mathematical distributions leads us to the fact that in reality we cannot measure the value of a physical entity at a point, but we can only measure mean or average values of the distribution or spread of a physical entity over sufficiently small neighbourhoods of the given point. Then, considering the system of these mean/average values of the (physical) distribution on the corresponding system of neighbourhoods of the point, if we can find the ‘limit’ of this system in an appropriate sense as the system of neighbourhoods shrinks to the given point, then this

86

Chapter 1 Schwartz distributions

limit will be called the value of the physical entity at the given point. As a simple example, we consider the case of a point load of intensity (magnitude) C1 acting at the point 0 2 R2 . D R2 /. In order to determine the load intensity at 0 2 R2 , we distribute uniformly (other non-uniform distributions are also possible) this unit load on a circular disc B" of radius " > 0 and centre at 0 2 R2 , i.e. 1 B" D ¹x W kxk D .x12 C x22 / 2  "º. Then, the mean/average load intensity F" over B" is given, 8" > 0, by: ´ 1 1 for kxk D .x12 C x22 / 2  " 2 " F" .x/ D (1.11.3) 0 for kxk > ": Then, ¹F" º">0 denotes the system of mean/average load intensities on the system ¹B" º">0 of neighbourhoods of 0 2 R2 with the following basic property: Z Z 1 1 F" .x/dx1 dx2 D dx1 dx2 D  "2 D 1; (1.11.4) 8" > 0; 2 "2 R2 B" " i.e. the integral of the mean intensity over the whole space R2 gives the total load of C1 for every choice of uniform distributions over B" with " > 0. Then, Z lim F" .x/dx1 dx2 D 1: (1.11.5) "!0C

R2

Now, we are interested in the load intensity at x D 0, i.e. for " ! 0C . Hence, the first intuitive approach suggests to take the ‘pointwise limit’ of ¹F" º in (1.11.3) as " ! 0C , i.e. ´ C1 for x D 0 lim F" .x/ D 0 for x ¤ 0; "!0C which is precisely what was done by Dirac to define the delta function ı.x/ in (1.1.1): ´ C1 for x D 0 lim F" .x/ D ı.x/ D : (1.11.6) 0 for x ¤ 0 "!0C Then, Z

Z lim F" .x/dx1 dx2 D

R2 "!0C

Z R2

ı.x/dx1 dx2 D

R2 ¹0º

0dx1 dx2 D 0;

(1.11.7)

H) the integral of the delta function ı.x/ D lim"!0C F" .x/ gives the total value of the load equal to 0, i.e. this ‘pointwise limit’ ı.x/ does not restore the total load C1. Thus, the delta function ı.x/ defined by (1.11.6) can not be accepted as the desired load intensity and the ‘pointwise limit’ of the system ¹F" º">0 of mean load intensities

87

Section 1.11 Physical distributions versus mathematical distributions

defined by (1.11.3) is not acceptable. Hence, we are to renounce this and consider the limit of the system ¹F" º">0 in a weaker sense, for example in the sense of convergence in D 0 .R2 / (1.8.1): 8 2 D.R2 /, Z Z F" .x/.x/d x D lim F" .x/.x/d x D .0/: lim hF" ; i D lim "!0C

"!0C

"!0C B"

R2

(1.11.8) In fact, for .x/ D .0/ C Œ.x/  .0/, Z Z Z .0/ 1 F" d x D d x C Œ.x/  .0/d x "2 B" "2 B" B" Z 1 D .0/ C 2 Œ.x/  .0/d x " B" Z Z 1 H) lim F" d x D .0/ C lim Œ.x/  .0/d x 2 "!0C B" "!0C " B" D .0/ since

8 2 D.R2 /;

ˇ ˇ Z ˇ 1 ˇ 1 ˇ Œ.x/  .0/d xˇˇ  max j.x/  .0/j  "2 ˇ "2 "2 x2B" B" D max j.x/  .0/j ! 0 as " ! 0C H)

lim

"!0C

1 "2

x2B"

Z

Œ.x/  .0/d x D 0

8 2 D.R2 /:

B"

Now, using (1.11.8) and the definition of Dirac distribution ı 2 D 0 .R2 /, hı; i D .0/, we have: 8 2 D.R2 /, Z F" .x/.x/d x D .0/ D hı; i lim hF" ; i D lim "!0C

(i.e. lim"!0C

R

R2

"!0C B"

F" .x/.x/d x ¤ D 0 .R2 /

R

R2 lim"!0C F" .x/.x/d x C 0 (see (1.8.1)).

D 0)

as " ! H) F" ! ı in In other words, the system ¹F" º of mean load intensities is not considered as a system of functions, but as a system of regular distributions (continuous linear functionals) in D 0 .R2 /, whose limit is the (singular) Dirac distribution ı 2 D 0 .R2 /, which is not a function. Thus, the point load of intensity C1 concentrated at the origin 0 2 R2 will be given by Dirac distribution ı 2 D 0 .R2 / (not by the delta function in (1.11.6)), if ı restores the total load C1. In fact, from (1.11.2), the total load is given by hı; 1i D C1, since hı; i D .0/ is well defined for  2 C 1 .R2 / with arbitrary support (i.e.  … D.R2 /) (see Section 5.6 for details) and consequently, for  D 1 2 C 1 .R2 /; hı; 1i D 1.0/ D 1.

88

Chapter 1 Schwartz distributions

1.11.3 Electrical charge distribution We consider another interesting example of physical distribution met with in electrostatics to find the corresponding mathematical distribution. Consider the electrical charge distribution on the real line corresponding to the dipole with electric/magnetic moment C1 placed at the origin 0 2 R. Two (concentrated) charges 1" at x D " and 1 " at x D 0 yield the same electric moment C1 with total charges equal to 0 8" > 0. Hence, the density of charges, which approximately corresponds to this dipole of electric/magnetic moment C1 at x D 0, is given by: 1 1 T" D ı"  ı " "

8" > 0;

(1.11.9)

where ı" (resp. ı0 D ı) is the Dirac distribution with unit charge concentrated at x D " (resp. x D 0). Then the dipole is defined as the limit of the system T" when " ! 0C . But 8" > 0, the system T" corresponds to the mathematical distribution defined by: 8" > 0C , 8 2 D.R/,  1 1 1 ."/  .0/ ı"  ı;  D Œhı" ; i  hı; i D " " " " ."/  .0/ D  0 .0/ 8 2 D.R/; lim hT" ; i D lim C C " "!0 "!0 

hT" ; i D H)

which suggests a mathematical definition of dipole as the distribution T 2 D 0 .R/: hT; i D  0 .0/

8 2 D.R/:

(1.11.10)

In this definition of dipole as the mathematical distribution T 2 D 0 .R/, we do not need to construct an approximating system T" and then take the limit as " ! 0C (see also Remark 2.3.3). Remark 1.11.1. T 2 D 0 .R/ defining the dipole of moment C1 at x D 0 will be related to Dirac distribution ı, and this relationship will give the density of charge distribution corresponding to the dipole of moment C1 at x D 0. But for this we are dı to use the results of Chapter 2 on the derivative ı 0 D dx of Dirac distribution ı given in (2.3.8). Instead of deferring the details we give them here, which the reader may read after going through the derivatives of Dirac distribution ı in the next chapter. Using (2.3.8), from (1.11.10), hT; i D  0 .0/ D hı;  0 i D hı 0 ; i D hı 0 ; i H)

T D ı 0 D 

dı dx

in D 0 .R/:

8 2 D.R/ (1.11.11)

89

Section 1.11 Physical distributions versus mathematical distributions

Thus, the required density of charge distribution corresponding to the dipole of moment C1 is ı 0 . Now, we are to check whether this density of charge distribution gives the total charge equal to 0 and the total moment equal to 1:   d1 Total charge D hT; 1i D hı 0 ; 1i D ı; D hı; 0i D 0I dx   dx 0 D hı; 1i D 1: Total moment D hı ; xi D ı; dx

(1.11.12) (1.11.13)

(For details, see Section 5.6 and the definition of ı 0 in (2.3.8).) Now, let us compute the density of charges corresponding to the dipole of moment C1 placed at the origin 0 2 R3 and oriented in the given direction of unit vector O D 1. O D .1 ; 2 ; 3 / 2 R3 with kk The density of charges corresponding to this dipole of moment C1 oriented in the direction of O is approximately given by: 1 1 T" D ı"O  ı " "

8" > 0;

(1.11.14)

where ı"O (resp. ı) is the Dirac distribution of charges with unit charge concentrated at "O 2 R3 (resp. 0 2 R3 ). Then the dipole is the limit of the system T" , when " ! 0C . But 8" > 0, system T" corresponds to the mathematical distribution defined by: 8 2 D.R3 /,  O  .0/ 1 1 1 ."/ hT" ; i D ı"O  ı;  D Œhı"O ; i  hı; i D " " " " 

8" > 0 (1.11.15)

H) lim hT" ; iD lim "!0C

"!0C

O  .0/ @ ."/ O R3 D .0/Dhr .0/; i " @

8 2 D.R3 /;

which suggests that this dipole of moment C1 and oriented in the direction of O should be defined mathematically as the distribution T directly by: hT; i D

@ O R3 .0/ D hr .0/; i @

8 2 D.R3 /;

(1.11.16)

where h  ;  iR3 denotes the inner product of vectors in R3 , r .  / D

@ @ @ .  /Oi1 C .  /Oi2 C .  /Oi3 ; @x1 @x2 @x3

hOik ; Oil i D ıkl ;

1  k; l  3:

This definition does not require the construction of the system ¹T" º">0 and taking its limit as " ! 0C .

90

Chapter 1 Schwartz distributions

Remark 1.11.2. Following Remark 1.11.1, we can write       @ı @ @ @ı hT; i D .0/ D ı; D ;  D  ;  8 2 D.R3 / @ @ @ @ @ı O D hr ı; i in D 0 .R3 / (1.11.17) H) T D  @ is the required density of charge distribution corresponding to the dipole of moment O C1 placed at x D 0 and oriented in the direction of . Total charge of the dipole is 0:     @ı @ O R3 i D hı; 0i D 0:  ; 1 D ı; 1 D hı; hr 1; i (1.11.18) @ @ Total moment due to this density of charge distribution is C1:     @ı @ O R3 D ı; .hx; i O R3 / D hı; hr .x1 1 C x2 2 C x3 3 /; i O R3 i  ; hx; i @ @ O i O R3 i D hı; kk O 2 i D hı; 1i D 1: D hı; h;

(1.11.19)

1.11.4 Simple layer and double layer distributions In electrostatics we meet with new types of physical distributions of charges called simple (or single) layer and double layer distributions on surfaces, which we will describe now and show the corresponding mathematical distribution T 2 D 0 .R3 /. Simple layer distributions Let  be a (piecewise) smooth, orientable (two-sided) surface (a Möbius strip is not admissible) in R3 and D .x/ be a continuous function on  defining the surface density of continuous electric charge distribution on surface . A generalization of the discrete Dirac distribution ıa with charges of intensity C1 concentrated at point a 2  gives Dirac distribution ı corresponding to charges concentrated on   R3 . Then ı 2 D 0 .R3 / is the volume or spatial density of distribution on R3 corresponding to charges concentrated on the surface  with continuous surface density . But corresponding to the physical distribution of charges over  with surface density , the mathematical distribution T 2 D 0 .R3 / is defined by: Z hT; i D

.x/.x/dS 8 2 D.R3 / (1.11.20) 

(dS being the surface area measure). This distribution T 2 D 0 .R3 / must not be confused with the distribution Tf 2 D 0 .R3 / defined by the volume density function f , where Tf is identified with the

91

Section 1.11 Physical distributions versus mathematical distributions

function f itself. But here T 2 D 0 .R3 / represents the volume/spatial density of charges in R3 corresponding to charges distributed on  with surface density . In fact, we will show that T D ı 2 D 0 .R3 / with T defined by (1.11.20) by means of generalization of point Dirac distributions ıa as follows:   is bounded in R3 : although this assumption is not necessary, for the sake of simplicity in presentation of all details, we assume that  is a bounded, S smooth surface subdivided into N distinct subsets S1 ; S2 ; : : : ; SN such that  D N kD1 Sk with SVk D ¹x W x 2 Sk ; x … @Sk ; @Sk is the boundary curve of Sk ºI SVj \ SVk D ;;

1  j ¤ k  NI

(1.11.21)

4Sk D surface area measure  of Sk D  .Sk / > 0, 1  k  N; for xk 2 SVk , 1  k  N , ıxk is the Dirac distribution with charges concentrated at xk . Then .xk /4Sk approximates the total charges on Sk and .xk /4Sk ıxk is the Dirac distribution corresponding to approximate total charges on Sk concentrated at the point xk 8k D 1; 2; : : : ; N . Then, 8N 2 N, the sum

 

N X

.xk /4Sk ıxk 2 D 0 .R3 /

(1.11.22)

kD1

represents the approximate total charges on  corresponding to discrete distributions of charges on S1 ; S2 ; : : : ; SN with intensities .x1 /4S1 ; .x2 /4S2 ; : : : ; .xN /4SN concentrated at the points x1 ; x2 ; : : : ; xN respectively. 0 3  The volume/spatial density ı  2 D .R / of charges corresponding to the continuous distribution of charges surface density is given P concentrated on surface 0 with 3 /, when N ! 1 such that by the limit of this sum N

.x /4S ı in D .R k k xk kD1 max1kN ¹4Sk º ! 0C , i.e.

ı D lim

N !1

H)

N X

.xk /4Sk ıxk

in D 0 .R3 /

as max ¹4Sk º ! 0C (1.11.23) 1kN

kD1

h ı ; i D lim

N !1

D lim

N !1

D

X N



.xk /4Sk ıxk ; 

kD1

X N



.xk /4Sk hıxk ; i

kD1

lim

N X

N !1 maxK ¹ SK º!0C kD1

Z D

.x/.x/dS 

8 2 D.R3 /

.xk /.xk /4Sk

8 2 D.R3 /;

92

Chapter 1 Schwartz distributions

since  is continuous on  and the limit exists by definition of surface integrals on , Z H) h ı ; i D

.x/.x/dS 8 2 D.R3 /: (1.11.24) 

Then, from (1.11.20) and (1.11.24) we have T D ı

in D 0 .R3 /;

(1.11.25)

which is called the simple layer distribution on the surface . To repeat again, it is the volume density of charges concentrated on the surface  with surface R R density . Then

ı must restore the total charges  .x/dS . In fact, h ı ; 1i D  .x/dS D total charges on , since h ı ; i is well defined for  2 C 1 .R3 / with arbitrary support (see Section 5.6) and hence, for  D 1 2 C 1 .R3 /, giving the result. Double layer distributions Let  be a smooth, orientable (two-sided) surface such that the unit normal nO to  is defined at each point on . Let  D .x/ denote the continuous surface moment density of the normally oriented dipole moment distribution on surface . Then a generalization of the discrete distribution of normally @ oriented dipole  @n .ıa / of unit moment concentrated at the discrete points a 2  @ gives a distribution  @n .ı / corresponding to normally oriented dipoles of moment @ intensity C1 distributed on the surface . Hence,  @n .ı / 2 D 0 .R3 / is the volume density of distribution corresponding to the distribution of normally oriented dipoles over the surface  with surface moment density . But corresponding to the physical distribution of normally oriented dipoles over  with surface moment density , the mathematical distribution T 2 D 0 .R3 / is defined by: Z @ .x/ .x/dS 8 2 D.R3 / (1.11.26) hT; i D @n  (dS is the surface area measure). @ Now, we will show that T D  @n .ı / in D 0 .R3 /. For this we are to use the @ results of Chapter 2 on the derivative @n .ıxk / of Dirac distribution ıxk 2 D 0 .R3 / in (2.3.8). Instead of deferring the details, we give them here, which the reader may read after going through derivatives in Chapter 2.   is bounded in R3 : although this assumption is not necessary, for the sake of simplicity in presentation of details we assume that  is a bounded, smooth surface subdivided into N distinct subsets S1 ; S2 ; : : : ; SN such that D

N [

Sk

with SVk D ¹x W x 2 Sk ; x … boundary @Sk of Sk º;

1  k  NI

kD1

(1.11.27)

93

Section 1.11 Physical distributions versus mathematical distributions   

SVj \ SVk D ; for 1  j ¤ k  N ; 4Sk = surface area measure of Sk D  .Sk / > 0; 1  k  N ; nO k is the unit normal to SVk at xk 2 SVk , 1  k  N ;

For 1  k  N; .xk /4Sk approximates the total surface moment acting on Sk and  @n@ ..xk /4Sk ıxk / with xk 2 SVk is approximately the normally oriented dipole of k moment .xk /4Sk concentrated at the point xk 2 SVk . PN @ 0 3  Then, 8N 2 N, the sum kD1  @n Œ..xk /4Sk ıxk / 2 D .R / approximates 

k

@ .ı / 2 D 0 .R3 / of the normally oriented dipole distributhe volume density  @n @ tion on  with surface moment density , and the (volume) density  @n .ı / is the PN @ 0 3 limit of this sum kD1  @n Œ..xk /4Sk ıxk / in D .R / when N ! 1 such that k max1kN ¹4Sk º ! 0C , i.e.

 X N @ @  .ı / D lim  ..xk /4Sk ıxk / N !1 @n @nk

in D 0 .R3 /

(1.11.28)

kD1

H)

   X N @ @  .ı /;  D lim  Œ.xk /4Sk ıxk ;  8 2 D.R3 / N !1 @n @nk kD1

N X

D lim

N !1

kD1

  @  Œ.xk /4Sk ıxk ;  @nk

 N  X @ D lim .xk /4Sk ıxk ; (see (2.3.9)) N !1 @nk kD1 N X

D lim

N !1

D lim

kD1 N X

N !1

kD1



@ .xk /4Sk ıxk ; @nk



@ .xk / .xk /4Sk D @nk

with max ¹ Sk º ! 0C 1kN

Z .x/ 

@ .x/dS; @n

since  @ is continuous on , and consequently the limit exists and the limit is the @n R dS by its definition, i.e. surface integral   @ @n  Z  @ @  dS 8 2 D.R3 /: (1.11.29)  .ı /;  D @n  @n @ .ı /; i 8 2 D.R3 /, Then, from (1.11.26) and (1.11.29), we get hT; i D h @n

H)

T D

@ .ı / 2 D 0 .R3 / @n

(1.11.30)

94

Chapter 1 Schwartz distributions

is called the double layer distribution of a normally oriented dipole over the surface  with continuous surface moment density .

1.11.5 Relation with probability distribution [7] A natural interesting question arises in our mind: is there any relation with probability distributions? Instead of entering this new domain of calculus of probability, which is outside the scope of the present treatment of topics based on the principle of determinism, we would like to indicate here only when a distribution T 2 D 0 .R/ is related to probability distribution at the most elementary level of the calculus of probabilities. Let f 2 L1 .1; 1Œ/ be an integrable function on 1; 1Œ with Z

1

f .x/  0;

f .x/dx D 1

(1.11.31)

1

such that f .x/ defines the probability distribution on .1; 1Œ/, and the probability P 2 Œ0; 1 of the variable x lying on .a; bŒ/ is given by: Z P .a < x < b/ D

b

f .x/dx: a

Hence, Z

1

P .1 < x < 1/ D 1

f .x/dx D 1 D hf; 1i D hTf ; 1i:

(1.11.32)

A distribution T 2 D 0 .R/ defines a probability distribution if and only if 1. T is a positive distribution (see Definition 1.7.2), i.e. T 0

.” hT; i  0 8 2 D.R/ with   0/I

(1.11.33)

2. hT; 1i is well defined in some sense, for example, T  0;

hT; 1i D sup hT; i

with  2 D.R/) and hT; 1i D 1 (1.11.34)

01

(see (1.11.1)) [8]. We complete our discussion with the following examples: (a) T D ıa D ı.x  a/ 2 D 0 .R/. Then, ıa  0, since hıa ; i D .a/  0 8 2 D.R/ with   0. hıa ; 1i D 1 (see (1.11.1)). Hence, ıa defines a probability distribution according to which the variable x can have only one value a with probability P D 1, i.e. the value x D a is certain.

95

Section 1.11 Physical distributions versus mathematical distributions

(b) T D 14 ıa C 34 ıb 2 D 0 .R/ with a ¤ b. Then 14 ıa C 34 ıb  0, since 

 1 3 3 1 3 1 ıa C ıb ;  D hıa ; i C hıb ; i D .a/ C .b/  0 4 4 4 4 4 4 8 2 D.R/ with   0:



 1 3 3 1 3 1 ıa C ıb ; 1 D hıa ; 1i C hıb ; 1i D  1 C  1 D 1 4 4 4 4 4 4

(see (1.11.1)):

Hence, 14 ıa C 34 ıb defines a probability distribution, according to which x can take only two values x D a and x D b with the probability 14 and probability 34 respectively. Remark 1.11.3. Laurent Schwartz [7] is of the opinion that we can not logically prove the analogy between physical and mathematical distributions in general. Then the natural question arises: What are these mathematical distributions, after all? The answer, according to him [7], is that mathematical distributions constitute the mathematically rigorous definitions of physical distributions.

Chapter 2

Differentiation of distributions and application of distributional derivatives

2.1

Introduction: an integral definition of derivatives of C 1 -functions

The notion of the derivatives of distributions is perhaps the most useful and important one in applications, especially in the study of Sobolev spaces and partial differential @T equations. This is due to the fact that the derivative @x of a distribution T on   i n R with respect to the variable xi , 1  i  n, is defined in such a way that if the distribution T is a function f with continuous partial derivatives on ,  being 1 0 the closure of  in Rn (i.e. R f 2 C ./, T D Tf D f 2 D ./ by (1.3.15) H) hf; i D hTf ; i D  f .x/.x/d x 8 2 D./), then we can retrieve the @f of the function f in the usual pointwise sense: partial derivative @x i

f .x1 ; : : : ; xi C xi ; : : : ; xn /  f .x1 ; : : : ; xn / @f .x/ D lim

xi !0 @xi xi 8x D .x1 ; x2 ; : : : ; xn / 2 ; (2.1.1) and importantly, unlike functions, distributions T on   Rn are infinitely differentiable on  and can be differentiated with respect to several variables in an arbitrary order of differentiation with respect to the variables. (For mixed partial derivatives of functions, such a result holds if Schwarz’s theorem on mixed partial derivatives of functions holds.) In other words, if a function f 2 C 1 ./ (i.e. has bounded and uni@T @f formly continuous partial derivatives @x , 1  i  n on ), then the derivative @xf i

i

of the distribution Tf associated with f 2 C 1 ./ is the distribution T @f associated @xi

with

@f @xi

2

C 0 ./,

i.e. 8 2 D./,     Z @Tf @f @f ;  D hT @f ; i D d x D ; ; @xi @xi @xi  @xi

(2.1.2)

@Tf @f on  is the function @x 2 C 0 ./ on  in the usual point@xi i @u wise sense (2.1.1), C 1 ./ D ¹u W function u and its partial derivatives @x , 1  i  n, i

H) the distribution

are bounded and uniformly continuous on , which have unique continuous extension to º.

Section 2.1 Introduction: an integral definition of derivatives of C 1 -functions

97

An integral definition of derivatives of functions of C 1 ./ @f Now we show that for any function f 2 C 1 ./ and its derivative @x 2 C 0 ./; .1  i i  n/, the following relation holds: 8 2 D./, Z Z @ @f .x/d x D  f .x/ .x/d x; d x D dx1 dx2 : : : dxn : (2.1.3) @x @x i i  

One dimensional case, n D 1 First of all, we consider the simplest case of functions of a single variable x: n D 1;  D R D 1; 1Œ with  D R D 1; 1Œ, 2 C 0 .1; 1Œ/. Then, by integrating by parts, 8 2 f 2 C 1 .1; 1Œ/, df dx D.1; 1Œ/, Z C1 Z C1 df d .x/.x/dx D Œf .x/.x/C1 (2.1.4)  f .x/ .x/dx: 1 dx dx 1 1 But  2 D.1; 1Œ/ H) 9 a bounded, closed interval Œa; b  R such that supp./  Œa; b H) .x/ D 0 8x lying outside Œa; b H) Œf .x/.x/C1 1 D 0. R R df d Hence, R dx .x/.x/dx D  R f .x/ dx .x/dx 8 2 D.R/. Similarly, for n D 1,  D a; bŒ  R with 1 < a < b < C1,  D Œa; b, f 2 C 1 .Œa; b/, df 2 C 0 .Œa; b/, we have dx Z b Z b df d .x/.x/dx D  (2.1.5) f .x/ .x/dx 8 2 D.a; bŒ/: dx dx a a The converse result also holds. Suppose that f 2 C 1 .Œa; b/. If 9g 2 C 0 .Œa; b/ such that Z b Z b d (2.1.6) g.x/.x/dx D  f .x/ .x/dx 8 2 D.a; bŒ/; dx a a D g in C 0 .Œa; b/, i.e. g is the usual derivative of f with respect to x on then df dx Œa; b. Proof. Since f 2 C 1 .Œa; b/; df 2 C 0 .Œa; b/.Then, from (2.1.5), we have dx Z b Z b Z b d df .x/.x/dx D f .x/ .x/dx D g.x/.x/dx 8 2 D.a; bŒ/  dx a a dx a Rb H) a . df  g/dx D 0 8 2 D.a; bŒ/. But df  g 2 C 0 .Œa; b/  L1loc .a; bŒ/. dx dx  g D 0 H) df D g in C 0 .Œa; b/. Hence, by Theorem 1.2.3A, df dx dx Conclusion For functions f 2 C 1 .Œa; b/, the derivative g D by the integral relation (2.1.5) instead of the definition: df f .x C x/  f .x/ .x/ D lim

x!0 dx x both the results being equivalent.

df dx

of f can be defined

8x 2 a; bŒ;

98

Chapter 2 Differentiation of distributions and application of distributional derivatives

x2 x1 = g (x2), c x2 d

d

x1 = h(x2), c x2 d c 0

x1

Figure 2.1 Domain  enclosed by x1 D g.x2 /; x1 D h.x2 /; c  x2  d , such that a line parallel to the x1 -axis intersects  at no more than two points

Two-dimensional case, n D 2 Now we consider   R2 with boundary  such that  D ¹.x1 ; x2 / : 8 fixed x2 2 R with c < x2 < d , h.x2 / < x1 < g.x2 /º, i.e.  is enclosed by two curves x1 D g.x2 / and x1 D h.x2 / as shown in Figure 2.1 such that any line parallel to the x1 -axis intersects  at no more than two points. (This assumption has been made for the sake of simplicity in presentation. For domains  not satisfying this assumption, the proof is slightly modified. For example,  may be subdivided into subdomains ¹i º, in each of which this assumption holds.) Suppose @f 2 C 0 ./. We prove (2.1.3) for i D 1. Transforming the double that f 2 C 1 ./, @x i integral over  into iterative (definite) integrals, we get Z Z d Z x1 Dg.x2 / @f @f .x1 ; x2 /.x1 ; x2 /dx1 dx2 D dx2 .x1 ; x2 /.x1 ; x2 /dx1 ;  @x1 c x1 Dh.x2 / @x1 in which x2 is held fixed on c; d Œ. Hence, we can integrate by parts the right-hand side integral with respect to x1 : 8 2 D./, Z x1 Dg.x2 / @f x Dg.x / .x1 ; x2 /.x1 ; x2 /dx1 D Œf .x1 ; x2 /.x1 ; x2 /x11 Dh.x22/ x1 Dh.x2 / @x1 Z x1 Dg.x2 / @  f .x1 ; x2 / .x1 ; x2 /dx1 ; @x 1 x1 Dh.x2 / @f / @ since @x  D @.f  f @x . @x1 1 1 But  2 D./ H) # D 0 H) .x1 ; x2 /jx1 Dg.x2 / D .x1 ; x2 /jx1 Dh.x2 / D 0 x Dg.x / H) Œf .x1 ; x2 /.x1 ; x2 /x11 Dh.x22/ D 0. Hence, 8 2 D./,

Z

x1 Dg.x2 / x1 Dh.x2 /

@f .x1 ; x2 /.x1 ; x2 /dx1 D  @x1

Z

x1 Dg.x2 /

f .x1 ; x2 / x1 Dh.x2 /

@ .x1 ; x2 /dx1 @x1

Section 2.1 Introduction: an integral definition of derivatives of C 1 -functions

99

and Z 

@f .x1 ; x2 /.x1 ; x2 /dx1 dx2 D  @x1

Z

Z

d

x1 Dg.x2 /

dx2 x1 Dh.x2 /

c

Z

D

f .x1 ; x2 /

f .x1 ; x2 / 

Hence, 8f 2 C 1 ./ with Z 

@f @x1

@f d x D  @x1

@ .x1 ; x2 /dx1 dx2 : @x1

2 C 0 ./, the relation (2.1.3) holds for i D 1:

Z f 

@ .x1 ; x2 /dx1 @x1

@ d x 8 2 D./ @x1

.d x D dx1 dx2 /:

Similarly, we can prove (2.1.3) for i D 2: Z 

@f d x D  @x2

Z f 

@ d x 8 2 D./: @x2

n-dimensional case Thus, we can prove that for f 2 C 1 ./, Rn , the relation (2.1.3) holds: for 1  i  n, Z 

@f d x D  @xi

Z f 

@ dx @xi

8 2 D./

@f @xi

2 C 0 ./  

.d x D dx1 dx2 : : : dxn /:

The converse result also holds. Suppose that f 2 C 1 ./ with bounded   Rn . If 9gi 2 C 0 ./, 1  i  n, such that Z

Z gi .x/.x/d x D  

f .x/ 

@ .x/d x @xi

8 2 D./;

(2.1.7)

@f then @x D gi in C 0 ./; 1  i  n, i.e. gi is the usual partial derivative of f with i respect to xi in . @f Proof. Since f 2 C 1 ./; @x 2 C 0 ./ for 1  i  n. Then, from (2.1.3), we have: i for 1  i  n, Z Z Z @f @  f .x/ .x/d x D .x/.x/d x D gi .x/.x/d x 8 2 D./ @xi   @xi 

H)

R

@f  . @xi

 gi /d x D 0 8 2 D./ H)

by Theorem 1.2.3A H) respect to xi in .

@f @xi

@f @xi

 gi D 0 in C 0 ./  L1loc ./

D gi in C 0 ./, i.e. gi is the usual derivative of f with

100

Chapter 2 Differentiation of distributions and application of distributional derivatives

Conclusion For functions f of n variables with f 2 C 1 ./,   Rn , partial deriva@f tives gi D @x on  can be defined by the integral relation in (2.1.3) instead of the i usual definition (2.1.1): gi .x/ D

@f .x/ @xi

D lim

xi !0

f .x1 ; x2 ; : : : ; xi C xi ; : : : ; xn /  f .x1 ; x2 ; : : : ; xn / xi

8x 2 ;

the results being the same. Remark 2.1.1. It is of general interest that Sobolev [30] extended the integral definition (2.1.3) to (discontinuous and unbounded) functions f 2 L2 ./ as follows: suppose that f 2 L2 ./ with   Rn . If 9gi 2 L2 ./, 1  i  n, such that Z Z @ gi d x D  f d x 8 2 C01 ./ D D./; (2.1.8) @x i   @f then gi D @x 2 L2 ./ is called the generalized derivative of f 2 L2 ./ with i respect to the variable xi . (Obviously, gi is not the partial derivative of f in the usual pointwise sense (2.1.1), which, in fact, does not exist in general in  for f 2 L2 ./.)

2.2

Derivatives of distributions

From (1.3.17), C 1 ./; C 0 ./  D 0 ./, we can identify f 2 C 1 ./ with Tf 2 @f 2 C 0 ./ with T @f 2 D 0 ./. Hence, we set f D Tf 2 D 0 ./, D 0 ./ and @x i

@xi

Z

H) H)

@f hf; i D hTf ; i D f .x/.x/d xI D T @f 2 D 0 ./ @xi @xi    Z @f @f hT @f ; i D ; D .x/.x/d x8 2 D./; (2.2.1) @xi @xi @x i 

where h  ;  i denotes the duality pairing h  ;  iD 0 ./D./ between D 0 ./ and D./. Consequently, using (2.2.1), we can rewrite (2.1.3) in the following form:     @f @ ;  D  f; 8 2 D./: (2.2.2) @xi @xi Hence, this relation (2.2.2) between function f 2 C 1 ./ and its derivative C 0 ./ D 0 ./

@T @xi

@f @xi

2

will be preserved if we can define the derivative of a distribution T 2 n on   R by the equation:     @T @ ;  D  T; 8 2 D./: (2.2.3) @xi @xi

101

Section 2.2 Derivatives of distributions

In order to prove that this definition (2.2.3) of

@T @xi

is meaningful, we are to show that

@ i @xi

@T S D @x defined by (2.2.3), hS; i D hT; 8 2 D./, is a continuous, linear i functional on D./. Linearity of S :       @ @1 @2 hS; ˛1 1 C ˛2 2 i D  T; .˛1 1 C ˛2 2 / D ˛1 T;  ˛2 T; @xi @xi @xi

D ˛1 hS; 1 i C ˛2 hS; 2 i

81 ; 2 2 D./:

Continuity of S : n !  in D./ H)

@n @ ! @x in D./ as n ! 1. But T 2 D 0 ./ is @xi i @ @ n n ! @x in D./ H) hT; @ i ! hT; @x i in R as continuous on D./. Hence, @ @xi @xi i i @n @ n ! 1 H) hT; @x i D hS; n i ! hT; @x i D hS; i in R as n ! 1. Thus, i i n !  in D./ H) hS; n i ! hS; i in R as n ! 1. H) S is a continuous, linear functional on D./ H) S 2 D 0 ./.

Hence, S D

@T @xi

2 D 0 ./ defined by (2.2.3) is a distribution on .

Second-order derivatives @2 T @xi @xj

of distributions T 2 D 0 ./ are defined using (2.2.3): 8 2 D./; 1  i  n,  2          @ T @ @T @T @ @2  ; D ; ; D  D T; I (2.2.4) @xi @xj @xi @xj @xj @xi @xj @xi         2   @T @ @2  @ T @ @T ; D  D T; : (2.2.5) ; D ; @xj @xi @xj @xi @xi @xj @xi @xj

But  2 D./ H) Schwarz’s theorem. @2 T

@2  @xi @xj

2

; @x@

j @xi

are continuous in  H) 2

Hence, 8 2 D./, hT; @x@

j @xi

@2 T

@2 T

@2  @xi @xj

i D 2

@2  by @xj @xi @2  hT; @x @x i H) i j in D 0 ./, i.e. any

D

T h @x @x ; i D h @x @x ; i 8 2 D./ ” @x @x D @x@ @x i j j i i j j i change of order of differentiation of a distribution T is permissible.

2.2.1 Higher-order derivatives of distributions T For ˛ D .˛1 ; ˛2 ; : : : ; ˛n / with integers ˛i  0 for i D 1; 2; : : : ; n, j˛j D ˛1 C ˛2 C    C ˛n 2 N,  ˛2   ˛n   @ @ @˛ 1 @j˛j @˛1 C˛2 CC˛n ˛ D @ D ˛1       D @xn˛n @x1 @x2˛2 @x1˛1 @x2˛2 : : : @xn˛n @x1˛1 @x2˛2 : : : @xn˛n denotes the partial differential operator of order j˛j 2 N. For example, @.1;2;3/ D

@6 I @x1 @x22 @x33

@.4;0/ D

@4 I @x14

@.0;4/ D

@4 I @x24

@.2;2/ D

@4 I @x12 @x22

etc.

102

Chapter 2 Differentiation of distributions and application of distributional derivatives @j˛j T ˛ ˛ ˛ @x1 1 @x2 2 :::@xn n ˛ defined by: h@ T; i

Then the derivative @˛ T D D 0 ./

tion T 2 equivalently, h

on  is

@j˛j T ˛1 ˛ ˛ @x1 @x2 2 :::@xn n

(of order j˛j 2 N) of the distribu-

D .1/j˛j hT; @˛ i 8 2 D./ or, j˛j ; i D .1/j˛j hT; ˛1 @ ˛2 ˛n i. @x1 @x2 :::@xn

Finally, we state the result:

Theorem 2.2.1. Every distribution T 2 D 0 ./ is infinitely differentiable, i.e. has successive derivatives of all orders that are distributions of D 0 ./, and the order of differentiation may be interchanged in an arbitrary manner. The derivatives @˛ T of T are defined by: 8j˛j 2 N; 8T 2 D 0 ./, h@˛ T; i D .1/j˛j hT; @˛ i where @˛ T D

2.3

@j˛jT ˛ ˛ ˛ , @x1 1 @x2 2 :::@xn n

@˛  D

8 2 D./;

@j˛j  ˛ ˛ ˛ , @x1 1 @x2 2 :::@xn n

(2.2.6)

j˛j D ˛1 C ˛2 C    C ˛n .

Derivatives of functions in the sense of distribution

Let f 2 L1loc ./  D 0 ./ (see (1.3.15)) be a locally summable (integrable) function on  (which, in particular, may belong to C 0 ./ or L2 ./ or Lp ./, 1  p  1) with f D Tf 2 D 0 ./. Definition 2.3.1. 8j˛j 2 N, the derivative @˛ f D i.e.

@j˛j f ˛ ˛ ˛ @x1 1 @x2 2 :::@xn n

defined by (2.2.6),

h@˛ f; i D h@˛ Tf ; i D .1/j˛j hTf ; @˛ i D .1/j˛j hf; @˛ i Z @j˛j .x/ j˛j D .1/ f .x/ ˛1 ˛2 d x 8 2 D./ @x1 @x2 : : : @xn˛n 

(2.3.1)

is called the derivative of the function f 2 L1loc ./ of order j˛j 2 N with respect to x1 ; x2 ; : : : ; xn in the sense of distribution, or is equivalently called the distributional derivative of f of order j˛j with respect to x1 ; x2 ; : : : ; xn . Some explanations on alternative notations used in (2.3.1) are in order. Since f 2 L1loc ./, weRset f D Tf 2 D 0 ./ and, hence, we write: hTf ; @˛ i D hf; @˛ i D  f .x/@˛ .x/d x on the right-hand side of (2.3.1). But f D Tf 2 D 0 ./ H) @˛ f D @˛ Tf 2 D 0 ./ 8j˛j 2 N and we have written h@˛ f; i D h@˛ Tf ; i, which can not be defined by an integral in general, and consequently we have not written an integral representation of h@˛ Tf ; i. In other words, for f 2 L1loc ./, @˛ f 2 D 0 ./ may be a singular distribution, and then h@˛ f; i D h@˛ Tf ; i cannot be represented by an integral on  and, hence, we have not used any integral representation of h@˛ f; i on the left-hand side of (2.3.1).

Section 2.3 Derivatives of functions in the sense of distribution

103

Remark 2.3.1. The derivatives of regular distributions may not be regular distributions, i.e. may be singular distributions. (2.3.2)



Every continuous function f 2 C 0 ./, or even locally summable function f 2 can be successively differentiated in the distribution sense, i.e. using the formula (2.3.1): 8 2 D./, 

L1loc ./,

h@˛ f; iD 0 ./D./ D h@˛ Tf ; iD 0 ./D./ D .1/j˛j hTf ; @˛ i Z @j˛j  j˛j D .1/ f .x/ ˛1 ˛2 d x: @x1 @x2 : : : @xn˛n  Hence, f is a function defining a regular distribution Tf , but @˛ f 2 D 0 ./ may not be a function, i.e. may be a singular distribution in D 0 ./ (see Example 2.3.2). The situation can be understood better by considering the fact that any algebraic equation, say, for example, ax 2 C bx C c D 0 with real a; b; c, has complex roots (i.e. may not have real roots). Similarly, any locally summable function has successive derivatives of all orders which are distributions (i.e. may not be functions). 

Suppose that the distributional derivative @˛ f of function f 2 L1loc ./ is also a function belonging to L1loc ./, i.e. f 2 L1loc ./  D 0 ./ with @˛ f 2 L1loc ./ H) @˛ f D T@˛ f D @˛ Tf 2 D 0 ./ Z ˛ ˛ H) h@ f; i D h@ Tf ; i D hT@˛ f ; i D @˛ f .x/.x/d x 8 2 D./:





(2.3.3) Then, for f 2 L1loc ./ with @˛ f 2 L1loc ./  D 0 ./, using (2.3.1) and (2.3.3), the equation h@˛ f; iD 0 ./D./ D .1/j˛j hf; @˛ iD 0 ./D./ 8 2 D./ can be rewritten in the following integral form: 8 2 D./, Z Z @j˛j f .x/ @j˛j .x/ j˛j .x/d x D .1/ f .x/ d x: (2.3.4) ˛1 ˛2 ˛n @x1˛1 @x2˛2 : : : @xn˛n  @x1 @x2 : : : @xn  Moreover, we have: Proposition 2.3.1. Let f 2 L1loc ./. Then, if 9 a function g˛ 2 L1loc ./ such that 8 2 D./, Z Z @j˛j .x/ j˛j g˛ .x/.x/d x D .1/ f .x/ ˛1 ˛2 d x; (2.3.5) @x1 @x2 : : : @xn˛n   then g˛ D @˛ f D

@j˛j f ˛ ˛ ˛ @x1 1 @x2 2 :::@xn n

2 L1loc ./ in the sense of distribution.

104

Chapter 2 Differentiation of distributions and application of distributional derivatives

0

Figure 2.2 Distributional derivative of jxj

Proof. 8 2 D./ Z

@j˛j .x/ d x D .1/j˛j hf; @˛ iD 0 ./D./ @x1˛1 @x2˛2 : : : @xn˛n  Z ˛ D h@ f; iD 0 ./D./ D g˛ d x D hg˛ ; iD 0 ./D./ j˛j

.1/

f .x/



” @˛ f D g˛ in D 0 ./. But g˛ 2 L1loc ./  D 0 ./ defines the unique distribution with g˛ D @˛ f 2 L1loc ./; @˛ f being in the sense of distribution.

Examples Example 2.3.1. For n D 1,   R D 1; 1Œ, ´ f .x/ D jxj D

x x

for x  0 for x < 0:

Then, 8 compact subsets Œa; b  R with a < b and 0 2 a; bŒ, Z

Z

0

b

xdx C a

xdx D 0

b 2 C a2 < C1 2

H) f 2 L1loc .R/. f is not differentiable in the usual pointwise sense at x D 0 (see Figure 2.2), H) f is not differentiable on R in the usual pointwise sense. But f 2 L1loc .R/

Section 2.3 Derivatives of functions in the sense of distribution

H)

df dx



105

2 D 0 .R/ is defined in the sense of distribution by, 8 2 D.R/,

df ; dx

 D 0 .R/D.R/

  Z 1 d d dx D  f; D f dx D 0 .R/D.R/ dx 1 Z 1 Z 0 d d D x dx  x dx dx dx 1 0 Z 0  D Œx.x/01  1  .x/dx  Œx.x/1 0C 1

Z

1

C

1  .x/dx

8 2 D.R/:

0

But  2 D.1; 1Œ/ H) supp./  1; 1Œ H) 9 a compact interval Œa; a with a > 0 outside which .x/ D 0  H) Œx.x/01 D 0  0 D 0, Œx.x/1 0 D 0  0 D 0. Hence,   Z 0 Z 1 df ; D 1  .x/dx C 1  .x/dx dx 1 0 D 0 .R/D.R/   Z 0 Z 1 .1/  .x/dx C .C1/  .x/dx D Z

1 1

D

0

g.x/.x/dx

8 2 D.R/;

1

where g.x/ D 1 for x > 0 and D 1 for x < 0 and belongs to L1loc .R/, since 8 Rb compact interval Œa; b  R with a < b; a jg.x/jdx D .b  a/ < C1. ; i D hg; i with g 2 L1loc .R/ 8 2 D.R/ H) df D g 2 L1loc .R/ H) h df dx dx in the sense of distribution by Proposition 2.3.1. Hence, both f and its derivative df D g in the sense of distribution are functions (locally summable on R). In fact, dx f 0 .x/ D g.x/ D 1 for x > 0 and D 1 for x < 0 is also in the usual pointwise sense (Figure 2.1). Example 2.3.2. Let   R  1; 1Œ and H.x/ be the Heaviside function defined by (1.1.5): H.x/ D 1 for x > 0 and H.x/ D 0 for x < 0. Then, 8 compact interval Œa; a  R with a > 0, Z

Z

a

a

H)

Z

0

H.x/dx D

0dx C a

H 2 L1loc .R/

a

1dx D a < C1 0

H)

dH 2 D 0 .R/; dx

106

Chapter 2 Differentiation of distributions and application of distributional derivatives

which is no longer a function and, in fact, dH 2 D 0 .R/ is the singular Dirac distridx bution ı D ı0 . We prove this now: 8 2 D.R/, 

   Z 1 Z 0 Z 1 d d d d dH ;  D  H; D dx  dx H.x/ dx D  0 1 dx dx dx dx dx 1 1 0 Z 1 d dx D Œ.x/1 D 0 D .0/; dx 0

since  2 D.R/ H) supp./  1; 1Œ H) 9 a compact interval Œa; a with a > 0 outside which .x/ D 0 H) .x/ D 0 for x D 1 H) h dH ; i D .0/ dx D hı; i 8 2 D.R/ (by the definition of Dirac distribution in (1.3.27)) ” dH D ı 2 D 0 .R/ in the sense of distribution H) dH … L1loc .R/ by Proposidx dx 1 tion 1.3.2, i.e. there does not exist any function g 2 Lloc .R/ such that h dH ; i D dx R1 1 1 g.x/.x/dx D .0/ 8 2 D.R/. Hence, H 2 Lloc .R/ is the regular distribuTH tion TH D H 2 D 0 .R/, but its derivative ddx D dH D ı 2 D 0 .R/ in the sense of dx distribution is a singular distribution on R. Remark 2.3.2. For the Heaviside function H in (1.1.5), which is discontinuous with jump J0 D H.0C /H.0 / D 10 D 1 at x D 0, we have two different distributions corresponding to two different notions of derivatives of H.x/. TH 1. The derivative dH D ddx 2 D 0 .R/ of H D TH 2 D 0 .R/ in the sense of dx d TH dH distribution, i.e. dx D dx D ı 2 D 0 .R/ in the sense of distribution. (We cannot write dH .x/ or ı.x/, since these are distributions and not functions, and dx consequently have no point values.)

2. The distribution T dH .x/ defined by the usual ordinary derivative dx

H.x C 4x/  H.x/ dH .x/ D lim 4x!0 dx 4x (in the pointwise sense), i.e. since 8 2 D.R/ Z hT dH .x/ ; i D dx

1 1

dH .x/ dx

dH .x/: dx

8x ¤ 0

D 0 8x ¤ 0 H) T dH .x/ D 0 2 D 0 .R/,

dH .x/.x/dx D dx

dx

Z

Z

0

1

0  dx C 1

0  dx D 0 0C

” T dH .x/ D 0 2 D 0 .R/. dx

.x/ to indicate that it is the usual Here, by abuse of notation, we have used dH dx ordinary derivative of H at x in the pointwise sense, whereas dH 2 D 0 .R/ is dx to be understood in the sense of distribution.

107

Section 2.3 Derivatives of functions in the sense of distribution

An equivalent notation for the distribution T dH .x/ is ΠdH .x/ 2 D 0 .R/ such that dx dx

 dH .x/ D 0 2 D 0 .R/ and T dH .x/ D dx dx   d TH dH dH D D .x/ C J0 ı in D 0 .R/ dx dx dx 

(2.3.6)

with J0 D 1 (see Chapter 3: Theorem 3.1.1). Then   dH dH Dı¤0D .x/ in D 0 .R/; dx dx

(2.3.7)

TH i.e. the distributional derivative dH D ddx 2 D 0 .R/ of the discontinuous Heaviside dx .x/ D T dH .x/ D 0 defined by function H on R is not equal to the distribution ΠdH dx dx

the usual ordinary derivative dH .x/ in the pointwise sense. dx dH 0 2 Œ dx .x/ 2 D .R/ corresponds to the null function 0 2 L1loc .R/, whereas dH dx 0 D .R/ is the Dirac distribution ı, which is not a locally summable function on R. In (2.3.6), the effect of the discontinuity of H with a finite jump J0 D 1 appears in the distributional derivative in the form of a point mass or force or charge as the case may be (see Section 3.1 in Chapter 3 for more details). Dirac’s result (1.1.5), dH D ı, is correct and is to be understood in the sense of dx distributional derivative and not in the usual pointwise sense of the derivative. In electrical engineering H.x/ is called the unit step function and Dirac distribution ı is called the impulse function. Example 2.3.3. Now we find the second-order distributional derivative df dx

d 2f dx 2

of the

function f .x/ D jxj in Example 2.3.1, where we have shown that D g.x/ D 1 1 D g.x/ D 1 for x < 0 is a function in L .R/. Here, we will for x > 0 and df loc dx show that

d 2f dx 2

D

dg dx

distribution on R, i.e. 

in the sense of distribution is not a function but is a singular d 2f dx 2

does not belong to L1loc .R/. In fact, 8 2 D./,

         d df d 2f dg d ;  D  g; ; D ; D dx 2 dx dx dx dx Z 1 d D g.x/ .x/dx (since g 2 L1loc .R/) dx 1 Z 1 Z 0 d d dx D .1/ dx  1 C dx dx 1 0 

D .x/j01  .x/j1 D .0/ C .0/ D 2.0/ D 2hı; i 0C

108

Chapter 2 Differentiation of distributions and application of distributional derivatives

(since .x/ D 0 for x ! ˙1 and .0/ D limx!0 .x/ D limx!0C .x/) 2 2 H) h ddxf2 ; i D h2ı; i 8 2 D.R/ ” ddxf2 D 2ı 2 D 0 .R/ is a singular distribution. Hence, the first-order distributional derivative df 2 L1loc .R/ is a regular distribudx 2

tion, but the second-order distributional derivative ddxf2 2 D 0 .R/ is a singular distribution on R, i.e. not a function (locally summable) on R.

Derivatives of Dirac distributions Example 2.3.4. Let ı 2 D 0 .R/ be the Dirac distribution on R defined by hı; i D .0/ 8 2 D.R/. Then the derivatives of Dirac distribution ı on R are defined by: 8k 2 N, 8 2 D.R/,   d k hı .k/ ; i D .1/k ı; D .1/k  .k/ .0/: dx k

(2.3.8)

.0/; for k D 2, hı (2) ; i D In particular, for k D 1, hı .1/ ; i D  .1/ .0/ D  d dx .1/2  (2) .0/ D

d 2 .0/, dx 2

and so on.

Remark 2.3.3. ı .1/ D ı 0 is the dipole of moment 1 at the origin 0. 2

Example 2.3.5. Now we find the second-order distributional derivative ddxH2 of the Heaviside function H of Example 2.3.2, dH D ı 2 D 0 .R/. Hence; 8 2 D.R/, dx 

         d 2H dH d d d dH ; D  ; D  ı; D hı 0 ; i ; D dx 2 dx dx dx dx dx

”

d 2H D ı 0 2 D 0 .R/; dx 2

i.e.

d 2H … L1loc .R/; dx 2

(2.3.9)

where ı 0 is the derivative of the Dirac distribution and the dipole of moment 1 at origin 0 and is defined by: hı 0 ; i D hı;  0 i D  0 .0/ (see Example 2.3.4). Similarly, 

   k1    k1  d d d H H d d kH ;  D ; ;  D  dx dx k1 dx k dx k1 dx   dH d k1  D    D .1/k1 ; dx dx k D .1/k1 hı;  .k1/ i D hı .k1/ ; i

”

d kH D ı .k1/ 2 D 0 .R/ dx k

.i.e.

8 2 D.R/

d kH … L1loc .R/ 8k 2 N/: dx k

(2.3.10)

Section 2.3 Derivatives of functions in the sense of distribution

109

Example 2.3.6. Since ln jxj (x ¤ 0) is a locally summable function in the neighbourhood of x D 0, ln jxj 2 L1loc .1; 1Œ/ and defines a regular distribution on R. R In fact, ln jxjdx D x ln jxj  x C C for x ¤ 0, lim x ln jxj D lim

x!0

ln jxj 1 x

x!0

1 x x!0 1 x2

D lim

D lim .x/ D 0; x!0

(2.3.11)

and Z

ˇ ˇ ˇ ln jxjˇdx D

´R

ln jxjdx R  ln jxjdx

for ln jxj  0 for ln jxj < 0:

Rb H) 8 compact interval Œa; b  R, a j ln jxjjdx < C1 H) ln jxj 2 L1loc .1; 1Œ/. Thus, ln jxj 2 D 0 .R/ defines a regular distribution on R H) the distributional d derivative dx lnjxj of ln jxj is a distribution on R defined by:   Z 1 d 0 ln jxj;  D hln jxj;  i D  ln jxj 0 .x/dx 8 2 D.R/ dx 1 Z 0 Z 1 0 D ln jxj .x/dx  ln jxj 0 .x/dx; (2.3.12) 1

Z

1 0

ln jxj 0 .x/dx D lim

Z

"!0C

0 1

.ln jxj/ 0 .x/dx

"

 1 .x/dx x "!0C "   Z 1 1 D lim .ln "/."/  .x/dx 8 2 D.R/: x "!0C " (2.3.13) R " R " Similarly, we have, 8" > 0, 1 ln jxj 0 .x/dx D .ln "/."/  1 x1 .x/dx 8 2 D.R/   Z 0 Z " 1 0 .x/dx : ln jxj .x/dx D lim .ln "/."/ H) 8 2 D.R/; "!0C 1 1 x (2.3.14)  Z 1 D lim .ln jxj/.x/j" 

1

Then, from (2.3.12)–(2.3.14), we get, 8 2 D.R/,   Z " Z 1 1 1 d ln jxj; i D lim .x/dx C .x/dx h dx x "!0C 1 x " C lim Œ.ln "/."/  .ln "/."/: "!0C

110

Chapter 2 Differentiation of distributions and application of distributional derivatives

But j.ln "/."/  .ln "/."/j D j.ln "/.."/  ."//j  j ln "j  j."/  ."/j  j ln "j  2" 

max

x2supp./

j 0 .x/j D 2

max

x2supp./

j 0 .x/j  j" ln "j ! 0

(using (2.3.11)). Hence,   Z 1 d 1 ln jxj;  D c:p:v: .x/dx dx x 1   1 D c:p:v: ; .x/ 8 2 D.R/ x

as " ! 0C

.by (1.3.45)/

d ” dx lnjxj D c:p:v: x1 in D 0 .R/, i.e. the distributional derivative c:p:v: x1 2 D 0 .R/ is not a regular distribution.

d dx

ln jxj D

Example 2.3.7 (Derivative of ln x.x ¤ 0/ in the sense of distribution). Now we are d in a position to show that Dirac’s remarkable formula for the derivative dx ln x of ln x.x ¤ 0/ in (1.1.6) is correct. Here, we allow x to be negative, and consequently we are to consider a logarithmic function of a complex variable z. From the theory of complex variables, for z D x C iy,   y 2 2 12 ln z D ln.x C iy/ D ln.x C y / C i arctan (2.3.15) x is analytic in the upper half-plane y > 0. Then, limy!0C ln.x C iy/ D ln x C i 0 with the properties: 



the first term 12 ln.x 2 C y 2 / on the right-hand side of (2.3.15) converges, decreasing monotonically to ln jxj, i.e. limy!0C 12 ln.x 2 C y 2 / D ln jxj. the second term i arctan. yx / has its absolute value bounded by , and as y ! 0C , it converges to: ´ i  for x < 0 HQ .x/ D i H.x/ D 0 for x > 0; i.e.

  y lim i arctan D HQ .x/ D i H.x/; x y!0C

where the Heaviside function H.x/ is defined by (1.1.5) and ´ 1 for x < 0 H.x/ D 0 for x > 0:

(2.3.16)

111

Section 2.3 Derivatives of functions in the sense of distribution

Hence, we write ln.x C i 0/ D lim ln.x C iy/ D ln jxj C i H.x/:

(2.3.17)

y!0C

Then, 

But

d dx

  d d ln.x C i 0/;  D ln jxj C i H.x/; i 8 2 D.R/ dx dx     d d ln jxj;  C i  H.x/;  : D dx dx

ln jxj D c:p:v: x1 , with   Z 1 Z .x/ .x/ 1 dx D lim dx c:p:v: ;  D c:p:v: C x x "!0 jxj" x 1

(see Example 2.3.6) and, 8 2 D.R/,   Z 1 d H.x/;  D hH.x/;  0 i D  H.x/ 0 .x/dx dx 1 Z 0 Z 1 D 1   0 .x/dx  0   0 .x/ 1

D

Œ.x/xD0 xD1

0

 0 D .0/ D hı; i D hı; i

(by the definition of Dirac distribution ı in (1.3.27)). Hence,     d 1 ln.x C i 0/;  D c:p:v: ;  C i hı; i dx x   1 D c:p:v:  i ı;  8 2 D.R/ x 1 d ln.x C i 0/ D c:p:v:  i ı in D 0 .R/: ” dx x Thus, writing ln x D ln.x C i 0/, we get Dirac’s formula for which x1 is to be understood as the c:p:v: x1 (see (1.3.38)).

d dx

(2.3.18)

(2.3.19)

ln x in (1.1.6), in

Example 2.3.8. Prove the following results: 1: 2:

1 d ln.x ˙ iy/ D 8 fixed y > 0 in D 0 .R/I dx x ˙ iy   1 d ln.x  i 0/ D c:p:v: C i ı in D 0 .R/: dx x

(2.3.20) (2.3.21)

112

Chapter 2 Differentiation of distributions and application of distributional derivatives

Proof. 1. 8 fixed y > 0, ln.x ˙ iy/ D 12 ln.x 2 C y 2 / ˙ i arctan. yx / (see (1.4.3)). Then, from the definition of the distributional derivative, 8 2 D.R/,   Z 1 d 0 ln.x C iy/;  D hln.x C iy/;  i D  ln.x C iy/ 0 .x/dx dx 1 Z 1 1 xDC1 .x/dx D Œln.x C iy/.x/xD1 C x C iy 1 Z 1 1 D .x/dx; 1 x C iy which is obtained by integrating by parts with: u D 12 ln.x 2 Cy 2 /Ci arctan. yx /, dv D  0 .x/dx,   x  iy x y 1 dx D 2 dx; du D  i dx D x2 C y2 x C y2 x C iy x 2 .1 C . yx /2 / v D .x/ and applying .x/ D 0 for x D ˙1. d 1 H) dx ln.x C iy/ D xCiy in D 0 .R/. Similarly,    d 1 d y 1 2 2 ln.x  iy/ D ln.x C y /  i arctan D dx dx 2 x x  iy

in D 0 .R/:

2. As in the case of ln.x C i 0/ defined by (2.3.17), we have ln.x  i 0/ D lim ln.x  iy/ D ln jxj  i H.x/; y!0C

(2.3.22)

with H.x/ D 1 for x < 0 and H.x/ D 0 for x > 0. Hence,     d d ln.x  i 0/;  D Œln jxj  i H.x/;  dx dx     1 D c:p:v:  i .ı/;  8 2 D.R/ x (see (2.3.18) and Example 2.3.6) d H) dx ln.x  i 0/ D c:p:v:. x1 / C i ı in D 0 .R/. Derivative of an unbounded and discontinuous function in R2 Example 2.3.9. Let ´ ln j ln rj u D u.x1 ; x2 / D 0

1

for 0 < r D .x12 C x22 / 2 < for 1e  r < 1:

1 e

(2.3.23)

113

Section 2.3 Derivatives of functions in the sense of distribution

0

Figure 2.3 Annular domain " enclosed by concentric circles  1 and " with radii e respectively

1 e

and "

Then, 1. u 2 L2 .R2 /; 2.

@u @xi

@u @xi

2 L2 .R2 /, i D 1; 2, where

is in the distributional sense.

(2.3.24)

Proof. 1. Since ju.x1 ; x2 /j ! 1 as r ! 0, u is unbounded and discontinuous in R2 . But u 2 L2 .R2 /. In fact, Z R2

u.x1 ; x2 /2 dx1 dx2 D

Z 0 1e

and belongs to L2 .R2 /, i.e.   @u .x/ 2 L2 .R2 /; @xi

1 e

(2.3.28)

i D 1; 2:

(2.3.29)

In fact, 2 Z  Z xi2 @u .x/ dx1 dx2 D dx1 dx2 4 2 R2 @xi 0 0,

" .x 2 C"2 /

lim

"!0C

defines a distribution in D 0 .R/ and

" Dı .x 2 C "2 /

lim

"!0C

in D 0 .R/I

  1 x D c:p:v: 2 2 x C" x

(2.9.5) in D 0 .R/:

(2.9.6)

3. Show that

  1 1 a lim lim  D ı 0 2 2 .x C h/2 C a2 h!0C a!0C 2h .x  h/ C a

in D 0 .R/: (2.9.7)

4. Prove that e ixt D .i 2/ı t!1 x  i 0 lim

in D 0 .R/:

(2.9.8)

Proof. 1. We have shown in Example 1.4.2 that limy!0C ln.x C iy/ D ln.x C i 0/ in D 0 .R/, i.e. Z 1 Z 1 ln.x C iy/.x/dx D Œln jxj C i H.x/.x/dx lim y!0C

1

Z

1 1

D

ln.x C i 0/.x/dx

8 2 D.R/;

1

where ln.x C i 0/ D ln jxj C i H.x/ 8x ¤ 0. But from (2.3.19), d ln.x C i 0/ D c:p:v:. x1 /  i ı in D 0 .R/. Hence, ln.x C iy/ ! ln.x C i 0/ in dx d d 0 D .R/ as y ! 0C H) dx ln.x C iy/ ! dx ln.x C i 0/ in D 0 .R/ as y ! 0C by Theorem 2.9.1. d 1 ln.x C iy/ D xCiy 8 fixed y > 0 in D 0 .R/ From (2.3.20), dx H)

lim

y!0C

1 d D lim ln.x C iy/ C x C iy y!0 dx

  1 d ln.x C i 0/ D c:p:v: D  i ı dx x

in D 0 .R/:

1 1 D limy!0C xCiy D c:p:v:. x1 /  i ı in D 0 .R/, i.e. Then we write xCi0 Z 1 Z 1 1 .x/ .x/dx D lim dx C y!0 1 x C i 0 1 x C iy Z 1 .x/ dx  hi ı; i D c:p:v: 1 x Z 1 .x/ dx  i .0/ 8 2 D.R/: D c:p:v: 1 x

Section 2.9 Continuity of differential operator @˛ W D 0 ./ ! D 0 ./

145

Similarly, 1 1 d D lim D lim ln.x  iy/ C x  i 0 y!0C x  iy dx y!0   1 d ln.x  i 0/ D c:p:v: C i ı in D 0 .R/; D dx x i.e. Z

1 1

1 .x/dx D lim x  i0 y!0C

Z

1

Z

1 1

D c:p:v: 1

.x/ dx x  iy .x/ dx C i .0/ x

8 2 D.R/:

" is locally summable on R and, consequently, defines .x 2 C"2 / R1 " " 0 a distribution in D .R/ by: h .x 2 C" 2 / ; i D 1 .x 2 C"2 / .x/dx 8 2 D.R/. " 1 1 1 But .x 2 C" 2 / D 2i Πxi"  xCi" 

2. 8" > 0, function

Z

H)

1

" .x/dx 2 C "2 / .x 1 Z 1  Z 1 1 .x/ .x/ D lim dx  dx "!0C 2i 1 x  i " 1 x C i "  Z 1 Z 1 .x/ .x/ 1 dx  dx D 2i 1 x  i 0 1 x C i 0   Z 1 Z 1 .x/ .x/ 1 D dx C i .0/  c:p:v: dx C i .0/ c:p:v: 2i 1 x 1 x 1 D 2i .0/ D hı; i 8 2 D.R/ 2i lim

"!0C

H) lim"!0C

" .x 2 C"2 /

D ı in D 0 .R/ H) lim"!0C

" .x 2 C"2 /

D ı in D 0 .R/.

From (2.9.3) and (2.9.4),       1 1 1 1 1 1 1 C lim C c:p:v: D D x 2 x C i0 x  i0 2 "!0C x C i " x  i " x in D 0 .R/ D lim 2 2 "!0C .x C " / i.e. Z

1

c:p:v: 1

.x/ dx D lim x "!0C

Z

1 1

.x 2

x .x/dx C "2 /

8 2 D.R/:

146

Chapter 2 Differentiation of distributions and application of distributional derivatives

3. Setting y D x  h, z D x C h 8 fixed h > 0, using (2.9.5) with a D " > 0, we have a a lim D lim 2 D ı.y/ 2 2 2 a!0C .x  h/ C a a!0C y C a D ı.x  h/ D ıh in D 0 .R/; a a D lim 2 D ı.z/ lim 2 2 2 C C a!0 .x C h/ C a a!0 z C a D ı.x C h/ D ıh Hence,

in D 0 .R/:

  a 1 1 lim lim  2 2 .x C h/2 C a2 h!0C a!0C 2h .x  h/ C a .ıh  ıh / ıh  ıh D lim D  lim D ı 0 2h 2h h!0C h!0C

in D 0 .R/:

In fact, lim

h!0C

1 1 hıh  ıh ; i D lim Œ.h/  .h/ C 2h h!0 2h D  0 .0/ D hı;  0 i D hı 0 ; i

ıh ıh 2h 1 From (2.9.4), xi0 D e ixt  i ı in D 0 .R/.

H) limh!0C

4.

8 2 D.R/

D ı 0 in D 0 .R/. c:p:v:. x1 / C i ı in D 0 .R/ H)

e ixt xi0

We are to show that    1 ixt lim e c:p:v: C lim .e ixt  i ı/ D i 2ı t!1 t!1 x

D e ixt c:p:v:. x1 / C

in D 0 .R/;

he ixt i ı; i D i hı; e ixt i D i .0/ D i hı; i D hi ı; i;

(2.9.9) (2.9.10)

8 2 D.R/, 8 fixed t > 0 H) H)

lim he ixt i ı; i D hi ı; i

t!1

lim e ixt i ı D i ı

t!1

in C

in D 0 .R/:

(2.9.11)

R ixt he ixt c:p:v:. x1 /; i D hc:p:v:. x1 /; e ixt i D lim"!0C jxj" e x.x/ dx. For  2 D.R/ with supp./  ŒA; A; A > 0 and for fixed t > 0,      Z " ixt  Z A ixt 1 e .x/ e .x/ ixt dx C dx e c:p:v: ;  D lim x x x "!0C A "   Z A ixt Z A ixt e .x/ e .x/ D lim dx  dx : x x "!0C " "

Section 2.9 Continuity of differential operator @˛ W D 0 ./ ! D 0 ./

147

But for  2 D.R/ with supp./  ŒA; A; A > 0; .˙x/ D .0/ ˙ x .˙x/ with 2 C 0 .R/,     1 ixt e c:p:v: ; x   Z A Z A ixt e  e ixt ixt ixt dx C Œe .x/ C e .x/dx D lim .0/ x "!0C " ƒ‚ … „" ƒ‚ … „ I1 ."/

I2 ."/

D lim .I1 ."/ C I2 ."//

for fixed t > 0:

"!0C

(2.9.12)

For fixed t > 0, Z

A

I1 ."/ D .0/ "

Z

2i sin xt dx D 2i .0/ x At

D 2i .0/ "t

But

sin y y

sin y dy y

Z

A "

sin xt d.xt / xt

.setting y D xt /:

has a removable discontinuity at y D 0. Hence, for fixed t > 0, Z At Z At sin y sin y dy D dy lim C y y "!0 "t 0 Z At sin y dy: (2.9.13) H) lim I1 ."/ D 2i .0/ C y "!0 0

Now, Z

A

lim I2 ."/ D

"!0C

Œe ixt .x/ C e ixt .x/dx:

(2.9.14)

0

Then, from (2.9.12)–(2.9.14),     Z 1 1 sin y lim e ixt c:p:v: ;  D 2i .0/ dy t!1 x y 0 Z A C lim Œe ixt .x/ C e ixt .x/dx t!1 0

 C 0Di .0/Dhi ı; i 8 2 D.R/; 2 R1 R At RA since lim t!1 0 sinyy dy D 0 sinyy dy D 2 and lim t!1 0 ¹sin xt; cos xt º .˙x/dx D 0 by the Riemann–Lebesgue Theorem (2.11.7i) [32] H) lim t!1 R A ˙ixt .˙x/dx D 0. Hence, 0 e   1 ixt lim e c:p:v: (2.9.15) D i ı in D 0 .R/: t!1 x D 2i .0/

148

Chapter 2 Differentiation of distributions and application of distributional derivatives

Finally, from (2.9.9), (2.9.11) and (2.9.15), the result (2.9.8) follows:    1 e ixt D lim e ixt c:p:v: C lim .e ixt i ı/ t!1 x  i 0 t!1 t!1 x lim

D i ı C i ı D 2i ı

in D 0 .R/:

Convergence of sequences of derivatives of functions of L1loc ./ Proposition 2.9.1. Let .fn / be a sequence of functions fn 2 L1loc ./ 8n 2 N,   Rn . Then, if 1. fn .x/ ! f .x/ a.e. on  in the ordinary sense, and 2. 9M > 0 such that jfn .x/j  M 8n 2 N a.e. on  2 L1loc ./ and jfn .x/j @˛ f in D 0 ./ as n !

(or 9g  0 such that g multi-index ˛, @˛ fn ! distributional sense. In particular, for ˛ D 0,

(2.9.16)

 g.x/ 8n 2 N a.e. on ), 8 1, where derivatives are in the

i.e. fn ! f in D 0 .R/:

@˛ fn D fn and @˛ f D f;

(2.9.17)

R Proof. fn 2 L1loc ./ H) hTfn ; i D hfn ; i D  fn d x 8 2 D./ 8n 2 N. By virtue of conditions (1) and (2), we can apply Lebesgue’s Theorem B.3.2.3 (Appendix B) on the convergence of a sequence of integrals and we have, for  2 D./ with supp./ D K, Z

Z fn d x D



Z fn d x !

K

Z f d x D

f d x

K



H) hfn ; i ! hf; i as n ! 1 8 2 D./ H) fn ! f in D 0 ./ as n ! 1 H) @˛ fn ! @˛ f in D 0 ./, by Theorem 2.9.1. In particular, we have Proposition 2.9.2. Let .fn / be a sequence in L1loc ./ such that fn ! f 2 L1loc ./ uniformly on every bounded subset of . Then @˛ fn ! @˛ f in D 0 ./ as n ! 1 8j˛j 2 N0 . Proof. For  2 D./ with supp./ D K  , Z hfn ; i D

Z fn d x D



Z fn d x !

K

f d x; K

Section 2.10 Delta-convergent sequences of functions in D 0 .Rn /

149

since ˇZ ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ f d x  f d x .f  f /d x D n n ˇ ˇ ˇ ˇ K K K Z jf .x/  fn .x/jdx ! 0  max j.x/j x2K

as n ! 1

K

H) hfn ; i ! hf; i as n ! 1 8 2 D./ H) fn ! f in D 0 ./ as n ! 1 H) @˛ fn ! @˛ f in D 0 ./ as n ! 1 8j˛j 2 N, by Theorem 2.9.1.

2.10

Delta-convergent sequences of functions in D 0 .Rn /

Definition 2.10.1. Let .fj / be a sequence of functions on Rn . Then, if fj ! ı in D 0 .Rn / as j ! 1, ı being the Dirac distribution with concentration at 0, i.e. if Z lim hfj ; i D lim fj .x/.x/d x D .0/ D hı; i 8 2 D.Rn /; (2.10.1) j !1

j !1 Rn

.fj / is called a delta-convergent sequence of functions in D 0 .Rn /. A characterization of delta-convergent sequences of functions in D 0 .Rn / Theorem 2.10.1. Let .fj / be a sequence of functions on Rn satisfying: 1

1. fj .x/  0 for kxkRn D .x12 C    C xn2 / 2  k, k > 0; 2. fj ! 0 uniformly as j ! 1 on every set ¹x W 0 < a  kxk  R 3. kxka fj .x/d x ! C1 as j ! 1 8a > 0.

1 a

< C1ºI

Then fj ! ı in D 0 .Rn / as j ! 1 (i.e. (2.10.1) holds). Remark 2.10.1. The property (1), fj .x/  0 for kxk  k, is not necessary and may be replaced by the following more general condition: 10 . For a suitably chosen k > 0, 9M > 0, independent of j , such that Z jfj .x/jd x  M 8j 2 N:

(2.10.2)

kxkk

Proof of Theorem 2.10.1. Z hfj ; i D fj .x/.x/d x Rn Z Z Z D fj .x/.0/d x C fj .x/Œ.x/  .0/d x C fj .x/.x/d x kxka kxka kxk>a „ ƒ‚ … „ ƒ‚ … „ ƒ‚ … I1 .j /

D I1 .j / C I2 .j / C I3 .j /:

I2 .j /

I3 .j /

(2.10.3)

150

Chapter 2 Differentiation of distributions and application of distributional derivatives

p Estimate for jI2 .j /j: j.x/.0/j D jhr ./; xij  kxk nM with M  max1in @ @ @ @ maxx2Rn j @x .x/j, r ./ D . @x ./; @x ./; : : : ; @x .// n 1 2 i p R H) jI2 .j /j  M na kxka fj .x/d x: Since properties (2) and (3) small a > 0, we can choose R hold for any sufficiently R a  k. Thus, for a  k, kxka fj .x/d x  kxkk fj .x/d x  C 8j 2 N H) p R p jI2 .j /j  M na kxka fj .x/d x  CM na. Hence, choose a 

" " p such that jI2 .j /j  . 3 3CM n

(2.10.4)

Estimate for jI3 .j /j: Now we are to make a final choice of a > 0 satisfying three conditions: 0 < a  k, a  3CM" pn and supp./  Œ a1 ; a1  for sufficiently small a. For such a choice of a; fj ! 0 uniformly as j ! 1 on ¹x W a  kxk  a1 º H)

I3 .j / ! 0 as j ! 1

H)

9j1 such that jI3 .j /j 

" 8j  j1 : 3 (2.10.5)

R Estimate for jI1 .j /  .0/j: Since kxka fj .x/d x ! 1 as j ! 1; I1 .j / ! .0/ as j ! 1 H) 9j2 such that jI1 .j /  .0/j  3" 8j  j2 . Now, choosing j0 D max¹j1 ; j2 º, jhfj ; i  .0/j D jI1 .j / C I2 .j / C I3 .j /  .0/j jI1 .j /  .0/jCjI2 .j /jCjI3 .j /j 

" " " C C D " 8j  j0 : 3 3 3

Hence, limj !1 hfj ; i D .0/ D hı; i 8 2 D.Rn / H) fj ! ı in D 0 .Rn / as j ! 1.  r 2 P Example 2.10.1. 8j 2 N, fj .x/ D "1n e "2 with " D j1 , r 2 D niD1 xi2 . (2.10.6) 8j 2 N, fj .x/  0 for all x 2 Rn H) property (1) holds 8k > 0. a2

For kxk D r  a, fj .x/  "1n e "2 ! 0 as " ! 0C H) fj ! 0 uniformly as j ! 1 for kxk  a H) property (2) holds 8a > 0. Changing variables: xi D "i , 1  i  n, with Jacobian J D "n , 8a > 0, " D j1 , Z

Z fj .x/d x D

kxka

fj ."/"n d  D

kkja

Z D

kkja

2

e  r d  !

Z

Z . kkja 2

1 "2 r 22 n " /" d  e "n

e  r d  D 1 Rn

Section 2.10 Delta-convergent sequences of functions in D 0 .Rn /

as j ! 1 (r 2 D 12 C    C n2 ) since Z

2

e  r d  D

Rn

Z

1

D Z

2

e

 12

1

Z

1

 1 „ ƒ‚

e  1 d 1

1 1

D

Z

d1 p 

Z

1

1 1

e

1 e

2

 k2

p

d k D

, 1  k  n,

2

e . 1 CC n / d 1 : : : d n

1

…

n times 1  22

e

Z

R1

151

 22

1

p p p    D p  p    p D 1:   

Z

1

d 2   

d2 p  

Z

2

e  n d n

1 1

k .setting k D p , 1  k  n/ 

2 dn e  n p  1

H) property (3) holds 8a > 0. Hence, by Theorem 2.10.1, fj ! ı in D 0 .Rn / as j ! 1. r 2 P Example 2.10.2. 8j 2 N, fj .x/ D n p1 n e .2"2 / with " D j1 , r 2 D niD1 xi2 . " . 2/ Following the steps of the proof of Example 2.10.1, we can show that all three properties (1)–(3) hold: jfj .x/j  0 8x 2 Rn , 8j 2 N; a2

jfj .x/j  8a > 0, Z

p1 e .2"2 / 2/n

"n .

a2

for kxk  a and

Z fj .x/d x D

kxka

kkja

p1 e .2"2 / 2/n

"n .

r 2 1 e 2 d ! p . 2/n

Z Rn

! 0 as " ! 0C ;

r 2 1 e 2 d D 1 p . 2/n

as j ! 1, since Z e

r 2 2

Z

1

d D

Rn

e 1 Z 1

D

2 1 2

Z

1

d 1

e

2 2 2

Z

1

2

e  1

Z p 2d 1

1

1

d 2   

e 1

1

2

e  2

p

2 n 2

Z

d n

1

2d 2   

1

p p p p D . 2/. 2/    . 2/ D . 2/n :

2p e  n 2d n

1

Hence, by Theorem 2.10.1, fj ! ı in D 0 .Rn / as j ! 1. Example 2.10.3. 8j 2 N, ´ fj .x/ D

n "n Sn

for kxk < " D

0

for kxk  " D

1 j 1 j;

(2.10.7)

152

Chapter 2 Differentiation of distributions and application of distributional derivatives n

where Sn D surface area of an n-dimensional unit sphere D

2. 2 / . n 2/

with S1 D 2,

S3 D 4; S4 D 2 2 etc.;     1 1 1p 3 D  D .1/ D 1;  : 2 2 2 2

S2 D 2;

  p 1  D ; 2

(2.10.8)

fj .x/  0 8x 2 Rn , 8j 2 N H) property (1) holds for all k > 0; fj .x/ D 0 for kxk  a with a > j1 D " H) fj ! 0 uniformly in every set ² ³ 1 1 x W 0 < < a  kxk  < 1 as j ! 1 j a H) property (2) holds; Z fj .x/d x D kxka

n "n Sn

Z kxk j1

for a> j1

dx D

n V" D 1 "n Sn

for all sufficiently large j 2 N, where V" is the volume of the n-dimensional ball n B.0I "/ with radius " D j1 , since V" D " nSn . R Hence, kxka fj .x/d x ! 1 as j ! 1 and property (3) holds. Consequently, by Theorem 2.10.1, fj ! ı in D 0 .Rn / as j ! 1. Alternative characterization of a delta-convergent sequence of functions Let .fj / be a sequence of functions on R satisfying the following two properties: 10 . For any M > 0 with jaj  M , jbj  M , 9K D K.M / > 0, independent of Rb a; b; j , such that j a fj .x/dxj  K 8j 2 N; 20 . 8 fixed a ¤ 0, b ¤ 0, ´ Z b 1 lim fj .x/dx D j !1 a 0

for a < 0 < b for a < b < 0 and 0 < a < b:

(2.10.9)

Then .fj / is a delta-convergent sequence in D 0 .R/, i.e. fj ! ı in D 0 .R/ as j ! 1 (see [1] for more details). Example 2.10.4. Let f t .x/ D

x 2 1 e 4t .t > 0/: p 2.  t /

(2.10.10)

Rb R 1 x2 4t dx D 1 In fact, f t .x/ > 0 8x 2 R, 8t > 0 and a f t .x/dx  p1 1 e R 1  2 R1 p p 2 t p 2 (since 1 e d  D ; setting x D 2 t, dx D 2 td , t > 0, 2p1. t/ 1 e  p R1 p 2 2 td  D p1 1 e  d  D p D 1/. H) property (10 ) holds with K D 1. 

Section 2.10 Delta-convergent sequences of functions in D 0 .Rn /

153

For a < 0 < b, Z

b

lim

t!0C

a

1 f t .x/dx D lim p C t!0 2  t 1 p 2 t

D lim

t!0C

1 Dp 

Z

1

e

Z

b

a

Z

b p t b p t

 2

1

x 2

e .4t / dx 2 p e  2 td 

p .setting x D 2 t/

p 1  d D p D 1 

H) property (20 ) holds for a < 0 < b. For 0 < a < b, x > a H) xa > 1, 1 p 2 t

Z

b

e

x 2 4t

a

Z 1 x 2 x 2t 1 dx e dx  p e 4t 2t a 2 t a a p x 2 2t t a2 1 D p Œe 4t a D p e 4t ! 0 as t ! 0C a  a2  t

1 dx  p 2 t

H) for 0 < a < b, lim t!0

p1 2 t

Z

Rb

1

x 2 4t

dx D 0. R b x 2 Similarly, for a < b < 0, lim t!0 p1 e 4t dx D 0. a 2 t 0 Thus, property (2 ) holds for a < 0 < b, 0 < a < b, a < b < 0. a

e

x 2 4t

Hence, .f t / is a delta-convergent sequence in D 0 .R/, and

2

x p1 e . 4t / 2 t

! ı in

D 0 .R/ as t ! 0C . But there are independent proofs of delta-convergent sequences in D 0 .R/. Example 2.10.5. Consider the sequence f" D Then f" ! ı proof).

in D 0 .R/ as "

!

0C

" .x 2 C"2 /

with " D j1 .

(2.10.11)

(see (2.9.5) in Example 2.9.2 for an independent

Example 2.10.6. Let fn D 1  sinxnx . (2.10.12) Then fn ! ı in D 0 .R/ as n ! 1 (see Example 1.8.2 for an independent proof). ixt

e 1 Example 2.10.7. Let f t .x/ D i2 for (t > 0). (2.10.13) .xi0/ Then, f t ! ı in D 0 .R/ as t ! 1 (see the proof of (2.9.8) in Example 2.9.2).

R n ixy 1 Example 2.10.8. Let Un .x/ D 2 dy 8n 2 N. n e Then, Un ! ı in D 0 .R/ as n ! 1.

(2.10.14)

154

Chapter 2 Differentiation of distributions and application of distributional derivatives

In fact, ˇ Z n sin nx e ixy ˇˇyDn e i nx  e i nx ixy D2 e dy D D ˇ ix yDn ix x n       Z n 1 2 sin nx 1 sin nx ixy ;  D lim ; H) lim e dy;  D lim n!1 2 n n!1 2 n!1  x x D hı; i H)

1 2

2.11

Rn

n e

ixy dy

8 2 D.R/

.see Example 2.10.6/

! ı in D 0 .R/ as n ! 1.

Term-by-term differentiation of series of distributions

P1 Theorem 2.11.1. Let in D 0 ./ P1 nD1 Tn be a 0convergent series of distributions P1 with its sum T D nD1 Tn ; T 2 D ./. Then the series nD1 Tn can be differentiated indefinitely term by term under the summation sign, i.e. 8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n /, @˛ T D

1 X

@ ˛ Tn

with @˛ D

nD1

@j˛j : @x1˛1 : : : @xn˛n

(2.11.1)

P P1 0 0 Proof.PSet SN D N nD1 Tn 2 D ./. Since nD1 Tn is convergent in D ./ with 1 0 0 ˛ T D nD1 Tn ; T 2 D ./, SN ! T in D ./ as N ! 1 H) @ SN ! @˛ T in P P1 ˛ ˛ D 0 ./ as N ! 1 by Theorem 2.9.1. H) @˛ SN D N nD1 @ Tn ! nD1 @ Tn D @˛ T in D 0 ./ as N ! 1. Remark 2.11.1. For a convergent series of functions (not treated as distributions), term-by-term differentiation (in the usual pointwise sense) of the series under the summation sign cannot be done unless some additional conditions are fulfilled. Hence, this is a nice property of distributions, which is extremely useful in applications. Convergence of trigonometric series in D 0 .R/ P i2kx be a trigonometric series in complex form. Theorem 2.11.2. Let 1 kD1 Ck e If the complex coefficients Ck satisfy the condition that 9 a constant M > 0 and an integer p 2 N0 such that jCk j  M jkjp 8k ¤ 0; (2.11.2) then the following hold: P i2kx converges in D 0 .R/ with its sum denoted by I. The series 1 kD1 Ck e T D

1 X kD1

Ck e i2kx I

(2.11.3)

155

Section 2.11 Term-by-term differentiation of series of distributions 1 1 X X dT d D ŒCk e i2kx  D .i 2k/Ck e i2kx I dx dx

II.

kD1

1 T D T;

III.

(2.11.4)

kD1

i.e. h1 T; i D hT; i

8 2 D.R/:

(2.11.5)

Proof. I. For fixed p 2PN0 satisfying (2.11.2), consider an auxiliary with the term PN series C Ck i2kx . Let S i2kx k e D C0 omitted: 1 N kD1 .i2k/pC2 kDN .i2k/pC2 e k¤0

k¤0

denote the sequence of the partial sums of the first 2N terms of this auxiliary series. Then, 8k ¤ 0, ˇ ˇ   X 1 ˇ ˇ M 1 C jCk j k ˇD ˇ  < C1:  2 and ˇ .i 2k/pC2 ˇ pC2 pC2 pC2 .2/ jkj .2/ jkj k2 k¤0

Hence, by Weierstrass’s M-test, the auxiliary series converges uniformly (and absolutely). Consequently, .SN / converges uniformly to a continuous function f on R H) .SN / converges to Tf D f in D 0 .R/ as N ! 1 H) .pC2/ d pC2 d pC2 S ! dx in D 0 .R/ as N ! 1 by Theorem 2.9.1 pC2 Tf D Tf dx pC2 N with ˛ D p C 2,  pC2 X  N d Ck i2kx H) lim e N !1 dx pC2 .i 2k/pC2 kDN k¤0

D lim

 X N

N !1

H)

P1

kD1 Ck e

Ck e

i2kx



kDN k¤0 i2kx

1 X

D

.pC2/

Ck e i2kx D Tf

in D 0 .R/

kD1 k¤0 .pC2/

D C0 C Tf

D T in D 0 .R/ with

.pC2/

.pC2/

; i D hC0 ; i C hTf ; i hT; i D hC0 C Tf Z C0 dx C .1/pC2 hTf ;  .pC2/ i D R Z D ŒC0  C .1/pC2 f  .pC2/ dx 8 2 D.R/: R

PN

II. Set TN D kDN Ck e i2kx , with TN ! T D as N ! 1. Hence,

P1

kD1 Ck e

i2kx

in D 0 .R/

N 1 X X d TN dT D D .i 2k/Ck e i2kx ! .i 2k/Ck e i2kx dx dx kDN

in D 0 .R/ as N ! 1 by Theorem 2.9.1.

kD1

156

Chapter 2 Differentiation of distributions and application of distributional derivatives

III. T is a periodic distribution with period 1, i.e. h1 T; i D hT; 1 i

.see (1.10.45b)/

D hT; .x C 1/i D hT; i

8 2 D.R/:

In fact, hT; .x C 1/i D lim hTN ; .x C 1/i N !1

N X

D lim

N !1

N !1

e i2kx .x C 1/dx

R

kDN N X

D lim

Z Ck Z Ck

e i2k ./d 

R

kDN

D lim hTN ; i D hT; i N !1

H)

8 2 D.R/

1 T D T inD 0 .R/:

(2.11.6)

Example 2.11.1. P i2kx converges in D 0 .R/. 1. Show that series 1 kD1 e P P1 i2kx . Show that .1e i2x /S D 0 and S D C 2. Let S D 1 kD1 e kD1 ık , C being a constant. Proof. P1 i2kx converges in D 0 .R/ by Theorem 2.11.2, since C D 1  jkjp 1. k kD1 e 8 fixed p 2 N0 , 8k ¤ 0. P i2kx in D 0 .R/. Then 2. Let S D 1 kD1 e

.1  e

i2x

/S D .1  e

D lim

N !1

i2x

/ lim

N !1

 X N kDN

e

N X

e i2kx

kDN

i2kx



N C1 X

e

i2kx

kDN C1

D lim Œe i2N x  e i2.N C1/x  D 0 N !1



in D 0 .R/;

157

Section 2.11 Term-by-term differentiation of series of distributions

since, 8 2 D.R/, he i2N x  e i2.N C1/x ; i Z Z i2N x D e .x/dx  e i2.N C1/x .x/dx R R Z Z 1 1 i2N x 0 e  .x/dx  e i2.N C1/x  0 .x/dx D i 2N R i 2.N C 1/ R H) jhe i2N x  e i2.N C1/x ; ij  8 2 D.R/.

1 2N

R

R j

0 .x/jdx

! 0 as N ! 1

Hence, .1  e i2x /S D 0 in D 0 .R/ with .1  e i2x / D 0 for x D 0; ˙1; ˙2; : : : ; ˙N; : : : , and, 8N 2 N, 8x 2 N; N Œ, we can write e i2x  1 D ˛.x/Œ.x  .N C 1//.x  .N C 2//     x  .x  1/    .x  .N  1// with ˛.x/ ¤ 0. Then ˛.x/.x.N C1//    .x.N 1//S D 0 in D 0 .N; N Œ/ with ˛.x/ ¤ 0 on N; N Œ (see Chapter 5 for more details on restrictions of distributions) H) .x  .N C 1//    .x  .N  1//S D 0 in D 0 .N; N Œ/ (see Sections 1.6 and 2.6) P 1 0 H) S D N kDN C1 Ck ık in D .N; N Œ/ (see .4/ in Example 1.6.1). Hence,  hS; i D

N 1 X kDN C1

 Ck ık ;  D

N 1 X

Ck .k/

8 2 D.N; N Œ/; 8N 2 N

kDN C1

P P1 P1 0 i2kx D H) S D 1 kD1 Ck ık in D .R/. Thus, S D kD1 e kD1 Ck ık in D 0 .R/. But h1 S; i D hS; 1 i D hS; .x C 1/i D hS; i

8 2 D.R/;

(2.11.7)

158

Chapter 2 Differentiation of distributions and application of distributional derivatives

since hS; .x C 1/i D lim

N !1

D lim

N !1

D lim

N !1

 e i2kx ; .x C 1/

 X N kDN

Z N X kDN

e i2kx .x C 1/dx

R

Z N X kDN

e i2k ./d 

.x D   1; e i2k D 1/

R

D lim hSN ; i D hS; i

8 2 D.R/ .SN D

N !1

N X

e i2k /:

kDN

Hence, for fixed k 2 Z, for  2 D.k  12 ; k C 12 Œ/ with .k/ D 1, using (2.11.7), we have h1 S; i D hS; 1 i D

 X 1

 Ck ık ; .x C 1/ D Ck1 .k/ D Ck1

kD1

D hS; i D

 X 1

 Ck ık ;  D Ck .k/ D Ck

kD1

H) Ck D Ck1 D C 8k 2 Z P H) S D C 1 kD1 ık . Fourier series of periodic functions and their convergence in the distributional sense in D 0 .R/ For periodic functions and distributions with period T > 0, we refer to Section 1.10, Chapter 1. L2.T /: Let L2.T / be a (complex) Hilbert space of (equivalence classes of) complexvalued functions f W R ! C which are periodic on R with period T > 0 and equipped with inner product h  ;  iT D T1 h  ;  iL2 .a;aCT Œ/ defined by 

1 1 hf; giT D hf; giL2 .a;aCT Œ/ D T T (see also Table B.3, Appendix B);

Z

aCT

f .x/g.x/dx a

(2.11.7a)

159

Section 2.11 Term-by-term differentiation of series of distributions 

and norm: kf

k2T

1 1 D hf; f iT D hf; f iL2 .a;aCT Œ/ D T T

Z

aCT

jf .x/j2 dx;

a

where a; a C T Œ is called a period interval on R 8a 2 R (see (1.10.45a)). Orthogonality in L2 .a; a C T Œ/ ” orthogonality in L2 .T /, since hf; giL2 .a;aCT Œ/ D 0 ” hf; giT D 0.



2 Orthonormal Systems in L2 .T /: .e i n!x /1 nD1 is an orthonormal system in L .T /, 2 i n!x i n!.aCT / i n!a since e is periodic with period T and ! D T , i.e. e De 8a 2 R and Z 1 aCT im!x i n!x im!x i n!x he ;e iT D e e dx T a Z 1 aCT i.mn/!x D e dx D ımn (2.11.7b) T a ´ 1 for m D n m; n 2 Z: D 0 for m ¤ n 

2 .e i n!x /1 nD1 is a complete (or total in French) orthonormal system in L .T / (see also Section A.12, Appendix A), since Z 1 aCT hf; e i n!x iT D f .x/e i n!x dx D 0 8n 2 Z H) f D 0 in L2 .T /: T a





L2 .T / ,! D 0 .R/, i.e. f 2 L2 .T / 2

fn ! f in L .T /

H) H)

Tf D f 2 D 0 .R/

and

0

(2.11.7c)

cn .f /e i n!x

(2.11.7d)

fn ! f in D .R/;

since hfn ; i ! hf; i 8 2 D.R/ as n ! 1. Definition 2.11.1. A trigonometric series 1 X

hf; e

i n!x

iT e

i n!x

D

nD1

1 X nD1

is called a Fourier series in complex form of the periodic function f 2 L2 .T / with 2 respect to the orthonormal system .e i n!x /1 nD1 in L .T /, and Z 1 aCT i n!x iT D f .x/e i n!x dx 8n 2 Z (2.11.7e) cn .f / D hf; e T a is called the Fourier coefficient of f 2 L2 .T /.

160

Chapter 2 Differentiation of distributions and application of distributional derivatives

For a D 0 (resp. a D ), T DP2, ! D 1, a; a C T Œ D 0; 2Œ (resp. ; Œ), i nx with Fourier coefficients we get the standard Fourier series 1 nD1 cn .f /e Z 2 Z  1 1 i nx f .x/e dx .resp. f .x/e i nx dx/ 8n 2 Z: cn .f / D 2 0 2  (2.11.7f) Important properties of Fourier coefficients Z 1 X 1 aCT 1. 2 2 jck .f /j  kf kT D jf .x/j2 dx T a

8f 2 L2 .T /

kD1

(called Bessel’s inequality), since, 8n 2 N, 2  n X   ik!x  0 ck .f /e  f  T

kDn

  n n X X ik!x ik!x D f  ck .f /e ;f  ck .f /e kDn

D    D kf k2T 

kDn n X

T

jck .f /j2

(2.11.7g)

kDn

(using the orthonormality of .e ik!x / and properties of the complex inner product). , 2. It is well known that for the complete orthonormal system .e ik!x /1 kD1 Bessel’s inequality becomes a strict equality, called Parseval’s relation, which holds 8f 2 L2 .T /, i.e. 1 X

jck .f /j2 D kf k2T

8f 2 L2 .T /;

(2.11.7h)

kD1

and the orthonormal system .e ik!x /1 is called closed in L2 .T /. kD1 3. ck .f /P ! 0 as jkj ! 1, since the general term jck .f /j2 of the convergent 2 series 1 kD1 jck .f /j must tend to 0 as jkj ! 1. 4. For periodic function (resp. L1loc .R/) f 2 L1 .R/ on R with period T , Fourier coefficients Z 1 aCT f .x/e ik!x dx (2.11.7i) ck .f / D T a are well defined for k 2 Z and ck .f / ! 0 as jkj ! 1 by the Riemann– Lebesgue Theorem.

Section 2.11 Term-by-term differentiation of series of distributions

161

5. Fourier coefficients ck .f / remain unaltered if the values f .x/ are changed on a set of points with measure 0. Convergence in L2 .T / Theorem 2.11.3. The Fourier series of f 2 L2 .T / converges to f in L2 .T / in the following sense:  2 n X   ik!x  lim  c .f /e f  k   n!1

1 n!1 T

aCT

D lim i.e. f D

P1

kD1 ck .f

T

kDn

Z a

ˇ ˇ2 n X ˇ ˇ ik!x ˇ ˇf .x/  ck .f /e ˇ ˇ dx D 0 kDn

/e ik!x in L2 .T /.

Proof. Since .e ik!x /1 is a closed orthonormal system in L2 .T /, by (2.11.7h), kD1 1 X

jck .f /j2 D kf k2T :

kD1

Set Sn D

Pn

kDn ck .f

/e ik!x . Then, from (2.11.7g), n X

kf  Sn k2T D kf k2T 

jck .f /j2

8n 2 N:

kDn

Hence, lim kf  Sn k2T D lim

n!1

n!1

  n X jck .f /j2 kf k2T 

D kf k2T 

kDn 1 X

jck .f /j2 D 0

kD1

H) f D limn!1 Sn D

P1

kD1 ck .f

/e ik!x in L2 .T /.

Remark 2.11.1A. For periodic f 2 L1 .R/ with period T > 0, Fourier series (2.11.7d) of f does not converge in general. In fact, Kolmogorov gave an example of a periodic f0 2 L1 .R/ whose Fourier series is divergent everywhere on R [30] (see also Property (4) of Fourier coefficients above). The convergence of Fourier series of f 2 L2 .T / in Theorem 2.11.3 is in the mean square sense on any period interval a; a C T Πon R, i.e. the convergence in L2 .T / is neither uniform nor pointwise in general. But we need uniform convergence of Fourier series (as a sufficient

162

Chapter 2 Differentiation of distributions and application of distributional derivatives

P ik!x is a continuous function. Moreover, condition) so that its sum 1 kD1 ck .f /e for term-by-term differentiation (resp. integration) of Fourier series, we require the uniform convergence of the resultant series (obtained by term-by-term differentiation (resp. integration)) so that its sum is continuous. Now, we will state some sufficient conditions for uniform convergence. For this, we define  C 0 .R/ D ¹f W f is a periodic complex-valued function on R with period T T and continuous on Rº  L2 .T /;  C m .R/ D ¹f W f and its derivatives f .1/ ; : : : ; f .m/ 2 C 0 .R/º 8m 2 N. T T Theorem 2.11.4. P ik!x of f converges uniI. For f 2 CT2 .R/, the Fourier series 1 kD1 ck .f /e formly and absolutely to f . P ik!x of f II. For f 2 CTm .R/ with m > 2, the Fourier series 1 kD1 ck .f /e can be differentiated l times term by term, and the resultant series converges uniformly and absolutely such that, 8x 2 R, f .l/ .x/ D

1 X kD1 k¤0

D

1 X kD1 k¤0

ck .f /

d l ik!x .e / dx l

ck .f /.i k!/l e ik!x D

1 X

ck .f .l/ /e ik!x

kD1 k¤0

with ck .f /  .i k!/l D ck .f .l/ / 8k 2 Z, i.e. the resultant series is the Fourier series of f .l/ 8l D 1; : : : ; m  2. Proof. I. Let f 2 CT2 .R/  L2 .T /. Then, integrating by parts and using the periodicity of f .x/e ik!x , we have, 8k ¤ 0, Z 1 aCT f .x/e ik!x dx ck .f / D T a Z ˇaCT 1 aCT 0 1 1  e ik!x f .x/ˇa  C f .x/e ik!x dx D T  .i k!/ i k! T a 1 D ck .f 0 /: i k! Again integrating by parts and using the periodicity of f 0 .x/e ik!x , we get, 8k ¤ 0,   Z 1 2 1 aCT 00 1 f .x/e ik!x dx D ck .f 00 /: ck .f / D i k! T a .i k!/2

163

Section 2.11 Term-by-term differentiation of series of distributions

For k D 0, c0 .f / D H)

1 T

jc0 .f /j 

R aCT a

1 T

Z

f .x/dx

aCT

jf .x/jdx  maxx2Œa;aCT  jf .x/j D M0 :

a

Then, 8k ¤ 0, 1 1 jck .f /j  2 2 k ! T with M1 D jf 00 .x/j

1 !2

Z

aCT

jf 00 .x/j  je ik!x jdx 

a

maxx2Œa;aCT  jf 00 .x/j  1 X

H)

jck .f /e

ik!x

kD1

1 T

R aCT a

M1 k2

1 !2

dx D

maxx2Œa;aCT 

1 X M1 j  M0 C k2 kD1 k¤0

 1 X  M2 1 C kD1 k¤0

M2 D max¹M0 ; M1 º, the majoring series M2 .1 C

 1 ; k2

P1

1 kD1 k 2 / k¤0

being a con-

vergent series of positive numbers. Weierstrass’s M-test with this P Hence, by ik!x majoring series, Fourier series 1 c .f /e converges uniformly and kD1 k P ik!x c absolutely on R. Hence, its sum g.x/ D 1 kD1 k .f /e P is a continuik!x ous function on R. Moreover, the uniform convergence of 1 kD1 ck e 2 2 to g implies its mean square convergence in L .T /, i.e. g 2 L .T / in the sense of Theorem 2.11.3. But f 2 CT2 .R/  L2 .T / H) by Theorem 2.11.3, P f D limn!1 nkDn ck .f /e ik!x in the mean square sense in L2 .T /. Hence, by virtue of the uniqueness of the limit, f D g. II. For f 2 CTm .R/  L2 .T /, we can repeatedly integrate by parts l times with l  m and using the periodicity of f .x/e ik!x and its derivatives of order  m 1 .l/ / for l  m. Then, for l D m and to get, for k ¤ 0, ck .f / D .ik!/ l ck .f k ¤ 0, jck .f /j 

1 jc .f .m/ /j jik!jm k



M jkjm j!jm

with jck .f .m/ /j  M D

maxaxaCT jf .m/ .x/j. But for 1  l  m  2, jck .f .l/ /j D j.i k!/l ck .f /j D jkjl j!jl jck .f /j  fD since M

M j!jml

f f M jkjl j!jl M M   ; jkjm j!jm k2 jkjml

and m  l  2 H) jkjml  k 2 H)

1 jkjml



1 . k2

164

Chapter 2 Differentiation of distributions and application of distributional derivatives

8l with 1  l  m  2, 1 X

jck .f /.i k!/l e ik!x j 

X

jck .f .l/ /j 

kD1 k¤0

1 X f M : k2

kD1 k¤0

P e M Again, by Weierstrass’s M-test with convergent majoring series 1 kD1 k 2 , the k¤0 P l e ik!x converges uniformly and absolutely with c .f /.i k!l/ series 1 k kD1 k¤0

sum gl .x/ as a continuous function on R for 1  l  m  2. Now, we are dl to prove that gl .x/ D dx l f .x/, 1  l  m  2. For this it is sufficient to prove this for l D 1, since the result can be similarly proved for other values of l  m  2, and we need the following lemma. n 1 n Lemma 2.11.1 ([33, p. 229]). Let . df / with df continuous on a; bŒ 8n condx nD1 dx verge uniformly to g on a; bŒ. If 9x0 2 a; bŒ such that limn!1 fn .x0 / D f .x0 /, d d then dx Œlimn!1 fn  D limn!1 Œ dx fn  D g on a; bŒ. Pn Pn ik!x 8n 2 N. Then, dSn D Proof. Set Sn D kDn Œck .f / kDn ck .f /e dx

k¤0

n D Sn0 ! g1 uniformly on R i k!e ik!x 8n 2 N with Sn ! f uniformly and dS dx 1 1 0 as n ! 1. Hence, the sequences .Sn /nD1 and .Sn /nD1 , which converge uniformly, satisfy all the hypotheses of Lemma 2.11.1 and we get g1 .x/ D limn!1 Sn0 .x/ D d d .limn!1 Sn / D dx f .x/ for x 2 R. dx

Remark 2.11.2. For weaker sufficient conditions for pointwise (resp. uniform) convergence of Fourier series, see, for example, [32], [33]. Fourier series in sines and cosines .1I cos n!xI sin n!x/1 nD1 is an orthogonal system but not an orthonormal system 2 p n!x I sin pn!x /1 , since in L .T /, the corresponding orthonormal system being .1I cos 1=2 1=2 nD1 p 8n 2 N, k cos n!xkT D k sin n!xkT D 1=2. 



2 .1I cos n!xI sin n!x/1 nD1 is a complete orthogonal system in L .T /, i.e.

hf; 1iT D 0;

hf; cos n!xiT D 0;

hf; sin n!xiT D 0

8n 2 N

(2.11.7j)

H) f D 0 in L2 .T /. Definition 2.11.2. The trigonometric series 1

X a0 C .an cos n!x C bn sin n!x/ 2 nD1

(2.11.7k)

Section 2.11 Term-by-term differentiation of series of distributions

165

is called the Fourier series in sines and cosines of periodic f 2 L2 .T / with respect 2 to the orthogonal system .1I cos n!xI sin n!x/1 nD1 in L .T /, with the Fourier coef1 ficients .a0 =2I an I bn /nD1 of f defined by: Z 2 aCT  a0 .f / D 2hf; 1iT D f .x/dxI (2.11.7l) T a 

8n 2 N, 2 an .f / D 2hf; cos n!xiT D T bn .f / D 2hf; sin n!xiT D

2 T

Bessel’s equality (or Parseval’s relation) isfy Bessel’s equality:

Z

aCT

f .x/ cos n!xdxI

(2.11.7m)

a Z aCT

f .x/ sin n!xdx: a

8f 2 L2 .T /, Fourier coefficients sat-

Z 1 X 1 aCT jak j2 C jbk j2 ja0 j2 2 C D kf kT D jf .x/j2 dx: 4 2 T a kD1

2 Hence, .1I cos n!xI sin n!x/1 nD1 is a closed orthogonal system in L .T /.

Convergence of Fourier series in L2 .T / Theorem 2.11.5. Fourier series (2.11.7k) of f 2 L2 .T / converges to f in L2 .T / in the mean square sense:  2  n X   a0  lim f  C .ak cos k!x C bk sin k!x/   D 0: n!1 2 T kD1

Proof. The proof is exactly similar to that of Theorem 2.11.3, since .1I cos n!xI 2 sin n!x/1 nD1 is a closed orthogonal system in L .T /. Uniform convergence of Fourier series Theorem 2.11.6. I. For f 2 CT2 .R/, Fourier series (2.11.7k) converges uniformly and absolutely on R. II. For f 2 CTm .R/ with m > 2, Fourier series (2.11.7k) can be differentiated l times term by term and the resultant series converges uniformly and absolutely,

166

Chapter 2 Differentiation of distributions and application of distributional derivatives

i.e. 8x 2 R, f .l/ .x/ D

 1  X dl dl .cos k!x/ C b .f /  .sin k!x/ ak .f /  k dx l dx l

kD1

D

1 X

.ak .f .l/ /  cos k!x C bk .f .l/ /  sin k!x/

kD1

for l D 1; : : : ; m  2. Remark 2.11.3. For weaker sufficient conditions for pointwise (resp. uniformly) convergence of (2.11.7k), see, for example, [32], [33]. Particular Cases For two important particular cases, a D  (resp. a D 0), T D 2, ! D 2 T D 1, a; a C T Œ D ; Œ (resp. 0; 2Œ/, the standard forms of Fourier series are obtained: R aC2 P1 i nx with c .f / D 1  f .x/e i nx dx, a D 0 (resp. a D n nD1 cn .f /e 2 a ) 8n 2 Z; R aC2 P1 a0 .f / a0 1  D 2 f .x/dx, nD1 .an cos nx C bn sin nx/ with a 2 C 2 an .f / D bn .f / D

1  1 

Z

aC2

f .x/ cos nxdx; Z

a aC2

f .x/ sin nxdx

8n 2 N; a D 0 .resp. a D /

a

and instead of L2 .2/, we may sometimes use the space L2 .; Œ/ (resp. L2 .0; 2Œ/) equipped with the usual inner product: Z  f .x/g.x/dx; hf; giL2 .;Œ/ D Z hf; giL2 .0;2Œ/ D

 2

f .x/g.x/dx: 0

In fact, f 2 L2 .a; a C T Œ/ can be given periodic extension to R with period T D 2 such that f 2 L2 .2/ equipped with the inner product h  ;  iT in (2.11.7a). (2.11.7n) Remark 2.11.4. 

For term-by-term integration of Fourier series, see the following Example 2.11.2. For Fourier series of periodic distributions, see Section 6.7, Chapter 6.

Section 2.11 Term-by-term differentiation of series of distributions 

P1

167

sin kx , k

which is the Fourier series of the sawtooth function P1 in Example 2.11.2, converges at the points of continuity of f , but kD1 cos kx D P1 d sin kx . k / diverges everywhere in the usual classical sense, whereas this kD1P dx 0 series 1 kD1 cos kx converges in the distributional sense in D .R/ (see Example 2.11.2), the sum being a distribution on R. kD1

Examples of Fourier series and their differentiation in D 0 .R/ See also (6.7.18)–(6.7.44). Example 2.11.2. Let f .x/ D .x/ for 0 < x < 2 be a periodic function (the 2 sawtooth function) with period 2. Prove that 1 X

1:

kD1

1 X 1 cos kx D  C  ı2k in D 0 .R/ with ı2k D ı.x  2k/I 2 kD1

(2.11.8) 1 X

2:

e ikx D 2

kD1

3:

1 X

l ikx

.i k/ e

kD1 .l/

1 X

ı2k in D 0 .R/I

(2.11.9)

kD1

D 2

1 X

.l/

ı2k in D 0 .R/;

(2.11.10)

kD1 l

d 0 where ı2k D dx l ı2k is the lth-order derivative of Dirac distribution ı2k 2 D .R/ with mass/charge/force etc. concentrated at points x D 2k, k 2 Z.

Proof. 1. f is a periodic function with period T D 2 on R and f 2 L2 .0; 2Œ/. Then, by virtue of (2.11.7l), f 2 L2 .2/ with T D 2. by Theorem 2.11.5, P Hence, sin kx f .x/ has Fourier series representation f .x/ D 1 kD1 k , since ˇ Z 1 1 2 ˇˇ2 1 2   x dx D .x  x /ˇ D 0; a0 D  0 2 2 2 0 Z 2  x 1 ak D cos kxdx  0 2 ˇ Z 2 1 sin kx ˇˇ2 1 D .  x/ C sin kxdx D 0 8k 2 N; 2 k ˇ0 2k 0 Z 1 2   x sin kxdx bk D  0 2 ˇ Z 2 1 cos kx ˇˇ2 1 1 D  .  x/ 8k 2 N  cos kxdx D ˇ 2 k 2k 0 k 0

168

Chapter 2 Differentiation of distributions and application of distributional derivatives

such that Sn D H)

Pn

kD1

sin kx k

! f in L2 .2/ ,! D 0 .R/ as n ! 1

Sn ! f in D 0 .R/ as n ! 1 .see (2.13.1) later/:

(2.11.11)

P sin kx Consider the auxiliary series 1 kD1  k 3 , which is obtained by formally inP1 1 sin kx tegrating each term twice. Since j sinkkx 3 j  k 2 8k 2 N, kD1  k 3 converges uniformly and absolutely to a continuous function F .x/ on R by Weierstrass’s M-test and FP .x C 2/ D F .x/, i.e. F is periodic on R with period n sin kx 2. Hence, Fn D ! F uniformly on every compact K as kD1  k 3 0 n ! 1 H) Fn ! F in D .R/ as n ! 1 by Proposition 2.9.2, since Pn sin kx d 2 Pn sin kx D Sn ! Fn 2 L1loc .R/, F 2 L1loc .R/ H) dx 2 kD1  k 3 D kD1 k d 2F dx 2

D F 00 in D 0 .R/ as n ! 1 by Theorem 2.9.1. By virtue of (2.11.11), P sin kx d P1 sin kx D f in D 0 .R/ H) dx D f D F 00 2 D 0 .R/. Hence, 1 kD1 k kD1 k P1 df 0 kD1 cos kx D dx in D .R/ by Theorem 2.11.1 on term-by-term differentia2 D 0 .R/ is the distributional derivative tion of series of distributions, where df dx of f on R. Since f is piecewise continuous on R with points of discontinuity at x D 2k 8k D 0; ˙1; ˙2; : : : , i.e. 8k 2 Z, where f has finite jump J2k D Œf .x C /  f .x  /xD2k D 2  . 2 / D  8k 2 Z and d x the usual derivatives Œ df .x/ D dx . 2 / D 1 2 , the distributional derivative dx df 0 .R/ is given by: 2 D dx   1 X df df D .x/ C J2k ı2k dx dx

(see Chapter 3, Theorem 3.1.1)

kD1

1 X 1 ı2k D C 2

in D 0 .R/:

kD1

P P1 P1 1 0 Hence, 1 kD1 cos kx D  2 C  kD1 ı2k inD .R/. (The series kD1 cos kx does not converge in the usual classical sense, but it converges in the distributional sense). 2. e ˙ikx D cos kx ˙ i sin kx H)

N X

cos kx D

kD1

N 1 X Cikx .e C e ikx / 2 kD1

D

N N 1 X ikx 1 X ikx 1 e D e  2 2 2 kDN k¤0

kDN

169

Section 2.11 Term-by-term differentiation of series of distributions

 X  1 N 1 1 X ikx ikx e D lim e N !1 2 2

H)

kD1

1 X

D

kDN

cos kx C

kD1 1 X

H)

1 D 2

1 X

ı2k

in D 0 .R/

.using (2.11.8)/

kD1 1 X

e ikx D 2

kD1

in D 0 .R/.

ı2k

kD1

3. By Theorem 2.11.1,  X  1 1 1 X dl dl X .l/ ikx e ı D 2 ı2k D 2 2k dx l dx l kD1

H)

1 X kD1

kD1

d l ikx .e / D dx l

1 X

in D 0 .R/

kD1

1 X

.i k/l e ikx D 2

kD1

.l/

ı2k

in D 0 .R/:

kD1

P1 P1 i2kx D C Example 2.11.3. Let S D kD1 kD1 ık as proved in ExamP1 e 1 i2kx on R, with its sum f .x/ D ple 2.11.1. Consider the series kD1 1Ck 2e P1 1 i2kx . kD1 1Ck 2 e 2

1. Show that ddxf2  4 2 f D 4 2 S in D 0 .R/. (2.11.12) 2. (a) Show that the distribution solutions of this equation for f in D 0 .0; 1Œ/ (resp. in D 0 .1; 0Œ/) are C 1 -functions. (b) Assuming the periodicity of f with a period of 1 (i.e. f .0/ D f .1/), f .x/ D C1 Œe 2x  C C2 e 2x

in D 0 .0; 1Œ/ (resp. in D 0 .1; 0Œ/) (2.11.13)

with C1 D d1 e  , C2 D d2 e  (resp. C1 D d1 e  , C2 D d2 e  /, di being constants, show that f is continuous on Œ1; 1. 3. Show that for continuous periodic f (i.e. f .0/ D f .1/ D f .1//, f 00 in the sense of distribution on 1; 1Œ is given by: f 00 D Œf 00 .x/ C 4d.e   e  /ı0 and    X 1 e  e  e   e  SD d d; ı0 D ı: (2.11.14) ık with C D   kD1

4. Considering the two representations (representation by the series and by R1 (2.11.13)) of f .x/ on 0; 1Œ, and computing 0 f .x/dx, show that constant dD

 ;  e  e 

i.e. S D

1 X kD1

ık D

1 X kD1

e i2kx in D 0 .R/: (2.11.15)

170

Chapter 2 Differentiation of distributions and application of distributional derivatives

Proof. P1 e i2kx 1 1 1. 8k 2 Z with k ¤ 0, 1Ck 2 < k 2 H) kD1 1Ck 2 converges uniformly and P 1 i2kx on R H) absolutely to a continuous function f .x/ D 1 kD1 1Ck 2 e P1 1 i2kx converges to T D f in D 0 .R/ with f kD1 1Ck 2 e 1 X

hTf ; i D hf; i D

kD1

1 1 C k2

Z

e i2kx .x/dx

8 2 D.R/

R

by Theorem 2.11.2,

H)

1 X d mf .i 2k/m i2kx D e dx m 1 C k2

in D 0 .R/ by Theorem 2.11.1

kD1 k¤0

H)

1 X d 2f 4 2 .1 C k 2 / C 4 2 i2kx D e D 4 2 f  4 2 S dx 2 1 C k2 kD1

H)

d 2f dx 2

 4 2 f D 4 2 S

in D 0 .R/.

P P1 i2kx D C 2a. Since S D 1 kD1 e kD1 ık , we have hS; i D 0 for  2 D.0; 1Œ/, i.e. 8 2 D.R/ with supp./  0; 1Œ (hC ık ; i D C .k/ with 2 .k/ D 0 8 2 D.0; 1Œ/ and 8k 2 Z). Hence, h ddxf2  4 2 f; i D 0 8 2 D.0; 1Œ/ H)

d 2f  4 2 f D 0 dx 2

in D 0 .0; 1Œ/:

(2.11.16)

Similarly, d 2f  4 2 f D 0 in D 0 .1; 0Œ/: dx 2

(2.11.17)

The distributional solutions of (2.11.16) (resp. (2.11.17)) are C 1 -functions and given by the usual solutions by Theorem 2.7.2: f .x/ D C1 e 2x C C2 e 2x in D 0 .0; 1Œ/ (resp. D 0 .1; 0Œ/), C1 and C2 being arbitrary constants. Setting C1 D d1 e  (resp. d1 e  ), C2 D d2 e  (resp. d2 e  ), we get f .x/ D 1 1 1 1 d1 e 2.x 2 / Cd2 e 2.x 2 / on 0; 1Œ (resp. f .x/ D d1 e 2.xC 2 / Cd2 e 2.xC 2 / on 1; 0Œ).

171

Section 2.11 Term-by-term differentiation of series of distributions

2b. Then, using the periodicity of f with period of 1: h1 f; i D hf; 1 i D hf; .x C 1/i   X N 1 i2kx e ; .x C 1/ D lim N !1 1 C k2 kDN

 X N

D lim

N !1

kDN

 X N

D lim

N !1

kDN

D hf; i

Z

1 1 C k2

e

i2kx

e

i2k

 .x C 1/dx

R

Z

1 1 C k2

 ./d 

R

8 in D.R/

(setting  D x C 1, e i2k. 1/ D e i2k ), we have f .0/ D f .1/ (resp. f .0/ D 1 1 f .1/) H) d1 D d2 D d H) f D d.e 2.x 2 / C e 2.x 2 / / in 0; 1Œ 1 1 (resp. f D d.e 2.xC 2 / C e 2.xC 2 / / in 1; 0Œ), such that f .0C / D f .0 / D d.e  C e  / D f .0/ D f .1/ D f .1/. Thus, f is continuous on Œ1; 1  R and periodic with period 1. 3. Then, 1

1

1

1

f 0 .x/ D d Œ2e 2.x 2 / C 2e 2.x 2 /  D g1 .x/ for 0 < x < 1I f 0 .x/ D d Œ2e 2.xC 2 / C 2e 2.xC 2 /  D g2 .x/

for  1 < x < 0:

J1 .0/ D jump of the first derivative f 0 .x/ at 0 D f 0 .0C /  f 0 .0 / D g1 .0C /  g2 .0 / D d Œe  .2  2/ C e  .2 C 2/ D 4d.e   e  /: 1

Œf 00 .x/ D the usual second-order ordinary derivative D 4 2 d Œe 2.x 2 / C 1 1 1 e 2.x 2 /  D 4 2 f on 0; 1Œ and 4 2 d Œe 2.xC 2 / C e 2.xC 2 /  D 4 2 f on 1; 0Œ H) Œf 00 .x/  4 2 f D 0 on 1; 0Œ [ 0; 1Œ. The second-order distributional derivative f 00 in D 0 .1; 1Œ/ is defined, 8 2 D.1; 1Œ/, by hf 00 ; i D hf 0 ;  0 i D  D f

0

 .x/.x/j01

Z

Z

0

C 1

Z

1

f 0  0 dx

0C

1

0

f 0  0 dx 

ˇ1 Z ˇ Œf .x/dx  f .x/.x/ˇˇ C C 00

0

0

1 0C

Œf 00 .x/dx

172

Chapter 2 Differentiation of distributions and application of distributional derivatives

Z D

1

Œf 00 .x/dx  f 0 .0 /.0/ C f 0 .0C /.0/

1 00

D hŒf .x/; i C Œf 0 .0C /  f 0 .0 /hı0 ; i D hŒf 00 .x/ C J1 .0/ı0 ; i with ı D ı0 H) f 00 D Œf 00 .x/ C J1 .0/ı0 D Œf 00 .x/ C 4d.e  C e  /ı0 in D 0 .1; 1Œ/ (see also Chapter 3, Theorem 3.1.1) H)

f 00  4 2 f D .Œf 00 .x/  4 2 f / C 4d.e   e  /ı0 D 0 C 4d.e   e  /ı0 D 4d.e   e  /ı0 :

P 2 0 But f 00 4 2 f D 4 2 S D 4 2 C 1 kD1 ık D 4 C ı0 in D .1; 1Œ/ 2   (see also Section 5.3, Chapter 5) H) 4 C ı0 D 4d.e e /ı0 H) C D      e e d D e e d.  P   d/ 1 Hence, S D . e e kD1 ık .  4. Since the defining series for f is uniformly convergent on R, term-by-term integration of the series is possible and we get Z

1 X

1

f .x/dx D 0

kD1

Z

1 1 C k2

dx C 0

Z

Z

1 0

ˇ 1 e i2kx ˇˇ1 D1C0D1 1 C k 2 i 2k ˇ0

1

1

.e 2.x 2 / C e 2.x 2 / /dx

0

 Dd H) 1 D d. e

e i2kx dx

0

kD1 k¤0 1

f .x/dx D d

1

1 X

1

D1

Z

 e 

 

1

/ H) d D



1

e 2.x 2 / e 2.x 2 / C 2 2

d D .e Finally, C D e e  P1 i2kx in D 0 .R/. kD1 e

1

 Dd

0

e   e  

 e  e  .

 e 



/

 e  e 

D 1 H) S D

P1



kD1 ık

D

Section 2.12 Convergence of sequences of C k ./ (resp. C k; .// in D 0 ./

2.12

173

Convergence of sequences of C k ./ (resp. C k; .// in D 0 ./

Proposition 2.12.1. Let .un / be a convergent sequence in Banach space C k ./ (resp. C k; ./, 0 <  < 1/ with k 2 N0 (see Appendix A, Sections A.4.1 and A.5.2) such that un ! u 2 C k ./ (resp. C k; ./). Then, 8 multi-index ˛ with j˛j 2 N0 , @˛ un ! @˛ u in D 0 ./

as n ! 1:

(2.12.1)

i.e. un ! u in D 0 ./:

(2.12.2)

In particular, for ˛ D 0, @˛ un D un ; @˛ u D u;

Proof. un ! u in C k ./ ” ku  un kC k ./ D max0jˇjk supx2 j@ˇ u.x/  @ˇ un .x/j. Hence, un ! u in C k ./ H) @ˇ un ! @ˇ u uniformly in every compact subset of  as n ! 1, 8jˇj  k and C k ./  L1loc ./ 8k 2 N0 H) un ! u in D 0 ./ as n ! 1 by Proposition 2.9.2 H) @˛ un ! @˛ u in D 0 ./ as n ! 1 8j˛j 2 N by Theorem 2.9.1. Similarly, for the space C k; ./ of Hölder continuous functions with k 2 N0 , index  2 0; 1Œ (see Appendix A, Definition A.5.3.1), the result can be proved. Imbedding results C k ./ ,! D 0 ./ (resp. C k; .// ,! D 0 ./, 0 <  < 1) 8k 2 N0 , the imbedding operator ,! being a continuous one, i.e. u 2 C k ./ (resp. u 2 C k; ./) H) Tu D u 2 D 0 ./ and un ! u in C k ./ (resp. C k; ./)

2.13

H)

un ! u in D 0 ./:

(2.12.3)

Convergence of sequences of Lp ./, 1  p  1, in D 0 ./

Theorem 2.13.1. Let .un / be a sequence in Lp ./, 1  p  1, which converges to u 2 Lp ./ strongly or weakly or in weak- sense as n ! 1 (see Appendix B, Section B.4). Then, 8 multi-index ˛ with j˛j 2 N0 , @˛ un ! @˛ u in D 0 ./ as n ! 1, i.e. h@˛ un ; i ! h@˛ u; i in R (resp. C)

as n ! 1 8j˛j 2 N0 :

(2.13.1)

174

Chapter 2 Differentiation of distributions and application of distributional derivatives strongly

Proof. Strong convergence in Lp ./, 1  p  1: un ! u in Lp ./ as n ! 1 ” ku  un kLp ./ ! 0 as n ! 1. Then, 8n 2 N, Z Z hTun ; i D hun ; i D un d x; hTu ; i D hu; i D ud x 8 2 D./ 



(2.13.2)

ˇZ ˇ ˇ ˇ ˇ jhu; i  hun ; ij D ˇ .u  un /d xˇˇ

H)



 ku  un kLp ./ kkLq ./ ! 0

as n ! 1

(by the Hölder inequality with p1 C q1 D 1, 1  p; q  1) H) hun ; i ! hu; i as n ! 1 8 2 D./ H) un ! u in D 0 ./ as n ! 1 H) @˛ un ! @˛ u in D 0 ./ by Theorem 2.9.1 8j˛j 2 N. weakly

Weak convergence in Lp ./, 1  p < 1, p1 C q1 D 1, 1 < q  1: un * u in Lp ./, 1  p < 1 Z Z un vd x ! uvd x 8v 2 Lq ./ (2.13.3) ”   ˇ ˇZ ˇ ˇ ˇ ” ˇ .u  un /vd xˇˇ ! 0 as n ! 1 8v 2 Lq ./; 1 < q  1 (2.13.4) 

R H) jhu; ihun ; ij D j  .uun /d xj ! 0 as n ! 1 8 2 D./  Lq ./ H) un ! u in D 0 ./ as n ! 1 H) @˛ un ! @˛ u in D 0 ./ by Theorem 2.9.1 8j˛j 2 N. Weak- convergence in L1 ./  .L1 .//0 : For 1 < p < 1, weak- convergence and weak convergence coincide; un

*

(2.13.5)

L1 ./

u in the weak- sense in Z Z ” un vd x ! uvd x 8v 2 L1 ./   ˇZ ˇ ˇ ˇ ˇ ” ˇ .u  un /vd xˇˇ ! 0 as n ! 1 8v 2 L1 ./:

(2.13.6)



R Hence, jhu; i  hun ; ij D j  .u  un /d xj ! 0 as n ! 1 8 2 D./, since D./  L1 ./ H) un ! u in D 0 ./ H) @˛ un ! @˛ u in D 0 ./ 8j˛j 2 N by Theorem 2.9.1. Remark 2.13.1. 8n 2 N, un 2 Lp ./, u 2 Lp ./, 1  p  1 H) un and u define regular distributions in D 0 ./. But @˛ un , @˛ u 2 D 0 ./ are distributions and do not belong to L1loc ./ in general. Hence, h@˛ un ; i, h@˛ u; i cannot be defined by integrals in general.

175

Section 2.14 Transpose (or formal adjoint) of a linear partial differential operator

Example 2.13.1. Let 8 ˆ n1

1 n

be a sequence in L1 .1; 1Œ/. Then un ! H in L1 .1; 1Œ/ as n ! 1, H.x/ D 1 for 0 < x < 1, H.x/ D 0 for 1 < x < 0 being the step function (Heaviside function). By Theorem 2.9.1, d u ! dH D ı 2 D 0 .1; 1Œ/ as n ! 1, since dx n dx 

 Z 1 Z 1 dH ; D  H.x/ 0 dx D   0 .x/dx dx 1 0 D .x/j10 D .0/ D hı; i

H)

dH dx

8 2 D.1; 1Œ/

D ı in D 0 .1; 1Œ/ is a singular distribution and does not belong to k

L1 .1; 1Œ/. Moreover, ddxukn … L1loc .1; 1Œ/ 8k  2, 8n 2 N, i.e. D 0 .1; 1Œ/ is also a singular distribution 8k  2, 8n 2 N.

Imbedding results

d k un dx k

2

Lp ./ ,! D 0 ./, 1  p  1, the imbedding ,! being strongly

a continuous one, i.e. u 2 Lp ./ H) Tu D u 2 D 0 ./ and un ! u or weakly

un * u or un * u (in the weak- sense) in Lp ./ H)

2.14

un ! u in D 0 ./

as n ! 1:

(2.13.7)

Transpose (or formal adjoint) of a linear partial differential operator

Let A, a linear partial differential operator of order m with coefficients a˛ D a.˛1 ;˛2 ;:::;˛n / 2 C 1 .Rn / 8 multi-index ˛, be defined by:

AD

X 0j˛jm

a ˛ @˛ D

X 0j˛jm

a.˛1 ;˛2 ;:::;˛n /

@j˛j : : : : @xn˛n

@x1˛1 @x2˛2

(2.14.1)

176

Chapter 2 Differentiation of distributions and application of distributional derivatives

Then, 8T 2 D 0 ./, AT is defined with AT 2 D 0 ./ such that  X   X  ˛ ˛ hAT; i D a˛ @ T;  D a˛ @ T;  0j˛jm

  ˛ @ T; a˛  D

X

D

0j˛jm

 D T;

0j˛jm

X

j˛j

.1/

  ˛ T; @ .a˛ /

0j˛jm

 .1/j˛j @˛ .a˛ /

X

8 2 D./:

0j˛jm

Definition 2.14.1. The operator A0 defined by, 8 2 D 0 ./, X

A0  D

X

.1/j˛j @˛ .a˛ / D

0j˛jm

.1/j˛j

0j˛jm

@j˛j .a˛ / @x1˛1 @x2˛2 : : : @xn˛n

(2.14.2)

such that hAT; i D hT; A0 i 8T 2 D 0 ./, 8 2 D./, is called the transpose or formal adjoint of the operator A defined by (2.14.1). Then hA0 T; i D hT; Ai 8T 2 D 0 ./, 8 2 D./. In fact, 8T 2 D 0 ./,  X  X hA0 T; i D .1/j˛j @˛ .a˛ T /;  D h.1/j˛j @˛ .a˛ T /; i 0j˛jm

X

D

ha˛ T; @˛ i D

0j˛jm

D hT;

X

0j˛jm

hT; a˛ @˛ i

0j˛jm

X

a˛ @ i D hT; Ai 8T 2 D 0 ./; ˛

8 2 D./:

0j˛jm 0

A is the transpose of A

H)

A00 D .A0 /0 D A:

(2.14.3)

Indeed, 8T 2 D 0 ./, hA00 T; i D h.A0 /0 T; i D hT; A0 i D hAT; i 8 2 D./ H) 8T 2 D 0 ./, A00 T D AT in D 0 ./ H) A00 D A. Example 2.14.1. A D D variables H)

A0 D .1/2

@2 @x12

C

@2 @x22

C  C

@2 2 @xn

is the Laplace operator in n

2 2 @2 2 @ 2 @ C .1/ C    C .1/ D D A: @xn2 @x12 @x22

Then, for the Laplace operator , we have h T; i D hT; i 8T 2 D 0 ./, 8 2 D./.

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

177

Example 2.14.2. Let A be the second-order linear partial differential operator with variable coefficients defined, 8T 2 D 0 ./, by: AT D 

n X n n X X @ @T @T .aij /C ai C a0 T; @xj @xi @xi iD1 j D1

iD1

where aij , ai , a0 2 C 1 .Rn /. Then the transpose or formal adjoint A0 of A is defined, 8 2 D./, by: A0  D 

n n X n X X @ @ @ .aj i / .ai / C a0 : @xj @xi @xi iD1 j D1

iD1

Proof.  X    X n n X n @ @T @T hAT ; i D  ai C a0 T ;  aij C @xj @xi @xi iD1 j D1

D

iD1

n X n  X iD1 j D1

D

n n X X iD1 j D1

D

    X n  @ @T @T ;  C ha0 T ; i aij ; C ai @xj @xi @xi iD1

 X   n  @ @T @  T;  aij ; .ai / C hT; a0 i @xi @xj @xi iD1

    X n @ @ @ .ai / C hT; a0 i T; aij  T; @xi @xj @xi

n X n  X iD1 j D1

iD1

    X n n X n X @ @ @ D T;  .ai / C a0  aij  @xi @xj @xi iD1 j D1

iD1

    X n X n n X @ @ @ D T;  .ai / C a0  aj i  @xj @xi @xi iD1 j D1

0

D hT; A i

2.15

iD1

8 2 D./:

Applications: Sobolev spaces H m ./; W m;p ./

2.15.1 Sobolev Spaces We will now show the most important application of distributions in the definition of Sobolev spaces, which are the basic tools in the study of boundary value problems of partial differential equations on   Rn . Sobolev spaces may be Hilbert spaces, usually denoted by H m ./ for m 2 N or H s ./ for s 2 R, or Banach spaces

178

Chapter 2 Differentiation of distributions and application of distributional derivatives

denoted by W m;p ./, 1  p  1, 8m 2 N, or W s;p ./ for s 2 R, 1  p  1, such that for p D 2; H m ./  W m;2 ./, H s ./  W s;2 ./ 8m 2 N, 8s 2 R respectively. Among the family of all these Sobolev spaces, the spaces H m ./ of integral order m 2 N and their subspaces are the most important ones owing to their nice Hilbert structure. Hence, we will begin by studying their elementary defining properties.

2.15.2 Space H m ./ Definition 2.15.1. Let  be an open subset of Rn . Then H m ./ is the set of all (equivalence classes Œu of) real-valued functions u 2 L2 ./ whose derivatives @˛ u in the distributional sense (2.3.1) also belong to L2 ./8 multi-index ˛ with j˛j D ˛1 C ˛2 C    C ˛n  m, i.e. H m ./ D ¹u W u 2 L2 ./; @˛ u 2 L2 ./

8j˛j  mº;

(2.15.1)

where the distributional derivatives @˛ u are defined by (2.3.1): Z

j˛j

˛

Z

u.x/@˛ .x/d x

@ u.x/.x/d x D .1/ 

8 2 D./:

(2.15.2)



Then the following properties hold for functions of H m ./: u D v in H m ./

”

u.x/ D v.x/; @˛ u.x/ D @˛ v.x/

8j˛j  m a.e. on I (2.15.3)

u D 0 in H m ./

”

u.x/ D 0; @˛ u.x/ D 0 8j˛j  m a.e. on I (2.15.4)

u; v D H m ./

”

.u/.x/ D u.x/

.u C v/.x/ D u.x/ C v.x/

a.e. on  8 2 R:

a.e. on I

(2.15.5) (2.15.6)

Consequently, H m ./ is a linear space, whose elements u 2 H m ./ are distributions on , (2.15.7) i.e. Z hu; i D

ud x; 

˛

Z

.@˛ u/d x

h@ u; i D

8 2 D./; 8j˛j  m:



(2.15.8)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

179

Proposition 2.15.1. H m ./ equipped with the inner product h  ;  iH m ./ D h  ;  im; defined, 8u; v 2 H m ./, by X

hu; vim; D hu; viL2 ./ C

< @˛ u; @˛ v >L2 ./

1j˛jm

Z D

X

u.x/v.x/d x C 

0j˛jm

@˛ u.x/@˛ v.x/d x



1j˛jm

Z

X

D

Z

@˛ u.x/@˛ v.x/d x;

(2.15.9)



where @˛ .  / D .  / for j˛j D 0, is an inner product space (see Appendix A, Section A.12, and also Table B.3 in Appendix B). Then the corresponding norm k  km; and semi-norm j  jm; are given, 8u 2 H m ./, by:  1 2 2 D kukL kukm; D hu; uim; 2 ./ C

X

2 k@˛ ukL 2 ./

 12

1j˛jm

Z D

ju.x/j d x C 

 D

 X

 12 j@ u.x/j d x I

Z

X

k@

2

˛



1j˛jm

0j˛jm

jujm; D

X

2

 12 j@ u.x/j d x

Z

2

˛

(2.15.10)



˛

2 ukL 2 ./

 12 D

 X Z

j˛jDm

j˛jDm

 12 j@ u.x/j d x : ˛

2

(2.15.11)



Theorem 2.15.1. 8m 2 N; H m ./ equipped with the inner product h  ;  im; defined in (2.15.9) is a Hilbert space. Proof. It is sufficient to show that every Cauchy sequence in H m ./ converges to an element in H m ./. Let (uk ) be a Cauchy sequence in H m ./, i.e. 2 kuk  ul k2m; D kuk  ul kL 2 ./ C

X

2 k@˛ uk  @˛ ul kL 2 ./ ! 0

1j˛jm

as k; l ! 1 ” kuk  ul kL2 ./ ! 0 and k@˛ uk  @˛ ul kL2 ./ ! 0 for 1  j˛j  m as k; l ! 1

180

Chapter 2 Differentiation of distributions and application of distributional derivatives

H) .uk / and .@˛ uk / are Cauchy sequences in L2 ./ 8˛ with 1  j˛j  m. But L2 ./ is a Hilbert space, i.e. a complete space (see Appendix B, Theorem B.4.1.2). Hence, 9u 2 L2 ./ and w˛ 2 L2 ./, 8˛ with 1  j˛j  m, such that uk ! u in L2 ./ and @˛ uk ! w˛ in L2 ./ as k ! 1:

(2.15.12)

Since L2 ./ ,! D 0 ./, the imbedding being a continuous one (see (2.13.7)), uk ! u in L2 ./ H) uk ! u in D 0 ./ by Theorem 2.13.1, i.e. huk ; i ! hu; i 8 2 D./; @˛ uk ! w˛ in L2 ./ H) @˛ uk ! w˛ in D 0 ./ by Theorem 2.13.1, i.e. h@˛ uk ; i ! hw˛ ; i

8 2 D./; 8˛ with 1  j˛j  m; as k ! 1: (2.15.13)

But @˛ W D 0 ./ ! D 0 ./ is continuous by Theorem 2.9.1. Hence, uk ! u in D 0 ./ H) @˛ uk ! @˛ u in D 0 ./ 8˛ with 1  j˛j  m, i.e. h@˛ uk ; i ! h@˛ u; i

8 2 D./:

(2.15.14)

Since the limit is unique, from (2.15.13) and (2.15.14), we have @˛ u D w˛ 2 D 0 ./ 8˛ with 1  j˛j  m. Therefore, u 2 L2 ./; @˛ u D w˛ 2 L2 ./ (by (2.15.12)) 8˛ with 1  j˛j  m. Hence, u 2 H m ./ and 2 ku  uk k2m; D ku  uk kL 2 ./ C

X

2 k@˛ u  @˛ uk kL as k ! 1. 2 ./ ! 0

1j˛jm

Thus, Cauchy sequence .uk / in H m ./ converges to u 2 H m ./. Hence, H m ./ is a complete space, i.e. a Hilbert space. Examples. 

L2 ./: We let m D 0 such that H 0 ./ D L2 ./ with Z hu; vi0; D hu; viL2 ./ D 1 2

kuk0; D hu; ui0; D

u.x/v.x/d x; 

Z 

 12 ju.x/j d x : 2

(2.15.15)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

181

H 1 ./: For m D 1;   Rn ; H 1 ./ is equipped with inner product h  ;  i1; , norm k  k1; and semi-norm j  j1; defined by:



hu; vi1; D hu; vi0; C

n  X @u iD1

kuk1;

@v ; @xi @xi

 0;

 Z  @u @v @u @v D C  C uv C d xI @x1 @x1 @xn @xn     12 n  X 1 @u @u 2 D hu; ui1; D hu; ui0; C ; @xi @xi 0;

(2.15.16)

iD1

juj1;

ˇ ˇ2 ˇ ˇ2   1 Z  2 ˇ @u ˇ ˇ @u ˇ 2 ˇ ˇ ˇ D .x/ˇ C    C ˇ .x/ˇˇ d x I ju.x/j C ˇ @x1 @xn  ˇ2  1 X X   12 n  n Z ˇ 2 ˇ ˇ @u @u @u ˇ dx : ˇ D ; D .x/ ˇ ˇ @xi @xi 0;  @xi iD1

(2.15.17) (2.15.18)

iD1

H 2 ./: For m D 2, n D 2,   R2 , H 2 ./ is equipped with inner product h  ;  i2; , norm k  k2; and semi-norm j  j2; defined, 8u; v 2 H 2 ./, by:



hu; vi2; D hu; vi1; C

 X  @2 u @2 v ; @xi @xj @xi @xj 0;

1i;j 2

Z  D

@u @v @2 u @2 v @u @v C C 2 2 @x1 @x1 @x2 @x2 @x1 @x1  2 2 2 2 @ v @ u @ u@ v C2 C 2 2 dx1 dx2 I @x1 @x2 @x1 @x2 @x2 @x2

uv C 

(2.15.19)

1 2 kuk2; D hu; ui2; ˇ ˇ ˇ ˇ ˇ ˇ Z  ˇ @u ˇ2 ˇ @u ˇ2 ˇ @2 u ˇ2 2 ˇ ˇ ˇ ˇ ˇ D Cˇ C ˇ 2 ˇˇ juj C ˇ @x1 ˇ @x2 ˇ @x1  ˇ 2 ˇ2 ˇ 2 ˇ2   12 ˇ @ u ˇ ˇ@ uˇ ˇ ˇ ˇ ˇ C 2ˇ C ˇ 2 ˇ dx1 dx2 I @x1 @x2 ˇ @x2 ˇ 2 ˇ2 ˇ 2 ˇ2   Z ˇ 2 ˇ2  12 ˇ @ u ˇ ˇ@ uˇ ˇ@ uˇ ˇ ˇ ˇ ˇ ˇ ˇ juj2; D ˇ @x 2 ˇ C 2ˇ @x @x ˇ C ˇ @x 2 ˇ dx1 dx2 : 1 2  1 2

(2.15.20)

(2.15.21)

182

Chapter 2 Differentiation of distributions and application of distributional derivatives

2.15.3 Examples of functions belonging to or not belonging to H m ./ Case of single variable (n D 1) Example 2.15.1. For  D 1; 1Œ, consider u.x/ D jxj8x 2 1; 1Œ. Show that 1. u 2 H 1 .1; 1Œ/, but 2. u … H 2 .1; 1Œ/. Proof. R1 R1 3 1. 1 ju.x/j2 dx D 1 x 2 dx D x3 j11 D 13 C 13 D 23 < C1 H) u 2 L2 .1; 1Œ/. From Example 2.3.1, the distributional derivative ´ du 1 D g.x/ D dx 1

for 0 < x < 1 for  1 < x < 0

and Z

1

Z

2

jg.x/j dx D 1

H) g 2 L2 .1; 1Œ/ H) u 2 H 1 .1; 1Œ/.

Z

0

1dx C 1

du dx

1

1dx D 2 < C1 0

2 L2 .1; 1Œ/. Thus, u; du 2 L2 .1; 1Œ/ H) dx 2

2. From Example 2.3.3, the second-order distributional derivative ddxu2 D 2ı … L1loc .1; 1Œ/ by Proposition 1.3.2, ı being the Dirac distribution (with mass/ 2 charge/force etc.) concentrated at 0. Hence, ddxu2 … L2 .1; 1Œ/, since L2 .1; 1Œ/  L1loc .1; 1Œ/. Thus u … H 2 .1; 1Œ/.

Case of two variables (n D 2) Example 2.15.2. For  D R2 , consider function u defined in (2.3.23) in Example 2.3.9: ´ 1 ln j ln rj for 0 < r D .x12 C x22 / 2 < u.x1 ; x2 / D 0 for 1e  r < 1:

1 e

Show that this unbounded and discontinuous function in R2 belongs to H 1 .R2 /. Proof. From (2.3.24), u 2 L2 .R2 / and

@u @u ; @x1 @x2

2 L2 .R2 /. Hence, u 2 H 1 .R2 /.

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

183

Example 2.15.3. For  D 0; 1Œ0; 1Œ  R2 with  D Œ0; 1Œ0; 1 (see Figure 2.4), consider the discontinuous, piecewise polynomial function u defined by: ´ 1 C 4x1  2x2 in TV1 D ¹.x1 ; x2 / W 0 < x1 < x2 < 1º u.x1 ; x2 / D 2 C 4x1 C 4x2 in TV2 D ¹.x1 ; x2 / W 0 < x2 < x1 < 1º with TVi D int.Ti /, T1 [ T2 D , T1 \ T2 D 0 , the diagonal of  joining .0; 0/ and .1; 1/. Then u … H 1 ./. For the proof and more details, we refer to Example 3.1.2 in Chapter 3.

u x2

T1

( x1, x2) = 1 + 4x1 - 2x2 a2 = (1, 1)

a3 = (0, 1)

0

T

T

T1

T2

u

T2

(x1, x2) = –2 + 4x1 + 4x2

a1 = (1, 0) x1

a4 = (0, 0)

Figure 2.4 Piecewise polynomial function u on  D T1 [ T2 with discontinuity across 0

0

Figure 2.5  D ¹.x; y/ W 0 < x < 1, 0 < y < x r , r > 0º

184

Chapter 2 Differentiation of distributions and application of distributional derivatives

Example 2.15.4. Let   R2 be defined by  D ¹.x; y/ W 0 < x < 1, 0 < y < x r , r > 0º (see Figure 2.5), and u.x; y/ D x ˛ 8.x; y/ 2 . Show that u 2 H 1 ./ if 2˛ C r > 1. Can ˛ be negative such that u 2 H 1 ./? Solution.   Z yDx r  Z Z 1  Z yDx r Z 1 2 2˛ 2˛ ju.x; y/j dxdy D x dy dx D x dy dx 

Z

yD0

0 1

D

0

yD0

x 2˛Cr dx < C1 if 2˛ C r > 1

0

H) u 2 L2 ./ if 2˛ C r > 1. For x > 0, u is a C 1 -function in  with @u D ˛x ˛1 , @u D 0 in , since the @x @y usual partial derivatives and distributional derivatives of u will coincide in . Hence, @u 2 L2 ./ 8˛; r, and @y  Z 1  Z yDx r Z ˇ ˇ2 ˇ @u ˇ 2 2˛2 ˇ ˇ dxdy D ˛ x dy dx ˇ ˇ  @x yD0 0 Z 1 D ˛2 x 2˛2Cr dx < C1 if 2˛  2 C r > 1 0 @u @x

or 2˛ C r > 1. Hence, 2 L2 ./ for 2˛ C r > 1. Thus, u; @u ; @u 2 L2 ./ for @x @y 2˛ C r > 1, H) u 2 H 1 ./ for 2˛ C r > 1. ˛ can be negative, if r is sufficiently large such that 2˛ C r > 1.

2.15.4 Separability of H m ./ Proposition 2.15.2. 8m 2 N, H m ./ is separable, i.e. 9 a countably dense subset of H m ./. The proof of Proposition 2.15.2 depends on the following well-known results: 1. The product of separable spaces is also separable. (For example, since L2 ./ is separable (see (B.4.3.5), Appendix B), H)

.L2 .//k D L2 ./  L2 ./      L2 ./ „ ƒ‚ … k times

is also separable.)

(2.15.21a)

2. If X is a Hilbert space and Y  X is a closed subspace of X , then the separability of X implies the separability of Y . (2.15.21b) In fact, let .xk /k2N be a dense sequence (hence countable) in X , and PY W X ! Y be the projection operator. Define yk D PY xk 8k 2 N. Then, .yk /k2N is a dense sequence in Y , from which its separability follows.

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

185

3. Let X; Y be Hilbert spaces and I W X ! Z  Y be an isometric isomorphism from X onto IX D Z  Y . If Y is separable, then I 1 Z D X is also separable. (2.15.21c) Proof of Proposition 2.15.2. Set N D N.n; m/ defined by: N.n; m/ D

X

² Card .˛1 ; ˛2 ; : : : ; ˛n / W 0  j˛j D

n X

³ ˛i  m; ˛i 2 N0 :

iD1

0j˛jm

D N: Define the product space .L2 .//N D L2 ./  L2 ./      L2 ./ : „ ƒ‚ … N times

By (2.15.21a), .L2 .//N is a separable product space. 8u 2 H m ./, define .@˛ u/0j˛jm by:    2      @u @ u @m u .@ u/0j˛jm D uI I I:::I : @xi 1in @xi @xj 1i;j n @x1˛1 : : : @xn˛n j˛jDm ˛

Then .@˛ u/0j˛jm 2 .L2 .//N 8u 2 H m ./. Let I W H m ./ ! .L2 .//N be defined, 8u 2 H m ./, by Iu D .@˛ u/0j˛jm 2 .L2 .//N with  2 kIuk.L2 .//N D kukL 2 ./ C

X

2 k@˛ ukL 2 ./

1=2 D kukm; :

1j˛jm

Hence, I is an isometry from H m ./ onto W D I.H m .// with W   .L2 .//N . Moreover, I is linear, continuous and injective from H m ./ onto W , and I 1 is also linear and continuous from W onto H m ./, i.e. I is an isometric isomorphism from H m ./ onto W , which is a closed subspace of the separable product space .L2 .//N . In fact, let .uk /k2N with uk D .@˛ uk /0j˛jm D Iuk 8k 2 N be a Cauchy sequence in W  I.H m .//. Then kuk  ul k.L2 .//N D kIuk  Iul k.L2 .//N D kuk  ul kH m ./ ! 0 as k; l ! 1. Hence, .uk /k2N is a Cauchy sequence in H m ./ H) 9u 2 H m ./ such that uk ! u in H m ./ as k ! 1 H) Iuk ! Iu in .L2 .//N with Iu 2 I.H m .//  W H) uk ! u 2 W with u D Iu D .@˛ u/0j˛jm . Hence, W is also separable by (2.15.21b). Then H m ./ D I 1 .W / is also separable by (2.15.21c). Reflexivity of H m ./ (i.e. H m ./  .H m .//00 ) follows from the fact that every Hilbert space is reflexive (see also the proof of reflexivity of W m;p ./ for 1 < p < 1 in Theorem 2.15.4, from which the result is obtained for p D 2).

186

Chapter 2 Differentiation of distributions and application of distributional derivatives

2.15.5 Generalized Poincaré inequality in H m ./ Theorem 2.15.1A. Let   Rn be a bounded domain with a sufficiently smooth boundary  (for example, a Lipschitz continuous boundary D)). Then, R P (see Appendix 8u 2 H m ./, 9C > 0 such that kuk2m;  C Œjuj2m; C j˛jm1 j  @˛ u.x/d xj2 . Proof. Suppose that the contrary holds, i.e. the inequality does not hold for any constant C . In other words, we can find a sequence .uk /1 in H m ./ with kuk km; D kD1 1 such that ˇ2   X ˇˇ Z ˇ 2 2 ˛ ˇ @ uk .x/d xˇˇ 1 D kuk km; > k juk jm; C ˇ j˛jm1

 H)

juk j2m;

C

X j˛jm1

H)

juk j2m;



ˇ2  ˇZ ˇ ˇ 1 ˛ ˇ @ uk .x/d xˇˇ < 8k 2 N ˇ k 

ˇZ ˇ2 ˇ ˇ ˛ ˇ ! 0 and ˇ @ uk .x/d xˇˇ ! 0 8j˛j  m  1 as k ! 1: 

(2.15.21d) Hence, juk j2m; ! 0 H) @˛ uk ! 0 in L2 ./ 8j˛j D m as k ! 1 and 1 ˛  @ uk .x/d x ! 0 as k ! 1 8j˛j  m  1. But .uk /kD1 is a bounded sequence with kuk km; D 1 8k 2 N in Hilbert space H m ./, which is reflexive. Hence, we from the sequence .uk /1 can extract a weakly convergent subsequence .ukl /1 lD1 kD1 m m in H ./. Let u 2 H ./ such that ukl * u weakly in H m ./ as l ! 1. But by the Rellich–Kondraschov Theorem 8.11.4, H m ./ ,!,! H m1 ./, i.e. H m ./ is compactly imbedded2 (see Section 8.11, Chapter 8 for more details) in H m1 ./. Hence, ukl * u weakly in H m ./ H) ukl ! u strongly in H m1 ./, i.e. ku  ukl km1; ! 0 as l ! 1 H) ukl ! u in L2 ./ and @˛ ukl ! @˛ u in L2 ./ for 1  j˛j  m  1. But ukl ! u in L2 ./ H) ukl ! u in D 0 ./ (by (2.13.1)) H) @˛ ukl ! @˛ u in D 0 ./ for 1  j˛j  m (by Theorem 2.9.1). (2.15.21e) From (2.15.21d), juk jm; ! 0 H) @˛ uk ! 0 in L2 ./ 8j˛j D m as k ! 1 H) @˛ ukl ! 0 in L2 ./ 8j˛j D m as l ! 1 H) @˛ ukl ! 0 in D 0 ./ 8j˛j D m as l ! 1. (2.15.21f) ˛ 2 Then, from (2.15.21e) and (2.15.21f), @ u D 0 in L ./8j˛j D m (by virtue of the uniqueness of the limit). Thus, @˛ ukl ! @˛ u in L2 ./ for 0  j˛j  m R

2 Let X and Y be Banach spaces and A W X ! Y be a linear operator from X into Y . Then A is called compact from X into Y if and only if xn * x weakly in X implies Axn ! Ax strongly in Y as n ! 1 (see Appendix A, Definition A.16.1.1). For X ,! Y , the imbedding operator ,!W X ! ,! X  Y is called compact from X into Y if and only if xn * x weakly in X implies xn ! x in Y as n ! 1, since xn 2 X 7! ,! xn D xn 2 Y; x 2 X 7! ,! x D x 2 Y . Then, for X ,! Y , the imbedding operator ,! is compact from X into Y , X is called compactly imbedded in Y and we write X ,!,! Y .

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

187

with @˛ u D 0 8j˛j D m H) ukl ! u in H m ./ as l ! 1 with @˛ u D 0 in L2 ./ 8j˛j D m. Then @˛ u D 0 in D 0 ./ 8j˛j D m and  is a connected set H) u 2 Pm1 , i.e. u is a polynomial of degree  m  1(by Proposition 2.8.1). But, R from (2.15.21d), 8j˛j  m  1; liml!1  @˛ ukl d x D 0 and, 8j˛j  m  1, ˇ ˇZ ˇ ˇZ Z ˇ ˇ ˇ ˇ ˛ ˛ ˛ ˛ ˇ ˇ ˇ @ ukl d x  @ ud xˇ D ˇ .@ ukl  @ u/d xˇˇ ˇ 





 k1kL2 ./ k@˛ ukl  @˛ ukL2 ./ ! 0 as l ! 1. R ˛ R ˛ Hence, l!1  @ ukl d x D  @ ud x D 0 for 0  j˛j  m1. Then u 2 Pm1 R lim with  @˛ ud x DR 0 for 0  j˛j  m  1 H) u D 0. In fact, 8j˛j D m  1, @˛ u D constant a˛ with  a˛ d x D 0 H) a˛ D 0 8j˛jR D m  1 ˛ H) u is a polynomial of degree  m2  @ ud x D 0 8j˛j D m2 H) u R with ˛ is a polynomial of degreeR m  3 with  @ ud x D 0 8j˛j D m  3 H)    H) for j˛j D 0, u D a0 with  a0 d x D 0 H) a0 D 0. Thus, uk ! u in H m ./ with u D 0 in H m ./ and we meet with a contradiction, since 0 D kukm; D liml!1 kukl km; D limk!1 kuk km; D 1. Hence, our original assumption is wrong and the inequality holds.

2.15.6 Space H0m ./ Definition 2.15.2. Let   Rn be an open subset of Rn . Then, 8m 2 N; H0m ./ is the closure of D./  C01 ./ in the norm k  km; of H m ./, i.e. H0m ./ D D./ D C01 ./

in H m ./:

(2.15.22)

In other words, D./  C01 ./ is dense in H0m ./ 8m 2 N. Alternative characterization of H0m ./ For domains  with sufficiently smooth boundary  (see Appendix D), there is an alternative characterization of H0m ./ with the help of trace theorems (see, for example, Theorem 8.9.11 for  D Rn with  D Rn1 ), for which we refer to Lions [13], Lions and Magenes [15], Neˇcas [16], Grisvard [17], [19], etc. For example, for circular, elliptic, polygonal domains   R2 with circular, elliptic, polygonal boundaries   R2 respectively, H01 ./ D ¹u W u 2 H 1 ./; u# D 0º  D./ in H 1 ./I ² ³ @u 2 2 H0 ./ D u W u 2 H ./; u# D 0; # D 0  D./ in H 2 ./: @n (2.15.23) Theorem 2.15.2. 8m 2 N; H0m ./ defined by (2.15.22) and equipped with inner product h  ;  im; is also a separable Hilbert space.

188

Chapter 2 Differentiation of distributions and application of distributional derivatives

Proof. By Definition 2.15.2, H0m ./ is a closed subspace of H m ./. But H m ./ is a Hilbert space by Theorem 2.15.1, and every closed subspace of Hilbert space H m ./ equipped with the inner product h  ;  im; is also a Hilbert space. Its separability follows from (2.15.21a), since H0m ./ is a closed subspace of the separable Hilbert space H m ./. Owing to its Hilbert space structure, H0m ./ is reflexive. Orthogonal complement of H0m ./ in H m ./ Proposition 2.15.3. The orthogonal complement of H0m ./ in H m ./ is the linear space of all u 2 H m ./ which satisfy the following equation: X .1/j˛j @2˛ u D 0 in D 0 ./; (2.15.23a) 0j˛jm

where partial derivatives @2˛ u are in the distributional sense. Proof. Let u 2 H m ./. Then u belongs to the orthogonal complement of H0m ./ if and only if hu; vim; D 0 8v 2 H0m ./ ” hu; im; D 0 8 2 D./; since the inner product hu;  im; is continuous on H0m ./ and D./ is dense in H0m ./. Thus, for u 2 H m ./ belonging to the orthogonal complement of H0m ./, 8 2 D./, X X Z 0 D hu; im; D h@˛ u; @˛ im; D @˛ u@˛ d x 0j˛jm

D

X

0j˛jm

.1/j˛j h@2˛ u; iD 0 ./D./ D

0j˛jm

”

P

j˛j 2˛ 0j˛jm .1/ @ u





X

.1/j˛j @2˛ u; 

 D 0 ./D./

0j˛jm

D 0 in D 0 ./.

Corollary 2.15.1. D./ is dense in L2 ./  H 0 ./. Proof. m D 0 H) ˛ D 0 H) @2˛ u D u. Hence, u 2 H 0 ./ D L2 ./ belongs to the orthogonal complement of H00 ./ D D./ in H 0 ./ ” hu; i0; D 0 P 8 2 D./ ” j˛jD0 .1/j˛j @2˛ u D u D 0 in D 0 ./ by Proposition 2.15.3; H) u D 0 in L2 ./ H) D./ is dense in L2 ./  H 0 ./. Norm equivalence in H0m ./ Theorem 2.15.3. The mapping  X 1=2 1=2  X Z k@˛ uk20; D j@˛ u.x/j2 d x u 2 H0m ./ 7! jujm; D j˛jDm

j˛jDm



Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

189

defines a norm in H0m ./ equivalent to the original norm kkm; induced by H m ./ 8m 2 N, i.e. 9C1 ; C2 > 0 such that C1 kukm;  jujm;  C2 kukm;

8u 2 H0m ./:

(2.15.23b)

.H0m ./I j  jm; / equipped with the new norm defined by j  jm; is a Hilbert space 8m 2 N. (2.15.23c) In particular, for m D 1, (H01 ./I j  j1; ) equipped with the norm defined by j  j1; (see Example 2.15.5 later) is a Hilbert space. (2.15.23d) 2 For m D 2, .H0 ./I j  j2; / equipped with the norm defined by j  j2; (see Example 2.15.5 later) is a Hilbert space. (2.15.23e) Proof of Theorem 2.15.3. The semi-norm j  jm; is a norm in H0m ./: First of all, we temporarily assume that (2.15.23b) holds. Then, for u 2 H0m ./ satisfying (2.15.23b), jujm; D 0 H) kukm;  C11 jujm; D 0 H) kukm; D 0 H) u D 0 in H0m ./. Hence, the semi-norm j  jm; is a norm in H0m ./. Now we prove (2.15.23b) for m D 1; 2, which will be met with in applications. Case m D 1: Let u 2 H01 ./. Then, juj1;  kuk1; with C2 D 1. (2.15.23f) It remains to prove that C1 kuk1;  juj1; with C1 > 0. For this we will apply the density of D./ in H01 ./ and the result (1.2.36) in Section 1.2. In fact, 9 a sequence .k /1 in D./ such that ku  k k1; ! 0 as k ! 1. Then kk k1;  kD1 ku  k k1; C kuk1; ! kuk1; as k ! 1, i.e. limk!1 kk k1; D kuk1; ; and jk j1;  ju  k j1; C juj1;  ku  k k1; C juj1; ! juj1; as k ! 1 (since (2.15.23g) ku  k k1; ! 0 as k ! 1) H) limk!1 jk j1; D juj1; . From (1.2.35) and (1.2.36), kk k20;  C kr k k20; D C jk j21; 8k 2 N. Hence, 1 kk k21; D jk j21; Ckk k20;  .1CC /jk j21; H) . 1CC /kk k21;  jk j21; H) p C1 kk k1;  jk j1; with C1 D 1=.1 C C / > 0. Thus, we have proved: 8k 2 N; 9C1 > 0 such that C1 kk k1;  jk j1; H) C1 limk!1 kk k1;  limk!1 jk j1; . Then, using (2.15.23g) and (2.15.23f) with C2 D 1, C1 kuk1;  juj1;  C2 kuk1; :

(2.15.23h)

Case m D 2: Let u 2 H02 ./. Then juj2;  kuk2; with C2 D 1. Again applying the density of D./ in H02 ./, (1.2.35), (1.2.36) and (2.15.23h), we get CQ1 kuk2;  juj2; . In fact, 9 a sequence . k /1 in D./ such that ku  k k2; ! 0 as kD1 k ! 1. Hence, limk!1 k k k2; D kuk2; and limk!1 j k j2; D juj2; (see the steps for the case m D 1). Moreover, ku  k k21;  ku  k k22; ! 0 as k ! 1 H) k ! u in H 1 ./ H) u 2 H01 ./ (by virtue of the density of D./ in H01 ./).Then, 8k 2 N; 8i D 1; 2; : : : ; n, @@xk 2 L2 ./, @x@ . @@xk / 2 L2 ./ i

j

i

190

Chapter 2 Differentiation of distributions and application of distributional derivatives

8i; j D 1; 2; : : : ; n    n  2 X  @ k 2  @ k 2     C  @x   @x @x  i 0; i j 0;

H)

j D1

H)

j

2 k j1;

  n  n X n  2 X X  @ k 2  @ k 2     D C D Cj  @x   @x @x  i 0; i j 0; iD1

2 k j2; :

iD1 j D1

(2.15.23i) Hence, 8k 2 N, k

2 k k2;

Dj

2 k j2;

Ck

2 k k1;

j

C j k j22; 2 C1   C D 1 C 2 j k j22; C1 j

H) CQ1 k

k k2;

2 k j2;

 j

C

k j2;

2 k j2;

C

1 j C12

2 k j1;

.using (2.15.23h)/

.by (2.15.23i)/

with CQ1 D

q 1=.1 C

C / C12

> 0 8k 2 N. Hence,

CQ1 limk!1 k k k2;  limk!1 j k j2; H) CQ1 kuk2;  juj2; H) 8u 2 H02 ./, CQ1 kuk2;  juj2;  CQ2 kuk2; with CQ2 D C2 D 1; CQ1 > 0. (2.15.23j) Similarly, by the method of induction, the general case of m 2 N can be easily proved. .H0m ./I j  jm; / is a Hilbert space: From Theorem 2.15.2, H0m ./ is a Hilbert space equipped with the original norm k  km; . Now, we will show that H0m ./ equipped with the new norm defined by the semi-norm j  jm; , i.e. .H0m ./I j  jm; /, in .H0m ./I is a Hilbert space. For this we consider a Cauchy sequence .uk /1 kD1 j  jm; /, i.e. juk  ul jm; ! 0 as k; l ! 1. Using norm equivalence (2.15.23b), kuk  ul km;  C11 juk  ul jl; ! 0 H) .uk /1 is also a Cauchy sequence kD1 in Hilbert space .H0m ./I k  km; / equipped with the original norm k  km; H) 9u 2 H0m ./ such that ku  uk km; ! 0 as k ! 1, i.e. limk!1 uk D u in .H0m ./I k  km; /. But ju  uk jm;  ku  uk km; ! 0 as k ! 1 H) limk!1 uk D u in .H0m ./I j  jm; /. Thus, every Cauchy sequence .uk /1 conkD1 verges to an element u 2 .H0m ./I j  jm; /, i.e. .H0m ./I j  jm; / is a Hilbert space. .H0m ./I j  jm; / is a Hilbert space and hence reflexive. The separability follows from the separability of H m ./, since every closed subspace of a separable Hilbert space is separable.

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

191

Example 2.15.5.

R @u 2 @u 2 1=2 1. For m D 1, juj1; D  .j @x j C    C j @x j / d x is a norm in H01 ./ n 1 equivalent to the original norm kuk1; given in (2.15.17). R 2 2u 2 j2 C j @ u2 j2 dx1 dx2 /1=2 is 2. For m D 2, n D 2, juj2; D .  Œj @ u2 j2 C 2j @x@1 @x 2 @x1

@x2

a norm in H02 ./ equivalent to the original norm kuk2; in (2.15.20).

2.15.7 Space H m ./ H m ./ is defined as the dual of H0m ./, i.e. H m ./  .H0m .//0 (see Section 4.3, Chapter 4 for full details and other results).

2.15.8 Quotient space H m ./=M Definition 2.15.2A. Let M be a closed subspace of H m ./. Then the quotient space H m ./=M (i.e. the quotient of H m ./ by M ) is the linear space of equivalence classes Œu of functions u 2 H m ./ satisfying the property: u; v 2 Œu

”

uv 2M

”

uDv

.mod M /;

i.e. Œu D u C M , u C M being the coset of u relative to M . H m ./=M is equipped with the usual quotient norm kŒukH m ./=M defined, 8Œu 2 H m ./=M , by: kŒukH m ./=M D inf kukH m ./ D inf ku C wkH m ./ : u2Œu





w2M

(2.15.24)

 W H m ./ ! H m ./=M is the linear mapping called canonical surjection from H m ./ onto H m ./=M and defined, 8u 2 H m ./, by: u D Œu D u C M 2 H m ./=M , u C M being the coset of u relative to M; (2.15.24a) and 8Œu D u C M 2 H m ./=M with u 2 H m ./, .u/ D Œu. Linearity of : .u C v/ D .u C v/ C M D .u C M / C .v C M / D u C v8u; v 2 H m ./, .˛u/ D .˛u/ C M D ˛.u C M / D ˛u

8˛ 2 R; 8u 2 H m ./I (2.15.24b)



Kernel of : Ker./= M , i.e. 8u 2 M , u D Œ0 2 H m ./=M ;



Continuity of :

(2.15.24c)

kukH m ./=M D kŒukH m ./=M D inf kukH m ./  kukH m ./ u2Œu

with u D Œu 2 H m ./=M .

8u 2 H m ./ (2.15.24d)

192

Chapter 2 Differentiation of distributions and application of distributional derivatives

Lemma 2.15.1. Let  W H m ./ ! H m ./=M be the canonical surjection defined above, and M ? be the orthogonal complement of M such that H m ./ D M ˚ M ? . Then the restriction of  to M ? W D #M ? W M ? ! H m ./=M is a linear bijection from M ? onto H m ./=M . Consequently, 1 W H m ./=M ! M ? is also a linear bijection. Proof. Let u 2 H m ./ with u D vCw, v 2 M , w 2 M ? . The linearity of follows from that of . Then, u D v C w D w 2 H m ./=M; since Ker./ D M (by (2.15.24c)) and v D Œ0 8v 2 M . Hence,  maps M ? onto H m ./=M , i.e.

.M ? / D H m ./=M . Now, we are to show that is one-to-one. Let w1 ; w2 2 M ? such that w1 D w1 2 H m ./=M and w2 D w2 2 H m ./=M . Then w1 D

w2 H) w1  w2 D Œ0 H) .w1  w2 / D Œ0 H) w1  w2 2 Ker./ H) w1 w2 2 M . But w1 w2 2 M ? H) w1 w2 2 M \M ? D ¹0º H) w1 w2 D 0 H) w1 D w2 . Thus, is one-to-one and the result follows. Two important quotient spaces are H m ./=H0m ./ and H m ./=Pm1 , where M D H0m ./ and Pm1 respectively, Pm1 is the (closed) subspace of polynomials of degree  m  1 in n variables x1 ; x2 ; : : : ; xn defined on . Since H m ./ is a Banach space and M is a closed subspace of H m ./, we have the classical result: Proposition 2.15.4A. The quotient space H m ./=M equipped with the quotient norm kŒ  kH m ./=M in (2.15.24) is a Banach space. In fact, H m ./=M is a Hilbert space: Proposition 2.15.4B. H m ./=M is a Hilbert space for the norm kŒ  kH m ./=M in (2.15.24). Proof. Since H m ./=M is a Banach space by Proposition 2.15.4A, it is sufficient to define an inner product hŒ  ; Œ  iH m ./=M in H m ./=M such that the corresponding norm satisfies (2.15.24). Since M is a closed subspace of Hilbert space H m ./, we can write H m ./ D M ˚ M ? (i.e. the direct sum of M and M ? ), where M ? is the orthogonal complement of M in H m ./. Then, 8u 2 H m ./, u D uM C uM ? with uM 2 M , uM ? 2 M ? such that huM ; uM ? im; D 0, the decomposition being a unique one. 8Œu 2 H m ./=M , 9 precisely one element uM ? 2 Œu with uM ? 2 M ? , since the mapping Œu 2 H m ./=M 7! uM ? 2 M ? is a bijection from H m ./=M onto M ? by Lemma 2.15.1. Hence, we can define an inner product in H m ./=M by: 8Œu; Œv 2 H m ./=M with u D uM C uM ? 2 Œu;

v D vM C vM ? 2 Œv;

hŒu; ŒviH m ./=M D huM ? ; vM ? iH m ./ :

(2.15.24e)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

193

Then 2 2 kŒukH m ./=M D kuM ? km;

8Œu 2 H m ./=M;

(2.15.24f)

with u D uM C uM ? 2 Œu. Now, we are to show that this norm (2.15.24f) satisfies (2.15.24). In fact, from (2.15.24), we have 2 2 2 kŒukH m ./=M D Œ inf .kukm; / D inf kukm; u2Œu

D

u2Œu

inf .kuM k2m; u2Œu

C kuM ? k2m; / D kuM ? k2m; ;

since the mapping Œu ! uM ? is one-to-one and the infimum is realized for uM D 0 and uM ? 2 M ? . Hence, Banach space H m ./=M equipped with the corresponding inner product hŒ  ; Œ  iH m ./=M in (2.15.24e) is a Hilbert space.

2.15.9 Quotient space H m ./=Pm1 Theorem 2.15.3A. Let   Rn be a bounded domain with Lipschitz continuous boundary  (see Appendix D). Then 9C1 ; C2 > 0 such that, 8Œu 2 H m ./=Pm1 with u 2 H m ./;

u D Œu;

C1 kŒukH m ./=Pm1  jujm;  C2 kŒukH m ./=Pm1 : (2.15.24g)

Another norm jjjŒujjjH m ./=Pm1 equivalent to the original quotient norm kŒukH m ./=Pm1 is given by: 1=2

jjjŒujjjH m ./=Pm1 D hŒu; ŒuiH m ./=Pm1 D

 X Z j˛jDm

D jujm; :

1=2 j@˛ u.x/j2 d x



(2.15.24h)

H m ./=Pm1 is a Hilbert space with inner product hŒ  ; Œ  iH m ./=Pm1 defined by: hŒu; ŒviH m ./=Pm1 D

X Z j˛jDm

@˛ u@˛ vd x:



Proof. See the proof of Theorem 2.15.7 with p D 2.

(2.15.24i)

194

Chapter 2 Differentiation of distributions and application of distributional derivatives

2.15.10 Other equivalent norms in H m ./ Theorem 2.15.3B. Let   Rn be a bounded domain with Lipschitz continuous boundary  (see Appendix D). Then we have the following results: I. Equivalent norm in H m ./: 8u 2 H m ./, 9C1 ; C2 > 0 such that  1=2 X 2 ˛ 2 C1 kukm;  kukL2 ./ C k@ ukL2 ./  C2 kukm; I j˛jDm

(2.15.25a) II. Equivalent norm in H 1 ./: For 0   H 1 ./, 9C1 ; C2 > 0 such that Z n Z X 2 ju.x/j dS C C1 kuk1;  0

iD1

with measure .0 / > 0, 8u 2

ˇ ˇ 1=2 ˇ @u ˇ2 ˇ ˇ dx  C2 kuk1; ; ˇ ˇ  @xi (2.15.25b)

which is also called Friedrichs’ inequality (Neˇcas [16]); III. Equivalent norm in H 2 ./: 8u 2 H 2 ./, 9C1 ; C2 > 0 such that Z 1=2 X Z C1 kuk2;  ju.x/j2 dS C j@˛ uj2 d x  C2 kuk2; : 

j˛jD2



(2.15.25c) (The same constants C1 ; C2 > 0 have been used to denote different values in the different inequalities (2.15.25a)–(2.15.25c).) An interesting counterexample has been given in [16]: Example 2.15.6. If   R3 is an open ellipsoid with boundary  in R3 and u 2 H 3 ./, an inequality analogous to (2.15.25c) for H 3 ./ with ellipsoidal domain  does not hold, since the left-hand side inequality becomes: Z 1=2 X Z 2 ˛ 2 C1 kuk3;  ju.x/j dS C j@ uj d x ; (2.15.25d) 

j˛jD3



which does not hold in this case. In fact, the right-hand side of inequality (2.15.25d) is not a norm in H 3 ./ for ellipsoidal . For u 2 H 3 ./ with u.x1 ; x2 ; x3 / D p2 .x1 ; x2 ; x3 / D ˛12 x12 C ˛22 x22 C ˛32 x32  1 D 0 (equation of an ellipsoidal surface ) with ˛1 ; ˛2 ; ˛3 > 0 8.x1 ; x2 ; x3 / 2 , p2 being a polynomial of degree 2 in three variables x1 ; x2 ; x3 , both the integrals in (2.15.25d) vanish and u D p2 ¤ 0 on . But if  is not an open ellipsoid and has a Lipschitz continuous boundary , inequality (2.15.25d) holds in H 3 ./.

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

195

Proof of Theorem 2.15.3B. We give the scheme of the proof, for example, for (I), as follows: first of all, it is shown that W1 D H m ./ equipped with the new norm defined by the right-hand side expression relative to the first inequality in (2.15.25a) is complete. Let W2 D H m ./ equipped with the original norm k  km; . Hence, W1 and W2 are Banach spaces. Then I W W1 ! W2 is a bijective linear, continuous mapping from W1 onto W2 and as a consequence (Corollary A.8.1.1, Appendix A) of the Open Mapping Theorem A.8.1.3, I 1 W W2 ! W1 is also continuous, i.e. I is an isomorphism from W1 onto W2 and the result follows.

2.15.11 Density results 

D./ is dense in H0m ./ by Definition 2.15.2.

(2.15.26a)



D.Rn / is dense in H m .Rn / 8m 2 N (see Theorem 8.9.6).

(2.15.26b)







For m2 < m1 , H0m1 ./ is dense in H0m2 ./(by virtue of (2.15.26a) and imbedding (2.15.27b). (2.15.26c) D./ is dense in L2 ./ (see Corollary 2.15.1 and Theorem 6.8.3). But D./ is not dense in H m ./ for  ¤ Rn with measure .Rn n/ > 0, 8m 2 N. See Proposition 2.15.7 for the proof with p D 2. See also (2.15.27f). (2.15.26d) D./ D ¹

W 9 2 D.Rn / such that

D # º (see Definition 8.10.2) (2.15.26e)

is dense in H m ./ for any  with the m-extension property (see Theorems 8.10.1 and 8.10.2 in Section 8.10, Chapter 8, which together prove the result). 



For arbitrary domain , D./ \ H m ./ (resp. C m ./ \ H m ./) is dense in H m ./ (see Theorem 2.15.8). (2.15.26f) For other density results, see Chapters 6 and 8.

2.15.12 Algebraic inclusions () and imbedding (,!) results For m1 ; m2 2 N with m1 > m2 , the following results hold: 1. H m1 ./ ,! H m2 ./, i.e. H m1 ./  H m2 ./ (algebraic inclusion) and 9C > 0 such that kukm2 ;  C kukm1 ; 8u 2 H m1 ./ (continuity of ,!), which follow from Definition 2.15.1 and (2.15.10) with C D 1; (2.15.27a) 2. H0m1 ./ ,! H0m2 ./, i.e. H0m1 ./  H0m2 ./ and kukm2 ;  C kukm1 ; with C D 1; 8u 2 H0m1 ./; (2.15.27b) 3. D./  H0m ./  H m ./  D 0 ./ 8m 2 N;

(2.15.27c)

4. for  D Rn , H0m .Rn /  H m .Rn / (see Theorem 8.9.6);

(2.15.27d)

5. for  ¨ Rn with measure .Rn n / > 0, H0m ./   H m ./.

(2.15.27e)

196

Chapter 2 Differentiation of distributions and application of distributional derivatives

But we have the counterexample, for n  2 and  D Rn n ¹0º with .Rn n / D .¹0º/ D 0; H01 ./ D H 1 ./ (see Brezis [26]). (2.15.27f) For more imbedding results, see Section 4.3, Section 8.9 (Theorems 8.9.4 and 8.9.5, (8.9.33), Proposition 8.9.2), Section 8.10 (Proposition 8.10.3), Sections 8.11 and 8.12. For compact imbedding (,!,!) results, see Sections 8.11 and 8.12, Chapter 8. Space H s ./ for arbitrary s 2 R For  D Rn , H s .Rn / with s > 0 defined with the help of Fourier transforms of tempered distributions, and its dual H s .Rn /, are studied in Section 8.9, Chapter 8. s For    Rn , H s ./ with s > 0 and their closed subspaces H0s ./, H00 ./, and s s 0 the dual spaces H ./, .H00 .// , are defined in Sections 8.10 and 8.11, Chapter 8.

2.15.13 Space W m;p ./ with m 2 N, 1  p  1 Definition 2.15.3. Let   Rn be an open subset of Rn . Then, 8m 2 N and 1  p  1, W m;p ./ is the set of all (equivalence classes Œu of) real-valued functions u 2 Lp ./ whose distributional derivatives @˛ u 2 Lp ./8j˛j  m, i.e. 

for 1  p < 1, W m;p ./ D ¹u W u 2 Lp ./; @˛ u 2 Lp ./ 8j˛j  mº;



(2.15.28)

for p D 1, W m;1 ./ D ¹u W u 2 L1 ./; @˛ u 2 L1 ./ 8j˛j  mº:

Then W m;p ./; 1  p  1, is a linear space. Proposition 2.15.5. 8m 2 N, 1  p  1, W m;p ./ is a normed linear space equipped with the norm k  km;p; and the semi-norm j  jm;p; defined, 8u 2 W m;p ./, by: 

for 1  p < 1,  kukm;p; D

p kukLp ./

X

C

k@

˛

p ukLp ./

 p1

1j˛jm

Z

jujp d x C

D 

X 1j˛jm

 p1 j@˛ ujp d x ;

Z 

(2.15.29)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./ 

197

for p D 1, X

kukm;1; D kukL1 ./ C

k@˛ ukL1 ./

1j˛jm

D ess sup ju.x/j C x2

kukm;1; D

X

.ess sup j@˛ u.x/j/

or

max ¹ess sup j@˛ u.x/jº:

0j˛jm

(2.15.30)

x2

1j˛jm

(2.15.31)

x2

The norms (2.15.30) and (2.15.31) are equivalent in W m;1 ./. For other equivalent norms in W m;p ./, see (2.15.36a) and (2.15.36b) later. ess sup is the essential supremum (see Appendix B, Definition B.2.1.2). 

for 1  p < 1, jujm;p; D

 X

k@

˛

p ukLp ./

 p1 D

 X Z

j˛jDm 

j˛jDm

 p1 j@ uj d x I (2.15.32) ˛

p



for p D 1, jujm;1; D

X

k@˛ ukL1 ./ D

j˛jDm

X j˛jDm

jujm;1; D max ¹ess sup j@˛ u.x/jº: j˛jDm

ess sup j@˛ u.x/j

or

x2

(2.15.33)

x2

Theorem 2.15.4. 8m 2 N, 1  p  1, W m;p ./ equipped with the norm k  km;p; defined in (2.15.29)–(2.15.31) is a Banach space. I. For 1  p < 1, W m;p ./ is separable. II. For 1 < p < 1; W m;p ./ is reflexive. Proof. Since Lp ./ is a Banach space for 1  p  1, the proof is exactly similar to that of Theorem 2.15.1 if L2 ./ is replaced by Lp ./ and ‘Hilbert space’ by ‘Banach space’. I. The proof is exactly similar to that given for the separability of H m ./  W m;2 ./ in Proposition 2.15.2 if we replace L2 ./ by Lp ./ with 1  p < 1 everywhere; W by Wp  .Lp .//N ; H m ./ by W m;p ./ with 1  p < 1; and use the separability of Lp ./ for 1  p < 1; the consequent separability of .Lp .//N for 1  p < 1; the separability of the closed subspace Wp  IŒW m;p ./ of .Lp .//N ; and, finally, the separability of I 1 .Wp /  W m;p ./, I being the isometric isomorphism (see Appendix A) from W m;p ./ onto Wp  .Lp .//N , 1  p < 1: kIuk.Lp .//N D kukm;p; 8u 2 W m;p ./.

198

Chapter 2 Differentiation of distributions and application of distributional derivatives

II. For the proof of the reflexivity of W m;p ./, 1 < p < 1, we prepare the following results: (a) Since Lq ./ D .Lp .//0 for 1 < p < 1,

1 p

C

1 q

D 1,

Œ.Lp .//N 0 D ŒLp ./      Lp ./0 D ŒLq ./      Lq ./ „ „ ƒ‚ … ƒ‚ … N times

q

N times

N

D Œ.L .// : Then, 8l 2 Œ.Lp .//N 0 , 9v D .vi /1iN 2 .Lq .//N such that 8u D .ui /1iN 2 ŒLp ./N , we have l.u/ D

N X

hui ; vi iLp ./Lq ./ D

iD1

N Z X

ui .x/vi .x/d x:

(2.15.33a)

iD1 

(b) Continuous linear functionals on W m;p ./, 1 < p < 1: Let L 2 .W m;p .//0 be a continuous linear functional on W m;p ./ with jL.u/j  kLk.W m;p .//0 kukm;p; 8u 2 W m;p ./. Define an isometric isomorphism (see Appendix A) I W W m;p ./ ! Wp  .Lp .//N (see details in the proof of Proposition 2.15.2 for the separability of H m ./  ˛ W m;2 ./) from W m;p ./ onto P Wp such that Iu D .@ u/0j˛jm 2 p N Wp  .L .// (N D 0j˛jm 1 D Card¹.˛1 ; : : : ; ˛n / W 0  j˛j  mº), kIuk.Lp .//N D k.@˛ u/0j˛jm k.Lp .//N D kukm;p; 8u 2 W m;p ./, Wp being a closed subspace of .Lp .//N . Then we can define a linear functional L0 on Wp by: L0 .Iu/ D L.u/ 8u 2 W m;p ./ with kL0 kWp0 D supkIuk p N 1 jL0 .Iu/j D supkukm;p; 1 jL.u/j D .L .// kLk.W m;p .//0 . Hence, L0 is a continuous, linear functional on the closed subspace Wp of .Lp .//N , and by Corollary A.7.3.1 of the Hahn–Banach Theorem A.7.2.1 (see Appendix A), L0 can be given a norm-preserving Q 0 .Iu/ D extension to .Lp .//N such that LQ 0 2 Œ.Lp .//N 0 and L m;p Q ./ with kL0 kŒ.Lp .//N 0 D kL0 kWp0 . L0 .Iu/ D L.u/ 8u 2 W Then 9v D .v˛ /0j˛jmP2 .Lq .//N such that 8u D .u˛ /0j˛jm 2 .Lp .//N , LQ 0 .u/ D 0j˛jm hu˛ ; v˛ iLp ./Lq ./ . Hence, 8u 2 W m;p ./, Q 0 .Iu/ D L Q 0 ..@˛ u/0j˛jm / L.u/ D L0 .Iu/ D L X h@˛ u; v˛ iLp ./Lq ./ D 0j˛jm

H)

L.u/ D

X 0j˛jm

Z

@˛ u.x/v˛ .x/d x

8u 2 W m;p ./:



(2.15.34)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

199

Let .uk /1 be a bounded sequence in W m;p ./, i.e. 9C > 0 such that kD1 kukm;p;  C 8k 2 N. For the proof of reflexivity of Banach space W m;p ./, by the Eberlein–Schmulyan Theorem A.11.1.2 in Appendix A (see [4]), for example, it is necessary and sufficient to show that we can extract a subsequence .ukl /1 of the sequence .uk /1 such that .ukl /1 converges weakly lD1 kD1 lD1 m;p ˛ p in W ./:8j˛j  m, 8k 2 N, @ uk 2 L ./ with k@˛ uk kLp ./  kuk km;p;  C H) 8j˛j  m, .@˛ uk /1 is a bounded sequence in the kD1 reflexive Banach space Lp ./ for 1 < p < 1. Hence, we can extract a subsequence .ukl /1 from .uk /1 such that ukl * u weakly in Lp ./ and lD1 kD1 @˛ ukl * w˛ weakly in Lp ./ for 1  j˛j  m. But ukl * u in Lp ./ H) ukl ! u in D 0 ./ (by (2.13.3) and (2.13.4)) H) @˛ ukl ! @˛ u in D 0 ./ (by Theorem 2.9.1). Again, @˛ ukl * w˛ in Lp ./ H) @˛ ukl ! w˛ in D 0 ./ 8 ˛ fixed ˛ with 1  j˛j  m. But the limit is unique. Hence, R w˛˛ D @ u 8j˛j  m. ˛ ˛ p RThus,˛ @ ukl * @ u in L q./ 8j˛j  m H)  @ ukl .x/v˛ .x/d x !  m, 1 < Rp; q < 1. Hence, from  @ u.x/v˛ .x/d x 8v˛ 2 L ./ with j˛jP m;p .//0 , L.u / D ˛ (2.15.34), 8L 2 .W k 0j˛jm  @ ukl .x/v˛ .x/d x ! l R ˛ P 0j˛jm  @ u.x/v˛ .x/d x D L.u/. Thus, ukl * u weakly in W m;p ./, and the reflexivity of W m;p ./ follows. In particular, for p D 2, W m;2 ./  H m ./ equipped with inner product (2.15.35a) h  ;  im; is a Hilbert space. For p ¤ 2, W m;p ./ cannot be equipped with an inner product, and hence is not a Hilbert space. (2.15.35b) We let m D 0 such that W 0;p ./  Lp ./ equipped with the norm kukLp ./ D kuk0;p; . 1  p < 1, Z  p1 p kukLp ./ D kuk0;p; D ju.x/j d x : (2.15.35c) 

p D 1, kukL1 ./ D kuk0;1; D ess sup ju.x/j:

(2.15.35d)

x2

Norm equivalence in W m;p ./, 1  p < 1 In many situations, equivalent norms in W m;p ./ are quite useful. For bounded   Rn , m 2 N, 1  p < 1, the following norms jjj  jjjm;p; and  m;p; defined, 8u 2 W m;p ./, by: 1.

p

p

jjjujjjm;p; D jjjujjjW m;p ./ D .kukLp ./ C jujm;p; /1=p  1=p X p p D kukLp ./ C k@˛ ukLp ./ I j˛jDm

(2.15.36a)

200

Chapter 2 Differentiation of distributions and application of distributional derivatives



2.

m;p;

D kukLp ./ C jujm;p; D kukLp ./ C

X

k@˛ ukLp ./ ;

j˛jDm

(2.15.36b) are equivalent to the original norm kukm;p; given by (2.15.29), j  jm;p; being the semi-norm in W m;p ./ defined by (2.15.32), i.e. 9C1 ; C2 > 0 such that C1 kukm;p;  jjjujjjm;p; (resp.  m;p; )  C2 kukm;p; 8u 2 W m;p ./ (it is understood that constants C1 and C2 have different values). For p D 2, the second equivalent norm  m;2; in (2.15.36b) is not a Hilbert norm, i.e. .W m;2 ./I  m;2; / is a Banach space, but not a Hilbert space, whereas the first equivalent norm jjjujjjm;2; in (2.15.36a) is a Hilbert norm and the corresponding Banach space .W m;2 ./I jjjujjjm;2; / is a Hilbert space with the inner product hh  ;  iim; such that  1=2 X 1=2 ˛ ˛ jjjujjjm;2; D hhu; uiim; D hu; uiL2 ./ C h@ u; @ uiL2 ./ : j˛jDm

(2.15.36c) Example 2.15.7. For m D 2, 1  p < 1, W 2;p ./ can be equipped with ˇ ˇ Z X Z ˇ @2 u.x/ ˇp 1=p p ˇ ˇ dx jjjujjj2;p; D ju.x/j d x C I (2.15.36d) ˇ ˇ   @xi @xj 1i;j n

Z u

2;p;

D

1=p ju.x/jp d x C



X 1i;j n

ˇ 1=p Z ˇ 2 ˇ @ u.x/ ˇp ˇ ˇ dx ; (2.15.36e) ˇ ˇ  @xi @xj

which are equivalent to the original norm: ˇ Z n Z ˇ X ˇ @u ˇp p ˇ ˇ C ju.x/j d x C kuk2;p; D ˇ ˇ   @xi iD1

X

Z ˇ 2 ˇ @ u ˇ ˇ @x @x

1i;j n 

i

j

ˇp 1=p ˇ ˇ dx : ˇ (2.15.36f)

(2.15.36g) For p D 2, jjj  jjj2;2; is a Hilbert norm in H 2 ./  W 2;2 P./ given in (2.15.36c) with hhu; viim; D hu; viL2 ./ C j˛jDm h@˛ u; @˛ viL2 ./ , whereas  2;2; is not a Hilbert norm. When there is no chance of confusion, we will use the same notation k  km;p; to denote any equivalent norm.

2.15.14 Space W0m;p ./, 1  p < 1 Definition 2.15.4. Let   Rn be an open subset of Rn . Then, 8m 2 N; 1  p < m;p 1, W0 ./ is the closure of D./  C01 ./ in the norm k  km;p; of W m;p ./,

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

201

i.e. m;p

W0

./ D D./ D C01 ./ in W m;p ./; 1  p < 1: m;p

In other words, D./ D C01 ./ is dense in W0 In particular, for p D 2; W0m;2 ./ D H0m ./. m;p

Alternative characterization of W0

(2.15.37)

./ for 1  p < 1, 8m 2 N0 . (2.15.38)

./, m 2 N, 1 < p < 1

As in the case of H0m ./ (see the examples in (2.15.23)), for domains with sufficiently m;p smooth boundary , there is an alternative characterization of W0 ./ with the help of trace theorems (see, for example, Theorem 8.9.11 for p D 2,  D Rn ;  D Rn1 ), for which we refer to Neˇcas [16], Grisvard [17], [18], [19], etc. For example, for m D 1; p D 2; W01;2 ./ D H01 ./ and m D 2; p D 2; W02;2 ./ D H02 ./ are characterized by (2.15.23). m;p

Theorem 2.15.4A. For m 2 N and 1  p < 1; W0 m;p space. For 1 < p < 1; W0 ./ is reflexive.

./ is a separable Banach

Proof. The proof is similar to that of Theorem 2.15.2 by virtue of (2.15.37), and Theorem 2.15.4. m;p

Null extension of functions of W0

./, 1  p < 1 m;p

Theorem 2.15.5. Let   Rn be an open set in Rn and u 2 W0 p < 1, and e u be its null extension ´ u.x/ for x 2  e u.x/ D 0 for x 2 Rn n : Then

./ with 1 

A e

u D .@˛ u/ in the distributional sense in Rn , where I. 8j˛j  m, @˛e ´ @˛ u.x/ for x 2  .@˛ u/.x/ D 0 for x 2 Rn n I II. uQ 2 W m;p .Rn / with kuk Q m;p;Rn D kukm;p; (isometry). In particular, for p D 2, u 2 H0m ./ H) uQ 2 H m .Rn / with kuk Q m;Rn D kukm; . m;p

Proof. By definition, D./ is dense in W0 ./ for 1  p < 1 and 8m 2 N. m;p Hence, 9 a sequence .k / in D./ such that k ! u in W0 ./ H) @˛ k ! @˛ u in Lp ./ 8j˛j  m, 1  p < 1 as k ! 1. But @˛ u 2 Lp ./ H) @˛ u 2 Lp .Rn / with k@˛ ukLp ./ D k.@˛ u/kLp .Rn / .

A

e

202

Chapter 2 Differentiation of distributions and application of distributional derivatives

I. From the definition of distributional derivatives in (2.3.1): 8 2 D.Rn /, 8j˛j  m, Z ˛ j˛j ˛ j˛j ˛ h@ u; Q i D .1/ hu; Q @ i D .1/ u.x/@ Q .x/d x Rn Z Z D .1/j˛j u.x/@˛ .x/d x D .1/j˛j lim k .x/@˛ .x/d x k!1 



strongly

(since k ! u in Lp ./ H) k * u weakly in Lp ./ 1 1 with @˛ 2 Lq ./, C D 1) p q Z k .x/@˛ .x/d x D lim .1/j˛j k!1  Z D lim @˛ k .x/ .x/d x k!1  Z D @˛ u.x/ .x/d x  strongly

(since @˛ k ! @˛ u in Lp ./ H) @˛ k * @˛ u weakly in Lp ./) Z @˛ u.x/ .x/d x D

e

Rn

e 8 2 D.R / ” @ uQ D .@eu/ in D .R / with .@eu/ 2 L .R / 8j˛j  m. X Z X Z kuk Q D j@ uj Q dx D j@euj d x n

D h@˛ u; i

˛

II.

0

˛

p m;p;Rn

n

˛

˛

0j˛jm

D

X 0j˛jm



n

p

Rn

Z

p

˛

0j˛jm

p

.by (I)/

Rn

p

j@˛ ujp d x D kukm;p;

H) kuk Q m;p;Rn D kukm;p; .

Proposition 2.15.6. Let  be a C 1 -regular bounded domain with boundary  (see Definition 8.10.4 and also Appendix D). Let u 2 W 1;p ./; 1 < p < 1, such that 1;p u# D 0. Then u 2 W0 ./. m;p

Norm equivalence in W0

./, 1  p < 1

m;p

The norm equivalence in W0

./ follows from the results given by:

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

203

Proposition 2.15.7. Let   Rn be a bounded domain. Then 9 C D C.; p/ > 0 1;p such that 8u 2 W0 ./, 1  p < 1,  X 1=p n   @u p 1.   D C juj1;p; : kukW 1;p ./  C kr uk.Lp .//n D C  @x  p i L ./

iD1

(2.15.39a) m;p

In general, 8m 2 N, 8u 2 W0 that: 2.

./, 1  p < 1, 9C D C.; m; p/ > 0 such

kukLp ./  C jujm;p; D C

 X

k@

˛

p ukLp ./

 p1 :

(2.15.39b)

j˛jDm m;p

3. The semi-norm jujm;p; is a norm in W0 k  km;p; defined by (2.15.29). m;p

4. .W0 1.

./ equivalent to the original norm (2.15.39c)

./I j  jm;p; / is a separable Banach space and reflexive for 1 < p < (2.15.39d)

Proof. The proof is similar to that of Theorem 2.15.3, if we replace L2 ./ by Lp ./ m;p and H0m ./ by W0 ./; 1  p < 1, m 2 N, and use the density of D./ in m;p W0 ./ along with (2.15.36a).

2.15.15 Space W m;q ./ W m;q ./ with m 2 N; 1 < q  1; 1  p < 1; p1 C m;p m;p of W0 ./, i.e. W m;q ./ D .W0 .//0 (see Section 4.3 for more details and other results).

1 q

D 1, is defined as the dual (2.15.40)

2.15.16 Quotient space W m;p ./=M for m 2 N; 1  p < 1 Definition 2.15.5. Let M be a closed subspace of Banach space W m;p ./ for m 2 N; 1  p < 1. Then, the quotient space W m;p ./=M (i.e. the quotient of W m;p ./ by M ) is a Banach space of equivalence classes Œu of functions u 2 W m;p ./ satisfying the property: u; v 2 Œu ” u  v 2 M ” u D v .mod M //, i.e:Œu D u C M; u C M being the coset of u relative to M and equipped with the usual quotient norm kŒ  kW m;p ./=M defined, 8Œu 2 W m;p ./=M , by: kŒukW m;p ./=M D inf kukm;p; D inf ku C wkm;p; : u2Œu

w2M

(2.15.41)

For canonical surjection  W W m;p ./ ! W m;p ./=M; (2.15.24a)–(2.15.24d) hold with H m ./ replaced by W m;p ./. For p D2, W m;2 ./=M H m ./=M is a Hilbert space (see Proposition 2.15.4B).

204

Chapter 2 Differentiation of distributions and application of distributional derivatives

The most important quotient space is W m;p ./=Pm1 with M D Pm1 , Pm1 being the N -dimensional (hence, closed) subspace of polynomials of degree  m  1 in n variables x1 ; : : : ; xn defined on , since this is used in many problems and specifically in error estimates for finite element approximations (see Bernadou [34], m;p Ciarlet [35]); the other one being the case M D W0 ./, which will not be dealt with here. For p D 2, see Proposition 2.15.4B. Let Pm1 be the linear space of polynomials  m  1 in n variables

of degree D N < C1. Hence, Pm1 x1 ; : : : ; xn defined on  with dim.Pm1 / D nC.m1/ m1 is a closed subspace of W m;p ./ for m 2 N; 1  p < 1. Then we have: Lemma 2.15.2. Let   Rn be a bounded domain with Lipschitz continuous boundm;p ./ such that 8q 2 ary . Then 9 continuous linear functionals ¹li ºN iD1 on W Pm1 , N X

jhli ; qijp D 0

”

q D 0 in Pm1 .

(2.15.42)

iD1 N 0 Proof. Let ¹qi ºN iD1 be a basis in Pm1 . Then 9 a unique dual basis ¹li ºiD1 in Pm1 such that

hli ; qj i D ıij ;

i  i; j  N:

(2.15.43)

m;p ./,

Since Pm1 is an N -dimensional closed subspace of W in which all norms are equivalent, li ’s are continuous on Pm1 in the norm k  km;p; of W m;p ./. Hence, by Corollary A.7.3.1 of the Hahn–Banach Theorem A.7.2.1 in Appendix A, each li ; i  i  N , can be extended to a continuous, linear functional on W m;p ./ (each extended continuous functional will still be denoted by li ; i  i  N ) such that (2.15.42) holds. P p In fact, for q 2 Pm1 , N iD1 jhli ; qij D 0 ” hli ; qi D 0, 1  i  N , for q 2 Pm1 . Hence, it is sufficient toPshow that hli ; qi D 0, 1  i  N ” q D 0 in Pm1 . For q 2 Pm1 ; q D jND1 ˛j qj . Then, for 1  i  N; hli ; qi D PN PN j D1 ˛j li .qj / D j D1 ˛j ıij D ˛i (by (2.15.43)) and hli ; qi D 0 ” ˛i D 0 for 1  i  N . Hence, q D 0 in Pm1 . .li /N , for iD1 can be constructed in different Rways. For example, for bounded R ˛ D ˛.i /, 0  j˛j  m  1; hl˛ ; qi D  @˛ q.x/d x, or hl˛ ; qi D  x˛ q.x/d x, etc. Theorem 2.15.6. Let   Rn be a bounded domain with Lipschitz continuous boundary  (see Definition D.2.3.1, Appendix D) and ¹li ºN iD1 be the continuous linear functionals on W m;p ./, m 2 N, 1  p < 1 (for the sake of simplicity, p D 1 is not considered) satisfying (2.15.42) and (2.15.43). Then 9C1 ; C2 > 0 such that  1=p N X p p C1 kukm;p;  jujm;p; C jhli ; uij  C2 kukm;p; : (2.15.44) iD1

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

205

Proof. The right-hand side inequality holds: Since li 2 .W m;p .//0 , 9CQ i > 0 such that jhl ; uij  CQ i kukm;p; 8u 2 W m;p ./. Hence, 8u 2 W m;p ./, PN PN i p p Q Q Qp iD1 jhli ; uij  C kukm;p; with C  iD1 Ci > 0 and jujm;p;  kukm;p; . Then, 

p jujm;p;

C

N X

jhli ; uij

p

1=p  C2 kukm;p;

(2.15.45)

iD1

with C2  .1 C CQ /1=p > 0. The left-hand side inequality holds: We give an indirect proof. Suppose that the lefthand side inequality does not hold for any C1 > 0. Then, it does not hold for C1 D k1 with k 2 N, i.e. 9uk 2 W m;p ./ such that after its normalization kuk km;p; D 1 8k 2 N,  1=p N X 1 1 p p kuk km;p; D > juk jm;p; C jhli ; uk ij k k

8k 2 N:

iD1

P p p p Then .juk jm;p; C N iD1 jhli ; uk ij / ! 0 as k ! 1 H) juk jm;p; ! 0 and PN p ˛ p iD1 jhli ; uk ij ! 0 as k ! 1 H) limk!1 @ uk D 0 in L ./ 8j˛j D m and limk!1 hli ; uk i D 0; 1  i  N . (2.15.46) m;p m1;p Since W ./ ,!,! W ./, i.e. the imbedding ,!,! is compact from W m;p ./ to W m1;p ./ by the Rellich–Kondraschov Theorem 8.11.4 in Section 8.11, Chapter 8, we can extract a subsequence .ukl /l2N of the bounded sequence .uk /k2N in W m;p ./ such that .ukl /l2N converges strongly in W m1;p ./, i.e. 9u 2 W m1;p ./ such that lim ku  ukl km1;p; D 0:

(2.15.47)

l!1

From (2.15.46), limk!1 @˛ uk D 0 in Lp ./ 8j˛j D m H) liml!1 @˛ ukl D 0 in Lp ./ 8j˛j D m. Then, from (2.15.47), @˛ ukl ! @˛ u in Lp ./ 8j˛j  m  1 and @˛ ukl ! 0 in Lp ./ 8j˛j D m H) @˛ ukl ! @˛ u in Lp ./ 8j˛j  m with @˛ u D 0 8j˛j D m H)

lim ukl D u

l!1

in W m;p ./ with @˛ u D 0 8j˛j D m.

(2.15.48)

Since  is a connected set and @˛ u D 0 8j˛j D m, u is a polynomial of degree  m  1 by Proposition 2.8.1, i.e. u 2 Pm1 , Pm1  W m;p ./ being a closed subspace of W m;p ./. ukl ! u in W m;p ./ with u 2 Pm1 H) ukl * u weakly in W m;p ./ with u 2 Pm1 H) 8i D 1; 2; : : : ; N , hli ; ukl i ! hli ; ui

H)

N X iD1

jhli ; ukl ijp !

N X iD1

jhli ; uijp

as l ! 1.

206

Chapter 2 Differentiation of distributions and application of distributional derivatives

P p ! 0 as l ! 1 and the limit is unique. But, from (2.15.46), N iD1 jhli ; ukl ij PN Hence, iD1 jhli ; uijp D 0 for u 2 Pm1 ” u D 0 in W m;p ./ by (2.15.42). Thus we have a contradiction, since 0 D kukm;p; D liml!1 kukl km;p; D limk!1 kuk km;p; D 1. Hence, our original assumption is wrong and the result follows. Norm equivalence in quotient space W m;p ./=Pm1 For W m;p ./=Pm1 equipped with the norm kŒ  kW m;p ./=Pm1 , the mapping Œu 2 W m;p ./=Pm1 7! jŒujW m;p ./=Pm1 D jujm;p; with u 2 Œu is a priori a seminorm jŒujW m;p ./=Pm1 . In fact, 8q 2 Pm1 ; @˛ q D 0 8j˛j D m H) 8u 2 W m;p ./, @˛ .u C q/ D @˛ u ˛

8q 2 Pm1 8j˛j D m

˛

H) k@ .u C q/kLp ./ D k@ ukLp ./ H)

ju C qjm;p; D jujm;p;

8j˛j D m

8u 2 Œu

with Œu 2 W m;p ./=Pm1 , 8q 2 Pm1 . Then, kŒukW m;p ./=Pm1 D D

inf

ku C qkm;p;

inf

 p jujm;p; C

q2Pm1

q2Pm1

X

p

k@˛ .u C q/kLp ./

1=p

1j˛jm1

 D jujm;p; C

inf

q2Pm1

X

p

k@˛ .u C q/kLp ./

1=p :

1j˛jm1

(2.15.49) This suggests the definition of the semi-norm jŒ  jW m;p ./=Pm1 should be: jŒujW m;p ./=Pm1 D jujm;p;

8Œu 2 W m;p ./=Pm1 with u 2 Œu. (2.15.50)

Now we show that the semi-norm jŒ  jW m;p ./=Pm1 is a norm in W m;p ./=Pm1 equivalent to the original norm kŒ  kW m;p ./=Pm1 . Theorem 2.15.7. Let   Rn be a bounded domain with Lipschitz continuous boundary  (see Definition D.2.3.1, Appendix D). Then 9C1 ; C2 > 0 such that, 8Œu 2 W m;p ./=Pm1 , m 2 N, 1  p < 1, with u 2 Œu, C1 kŒukW m;p ./=Pm1  jujm;p; D jŒujW m;p ./=Pm1  C2 kŒukW m;p ./=Pm1 : (2.15.51)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./

207

For p D 2, W m;2 ./=Pm1  H m ./=Pm1 is a Hilbert space equipped with the inner product hŒ  ; Œ  iH m ./=Pm1 : 8Œu; Œv 2 H m ./=Pm1 with u 2 Œu, v 2 Œv, X X Z ˛ ˛ h@ u; @ viL2 ./ D @˛ u.x/@˛ v.x/d x: hŒu; ŒviH m ./=Pm1 D j˛jDm

j˛jDm



(2.15.52) Proof of Theorem 2.15.7. 8u 2 W m;p ./, the inequality (2.15.44) holds with li 2 .W m;p .//0 , hli ; qj i D ıij , 1  i; j  N D dim.Pm1 /, .qi /N iD1 being a basis in m;p ./, 9qu 2 Pm1 such that hli ; u C qu i D 0 for 1  i  Pm1 . But 8u 2 W N. (2.15.53) P In fact, we define qu D  jND1 hlj ; uiqj 2 Pm1 . Then, for 1  i  N , hli ; qu i D 

N X

hlj ; uihli ; qj i D 

j D1

N X

hlj ; uiıij D hli ; ui

j D1

H) hli ; qu i C hli ; ui D 0 for 1  i  N H) hli ; u C qu i D 0 8i D 1; 2; : : : ; N . Then, for qu 2 Pm1 satisfying (2.15.53), hli ; u C qu i D 0, we get from (2.15.44): 9C1 > 0 such that 8Œu 2 W m;p ./=Pm1 with u 2 Œu, C1 kŒukW m;p ./=Pm1 D C1 .

inf

w2Pm1

ku C wkm;p; /  C1 ku C qu km;p;

p

 .ju C qu jm;p; C 0/1=p D ju C qu jm;p; D jujm;p; ; since for qu 2 Pm1 , ju C qu jm;p; D jujm;p; . Thus, we have proved the left-hand side inequality: C1 kŒukW m;p ./=Pm1  jujm;p; D jŒujW m;p ./=Pm1  kŒukW m;p ./=Pm1

(by (2.15.50))

8Œu 2 W m;p ./=Pm1 with u 2 Œu;

the right-hand side inequality holds with C2 D 1 > 0 and the result follows. Since the semi-norm jŒujW m;p ./=Pm1 D jujW m;p ./ is a norm in W m;p ./= Pm1 equivalent to the original norm kŒukW m;p ./=Pm1 , for p D 2, the result follows immediately.

2.15.17 Density results 

m;p

D./ is dense in W0 by Definition 2.15.4.

./ for 1  p < 1; m 2 N and arbitrary  (2.15.54a)

208 





Chapter 2 Differentiation of distributions and application of distributional derivatives

D.Rn / is dense in W m;p .Rn / for 1  p < 1, m 2 N0 by Theorem 6.8.9. (2.15.54b) m1 ;p

For fixed p with 1  p < 1 and m2 < m1 , W0

m ;p

./ is dense in W0 2 ./. (2.15.54c)

D./ defined by (2.15.26e) (see also Definition 8.10.2) is dense in W m;p ./ for  with the m-extension property (for example,  with a Lipschitz continuous boundary, or  a C m -regular domain (see Definition 8.10.4 and Appendix D)), 1  p < 1, m 2 N (see also (8.10.98d) Section 8.10). (2.15.54d)

Theorem 2.15.8 (Meyers and Serrin (see Adams [12, p. 52])). For arbitrary domain   Rn , D./ \ W m;p ./ (resp. C m;p ./ \ W m;p ./), 1  p < 1, m 2 N, is dense in W m;p ./, i.e. 8u 2 W m;p ./, 9 a sequence .uk /k2N in D./\W m;p ./ (resp. C m;p ./ \ W m;p ./) such that uk ! u in W m;p ./ as k ! 1.

2.15.18 A non-density result Proposition 2.15.8. For  ¤ Rn with measure .Rn n / > 0, D./ is not dense in W m;p ./ 8m 2 N; 1  p < 1. (Precisely speaking, for  with { D Rn n  not .mI q/-polar, p1 C q1 D 1 (see Lions [13]).) In particular, for p D 2, D./ is not dense in H m ./. (2.15.54e) Proof. Suppose that the contrary holds, i.e. D./ is dense in W m;p ./. Consider the continuous, linear mapping  2 D./ 7! Q 2 D.Rn /; Q being the null extenQ m;p;Rn D kkm;p; . Since, sion to Rn of  as defined in Theorem 2.15.5 with kk m;p by our assumption, D./ is dense in W ./, this linear continuous mapping can be extended by continuity to a continuous linear mapping u 2 W m;p ./ 7! uQ 2 W m;p .Rn /, uQ being the null extension to Rn of u, from W m;p ./ into W m;p .Rn / with kuk Q m;p;Rn D kukm;p; , which is impossible if we choose, for example,  D B.0I 1/ D unit ball in Rn with boundary  and u D 1 in . Then .Rn n / D .Rn n B.0I 1// > 0 and u D 1 2 W m;p ./8m 2 N; 1  p < 1, but its null exten@u Q sion uQ … W m;p .Rn /. In fact, the distributional derivative @x 2 D 0 .Rn / of disconk tinuous uQ is given by an expression containing the Dirac distribution ı concentrated @u Q on the boundary . (For n D 2, see (3.1.29) in Chapter 3 with Œ @x .x/ D 0, 0 D , k

@u Q @u Q … L1loc .Rn / H) @x … Lp .Rn / jump J0 D 0  1 D 1, ‚k D n  ik .) Hence, @x k k H) uQ … W 1;p .Rn / H) uQ … W m;p .Rn /, which contradicts that uQ 2 W m;p .Rn / 8m 2 N; 1  p < 1. Hence, our original assumption that D./ is dense in W m;p ./ is wrong and the result follows. 

For bounded domain   Rn with a sufficiently smooth (for example, C m regular or Lipschitz continuous (see Definition D.2.3.1 in Appendix D)) boundary, D./ is not dense in W m;p ./, 1  p  1, m 2 N. (2.15.54f)

Section 2.15 Applications: Sobolev spaces H m ./; W m;p ./ 



209

For arbitrary   Rn , D./ is not dense in W m;1 ./ for any m 2 N0 . (2.15.54g) For arbitrary domain   Rn , D./ is not dense in W m;p ./, 1  p < 1, m 2 N (see also Theorem 2.15.8). (2.15.54h) 2 For example, for   R in the example in Remark 8.10.2, which does not possess the m-extension property for m D 1, D./ is not dense in W 1;p ./, 1  p < 1. In fact, for  D ¹.x1 ; x2 / W 0 < jx1 j < 1, 0 < x2 < 1º and u 2 W 1;2 ./  H 1 ./ defined by u.x1 ; x2 / D 1 for 0 < x1 < 1 and D 0 for 1 < x1 < 0, x2 2 0; 1Œ, (see (8.10.26a)), À any sequence .uk /k2N in D./ such that uk ! u in W 1;2 ./ as k ! 1.

For other density results, see Section 8.10, Chapter 8.

2.15.19 Algebraic inclusion  and imbedding (,!) results 1. For fixed p with 1  p  1 and m1 > m2 , W m1 ;p ./ ,! W m2 ;p ./, i.e. W m1 ;p ./  W m2 ;p ./ (algebraic inclusion) and 9C > 0 such that kukm2 ;p;  C kukm1 ;p; 8u 2 W m1 ;p ./ (continuity of ,!), which follows from Definition 2.15.3 and (2.15.29)–(2.15.31). with C D 1; (2.15.55a) m ;p

m ;p

2. For fixed p with 1  p < 1 and m1 > m2 , W0 1 ./ ,! W0 2 ./, i.e. m ;p m ;p W0 1 ./  W0 2 ./ and 9C > 0 such that kukm2 ;p;  C kukm1 ;p; m1 ;p 8u 2 W0 ./. (2.15.55b) m;p

3. D./ ,! W0

./ ,! Wm;p ./ ,! D 0 ./ 8m 2 N, 1  p < 1. (2.15.55c)

m;p

4. For  D Rn , W0 Rn

.Rn /  W m;p .Rn / (see Theorem 8.9.6). .Rn

m;p W0 ./

5. For    with measure n / > 0, 1, m 2 N (see counterexample in (2.15.27f)).

(2.15.55d)

  Wm;p ./, 1  p < (2.15.55e)

For more imbedding (,!) results, see Sections 4.3, 8.10 and 8.12. For compact imbedding (,!,!) results, see Sections 8.11 and 8.12, Chapter 8.

2.15.20 Space W s;p ./ for arbitrary s 2 R s;p

s;p

For W s;p ./ with s > 0, 1  p < 1, and its closed subspaces W0 ./, W00 ./ s;p for s > 0, 1 < p < 1, and their dual spaces W s;q ./, .W00 .//0 , see Sections 8.10–8.12, Chapter 8. Remark 2.15.1. Since we will be primarily concerned with real-valued functions, we have not considered Sobolev spaces H m ./ (resp. W m;p ./; 1  p  1) of (equivalence classes Œu of) complex-valued functions u 2 L2 ./, @˛ u 2 L2 ./ 8j˛j  m (resp. u 2 Lp ./, @˛ u 2 Lp ./). Complexification of H m ./ (resp. W m;p ./; 1  p  1) is a straightforward procedure. For example, for H m ./ D

210

Chapter 2 Differentiation of distributions and application of distributional derivatives

¹u W u.x/ is a complex-valued function of real variables x, u 2 L2 ./, @˛ u 2 L2 ./ 8j˛j  mº, (2.15.56a) where L2 ./ is a (complex) Hilbert space with complex inner product h  ;  iL2 ./ such that Z u.x/v.x/d x 8u; v 2 L2 ./: (2.15.56b) hu; viL2 ./ D 

Then, (2.15.2)–(2.15.6) hold, and u 2 H m ./ H) u 2 H m ./8 2 C. Hence, is a complex vector space. H m ./ equipped with complex inner product h  ;  im; defined, 8u; v 2 H m ./, by: X hu; vim; D hu; viL2 ./ C h@˛ u; @viL2 ./ H m ./

Z D

1j˛jm

u.x/v.x/d x C 

X 1j˛jm

Z

@˛ u.x/@˛ v.x/d x;

(2.15.56c)



where v.x/; @˛ v.x/ are the complex conjugates of v.x/ and @˛ v.x/ respectively, i.e.

H)

u.x/ D u1 .x/ C i u2 .x/

with u1 .x/ D ReŒu.x/; u2 .x/ D ImŒu.x/; (2.15.57a)

v.x/ D v1 .x/ C iv2 .x/

with v1 .x/ D ReŒv.x/; v2 .x/ D ImŒv.x/ (2.15.57b)

v.x/ D v1 .x/ C iv2 .x/ D v1 .x/  iv2 .x/

a.e. on ;

@˛ v.x/ D @˛ v1 .x/  i @˛ v2 .x/ a.e. on , is a (complex) Hilbert space.

(2.15.57c) (2.15.57d)

All other formulae and results remain unchanged or undergo minor changes owing to (2.15.56a)–(2.15.57d). Hence, these minor changes (2.15.56a)–(2.15.57d) due to complexification will be introduced in later chapters, if necessary, without any accompanying explanation.

Chapter 3

Derivatives of piecewise smooth functions, Green’s formula, elementary solutions, applications to Sobolev spaces

3.1

Distributional derivatives of piecewise smooth functions

Motivation Piecewise smooth functions in general, and piecewise polynomials in particular, are the workhorses needed for efficient and elegant solution of interpolation and approximation problems. In many methods of approximation (for example, the finite element method for elliptic boundary value problems [34], [35], [36]), construction of finite element subspaces of Sobolev spaces H 1 ./; H 2 ./, etc. with the help of piecewise polynomials is essential. For this, the distributional derivatives of these piecewise smooth functions must belong to L2 ./, which suggests studying their differentiation in detail. (For the construction of subspaces of H m ./, m D 1; 2, see Section 3.4 at the end of the chapter).

3.1.1 Case of single variable (n D 1) Let ¹xi ºniD1 be a set of n points on a; bŒ; a < x1 < x2 <    < b. Let f be a piecewise smooth function having continuous ordinary derivatives f 0 .x/; f 00 .x/; : : : ; f .m/ .x/ in the usual pointwise sense everywhere on a;bŒ except at the points ¹xi ºniD1 , where f .x/; f 0 .x/; f 00 .x/; : : : ; f .m/ .x/ have discontinuities of the first kind, i.e. the right-hand side and left-hand side limits of f .x/; f 0 .x/; : : : ; f .m/ .x/ at ¹xi ºniD1 exist: for 0  k  m, f .k/ .xiC / D lim f .k/ .x/; x!xiC

f .k/ .xi / D lim f .k/ .x/; x!xi

f .0/ .x/ D f .x/; .k/

such that f .k/ .x/ has finite jumps Ji .k/

Ji

(3.1.1)

at xi :

D f .k/ .xiC /  f .k/ .xi /I

.0/

Ji

D Ji :

(3.1.2)

212

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions k

The kth-order derivative of f in the sense of distribution will be denoted by ddxfk or f .k/ 2 D 0 .a; bŒ/, and the distributions defined by the usual ordinary derivatives k dkf .x/ or f .k/ .x/ in the pointwise sense will be denoted by Œ ddxfk .x/ or Œf .k/ .x/ 2 dx k D 0 .a; bŒ/ for 1  k  m, i.e. 

D

d k Tf dx k

2 D 0 .a; bŒ/ with

  k  d Tf d kf ; D ;  D .1/k hTf ;  .k/ i D .1/k hf;  .k/ i dx k dx k Z b d k k D .1/ f .x/ k .x/dx 8 2 D.a; bŒ/; dx a

k

and Πddxfk .x/ D T 

dkf dx k

kf dx k

Œd

.x/

(3.1.3)

2 D 0 .a; bŒ/ with

    Z b k  d kf d f .x/ ;  D T d k f ; D .x/ .x/dx Πk .x/ dx k dx k a dx

8 2 D.a; bŒ/: (3.1.4)

k

Note that for 1  k  m, Œ ddxfk .x/ 2 D 0 .a; bŒ/ is a regular distribution, whereas dkf dx k

2 D 0 .a; bŒ/ is not a regular distribution in general. In fact, they are related by:

Theorem 3.1.1. Let f be a piecewise smooth function having derivatives f 0 .x/; f 00 .x/, : : : ; f .m/ .x/ (in the usual pointwise sense) which are continuous everywhere on a; bŒ except at the points ¹xi ºniD1 of discontinuity (of the first kind) of f .k/ .x/, 0  k  m, where conditions (3.1.1) and (3.1.2) hold (see Figure 3.1). Let the kth-order distributional derivative of f on a; bŒ, dkf dx k

k and Πddxfk

dkf dx k

be

.x/ be the distribution

defined by the ordinary derivative .x/ in the usual pointwise sense in (3.1.3) and (3.1.4) respectively. k k Then ddxfk and Œ ddxfk .x/ in D 0 .a; bŒ/ are related by:

k D 1W

  X n df df D .x/ C Ji ıxi I dx dx iD1

k D 2W :: :

d 2f dx 2

 D :: :

 X n n X .1/ .1/ .x/ C J ı C Ji ıxi I i xi 2

d 2f dx

iD1

iD1

(3.1.5)

213

Section 3.1 Distributional derivatives of piecewise smooth functions

0 Figure 3.1 Piecewise smooth function f with finite jumps Ji at points of discontinuity of the first kind at x1 < x2 <    < xn

 k  X n n X d f d kf .1/ .k1/ D .x/ C J ı C Ji ıx.k2/ C  i xi i dx k dx k iD1 iD1 C

n X

.k1/

Ji

ıxi ;

1  k  m;

iD1

where ıxi 2 D 0 .a; bŒ/ is the singular Dirac distribution with mass/charge/force etc., as the case may be, concentrated at xi and defined by hıxi ; i D .xi / 8 2 .k/ D.a; bŒ/, 1  i  n (see (1.3.28)); ıxi 2 D 0 .a; bŒ/ is the kth-order derivative of .k/ ıxi defined by hıxi ; i D .1/k hıxi ;  .k/ i D .1/k  .k/ .xi / 8 2 D.a; bŒ/ (see (2.3.8)). Proof. We give the proof for k D 1, since for k  2 the proof is similar. From the definition (3.1.3), we have:     Z b d df d ;  D  f; D (3.1.6) f .x/ .x/dx 8 2 D.a; bŒ/; dx dx dx a since f is piecewise continuous on a; bŒ with finite jumps Ji D f .xiC /  f .xi / at the points xi of discontinuity, 1  i  n, and consequently f 2 L1loc .a; bŒ/. Z x Z x Z x Z b Z b 1 i iC1 0 f .x/ .x/dx D .: : : / C    C .: : : / C .: : : / C    C .: : : /: a

a

C xi1

xiC

C xn

(3.1.7) Integrating by parts and applying the properties of  2 D.a; bŒ/ (.xi / D .xiC / D .xi / by virtue of the continuity of  at xi ; .a/ D .b/ D 0), we have Z x Z x 1 1 f .x/ 0 .x/dx D f .x1 /.x1 /  f 0 .x/.x/dx; a

a

214

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

and for .1  i  n  1/, Z Z x iC1 C 0  f .x/ .x/dx D f .xiC1 /.xiC1 /  f .xi /.xi /  xiC

Z

b

C xn

f .x/ 0 .x/dx D f .xnC /.xn / 

Z

b

C xn

f 0 .x/.x/dx

 xiC1

xiC

f 0 .x/.x/dx;

..b/ D 0/: (3.1.8)

.1/

f 0 .x/ is piecewise continuous on a; bŒ with finite jumps Ji D f 0 .xiC /  f 0 .xi / at the points xi of the discontinuity of f 0 .x/; 1  i  n (see (3.1.2)) H) f 0 .x/ 2 L1loc .a; bŒ/. Hence, we can combine the integrals in (3.1.8) into a single integral on a; bŒ and write (3.1.7) as follows: 8 2 D.a; bŒ/, Z

b a

d f .x/ .x/dx D  dx

Z

b

f 0 .x/.x/dx 

a

n X Œf .xiC /  f .xi /.xi /: iD1

But from the definition of the distribution Œ df .x/ 2 D 0 .a; bŒ/ in (3.1.4), we have dx 

  Z b df .x/ ;  D f 0 .x/.x/dx dx a

8 2 D.a; bŒ/:

Hence, Z

b

f .x/ a

   X  X   n n df d df dx D  .x/ ;   .x/ C Ji hıxi ; i D  Ji ıxi ;  : dx dx dx iD1

iD1

Finally, from (3.1.6), we get: 

   X  n df df ; D .x/ C Ji ıxi ;  8 2 D.a; bŒ/ dx dx iD1



”

 X n df df D .x/ C Ji ıxi dx dx

in D 0 .a; bŒ/.

iD1

Remark 3.1.1. 



The effect of the discontinuity of f at xi , 1  i  n, appears in the form of a point mass/charge/force, as the case may be, concentrated at xi in the derivative of f in the distributional sense. Œ df .x/ 2 D 0 .a; bŒ/ is a regular distribution defined by the ordinary derivative dx 2 f 0 .x/ in the usual pointwise sense, whereas the distributional derivative df dx 0 D .a; bŒ/ of f is a singular distribution, i.e. the distributional derivative and the usual derivative in the pointwise sense do not coincide here!

Section 3.1 Distributional derivatives of piecewise smooth functions

215

x Example 3.1.1. Consider the sawtooth function f .x/ D 12  2 for x 2 0; 2Œ such that f is 2-periodic on 1; 1Œ, i.e. f .x ˙ 2/ D f .x/. Then f has discontinuities at x D 0; ˙2; ˙4; : : : with jump Jk D 12  . 12 / D 1 at xk D ˙2k with k D 0; 1; : : : ; 1 (see FigureP3.2). P1 1 Hence, f 0 D Œ df .x/ C 1 kD0 1  ı˙2k D  2 C kD1 ı2k , where ı2k D dx ı.x  2k/ is Dirac distribution with unit mass/charge/force concentrated P1 at x D 2k, k D 0; ˙1; ˙2; : : : ; ˙1. In Example 2.11.2, it is shown that kD1 ı2k P converges in D 0 .1; 1Œ/. Moreover, the series 1 ı is a periodic Dirac kD1 2k distribution on R with period 2 (see Section 1.10).

f (x) /

x

0 /

x Figure 3.2 Sawtooth function f .x/ D 12  2 ; x 2 0; 2Œ, with periodic extension f .x ˙ 2k/ D f .x/ and jumps Jk D 1 at xk D ˙2k; k D 0; 1; 2; : : :

3.1.2 Case of two variables (n D 2) Let   R2 be a bounded domain in R2 with a piecewise smooth boundary  such that  D  [  and the unit vector nO normal to  and exterior to  is defined almost everywhere on . The (positive) direction of the unit vector O tangent to  is obtained by rotating the exterior unit normal nO through 2 in the anticlockwise O Oi1 /Oi1 C cos.n; O Oi2 /Oi2 , where .n; O Oik / is the direction (see Figure 3.3), with nO D cos.n; angle between nO and Oik measured in the anticlockwise direction, Oik being the unit vector in the positive direction of xk -axis. Measuring angles in the anticlockwise O Oi1 /, O Oi2 / D .n; direction, we get .; O Oi2 / O Oi1 / D  C .n; .;

H)

O Oi2 /; cos.; O Oi1 /; O Oi1 / D  cos.n; O Oi2 / D cos.n; cos.; (3.1.9)

O Oi2 /ds; O Oi1 /ds D  cos.n; dx1 D cos.;

where ds is the arc length measure.

O Oi1 /ds; O Oi2 /ds D cos.n; dx2 D cos.; (3.1.10)

216

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

x2 i i

i

(i = 1,2)

n̂ 1 (resp. n̂n2) unit normal exterior to

(resp.

)

n̂n

n̂

̂

0

x1

Figure 3.3 Piecewise smooth curve  0 subdividing domain  into two subdomains 1 and 2 with closed boundaries  1 and  2 respectively

Let 0   be a piecewise smooth curve which subdivides  into two subdomains 1 and 2 with boundaries  1 and  2 respectively such that: 

 D 1 [ 2 [ 0 ;  D 1 [ 2 , i D i [  i (see Figure 3.3);



 i D i [ 0 (i D 1; 2),  1 \  2 D 0 ,



 D 1 [ 2 , meas.1 \ 2 / D 0. 2f

@f Let f , Π@x .x/, Π@x@

k @xl

k

(3.1.11)

.x/ 2 C 0 ( n 0 ) with  n 0 D   0 be bounded

and continuous functions in the complement of 0 in  (i.e. discontinuous with finite 2f @f jumps across 0 ), Π@x .x/ and Π@x@ @x .x/ being partial derivatives of f in the usual k k l pointwise sense: for example, for k D 1, 

 @f f .x1 C x1 ; x2 /  f .x1 ; x2 / .x/ D lim :

x1 !0 @x1 x1

Define fi D f #i 0 with fi .x/ D f .x/ 8x 2 i  0

(3.1.12)

217

Section 3.1 Distributional derivatives of piecewise smooth functions 2

@fi such that fi and its partial derivatives Π@x .x/; Π@x@ f@xi .x/ (in the usual pointwise k k l sense) can be continuously extended to 0 as follows: 8 2 0 ,

f1 ./ D

lim

x!;x21

f .x/ D f .  /I

f2 ./ D

lim

x!;x22

f .x/ D f . C /I (3.1.13)



     @f1 @f @f  ./ D lim .x/ D . / I @xk @xk x!;x21 @xk       @f2 @f @f ./ D lim .x/ D . C / I @xk @xk x!;x22 @xk  2    2   2 @ f1 @ f @ f ./ D lim .x/ D .  / I @xk @xl @xk @xl x!;x21 @xk @xl    2   2  2 @ f @ f @ f2 ./ D lim .x/ D . C / : @xk @xl @xk @xl x!;x22 @xk @xl

(3.1.14)

(3.1.15)

@f .x/ and Then the jumps J0 , Jk (k D 1; 2), Jkl D Jlk (k; l D 1; 2) of f , Π@x k

2f

Π@x@

k @xl

.x/ across 0 respectively are defined, 8x 2 0 , by:

J0 D f .xC /  f .x / D f2 .x/  f1 .x/I (3.1.16)         @f C @f  @f2 @f1 Jk D .x /  .x / D .x/  .x/ I (3.1.17) @xk @xk @xk @xk  2   2   2   2  @ f @ f @ f2 @ f1 C  Jkl D .x /  .x / D .x/  .x/ I @xk @xl @xk @xl @xk @xl @xk @xl (3.1.18) such that J0 ; Jk ; Jkl are functions continuous on 0 . 2f @f .x/; Œ @x@ @x .x/ 2 C 0 . n 0 / satisfying (3.1.12)–(3.1.15) will belong to f; Œ @x k k l L1loc ./ (since two-dimensional (area) measure .0 / D 0) and will define regular distributions Tf , TŒ @f .x/ ; T @2 f 2 D 0 ./ respectively, 8 2 D./, by: @xk

Π@x

k @xl

.x/

Z hTf ; i D hf; i D f .x/.x/dx1 dx2 I     Z @f @f hTΠ@f .x/ ; i D .x/ ;  D .x/.x/dx1 dx2 I @xk @xk  @xk   Z   2  @ f @2 f ;  D .x/ ;  D .x/.x/dx1 dx2 : T @xk @xl @2 f  @xk @xl .x/ @xk @xl

(3.1.19) (3.1.20) (3.1.21)

218

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions @2 T

@T

2

@f f f Let @xf D @x 2 D 0 ./ and @x @x D @x@ @x 2 D 0 ./ denote the first- and k k k l k l second-order distributional derivatives of f defined, 8 2 D./, by:       Z @Tf @f @ @ ; D ;  D  f; f dx1 dx2 I (3.1.22) D @xk @xk @xk @xk   2   2    Z @ Tf @ f @2  @2  ; D ;  D f; f dx1 dx2 : (3.1.23) D @xk @xl @xk @xl @xk @xl @xk @xl  2

@f f Since f and its partial derivatives Π@x .x/; Π@x@ @x .x/ in the usual pointwise sense k k l are discontinuous across 0 with finite jumps J0 ; Jk ; Jkl defined by(3.1.16), (3.1.17) @2 T

@T

2

@f f f ; D @x@ @x and (3.1.18) respectively, the distributional derivatives @xf D @x k k @xk @xl k l defined by (3.1.22) and (3.1.23) will not be equal to the regular distributions 2f @f TΠ@f .x/ D Π@x .x/ 2 D 0 ./, T @2 f D Π@x@ @x .x/ 2 D 0 ./ defined by @xk

k

Π@x

k @xl

.x/

l

k

(3.1.20) and (3.1.21) respectively as in the case of a single variable (see Theorem 3.1.1). The relations between the distributional derivatives 2

@Tf @xk

D

@2 Tf @f , @xk @xk @xl

D

@2 f @xk @xl

@f f and the partial derivatives Œ @x .x/ and Œ @x@ @x .x/ in the usual pointwise sense will k k l now be established, but for this we need Green’s Theorem from calculus.

Theorem 3.1.2 (Green’s Theorem [32]). Let D  R2 be a domain with a piecewise @P smooth boundary @D and D D D [ @D (see Figure 3.4). Let P; Q; Œ @x .x/ and 2 @Q Œ @x .x/ 2 C 0 .D/. Then 1  Z  I @Q @P .x/  .x/ dx1 dx2 D P dx1 C Qdx2 (3.1.24) @x2 D @x1 @D  I  O Oi2 / C Q cos.n; O Oi1 / ds; P cos.n; D @D

O Oi2 /ds, dx2 D cos.n; O Oi1 /ds; nO being the unit normal exterior to where dx1 D  cos.n; D, and the line integral over @D is to be understood in the anticlockwise direction so that the domain D remains on the left-hand side while moving along @D.

D

D

Figure 3.4 Domain D with piecewise smooth boundary @D

219

Section 3.1 Distributional derivatives of piecewise smooth functions

Theorem 3.1.3. Let   R2 be a domain with a piecewise smooth boundary  and 0   be a piecewise smooth curve subdividing  into 1 and 2 with boundaries 2f @f  1 and  2 respectively such that (3.1.9)–(3.1.11) hold. Let f; Œ @x .x/; Œ @x@ @x .x/ 2 k

k

l

C 0 . n 0 / satisfy (3.1.12)–(3.1.15) (consequently (3.1.19)–(3.1.21)). Then the dis2 @2 Tf 2 D 0 ./ of f and the partial tributional derivatives @f D fQ and @ f D @xk 2f @f .x/ Œ @x .x/; Œ @x@ @x k k l

@xk @xl

@xk @xl

of f in the usual pointwise sense are related, for

derivatives 1  k; l  2, by:

@Tf D TŒ @f .x/ C J0 cos k ı0 in D 0 ./; @xk @xk

(3.1.25)

@2 Tf @ D T @2 f C .J0 cos k ı0 / C Jk cos l ı0 Œ @x @x .x/ @xk @xl @x l k l

in D 0 ./; (3.1.26)

which can be written equivalently as follows: for 1  k; l  2,   @f @f D .x/ C J0 cos k ı0 in D 0 ./; @xk @xk  2  2 @ f @ f @ D .x/ C .J0 cos k ı0 / C Jk cos l ı0 @xk @xl @xk @xl @xl

(3.1.27) in D 0 ./; (3.1.28)

where TŒ

@f @xk

.x/

;T

2f k @xl

Π@x@

.x/

2 D 0 ./ are regular distributions defined by (3.1.21)

and (3.1.22) respectively; the jumps J0 ; Jk are continuous functions of x 2 0 and O Oik /; nO being the unit normal to defined by (3.1.16) and (3.1.17) respectively; k D .n; 0 and exterior to 1 (see Figure 3.3); Z hJ0 cos k ı0 ; i D J0 cos k ds 8 2 D./; (3.1.29) 0

Z hJk cos l ı0 ; i D

0

Jk cos l ds

8 2 D./

(3.1.30)

define Dirac distributions corresponding to mass or charge placed on 0 with a linear density J0 cos k and Jk cos l respectively. In particular, for l D k, we get the important formula: @2 Tf @xk2

D T @2 f @x 2 k

C @ .J0 cos k ı / C Jk cos k ı 0 0 @xk

.x/

or, equivalently,  2  @ f @2 f @ D .x/ C .J0 cos k ı0 / C Jk cos k ı0 @xk @xk2 @xk2

in D 0 ./

(3.1.31)

in D 0 ./:

(3.1.32)

220

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions @T

@f Proof. The distributional derivative @xf D @x 2 D 0 ./ of f discontinuous on 0 k k is given, 8 2 D./, by:       Z @Tf @ @ @ ;  D  Tf ; f dx1 dx2 D  f; D @xk @xk @xk @x  k Z Z @ @ D f1 dx1 dx2  f2 dx1 dx2 @xk @xk 1 2  2 Z X @ D fi dx1 dx2 @xk i iD1

D

²

2 Z X iD1 i

D



2 Z X iD1 i

   ³ @fi @.fi / .x/  .x/  dx1 dx2 @xk @xk

   2 Z X @fi @.fi / .x/ dx1 dx2  .x/ dx1 dx2 ; (3.1.33) @xk @xk i iD1

@fi .x/ 2 C 0 .i / where fi D f #i 0 (i D 1; 2) satisfies (3.1.12)–(3.1.18), and Œ @x k

with i D i [  i ,  i D i [ 0 (see Figure 3.3). Now we will transform the second double integral over i in (3.1.33) into a line integral along the boundary  i of i in the anticlockwise direction (i D 1; 2) using (3.1.24). For this, set i D D,  i D i [ 0 D @D with i D 1; 2, P Q D fi  with  2R D./, # D 0, and R D 0, @ k D 1, and we have, from (3.1.24): i Π@x1 .fi /dx1 dx2 D  i fi dx2 such that Z Z

Z 1

2

Z

f1 dx2 D

f1 dx2 C Z

1

f2 dx2 D

O Oi1 /ds; f1  cos.n;

f1 dx2 D 0

0

Z f2 dx2 C

2

Z

0

Z

O Oi1 /ds; f2  cos.n;

f2 dx2 D 

(3.1.34)

0

R where #i D 0 H) i fi dx2 D 0, the curve 0 traversed in the opposite direction is denoted by 0 , nO is the unit normal to 0 , the exterior to 1 (see Figure 3.3), O Oi1 /ds (see (3.1.10)). Hence, from (3.1.34), dx2 D cos.n; 

2 Z X iD1 i



 Z @ O Oi1 /ds .fi .x// dx1 dx2 D Œf2 .x/  f1 .x/.x/ cos.n; @x1 0 Z O Oi1 /ds .by (3.1.16)/; D J0  cos.n; 0

(3.1.35) where J0 D J0 .x/ D f2 .x/  f1 .x/ 8x 2 0 is defined by (3.1.16).

221

Section 3.1 Distributional derivatives of piecewise smooth functions

RFor Q@ D 0, P D fi  with R  2 D./ and k D 2, we have, from (3.1.24),  i Π@x2 .fi /.x/dx1 dx2 D  i fi dx1 such that Z Z

Z 1

2

Z

f1 dx1 D

Z

f1 dx1 C 1

Z f2 dx1 D

Z f2 dx1 C

2

O Oi2 /ds; f1  cos.n;

f1 dx1 D  0

0

Z

(3.1.36)

0

f2 dx1 D

O Oi2 /ds; f2  cos.n;

0

R

O Oi2 /ds (see (3.1.10)), # D 0 H) where dx1 D  cos.n; R R   .  /dx1 D  0 .  /dx1 . Hence, from (3.1.36),

i

fi dx1 D 0;

0



2 Z X



iD1 i

  Z  @ O Oi2 /ds .fi /.x/ dx1 dx2 D f2 .x/  f1 .x/ .x/ cos.n; @x2 0 Z O Oi2 /ds (by (3.1.16)): D J0  cos.n; 0

(3.1.37) Thus, from (3.1.33), (3.1.35) and (3.1.37), we have, for k D 1; 2,  Z @ O Oik /dsI .fi /.x/ dx1 dx2 D J0  cos.n; (3.1.38) @xk   0 i iD1  Z     Z  Z @Tf @f1 @f2 ; D .x/ dx1 dx2 C .x/ dx1 dx2 C J0  cos k ds @xk 1 @xk 2 @xk 0  Z Z  @f .x/ dx1 dx2 C J0  cos k ds D  @xk 0 

D hTŒ D hTŒ

”

@Tf @xk



2 Z X

D TŒ

@f @xk

.x/

@f @xk

.x/

@f @xk

.x/

; i C hJ0 cos k ı0 ; i C J0 cos k ı0 ; i

8 2 D./

C J0 cos k ı0 in D 0 ./, which can be written equivalently in

the form (3.1.27). 2f Now we prove (3.1.26)/(3.1.28). Since the partial derivative Π@x@ @x .x/ in the usual k

l

pointwise sense is continuous in   0 , the order of differentiation can be inter-

222

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions 2f

2

changed, i.e. Π@x@

k @xl







f .x/ D Π@x@ @x .x/ 8x 2   0 . Then, l

k

    @Tf @ @ @Tf ; D ; ; D  @xk @xl @xl @xk @xk @xl   @ (using (3.1.25)) D  TŒ @f .x/ C J0 cos k ı0 ; @xk @xl     @ @ D  TŒ @f .x/ ;  J0 cos k ı0 ; @xk @xl @xl    Z  @ @ @f .x/ dx1 dx2 C .J0 cos k ı0 /;  D @xl @xl  @xk @2 Tf



8 2 D./:

(3.1.39)

But     2  ³ Z  Z ² @f @ @f @ @ f  .x/ dx1 dx2 D  .x/  .x/  dx1 dx2 @xl @xk @xl  @xk  @xl @xk    Z  2 Z @ f @ @f D .x/ dx1 dx2  .x/ dx1 dx2 8 2 D./:  @xk @xl  @xl @xk (3.1.40) @f @fi Replacing ‘f ’ by ‘Œ @x .x/’, ‘fi ’ by ‘Œ @x .x/’, ‘xk ’ by ‘xl ’ and ‘ k ’ by ‘ l ’ in k k (3.1.35)–(3.1.38), and using the definition of Jk in (3.1.17), we get:      Z Z  @f1 @f2 @ @f  .x/.x/ dx1 dx2 D .x/  .x/  cos l ds @xk @xk  @xl @xk 0 Z Jk cos l ds D hJk cos l ı0 ; i 8 2 D./; D 0

(3.1.41) @f2 @f1 where Jk D Jk .x/ D Œ @x .x/  Œ @x .x/ 8x 2 0 . Then, from (3.1.39)–(3.1.41), we k k get:  Z  2     2 @ Tf @ @ f ; D .x/ dx1 dx2 C .J0 cos k ı0 /;  @xk @xl @xl  @xk @xl

C hJk cos l ı0 ; i   @ ; i C .J0 cos k ı0 /;  C hJk cos l ı0 ; i D hT @2 f Œ @x @x .x/ @xl k l H)

@2 Tf @ D T @2 f C .J0 cos k ı0 / C Jk cos l ı0 Œ .x/ @xk @xl @xl @xk @xl

which can be written equivalently in the form (3.1.28).

8 2 D./ in D 0 ./;

223

Section 3.1 Distributional derivatives of piecewise smooth functions 2f @xk2

Corollary 3.1.1. Under the assumptions of Theorem 3.1.3, for f; Π@ to

C 0 .

.x/ belonging

n 0 / and satisfying (3.1.12)–(3.1.21), Tf D TŒ f .x/ C Jn ı0 C

@ .J0 ı0 / @n

in D 0 ./;

(3.1.42)

f D Œ f .x/ C Jn ı0 C

@ .J0 ı0 / @n

in D 0 ./;

(3.1.43)

or, equivalently,

where Tf D f D

@2 f @x12

C

@2 f @x22

2 D 0 ./ is the Laplacian of (discontinuous) f in 2f @x12

the sense of distribution; Πf .x/ D Π@

2f @x22

.x/ C Π@ 2f

of f in  n 0 in the usual pointwise sense, Π@

@xk2

.x/ 2 D 0 ./ is the Laplacian

.x/ being the second order partial

derivative of f with respect to xk in the usual pointwise sense;         @f1 @f C @f  @f2 .x /  .x / D .x/  .x/ 8x 2 0 Jn D @n @n @n @n

(3.1.44)

is the jump of the normal derivative of f in the direction of nO in crossing 0 , nO being the unit normal exterior to 1 (see Figure 3.3), such that the distribution Jn ı0 2 D 0 ./ is defined by: Z hJn ı0 ; i D Jn ds 8 2 D./I (3.1.45) 0

@ .J ı / @n 0 0

2 D 0 ./ is the distribution defined, 8 2 D./, by:     Z @ @ @ .J0 ı0 /;  D  J0 ı0 ; J0 ds: D @n @n @n 0

(3.1.46)

R Remark 3.1.2. The distribution hT; i D  0 J0 @ ds 8 2 D./ in (3.1.46) is @n formed by dipoles directed along the normal nO with a linear moment density J0 @ along 0 . Obviously, this distribution can also be represented by @n .J0 ı0 / as in (3.1.46) (for details, see Section 1.11 in Chapter 1). Proof of Corollary 3.1.1. From (3.1.31)–(3.1.32), we have, for k D 1; 2, @2 Tf @xk2

DT

2f @x 2 k

Œ@

.x/

C

@ .J0 cos k ı0 / C Jk cos k ı0 @xk

 2  @ f @2 f @ D .x/ C .J0 cos k ı0 / C Jk cos k ı0 @xk @xk2 @xk2

in D 0 ./

or

in D 0 ./:

224

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Hence,

Tf D

2 X @2 Tf

D

(3.1.47)

@xk2

kD1 2 X

T

kD1

X 2 kD1

2f @x 2 k

Œ@

C

.x/

2 2 X X @ .J0 cos k ı0 / C Jk cos k ı0 @xk

kD1

in D 0 ./:

kD1

 T

2f @x 2 k

Œ@

.x/

;  D hT

2f 2 @x1

Œ@

Z  D Z



D

.x/

CT

2f 2 @x2

Œ@

.x/

; i

8 2 D./

  Z  2 @2 f @ f .x/ dx1 dx2 C .x/ dx1 dx2 2 @x12  @x2

Πf .x/dx1 dx2

8 2 D./

 2 X

H)

T

kD1 2 X kD1

2f @x 2 k

Œ@

.x/

in D 0 ./:

D TΠf .x/

(3.1.48)

  ³  2 ² X @f  @f C Jk cos k D .x /  .x / cos k @xk @xk kD1

  2  2  X X @f C @f  .x / cos k  .x / cos k D @xk @xk kD1

D H)

2 X

kD1

@f C @f  .x /  .x / D Jn @n @n

Jk cos k ı0 D Jn ı0

in D 0 ./

(3.1.49)

kD1

is the distribution defined by hJn ı0 ; i D X 2 kD1

R

0

Jn ds 8 2 D./.

  2  X @ @ .J0 cos k ı0 /;  D  J0 cos k ı0 ; @xk @xk kD1

D

2 Z X kD1

0

J0 cos k

@ ds @xk

Section 3.1 Distributional derivatives of piecewise smooth functions

225

 Z @ @ D J0 cos k ds D  J0 ds @xk @n 0 0 kD1     @ @ D  J0 ı0 ; D .J0 ı0 /;  8 2 D./ @n @n Z

H)

X 2

2 X @ @ .J0 ı0 / .J0 cos k ı0 / D @xk @n

in D 0 ./:

(3.1.50)

kD1

Then, combining (3.1.47)–(3.1.50), we get the result (3.1.42)/(3.1.43). Remark 3.1.3. 

The jump Jn of the normal derivative @f in crossing 0 is independent of the @n choice of the normal nO to 0 . If the direction is reversed, then the sign of Jk changes and the sign of the normal derivative itself also changes, so that the jump Jn of the normal derivative ultimately remains unchanged.



O If the The formula (3.1.50) does not depend on the choice of the normal n. direction of the normal is reversed, the signs of both J0 and @ change. @n



The essential point is to use the same sense in the evaluation of the jump Jk and the choice of the normal nO to 0 .

Remark 3.1.4. Let nO 1 and nO 2 be unit normals to 0 exterior to 1 and 2 respectively such that nO 1  nO 2 D 1 (see Figure 3.3). Then, without introducing a sense @ .J0 ı0 / in (3.1.50) can be in crossing 0 , the expressions for Jn in (3.1.49) and @n replaced respectively by: 

   @f @f Jn D .x/ C .x/ @n1 @n2

.see (3.1.44)/;

(3.1.51)

and @ @ @ .J0 ı0 / D .f1 ı0 / C .f2 ı0 /: @n @n1 @n2

(3.1.52)

Then the formula (3.1.42)/(3.1.43) can be rewritten equivalently as: 

   @f @f Tf D f D Œ f .x/ C .x/ C .x/ ı0 @n1 @n2 @ @ C .f1 ı0 / C .f2 ı0 / in D 0 ./. @n1 @n2

(3.1.53)

226

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Example 3.1.2. Let  D 0; 1Œ  0; 1Œ  R2 be the open unit square with the square boundary  such that  D TV1 [ TV2 [ V0 , where TVi D int.Ti /, T1 and T2 being two closed triangles, V 0 D 0  ¹a2 ; a4 º: T1 D ¹.x1 ; x2 / W 0  x1  x2  1º

with vertices a2 .1; 1/; a3 .0; 1/; a4 .0; 0/I

T2 D ¹.x1 ; x2 / W 0  x2  x1  1º

with vertices a1 .1; 0/; a2 .1; 1/; a4 .0; 0/I

0 D T1 \ T2 D ¹.x1 ; x2 / W 0  x1 D x2  1º D Œa4 ; a2  D join of a4 and a2 I  D  [  D T 1 [ T2

.see Figure 3.5/:

x2 a2 = (1, 1)

a3 = (0, 1) T1

= [a4, a2] join of a4 and a2

T2 T1 T2 a4 = (0, 0)

a1 = (1, 0)

x1

Figure 3.5 Piecewise polynomial function u D u.x1 ; x2 / on  with discontinuity across 0 (see also Figure 2.3)

Let f be a function discontinuous across 0 defined by: ´ f .x1 ; x2 / D

p.x1 ; x2 / D 1 C 4x1  2x2 q.x1 ; x2 / D 2 C 4x1 C 4x2

in TV1 in TV2

(3.1.54)

such that p and q are given natural polynomial extensions to T1 and T2 . Let 

 ´ @p .x/ @f i .x/ D @x @q @xi .x/ @x i

in TV1 in TV2

.1  i  2/

(3.1.55)

Section 3.1 Distributional derivatives of piecewise smooth functions

227

@f @xi

be the distributional

be the usual partial derivative of f with respect to xi , and derivative of f with respect to xi . Then, 1. f 2 L2 ./;

(3.1.56)

@f .x/ 2 L2 ./; 2. Π@x

(3.1.57)

i

3.

@f @xi

… L2 ./.

(3.1.58)

Solution. Z

1.

jf .x1 ; x2 /j2 dx1 dx2 D

Z TV1



jp.x1 ; x2 /j2 dx1 dx2

Z

C

TV2

jq.x1 ; x2 /j2 dx1 dx2 < C1

H) f 2 L2 ./. 2.

 ´ @ .1 C 4x1  2x2 / D 4 in TV1 @f .x/ D @x@1 @x1 .2 C 4x1 C 4x2 / D 4 in TV2 @x1   @f H) .x/ D 4 in : @x1 2 Z Z  @f .x/ dx1 dx2 D 42 dx1 dx2 D 16 < C1 H)  @x1    @f H) .x/ 2 L2 ./: @x1  ´  @f 2 in TV1 .x/ D @x2 4 in TV2 2 Z  Z Z @f H) .x/ dx1 dx2 D 4dx1 dx2 C 16dx1 dx2 TV1 TV2  @x2



D4  H)

 @f .x/ 2 L2 ./: @x2

@f Hence, Π@x .x/ 2 L2 ./, k D 1; 2. k

1 1 C 16  < C1 2 2

228

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

3. f 2 L2 ./ H) f 2 L1loc ./  D 0 ./ with f D Tf 2 D 0 ./. Then the distributional derivative

@f @xk

D

@ T @xk f

is defined, 8 2 D./, by:

  Z @ @ D  f; f dx1 dx2 I (3.1.59) D @x @x 0  k k D ./D./   Z @f @ ; D f dx1 dx2 @x1 @x 1  D 0 ./D./   2 Z  2 Z  X X @ @f D .f /.x/ dx1 dx2 C .x/ dx1 dx2 : (3.1.60) TVi @x1 TVi @x1



@f ; @xk



iD1

iD1

Let @Ti be the triangular boundary of TVi such that Ti D TVi [ @Ti (i D 1; 2), and 0  @Ti , .@Ti  0 /  . Then, applying Green’s Theorem 3.1.2, we get   Z  Z  @ @ .f /.x/ dx1 dx2 D .p/.x/ dx1 dx2 TV1 @x1 TV1 @x1 Z Z Z D pdx2 D pdx2 C pdx2 Z

@T1

D

pdx2

@T1 0

0

8 2 D./;

0

since  2 D./ H) # D 0 H) #@T1 0 D 0. Hence,  Z  Z Z @ O Oi1 /ds; .p/.x/ dx1 dx2 D pdx2 D p cos.n; TV1 @x1 0 0 O Oi1 /Oi1 C cos.n; O Oi2 /Oi2 is the unit normal exterior to 0 . Again, where nO D cos.n; applying Green’s Theorem 3.1.2, we get   Z  Z  @ @ .f /.x/ dx1 dx2 D .q/.x/ dx1 dx2 TV2 @x1 TV2 @x1 Z Z Z O Oi1 /ds; D qdx2 D  qdx2 D  q cos.n; 0

0

0

where the orientation of 0 is from a2 to a4 , i.e. opposite to that of 0 . Hence 

2 Z X V iD1 Ti



 Z @ O Oi1 /ds .f /.x/ dx1 dx2 D .q  p/ cos.n; @x1 0 Z O Oi1 /ds; D J cos.n; (3.1.61) 0

229

Section 3.1 Distributional derivatives of piecewise smooth functions

where J D jump of f across 0 D .2 C 4x1 C 4x2 /  .1 C 4x1  2x2 / D 3 C 6x2

8.x1 ; x2 / 2 0 : (3.1.62)

Then, from (3.1.60), (3.1.61) and (3.1.62) we get 

@f ; @x1



 Z @f O Oi1 /ds .x/ dx1 dx2 C J cos.n; Vi @x1 T  0 iD1  Z  @f O Oi1 /ı0 ; i D .x/ dx1 dx2 C hJ cos.n;  @x1    @f O Oi1 /ı0 ; i D .x/ ;  C hJ cos.n; @x1    @f O Oi1 /ı0 ;  8 2 D./; .x/ C J cos.n; D @x1

D D 0 ./D./

where hı0 ; i D

R

0

2 Z X



.x1 ; x2 /ds 8 2 D./ is the Dirac distribution with

O Oi1 / D 1=2; J.x/ D 3 C mass/charge/force etc. concentrated along 0 ; cos.n; 6x2 (by (3.1.50)); Z O Oi1 /i D O Oi1 /ds 8 2 D./ D hı0 ; J cos.n; J cos.n; 0

@f @f O Oi1 /ı0 2 D 0 ./ (see (3.1.27)). Œ @x D Œ @x .x/ C J cos.n; .x/ 2 1 1 1 2 2 O Oi1 /ı0 … L2 ./, L ./, but ı0 … Lloc ./ H) ı0 … L ./ H) J cos.n; O Oi1 /ı … L2 ./. O Oi1 / D 1 ). Consequently, Œ @f .x/ C J cos.n; (J ¤ 0, cos.n;

H)

@f @x1

2

@x1

0

@f Hence, @x … L2 ./. 1 For k D 2, similarly applying Green’s Theorem 3.1.2, we have  Z  Z Z @ O Oi2 /ds;  .f /.x/ dx1 dx2 D pdx1 D  p cos.n; TV1 @x2 0 0  Z Z Z  @ .f /.x/ dx1 dx2 D qdx1 D  qdx1  TV2 @x2 0 0 Z O Oi2 /ds D q cos.n; 0

H) 

2 Z X V iD1 Ti



 Z @ O Oi2 /ds .f /.x/ dx1 dx2 D .q  p/ cos.n; @x2 0 Z O Oi2 /ds: D J cos.n; 0

(3.1.63)

230

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Following the steps of the proof for k D 1, finally, from (3.1.63), we have    Z  Z @f @f O Oi2 /ds ; D .x/ dx1 dx2 C J cos.n; @x2  @x2 0 D 0 ./D./    @f O O i2 /ı0 ;  8 2 D./ .see (3.1.27)/ D .x/ C J cos.n; @x2 H)

@f @x2

@f O Oi2 /ı0 … L2 ./ (see the steps for the case D Œ @x .x/ C J cos.n; 2

k D 1). Hence, the distributional derivative

@f @xk

… L2 ./ for k D 1; 2.

3.1.3 Case of three variables (n D 3) Let   R3 be a domain in R3 bounded by an orientable piecewise smooth boundary O Oi2 /Oi2 C cos.n; O Oi3 /Oi3 to O Oi1 /Oi1 C cos.n; surface  such that the unit normal nO D cos.n;  and exterior to  is defined almost everywhere on  and 0   be an orientable piecewise smooth surface subdividing  into subdomains 1 and 2 with boundary surfaces  1 D 1 [ 0 and  2 D 2 [ 0 such that conditions analogous to (3.1.11) for n D 2 also hold for n D 3:  D 1 [ 2 [ 0 ;  1 \  2 D 0 ;

i D  i [  i ;

 D 1 [ 2 ;

 D 1 [ 2 ; (surface area)

 i D i [ 0 ;

meas.1 \ 2 / D 0:

(3.1.64)

2

@f f .x/; Œ @x@ @x .x/ 2 C 0 . n 0 / with finite jumps J0 ; Jk ; Jkl Let functions f; Œ @x k k l across 0 respectively, as defined by formulae (3.1.16), (3.1.17) and (3.1.18) respec@fi i tively such that fi D f #0 , Œ @x .x/, Œ @x@f@x .x/, i D 1; 2, can be continuk k l ously extended to 0 using (3.1.12)–(3.1.15) with x D .x1 ; x2 ; x3 / 2 i (resp.  D .1 ; 2 ; 3 / 2 0 /. 2f @T @2 Tf @f D @xf 2 D 0 ./ and @x@ @x D @x @x @xk k k l k l @f @2 f 0 Œ @x .x/ D TŒ @f .x/ 2 D ./, Œ @x @x .x/ k k l @x

Then the distributional derivatives D 0 ./, and the partial derivatives T

2f k @xl

Π@x@

.x/

2

D 0 ./

2 D

k

in the usual pointwise sense, are related by the same types of

formulae as in (3.1.25) and (3.1.26) for 1  k, l  3, i.e.: 2

@f f Theorem 3.1.4. For functions f; Π@x .x/; Π@x@ @x .x/ 2 C 0 .n0 / with finite jumps k k l J0 ; Jk ; Jkl across 0 respectively, the following formulae hold for 1  k; l  3:

@Tf D TŒ @f .x/ C J0 cos k ı0 @xk @xk

in D 0 ./;

@2 Tf @ D T @2 f C .J0 cos k ı0 / C Jk cos l ı0 Œ .x/ @xk @xl @xl @xk @xl

(3.1.65) in D 0 ./; (3.1.66)

231

Section 3.1 Distributional derivatives of piecewise smooth functions

which can be written equivalently as:   @f @f D .x/ C J0 cos k ı0 in D 0 ./; @xk @xk  2  @2 f @ f @ D .x/ C .J0 cos k ı0 / C Jk cos l ı0 @xk @xl @xk @xl @xl

(3.1.67) in D 0 ./; (3.1.68)

where TŒ

@f @xk

.x/

and T

2f k @xl

Π@x@

.x/

2 D 0 ./ are regular distributions defined by (3.1.20)

and (3.1.21) with   R3 , x D .x1 ; x2 ; x3 / and ‘dx1 dx2 ’ replaced by ‘dx1 dx2 dx3 ’; O Oik /; 8x 2 0 ,

k D .n; J0 D f .xC /  f .x / D f2 .x/  f1 .x/;     @f C @f  Jk D .x /  .x / (3.1.69) @xk @xk @f1 @f2 .x/  .x/ .see also (3.1.16)–(3.1.17)/; D @xk @xk Z J0 cos k dA 8 2 D./; (3.1.70) hJ0 cos k ı0 ; i D 0

Z hJk cos l ı0 ; i D

0

Jk cos l dA

8 2 D./;

(3.1.71)

define Dirac distributions corresponding to mass or charge or load placed on 0 with a surface area density J0 cos k in (3.1.70) and a surface area density Jk cos l in (3.1.71), dA being surface area measure in (3.1.70) and (3.1.71). In particular, for l D k in (3.1.66), we get, for 1  k  3, @2 Tf @xk2

DT

2 Π@ f2 @x k

.x/

C

@ .J0 cos k ı0 / C Jk cos k ı0 @xk

 2  @ f @2 f @ D .x/ C .J0 cos k ı0 / C Jk cos k ı0 2 2 @xk @xk @xk

in D 0 ./ in D 0 ./:

or (3.1.72)

Consequently, Tf D TΠf .x/ C

3 X

.Jk cos k ı0 / C

kD1

D TŒ f .x/ C Jn ı0 C

3 X @ .Jk cos k ı0 / @xk

kD1

@ .J0 ı0 / @n

in D 0 ./

(3.1.73)

232

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

(using a formula for n D 3 analogous to (3.1.49) for n D 2), or f D Œ f .x/ C Jn ı0 C

@ .J0 ı0 / @n

in D 0 ./

(3.1.74)

with 3 X

Jk cos k ı0 D Jn ı0 ;

kD1

3 X @ @ .J0 ı0 / .J0 cos k ı0 / D @xk @n

(3.1.75)

kD1

(see (3.1.49) and (3.1.50) for n D 2), where Tf D f D

P3

@2 f kD1 @x 2 is the Laplak  R3 , and Πf .x/ D

cian of (discontinuous) f in the distributional sense in  2 @2 f .x/ @2 f .x/ Π@ f .x/ 2  C Π2  C Π2  is the Laplacian of f in the usual pointwise sense in @x1

  0 .

@x2

@x3

Proof.       Z @Tf @ @ @ ;  D  Tf ; f dx1 dx2 dx3 D  f; D @xk @xk @xk @xk  Z Z @ @ D f1 dx1 dx2 dx3  f2 dx1 dx2 dx3 @xk @xk 1    2 Z 2 Z X X @fi @ D .fi /.x/dx1 dx2 dx3 C .x/ dx1 dx2 dx3 : i @xk i @xk iD1

iD1

(3.1.76) We need the Divergence Theorem of Gauss–Ostrogradski: Theorem 3.1.5 ([32]). Let D  R3 be a bounded (volume) domain in R3 with piecewise smooth orientable boundary surface @D and D D D [ @D. Then, for k Pk D Pk .x1 ; x2 ; x3 /, 1  k  3, with Pk ; @P 2 C 0 .D/, @x k

Z  D

 Z @P2 @P3 @P1 C C .P1 cos 1 CP2 cos 2 CP3 cos 3 /dA; dx1 dx2 dx3 D @x1 @x2 @x3 @D (3.1.77)

O Oik / D nO  Oik , 1  k  3. where cos k D cos.n; Now we continue with the proof.R We transform the triple integral i @x@ .fi ; /dx1 dx2 dx3 in (3.1.76) into a surk face integral over  i D i [ 0 , i D 1; 2, for k D 1; 2; 3, as in the case of two variables in steps (3.1.34)–(3.1.37) and get the result. In fact, for this we set i D D,  i D @D in (3.1.77). Then, for fixed k D 1; 2; 3, Pk D fi  2 C 0 .i /,

Section 3.1 Distributional derivatives of piecewise smooth functions

233

@Pk @xk

D @x@ .fi / 2 C 0 .i /; i D i \  i with  2 D./; # D 0 and Pj D 0 k for 1  j ¤ k  3, from (3.1.77) we get, for i D 1; 2: 

Z  i

 Z @ .fi /.x/ dx1 dx2 dx3 D  .fi / cos.nO i ; Oik /dA @xk  i Di [0 Z D .fi / cos.nO i ; Oik /dA .since #i D 0/; 0

(3.1.78) O Hence, from where nO i is the unit normal exterior to i with nO 2 D nO 1 , nO 1 D n. (3.1.76) and (3.1.78), 

 X   2 Z @Tf @fi ; D .x/ dx1 dx2 dx3 @xk @xk iD1 i Z  Z  f1  cos.nO 1 ; Oik /dA  f2  cos.nO 2 ; Oik /dA Z  D 

Z

0

 @f .x/ dx1 dx2 dx3 @xk

C 0

D hTŒ

@f @xk

0

Œf2 .x/  f1 .x/.x/ cos.nO 1 ; Oik /dA Z ; i C .x/

0

O Oik /dA J0 cos.n;

.nO 2 D nO 1 /

8 2 D./

.nO D nO 1 ; see (3.1.69) for J0 / @Tf O Oik //: (3.1.79) D TŒ @f .x/ C J0 cos k ı0 in D 0 ./ . k D cos.n; @xk @xk         2 @Tf @ @ Tf @ @Tf ; D ; ; D  @xk @xl @xl @xk @xk @xl   @ D  TŒ @f .x/ C J0 cos k ı0 ; @xk @xl     @ @ D  TŒ @f .x/ ;  J0 cos k ı0 ; @xk @xl @xl  Z  @ @f .x/ dx1 dx2 dx3 D @x @x  k l   @ C .J0 cos k ı0 /;  8 2 D./: (3.1.80) @xl H)

234

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

But   Z  2 Z  @f @ f @ .x/ dx1 dx2 dx3 D .x/ dx1 dx2 dx3  @xl  @xk  @xk @xl    Z @f @  .x/  dx1 dx2 dx3 : (3.1.81) @xk  @xl Now, weRwill transform P theRlast volume integral in (3.1.81) into a surface integral. Since  .: : : / D 2iD1 i .: : : /, applying the results of (3.1.78) and (3.1.79) with @fi @f2 ‘fi ’ replaced by ‘Œ @x .x/’ and subscript ‘k’ by ‘l’, and using Jk D Œ @x .x/  k

k

@f1 Π@x .x/ on 0 , we get 8 2 D./ (#i D 0): k

Z  

D

@ @xl



     2 Z X @f @fi @ .x/  d x D  .x/  d x @xk @xk i @xl iD1

2 Z X iD1

 i Di [0



  @fi O .x/  cos.nO i ; il /dA @xk

  @f1 @f2 .x/ cos.nO 1 ; Oil / C Œ .x/ cos.nO 2 ; Oil / dA @xk @xk 0    Z  @f2 @f1 .x/  .x/  cos.nO 1 ; Oil /dA D @xk @xk 0 Z  Z Jk cos l dA D hJk cos l ı0 ; i .: : : /dA D 0 ; D Z



D

0

(3.1.82)

i

where nO D nO 1 , nO 2 D nO 1 D nO have been used. Combining (3.1.80)–(3.1.82), we get the result, 8 2 D./:  2  Z  2    @ Tf @ @ f ; D .x/ dx1 dx2 dx3 C .J0 cos k ı0 /;  @xk @xl @xl  @xk @xl C hJk cos l ı0 ; i   @ D T @2 f C .J0 cos k ı0 / C Jk cos l ı0 ;  Œ @x @x .x/ @xl k l ”

@2 Tf @ D T @2 f C .J0 cos k ı0 / C Jk cos l ı0 in D 0 ./ or Œ @x @x .x/ @xk @xl @x l k l  2  2 @ f @ @ f D .x/ C .J0 cos k ı0 / C Jk cos l ı0 in D 0 ./: @xk @xl @xk @xl @xl

Following the proof of Corollary 3.1.1 and Remarks 3.1.3 and 3.1.4, with necessary modifications for n D 3, we get the result (3.1.73)/(3.1.74).

Section 3.2 Unbounded domain   Rn , Green’s formula

235

Remark 3.1.5. Since Remarks 3.1.3 and 3.1.4 can be extended to the case n D 3 with the necessary modifications, formulae (3.1.51) and (3.1.52) hold with the necessary modifications for n D 3, and formula (3.1.74) can be rewritten equivalently, for nO 1 :nO 2 D 1, as:     @f @f f D Œ f .x/ C .x/ C .x/ ı0 @n1 @n2 @ @ C .f1 ı0 / C .f2 ı0 / in D 0 ./: (3.1.83) @n1 @n2

3.2

Unbounded domain   Rn , Green’s formula 2

@f f Let f; Π@x .x/; Π@x@ @x .x/ be continuous in the complement Rn n 0 (n D 2; 3) k k l of 0 with finite jumps J0 ; Jk ; Jkl defined by (3.1.16),(3.1.17) and (3.1.18) respectively for the case n D 2 (similar formulae for the case n D 3 are obtained with x D .x1 ; x2 ; x3 /,  D .1 ; 2 ; 3 /), 0 being a piecewise smooth curve in R2 (resp. piecewise smooth surface in R3 ). Then Theorem 3.1.3, Corollary 3.1.1 and Remarks 3.1.2, 3.1.4 for n D 2 (resp. Theorem 3.1.4, Remark 3.1.5 for n D 3) will hold, and we have

@ Tf D TŒ f .x/ C Jn ı0 C .J0 ı0 / @n     @f @f D TŒ f .x/ C .x/ C .x/ ı0 @n1 @n2   @ @ C .f1 ı0 / C .f2 ı0 / in D 0 .Rn /; @n1 @n2 @ .J0 ı0 / f D Œ f .x/ C Jn ı0 C @n     @f @f D Œ f .x/ C .x/ C .x/ ı0 @n1 @n2 @ @ C .f1 ı0 / C .f2 ı0 / in D 0 .Rn /; @n1 @n2

(3.2.1)

(3.2.2)

@ where the distributions Jn ı0 ; @n .J0 ı0 / are defined by (3.1.75) for n D 3 (resp. (3.1.49) and (3.1.50) for n D 2); for nO 1  nO 2 D 1, using (3.1.53) for n D 2 and (3.1.83) for n D 3, the alternative expression in (3.2.1)/(3.2.2) is obtained.

Green’s formula Let   Rn (n D 2; 3) be a domain in Rn with an orientable smooth boundary  such that the unit vector nO normal to  and exterior to  is defined everywhere on . Let f 2 C 2 ./ \ C 1 ./ such that f .x/ D 0 8x 2 Rn n , i.e. f

236

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

is defined on Rn such that f and its normal derivative (in the pointwise sense) Π@f .x/ @n have finite jumps J0 and Jn across  defined, 8x 2 , by: J0 D f .xC /  f .x/ D 0  f .x/ D f .x/;         @f @f @f C @f .x /  .x/ D 0  .x/ D  .x/ : Jn D @n @n @n @n

(3.2.3)

Then we have: Theorem 3.2.1 (Green’s formula). 8 2 D.Rn /,   Z Z  @f @  .x/ dS .f   Œ f .x//d x D f @n @n         @ @f D .f ı /;  C  .x/ ı ;  ; @n @n

(3.2.4)

where, 8 2 , f ./ D limx!;x2 f .x/, Π@f ./ D limx!;x2 Π@f .x/ is @n @n the normal derivative of f from within  in the usual pointwise sense; Πf .x/ D Pn @2 f kD1 Π2 .x/ is the Laplacian of f in the usual pointwise sense; d x D dx1 dx2 @xk

and dS D ds for n D 2 (resp. dS D dA and d x D dx1 dx2 dx3 for n D 3). Proof. From (3.1.43) for n D 2 (resp. (3.1.74) for n D 3) with  D 0 , J0 D f .x/, Jn D Œ @f .x/ (3.2.3), we have, 8 2 D.Rn /, @n    @ @f .f ı /;  h f; i D hŒ f .x/; i C h .x/ı ;  C @n @n  ³ Z ² Z @f @ .x/   f Œ f .x/d x  D dS @n @n   (dS D ds for n D 2, dS D dA for n D 3), since f .x/ D 0 outside . But 8 2 D.Rn ), Z f d x; h f; i D hf; i D 

L1loc .Rn /

and f .x/ D 0 outside . since f 2 Hence,   Z Z Z  @f @  .x/ dS 8 2 D.Rn / f d x D Œ f .x/d x C f @n @n      Z Z  @f @ H)  .x/ dS .f   Œ f .x//d x D f @n @n         @ @f .f ı /;  C  .x/ ı ;  ; D @n @n

Section 3.2 Unbounded domain   Rn , Green’s formula

where

237



   Z @ @ @ .f ı /;  D f ı ; dS I f D @n @n @n       Z  @f @f  .x/ ı ;  D   .x/ dS: @n @n 

Proposition 3.2.1. 8" > 0, let " D ¹x W x 2 Rn ; kxk > "º and " D ¹x W kxk D "º be its boundary with radius ". Then, 8f 2 C 1 .Rn n ¹0º/, 8 2 D.Rn /,  Z  Z @ @f ¹f .x/ .x/  Œ f .x/.x/ºd x D .x/ .x/  f .x/ .x/ dS; @r @r " " (3.2.5) @ @ where Œ f .x/ is the Laplacian of f in the usual pointwise sense; @r D  @n@ " , @r ./ being the derivative in the radial direction nO " , nO " being the exterior unit normal to " ; d x D dx1 dx2 dx3 and dS D dA (surface area measure) for n D 3 (resp. d x D dx1 dx2 and dS D ds (arc length measure) for n D 2).

Proof. For  2 D.Rn /, 9R > 0 such that supp./  ¹x W kxk < Rº. Define ";R D ¹x W " < kxk < Rº enclosed by " and R with radius " and R respectively. Since f ,  2 C 1 .";R /, the classical Green’s formula of calculus [32] holds:  Z Z  @f @ ¹f .x/ .x/  Œ f .x/.x/ºd x D .x/  .x/ .x/ dS f .x/ @n" @n" ";R "  Z  @f @ C .x/  .x/ .x/ dS; f .x/ @nR @nR R (3.2.6) where nO R is the unit exterior normal R to R . R R @ Since #R D @n # D 0, " ¹: : : ºd x D ";R ¹: : : ºd x and R .: : : / D 0, R R and

@ ./ @n"

@ D  @r .  / in (3.2.6), the result (3.2.5) follows from (3.2.6).

Example 3.2.1. For r D kxk D .x12 C x22 C    C xn2 /1=2 and  2 D.Rn /, prove that 1. for n D 2, Z



Z ln r dx1 dx2 D

" Wkxk>"

2. for n D 3, Z " Wkxk>"

1 dx1 dx2 dx3 D r

" WkxkD"

Z

 @ 1 .x/  ln " dsI " @r

  1 1 @ dS: .x/ 2  " " @r " WkxkD"

(3.2.7)

(3.2.8)

238

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Solution. 1.

2 2x 2 @ @r .ln r/ D 1r  @x D rx2i ; @ 2 .ln r/ D @x@ . rx2i / D r12  r 4i ; H) Œ ln r.x/ D @xi @xi i i P2 2 2r 2 @2 1 2 iD1 @x 2 ln r D r 2  r 4 D 0 8x ¤ 0. Since ln r 2 C .R n ¹0º/ and i

Œ ln r.x/ D 0 8x ¤ 0, from (3.2.5),  Z Z  @ @ ln r .x/  ln r .x/ ds ln r dx1 dx2 D .x/ @r @r " "  Z  @ 1 D .x/  ln " .x/ ds; " @r " since

@ @r

ln r#" D 1r #" D

2. For n D 3,

@ 1 . / @xi r

H) Π. 1r /.x/ D

1 " @r  r12  @x i

and ln r#" D ln ".

D P3

D  rx3i ,

@2 1 iD1 @x 2 . r / i

@2 1 . / @xi2 r

D

@ . rx3i / @xi

D  r13 C 3  rx4i  xri

2

D  r33 C 3 rr 5 D 0 8x ¤ 0.

Since 1r 2 C 1 .R3 n ¹0º/ and Œ . 1r /.x/ D 0 8x ¤ 0, we get the result (3.2.8) @ 1 . r /#" D  r12 #" D  "12 and 1r #" D 1" . from (3.2.5) by writing @r

3.3

Elementary solutions

1 Laplacian . r n2 / of

1 r n2

in the distribution sense (n  3)

1 r n2

is a harmonic function in Rn ¹0º for n ¤ 2 Every constant function f .x/ D C 8x 2 Rn is harmonic everywhere in Rn , since Œ f .x/ D 0 8x 2 Rn . Now we have to show that a non-constant function f D f .r/ with ´ X  12 n 1 C C2 for n ¤ 2 .r ¤ 0/; C1 r n2 2 r D kxk D xi and f .r/ D 1 C1 ln r C C2 for n D 2 .r ¤ 0/; iD1 where C1 , and C2 are constants, is harmonic in Rn  ¹0º, i.e. Œ f .x/ D 0 8x 2 .Rn  ¹0º/. Since, for r ¤ 0,   @f df @r df xi   ; .x/ D D @xi dr @xi dr r       2   d f @ df xi df 1 xi2 d 2 f xi2  .x/ D C ; (3.3.1) D @xi dr r dr 2 r 2 dr r r3 @xi2  X   n  2 n  2  2 X @ f d f xi df 1 xi2  3 .x/ D C Œ f .x/ D dr 2 r 2 dr r r @xi2 iD1 iD1    2  d f n  1 df .r/ .r/ C D 2 dr r dr

239

Section 3.3 Elementary solutions

for r ¤ 0. Hence, f is a harmonic function of r for r ¤ 0 H) Œ f .x/ D 0  d 2f n  1 df .r/ D 0: .r/ C f must satisfy dr 2 r dr 

H)

(3.3.2)

, we get, from (3.3.2), dg C n1 Setting g D df r g D 0, which has the general solution dr dr R 1 1 g D C1 r n1 H) f D C1 r n1 dr C C2 H)

´ 1 C C2 with constants C1 ; C2 C1 r n2 f D 1 C1 ln r C C2 with constants C1 ; C2

for n ¤ 2 .r ¤ 0/; for n D 2 .r ¤ 0/

is the general solution of (3.3.2), since ln r and ln 1r D  ln r are linearly dependent solutions of (3.3.2) for n D 2. Properties of

1 r n2

for n  3

C 1 -function

in

Rn

Function

1 r n2

is

 ¹0º with a singularity at the origin r D 0;



a



a harmonic function:     1 n2 .x/ D 0 r

8x 2 Rn  ¹0ºI

(3.3.3)

1 a locally summable function in Rn , i.e. r n2 2 L1loc .Rn /, since n  2 < n; R 1 1 H) kxkR r n2 d x < C1 8R > 0. As r n2 2 L1loc .Rn /, it defines a regular 1 distribution T 1 2 D 0 .Rn /. But the Laplacian Π. r n2 /.x/ D 0 in the pointwise 

r n2

1 sense 8x 2 .Rn  ¹0º/, and x D 0 is a singularity of r n2 as a function. Hence, we 0 n expect that T 1 2 D .R / will be a singular distribution, which will involve the r n2

Dirac distribution concentrated at the origin x D 0. For the sake of simplicity, we establish the result for the most important case of n D 3, i.e. for T 1 : 8 2 D.R3 /, r

 h T 1 ; i D hT 1 ; i D r

r

 Z Z 1 1 1 ;  D d x D lim d x: C 3 r r "!0 R kxk>" r (3.3.4)

Define an auxiliary function f" by: ´ f" .r/ D with " > 0.

0 1 r

for r < "I for r > "

(3.3.5)

240

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Then f" 2 L1loc .R3 / is discontinuous on the spherical surface S" with radius r D ", @ normal nO " D rO ; rO being the unit vector in the direction of r D r  rO such that @n@ " D @r . Hence, the jumps J0 and Jn" D Jr of f" and are given by:

@f" @n"

D

@f" @r

respectively in crossing S"

1 1 0D ; r r   @f" C @f"  @ 1 1 ." /  ." / D Jr D  0 D  2: @r @r @r r r

J0 D f" ."C /  f" ." / D

(3.3.6)

f" is harmonic in R3 n S" H) Πf" .x/ D 0 8x 2 R3 n S" Z H) R3

8 2 D.R3 /:

Πf" .xd x D 0

(3.3.7)

f" 2 L1loc .R3 / H) hTf" ; i D h Tf" ; i 8 2 D.R3 /. Z hTf" ; i D hf" ; i D Z

H)

Z R3

f" d x D kxk>"

1 d x r

1 d x D lim hTf" ; i D lim h Tf" ; i lim "!0C kxkDr>" r "!0C "!0C    Z @ Œ f" .x/d x C .J0 ıS" /;  C hJn" ıS" ; i D lim @n" "!0C R3

O Applying (3.3.4)–(3.3.7), (using (3.2.1)/(3.2.2) with S" D 0 , nO " D n).   @ 1 . ıS /;  C hJr ıS" ; i h T 1 ; i D lim r @r r " "!0C     1 @ 1 D lim  ıS" ; 8 2 D.R3 / C h 2 ıS" ;  C r @r r "!0    Z Z 1 1 @ D lim  dS C lim  2 dS ; "!0C "!0C S" " @r S" " 

since r D " on S" .

(3.3.8)

241

Section 3.3 Elementary solutions

Now we will show that, 8 2 D.R3 /, the first and second surface integrals in (3.3.8) tend to 0 and 4.0/ respectively as " ! 0C . 8 2 D.R3 /, ˇ Z ˇ ˇZ ˇ ˇ ˇ ˇ ˇ @ ˇ ˇ @ ˇ 1 1 @ ˇˇ 1 ˇ ˇ ˇ ˇ dS ˇ  max ˇ .x/ˇ dS D max ˇ .x/ˇˇ  4"2 ˇ " x2S" @r " x2S" @r S" " @r S" ˇ ˇ ˇ @ ˇ  max ˇˇ .x/ˇˇ  4" ! 0 as " ! 0C x2S" @r   Z 1 @ dS D 0I (3.3.9) H) lim  "!0C S" " @r Z Z Z 1 1 1  2 dS D  .0/dS C  2 ..x/  .0//dS: (3.3.10) 2 S" " S" " S" " But Z  S"

.0/ 1 .0/dS D  2 2 " "

Z dS D  S"

.0/ 4"2 D 4.0/: "2

(3.3.11)

Using the mean-value theorem and Schwarz’s inequality (see (B.4.3.1) in Appendix B), for x 2 S" with kxk D ", ˇ 3 ˇ ˇ X @ ˇ ˇ j.x/  .0/j D ˇ xi ./ˇˇ D jhx; r ./ij @x i

iD1

ˇ ˇ ˇ ˇ @ ˇ max ˇ .x/ˇˇ 1i3 x2supp./ @xi

p  kxkkr ./k  " 3 max

8 2 D.R3 /;

since ˇ2 ˇ2 ˇ  3 ˇ X ˇ ˇ ˇ @ ˇ @ ˇ ˇ ˇ ˇ ./  3 max ./ kr ./k D ˇ ˇ ˇ @x 1i3 ˇ @x 2

iD1

i

i

ˇ 2 ˇ ˇ ˇ @ ˇ max ˇ .x/ˇˇ : 1i3 x2supp./ @xi

  3 max Hence, ˇZ ˇ ˇ ˇ

ˇ ˇ ˇ Z  p ˇ @ ˇ ˇ 1 ˇ ˇ  1 " 3 max ˇ ..x/  .0//dS .x/ dS max ˇ ˇ ˇ 2 2 1i3 x2supp./ @xi " S" " S" ˇ ˇ  p ˇ 1 ˇ @  4"2 ! 0 as " ! 0C 8 2 D.R3 / .x/ˇˇ D 3 max max ˇˇ 1i3 x2supp./ @xi " Z 1  2 ..x/  .0//dS D 0: H) lim C "!0 S" " 

242

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Then, from (3.3.10) and (3.3.11), Z lim "!0C

 S"

1 dS D 4.0/: "2

(3.3.12)

Finally, from (3.3.8), (3.3.9) and (3.3.12), we get, 8 2 D.R3 /: h T 1 ; i D 4.0/ D 4hı; i D h4ı; i r

H) T 1 D 4ı, ı 2 D 0 .R3 / being the Dirac distribution concentrated at the r

singular point x D 0 2 R3 . Similarly, .T

1 r n2

/ can be evaluated 8n ¤ 2.

For n D 2, Laplacian of ln r in the distributional sense in R2 lnRr is locally integrable in R2 , i.e. ln r 2 L1loc .R2 /: for this, it is sufficient to show that kxk1 j ln rj R d x < C1, since 1kxkR j ln rjd x < C1 8R > 1. Z

1 Z 2

Z j ln rjd x D

kxk1

Z j ln rjrdrd D 

0

0

Z



1

r ln rdr D 2 0

1

d

0

D 2

Z

2

r ln rdr 0

ˇ1 Z 1 2  ˇ r2 r ln r ˇˇ  dr 2 0 2r 0

ˇ  r 2 ˇˇ1  D 2 D < C1: 4 ˇ0 2 

Hence, ln r 2 D 0 .R2 /, and the Laplacian of ln r in theRdistributional sense is defined by, 8 2 D.R2 /, h ln r; i D hln r; .x/i D R2 .ln r/ .x/d x. ln r 2 L1loc .R2 /, but its distributional derivatives of order two are not locally integrable. Hence, we cannot apply integration by parts. Since ln r 2 L1loc .R2 /, we can apply Lebesgue’s Theorem and write Z Z h ln r; i D .ln r/ .x/d x D lim .ln r/ .x/d x D lim I1 ."/; R2

"!0C

" Wkxk>"

"!0C

where " D ¹x W x 2 R2 ; kxk > "º with boundary " D ¹x W kxk D "º, and, from (3.2.7),   Z 1 @ .x/   ln "  I1 ."/ D ds " @r " WkxkD" Z Z .x/ @ ds  ln " ds D I2 ."/  I3 ."/ D @r " " " H) lim"!0C I1 ."/ D lim"!0C I2 ."/  lim"!0C I3 ."/.

243

Section 3.3 Elementary solutions

For finding the right-hand side limits, " is represented by x1 D " cos , x2 D " sin . R 2 Q R 2 Q Q Then I2 ."/ D 0 .";/ " "d D 0 ."; /d with ."; / D ." cos ; " sin /. But Z 2 Z 2 Q Q /d

lim I2 ."/ D lim ."; /d D lim ."; "!0C

"!0C

Z

0

2

Q /d D .0;

D 0

0

Z

"!0C

Z

2

2

.0; 0/d D .0/ 0

d D 2.0/; 0

Q / D .0  cos ; 0  sin / D .0; 0/ D .0/. since .0; Again, ˇ ˇ ˇ ˇ p ˇ @ ˇ ˇ ˇ ˇ ˇ D jr   .nO " /j  kr k k  nO " k  2 max max ˇ @ .x/ˇ D C1 : ˇ @r ˇ ˇ ˇ 1i2 supp./ @xi Then ˇ ˇZ ˇ @ ˇˇ ˇˇ 2 @ ˇˇ ln " ds ˇ D ˇ .ln "/" d ˇ @r @r " 0 Z 2 d D 2 C1 j" ln "j ! 0 as " ! 0C  j" ln "jC1

ˇZ ˇ jI3 ."/j D ˇˇ

0

H) lim"!0C I3 ."/ D 0. Hence, lim I1 ."/ D lim I2 ."/  lim I3 ."/ D 2.0/

"!0C

"!0C

"!0C

H)

h ln r; i D 2.0/ D h2ı; i

H)

ln r D 2ı

8 2 D.R2 /

in D 0 .R2 /:

(3.3.13)

The Laplacian   1 ln D  ln r D 2ı r

in D 0 .R2 /:

(3.3.14)

Theorem 3.3.1. For n ¤ 2,  T

1 r n2

D

1 r n2

 D .n  2/Sn ı

in D 0 .Rn /;

(3.3.15)

where n

2. 2 / Sn D D surface area of the unit n-sphere . n2 /

X n iD1

xi2

 D1

(3.3.16)

244

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

p with . 12 / D , . n2 C 1/ D . n2 /. n2 /; ı is the (n-dimensional) Dirac distribution concentrated at 0 2 Rn . For n D 2,   1 Tln 1 D ln D 2ı in D 0 .R2 /: (3.3.17) r r For n D 1, .Tjxj / D .jxj/ D 2ı

in D 0 .R/:

(3.3.18)

  1 T 1 D D 4ı r r

in D 0 .R3 /:

(3.3.19)

For n D 3,

Elementary (fundamental) solution of Laplace operator Definition 3.3.1. A function E 2 C 1 .Rn n ¹0º/ which is locally integrable on Rn (i.e. E 2 D 0 .Rn / is a distribution on Rn ) is called an elementary or fundamental P 2 solution of the Laplace operator D niD1 @ 2 if it satisfies the equation: @x1

E D ı

in D 0 .Rn /:

(3.3.20)

Elementary solution E of the Laplace operator is not unique. (3.3.21) In fact, any function E0 harmonic in Rn (i.e. E0 D 0 in Rn ), which is a C 1 function in Rn , is also an elementary solution of . Thus, E C E0 is also an elementary solution of , i.e. .E C E0 / D E C E0 D ı C 0 D ı in D 0 .Rn /. Theorem 3.3.2. En D 

1 1  n2 .n  2/Sn r

1

for n ¤ 2; r D .x12 C    C xn2 / 2

(3.3.22)

and E2 D 

1 1 ln 2 r

1

for n D 2; r D .x12 C x22 / 2

(3.3.23) n

are elementary solutions of the Laplace operator , where Sn D

2. 2 / . . n 2/

1 Proof. En 2 C 1 .Rn n ¹0º/ and En 2 L1loc .Rn /, since for n ¤ 2, r n2 2 C 1 .Rn n 1 1 1 n 1 2 2 ¹0º/ and r n2 2 Lloc .R /, and ln r 2 C .R n¹0º/, ln r 2 Lloc .R / for n D 2. Con1 1 . r n2 / D sequently, En 2 D 0 .Rn / and, from Theorem 3.3.1, En D  .n2/S n 1 n ı/  ..n2/S D ı in D 0 .Rn / for n ¤ 2 and E2 D  2 .ln 1r / D  2ı 2 D ı in .n2/Sn 0 2 D .R / for n D 2.

245

Section 3.3 Elementary solutions

Elementary solution of ordinary differential operator linear ordinary differential operator defined by: Ln 

Let Ln be the nth-order

dn d n1 C a1 .t / n1 C    C an .t /; n dt dt

(3.3.24)

where ak 2 C 1 .1; 1Œ/ 8k D 1; 2; : : : ; n. Definition 3.3.2. A distribution E 2 D 0 .1; 1Œ/ is called an elementary or fundamental solution of Ln on 1; 1Œ if and only if Ln E D ı

in D 0 .1; 1Œ/:

Elementary solution E 2 D 0 .1; 1Œ/ of (3.3.24) is not unique.

(3.3.25) (3.3.26)

Construction of elementary solution E 2 D 0 . 1; 1Œ/ of (3.3.24) Let y D y.t / with y 2 C 1 .Œ0; 1Œ/ be the unique solution of the Cauchy problem Ln y D 0, y.0/ D y 0 .0/ D    D y .n2/ .0/ D 0; Let H D H.t / be the Heaviside function. Set ´ y.t / E.t / D H.t /y.t / D 0

y .n1/ .0/ D 1:

for t > 0I for t < 0:

(3.3.27)

(3.3.28)

Then E 2 D 0 .1; 1Œ/ is an elementary solution of Ln , i.e. Ln E D ı. In fact, 8 2 D.1; 1Œ/,   Z 1 Z 1 d d hE 0 ; i D  E; H.t /y.t / dt D  y.t / 0 .t /dt D C dt dt 1 0 Z 1 Z 1 C y 0 .t /.t /dt D H.t /y 0 .t /.t /dt; D y.t /.t /j1 0C 0C

1

since y.0C / D y.0/ D 0, .t / D 0 for t D 1 H) E 0 .t / D H.t /y 0 .t / 8t ¤ 0. Similarly, E .k/ .t / D H.t /y .k/ .t / 8k D 2; 3; : : : ; n  1: Then hE

.n/

(3.3.29)

  Z 1 .n1/ d ; i D  E ; y .n1/ .t / 0 .t /dt D dt 0C Z 1 C H.t /y .n/ .t /.t /dt D y .n1/ .t /.t /j1 0C 1

Dy

.n1/

.0/.0/ C hH.t /y .n/ ; i D hı; i C hH.t /y .n/ ; i;

D hı C H.t /y .n/ ; i

8 2 D.R/

246

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

since y .n1/ .0C / D y .n1/ .0/ D 1, hı; i D .0/, H)

E .n/ D ı C H.t /y .n/

in D 0 .R/:

(3.3.30)

Then, from (3.3.29) and (3.3.30), Ln E D E .n/ C a1 E .n1/ C    C an E D ı C H.t /y .n/ C a1 H.t /y .n1/ C    C an H.t /y D ı C H.t /ŒLn y D ı

in D 0 .R/;

since, by definition of y.t /, Ln y D 0. H) Ln E D ı in D 0 .1; 1Œ/, i.e. E D H.t /y.t / is an elementary solution of (3.3.24). In particular, 



for n D 1, a1 D a, y D e at is the solution of dy C ay D 0 with y.0/ D 1, dt d at is an elementary solution of dt C a; (3.3.31) and E1 .t / D H.t /e for n D 2, y D

sin at a

is the solution of

d 2y dt 2

C a2 y D 0 with y.0/ D 0; y 0 .0/ D

1, and E2 .t / D H.t / sinaat is an elementary solution of

d2 dt 2

C a2 .

(3.3.32)

Elementary solution of C k2 , k > 0 Example 3.3.1. Show that  ikr  e e ikr ; E .x/ D  C .1  / 4 r 4 r

(3.3.33)

with  2 R, r D .x12 C x22 C x32 /1=2 , is an elementary solution of C k 2 , D P3 e ikr @2 2 , k > 0, E0 .x/ D  4 r being the only elementary solution satisfying the j D1 @xj

Sommerfeld condition [2, p. 315]. Proof. e ¹0º/ and

˙ikr

D cosrkr ˙i sinrkr is a C 1 -function in R3 n¹0º, since cosrkr 2 C 1 .R3 n r sin kr 2 C 1 .R3 /; r D 0 being a removable singular point. Hence Œ. C r ˙ikr k 2 /. e r /.x/ in the usual pointwise sense can be found as follows: 8x 2 R3 n ¹0º,     @ ikr @ 1 i kxj ikr xj e I e .x/ D .x/ D  3 I @xj r r @xj r       2 2 k 2 xj2 ikr 2i k i k i kxj @ ikr ikr 2  3  2 e I Œ .e /.x/ D  k e ikr I e .x/ D 2 r r r r @xj



247

Section 3.3 Elementary solutions

   ikr        3  X e 1 @ 1 @ ikr ikr .e / .x/ .x/ D e .x/ C 2 .x/  r r @xj r @xj j D1

1 C Πe ikr .x/ r    3 ikr X i kxj2 1 2i k ikr 2 2e 0C2 D k De  k C r4 r r r j D1

(3.3.34) H) H)

  ikr  e 2 . C k / .x/ D 0 8x 2 R3 n ¹0º r     cos kr sin kr . C k 2 / .x/ D 0; . C k 2 / .x/ D 0 r r

(3.3.35) 8x 2 R3 n ¹0º;

but sin kr 2 C 1 .R3 / r

H)

  2 sin kr .x/ D 0 . C k / r

H) . C k 2 / sinrkr D 0 in D 0 .R3 / in the distributional sense. 8R > 0, Z

8x 2 R3 :

(3.3.36)

ˇ ˇ ˇ ˇ Z Z ˇ cos kr ˇ ˇ cos kr ˇ 1 ˇd x D lim ˇd x  lim ˇ ˇ dx ˇ ˇ r ˇ ˇ C C r "!0 "!0 kxkR "kxkR "kxkR r Z R Z  Z 2 R2 1 2 r sin drd d D 4  < C1 D lim r 2 "!0C " 0 0

H)

cos kr r

2 L1loc .R3 /. Now we will find . C k 2 / cosrkr in D 0 .R3 / 8 2 D.R3 /:

        cos kr 2 cos kr 2 cos kr ; D ; . C k / ; C k r r r       cos kr cos kr 2 cos kr D ;  C ;k  D ; . C k 2 / r r r Z cos kr D lim . C k 2 /d x D lim I1 ."/: r "!0C kxk" "!0C „ ƒ‚ … I1 ."/

(3.3.37)

248

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions cos kr r

2 C 1 .R3 n ¹0º/, using (3.2.5) we have   ²  ³ Z cos kr cos kr 2 2 . C k /  . C k / .x/.x/ d x r r kxk"    ³ ² Z cos kr cos kr   .x/.x/ d x D r r kxk"       Z @.  / @ @ cos kr cos kr @ D ./ ; D .x/  dS; @r r r @r @r @n" " WkxkD"

Since

R and using (3.3.35): kxk" Œ. C k 2 / cosrkr .x/.x/d x D 0. Hence, 8" > 0, from (3.3.37), Z cos kr I1 ."/ D . C k 2 /d x r kxk"  ²    ³ Z cos kr kr sin kr  cos kr @ 2  D .x/ " d! r2 r " WkxkD" rD" rD" @r Z Z Z @ d! .x/d! cos k" .x/d! " cos k" D  k" sin k" " WkxkD" " WkxkD" " WkxkD" @r ƒ‚ … „ ƒ‚ … „ ƒ‚ … „ I2 ."/

I3 ."/

I4 ."/

D I2 ."/  I3 ."/  I4 ."/; where dS D "2 d!; d! being the surface area measure on the unit sphere. ˇ ˇ ˇ ˇ ˇ ˇ  p ˇ ˇ ˇ @ ˇ @ ˇ ˇ ˇ ˇ D ˇr   .nO " /ˇ  kr .x/kknO " k  3 max max ˇ .x/ˇˇ D C I ˇ ˇ ˇ @r ˇ ˇ 1i3 x2supp./ @xi Z  Z 2 Z d! D sin d

d D . cos j 0 /2 D 4I kxkD1

0

 Z jI4 ."/j  "j cos k"j C

 d!

0

 .4 C "/ ! 0

as " ! 0C

kxkD1

 jI2 ."/j  k"j sin k"j

H) lim I4 ."/ D 0I "!0C  d!  .4k max j.x/j/" ! 0

Z max j.x/j x2supp./

kxkD1

H) Z



Z

I3 ."/ D cos k"

lim I2 ."/ D 0I

"!0C

2

." sin cos ; " sin sin ; " cos / sin d d 0

0

with x1 D " sin cos , x2 D " sin sin , x3 D " cos , 0   , 0    2.

249

Section 3.3 Elementary solutions

But lim"!0C ." sin cos ; " sin sin ; " cos / D .0; 0; 0/ D .0/, and lim"!0C cos k" D 1, Z H)



Z

2

lim I3 ."/ D lim cos k"  lim

"!0C

"!0C

Z

"!0C



Z

Z

"!0C 0  Z 2

.  ;  ;  / sin d d 0

0

2

lim .  ;  ;  / sin d d

D1 0

D .0/

sin d d D 4.0/: 0

0

Hence, lim I1 ."/ D  lim I2 ."/  lim I3 ."/  lim I4 ."/

"!0C

"!0C

"!0C

D 4.0/

"!0C

3

8 2 D.R /:

From (3.3.37), h. C k 2 / cosrkr ; iD4.0/Dh4ı; i H) . Ck 2 / cosrkr D ikr

4ı in D 0 .R3 / H) . C k 2 / e r

D 4ı in D 0 .R3 /, since . C k 2 / sinrkr D 0 ikr

in D 0 .R3 / (by (3.3.36)). Hence, E D  e4 r is an elementary solution of C k 2 in D 0 .R3 /. Replacing ‘i ’ by ‘i ’ in (3.3.34) and (3.3.35) and following the subsequent ikr steps with the necessary modifications, we can show that EN D  e4 r is also an ikr

ikr

e e elementary solution of C k 2 . Hence, 8 2 R, E D . 4 r / C .1  / 4 r is 2 also an elementary solution of C k . cos kr 2 Obviously,  4 r is an elementary solution of C k in real form. (3.3.38)

Construction of elementary solution with the help of pseudofunctions Pf.r  / 1 Since a detailed analysis of Pf.r  / for  2 C and r D .x12 C    C xn2 / 2 in the distributional sense with proofs and justifications is prohibitively long and involved, and, moreover, such an analysis will not be required later for other topics, we have decided to state only the final results, without any proof, for which we refer to [1] and [8]. But we have shown how these interesting results can be used to construct elementary solutions of operators encountered in real applications. For this, we begin with the case n D 1, i.e. with the distribution defined by the pseudo-function (finite  part) Pf.xC / D Pf.x  /x>0 for  2 C [8]. 8 2 D.R/, 

Z

1

hPf.x /x>0 ; i D Pf

Z



1

x .x/dx D lim

"!0C

0

Z

1

D lim

"!0C



x .x/dx C "

 x .x/dx  I."/ 

"

kDk./ X j D0

  .j / .0/ "Cj C1 ; (3.3.39) jŠ  C j C 1

250

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

where I."/ is the infinite part of the divergent integral (see also (1.4.12) and (1.4.18)); Pf.x  /x>0 D .x  /x>0

for Re./ > 1I

(3.3.40)

k D k./ in (3.3.39) depends on the value of , for  C j C 1 D 0 (i.e. for 0  D m with m 2 N and m C j C 1 D 0), the term "0 must be replaced by ln " (see (1.4.22)). (3.3.41) Distribution defined by a pseudofunction (or finite part) Pf.r  / Now we will extend the definition in (3.3.39)–(3.3.41) to Pf.r  / with r D r.x/ D .x12 C x22 C    C xn2 /1=2 8x D .x1 ; x2 ; : : : ; xn / 2 Rn ,  2 C, as follows. Let r D " denote an n-dimensional sphere with radius " > 0. Then, for  2 C, the distribution Pf.r  / is defined similarly by [8] (see also [1]): 8 2 D.Rn /,  Z Z r  .x/d x D lim r  .x/d x  I."/ hPf.r  /; i D Pf "!0C

Rn

Z



D lim

"!0C

where

r .x/d x C r"

p . /n Hk D 2k1 2 kŠ. n2 C k/ D

r"

X k

 "CnC2k ; Hk .0/  C n C 2k k

p 2. /n I with H0 D .n=2/

(3.3.42)

(3.3.43)

@2 @2 @2 C C    C I @xn2 @x12 @x22

.0/ D hı; .x/i D h ı; i

8 2 D.Rn /I

the integer k depends on ; for  C n C 2k D 0 (i.e. for negative integer  D m 0 with m 2 N and m C n C 2k D 0), the term "0 must be replaced by ln ". (3.3.44) For the deduction of the ‘exotic’ expressions in square brackets Œ: : : in (3.3.42) and of Hk in (3.3.43), the analyticity of F ./ D hPf.r  /; i, except for non-positive, even integral values of  C n, and its Laurent series expansion have been used to find the residue at simple poles, along with other artifices (see [1], [11, p. 293]). For Re./ > n, Pf.r  / D r  , i.e., 8 2 D.Rn /, Z   r  .x/d x: (3.3.45) hPf.r /; i D hr ; i D Rn

Laplacian ŒPf.r m / for integral values of  [8] For  D m 2 Z with m C n D 2p C 2, p 2 N0 (i.e. p  0, m C n  2 D 2p), p .2  n  4p/. /n p m m2 ı; / C 2p1 (3.3.46) ŒPf.r / D m.m C n  2/ Pf.r 2 pŠ. n2 C p/ where ı is the Dirac distribution with concentration at 0.

251

Section 3.3 Elementary solutions

For m C n ¤ an even integer  2, ŒPf.r m / D m.m C n  2/ Pf.r m2 /

Œ8:

(3.3.47)

The most important particular case of (3.3.45)/(3.3.46) is m C n D 2, i.e. p D 0 and m D .n  2/ > n with n ¤ 2. Then, from (3.3.45) and (3.3.46), for n ¤ 2, we have   1 1 Pf n2 D n2 (i.e. the symbol Pf is useless), and r r p   .n  2/2. /n 1 ı D .n  2/H0 ı D .n  2/Sn ı; (3.3.48) n2 D r . n2 / p where H0 D Sn D 2. /n = .n=2/ D surface area of the n-dimensional unit sphere (see (3.3.43) and Theorem 3.3.1 for n ¤ 2). 1 Case n D 2: .n  2/H0 D 0 and we are to replace r n2 by ln 1r , i.e., for n D 2, 1 . ln 1r plays the role of r n2   1 ln D 2ı r

for n D 2

(3.3.49)

(see the independent proof of (3.3.13)–(3.3.14)). Iterated Laplacian k ŒPf.r m / ; k Œr 2kn ln r Using (3.3.45)–(3.3.48), the iterated Laplacian of the distribution Pf.r m / can be found. Here, we state the following important final results, which will be of use in the construction of an elementary solution of iterated Laplace operator k . We consider the following two cases: 1. For k 2 N, 2k  n > n, from (3.3.48), Pf.r 2kn / D r 2kn . Then k .r 2kn / is given in [8, p. 47] (see also [1] for more details): for 2k n < 0 or 2k n > 0 and odd n, k .r 2kn / D .2k  n/.2k  2  n/    .4  n/.2  n/2k1 .k  1/ŠH0 ı; (3.3.50) where H0 is given by (3.3.43). 2. For 2k  n  0 and even n, we give the formula for k Œr 2kn ln r (valid for this case only): k Œr 2kn ln r D Œ.2k  n/.2k  2  n/    .4  n/.2  n/2k1 .k  1/ŠH0 ı; (3.3.51) where the factor 0 of the expression in square brackets Œ: : : in (3.3.51) must be omitted. (From (3.3.51), ln r D  ln 1r D 2ı in (3.3.49) can be retrieved

252

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

with the choice of k D 1 and n D 2). For example, for k D 2, n D 2, 2k  n D 2, we are to omit the factor 2  n D 2  2 D 0 in square brackets Œ: : : in (3.3.51) and we get, from (3.3.51): 2 .r 2 ln r/ D Œ.4  2/221 .2  1/ŠH0 ı D 4H0 ı

with H0 D 2: (3.3.52)

Elementary solution E of iterated Laplace operator k From (3.3.50) and (3.3.51), we get an elementary solution E of k , i.e. k E D ı in D 0 .Rn /, as follows: 1. For 2k  n < 0 or for 2k  n  0 and odd n, 9 a constant Bk;n such that k .Bk;n r 2kn / D ı;

(3.3.53)

i.e. E.x/ D Bk;n r 2kn is an elementary solution of k for 2k  n < 0, or for 2k  n  0 and odd n, Bk;n is determined from (3.3.50). 2. For 2k  n  0 and even n, 9 a constant Ak;n such that k ŒAk;n r 2kn ln r D ı;

(3.3.54)

i.e. E.x/ D Ak;n r 2kn ln r is an elementary solution of k for 2k  n  0 and even n, Ak;n is determined from (3.3.51). Combining (3.3.53) and (3.3.54) for arbitrary k 2 N and n, 9 constants Ak;n and Bk;n , one of these two constants being equal to 0 in each case, such that k Œr 2kn .Ak;n ln r C Bk;n / D ı;

(3.3.55)

from which (3.3.53) (resp. (3.3.54)) is obtained for Ak;n D 0 (resp. Bk;n D 0). For details, we refer to [1], [8]. Hence, ´ Ak;n r 2kn ln r; if 2k  n  0 and n is evenI E.x/ D (3.3.56) Bk;n r 2kn if 2k  n < 0; or if 2k  n  0 and n is odd is an elementary solution of k , from which we can retrieve an elementary solution of the Laplace operator with k D 1, i.e. 



1 1 for k D 1, n ¤ 2, E.x/ D B1;n  r n2 with B1;n D  .n2/S (since for n  3, n 2k  n D 2  n < 0, and for n D 1, 2k  n D 2  1 > 0, n being an odd number),

for k D 1, n D 2, E.x/ D A1;2 ln r with A1;2 D and n is even),

1 2

(since 2k  n D 2  2 D 0

are elementary solutions of the Laplace operator given in Theorem 3.3.2.

253

Section 3.3 Elementary solutions

 k Elementary solution of the operator .1  4 Here we will state, as usual, 2/ the final results given in [8], which depend heavily on the use of results of classical Bessel function theory for the evaluation of the involved integrals obtained by the introduction of n-dimensional spherical coordinates (see, for example, [7, p. 334]). Following [8, p. 47], we define the distributions Lm 2 D 0 .Rn / by: p mn 2. /m PfŒr 2 K nm .2 r/ for m ¤ 2k with k 2 N0 I (3.3.57) Lm D 2 .m=2/   k L2k D 1  ı for m D 2k with k 2 N0 I (3.3.58) 4 2

L0 D ı;

(3.3.59)

where K is the classical function from the theory of Bessel functions defined, 8 2 R, by:   X 1 . x2 /2k ŒI .x/  I .x/ x K .x/ D  ; with I .x/ D ; 2 sin  2 kŠ. C k C 1/ kD0

(3.3.60) which converges exponentially to 0 at infinity. For Lm defined in (3.3.57)–(3.3.59), the following recurrent relation holds [8, p. 47]: for m ¤ 2k with k 2 N0 ,     2 Lm D 1  (3.3.61) Lm2 D Lm4 ; : : : 1 4 2 4 2 Hence, 

1 4 2

k Lm D Lm2k

In particular, for m D 2k, .1  H)

From (3.3.57),

.m ¤ 2k with k 2 N0 /:

k / L2k 4 2

L2k D

is an elementary solution of .1 

(3.3.62)

D L0 D ı (by (3.3.59) and (3.3.57))

n 2 k PfŒr k 2 K n2 k .2 r/ .k  1/Š

(3.3.63)

k / . 4 2

Elementary solution E of the biharmonic operator 2 Example 3.3.2. Find an elementary solution E of the biharmonic operator 2 . d4 Solution. Case n D 1: 2 D dx 4 , r D jxj 8x 2 R, k D 2 H) 2kn D 41 > 0, but n is odd. From (3.3.50) and (3.3.53), E.x/ D B2;1 r 41 D

1 jxj3 ; 12

(3.3.64)

254

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions p 2  .1=2/

since from (3.3.43), H0 D

D 2, and from (3.3.50), B2;1 D

1 .41/.21/21Š2

D

1 12 .

In fact,

1 D 12 But Z 0 1

Z

0

Z

1

1 d 4 jxj3 4 dx dx 1 12  Z 1 3 (4) 3 (4) .x/  .x/dx C x  .x/dx 8 2 D.R/:

h 2 E; i D hE; 2 i D

1

0

x 3  (4) .x/dx D x 3  .3/ .x/j01 C 3 D 3x 2  2 .x/j01  6 D 6x

.1/

.x/j01

Z

Z

0

x 2  (3) .x/dx

1 0

x .2/ .x/dx

1

Z

0

 .1/ .x/dx D 6.x/j01 D 6.0/;

C6 1

since x k  .j /R.x/ D 0 for x D 0, 1 . 2 D.R/ H)  .k/ .˙1/ D 0 8k/. 1 Similarly, 0 x 3  (4) .x/dx D 6.0/. 1 2 Hence, h E; i D 12 Œ6.0/ C 6.0/ D .0/ D hı; i 8 2 D.R/ H) 2 E D Case n D 2:

d4 E dx 4

2 D D

D ı with E D

1 3 12 jxj

8x 2 R:

@4 @4 @4 C 2 2 2 C 2; 4 @x1 @x1 @x2 @x4

r D .x12 C x22 /1=2 ;

kD2

H) 2k  n D 4  2 > 0 and n is even. Hence, from (3.3.51) and (3.3.54), an elementary solution E of is given by [37]: E.x/ D A2;2 r 2kn ln r D since from (3.3.43), H0 D

p 2. /2 . 22 /

1 2 r ln r; 8

D 2, and from (3.3.52), A2;2 D

(3.3.65) 1 8 .

In fact, r 2 2 C 1 .R2 /; ln r 2 L1loc .R2 / with its distributional derivative the usual partial derivative Π@x@ k formula (2.5.2) with f D r 2 , .r 2 ln r/ D

ln r.x/ D

xk r2

2

L1loc .R2 /.

@ @xk

ln r D

Hence, applying Leibniz’s

2 2 X X @2 2 @r 2 @ ln r 2 .r ln r/ D ln r r C 2  C r 2 ln r @xk @xk @xk2

kD1

D 4 ln r C 2

kD1

2 X kD1

2xk 

xk C r 2 2ı D 4 ln r C 4 C 0 in D 0 .R2 /; r2

255

Section 3.3 Elementary solutions

since ln r D 2ı (by (3.3.14)). hr 2 2ı; i D 2hı; r 2 i D 2.r 2 /.0/ D 2  0 D 0

8 2 D.R2 /

H) r 2 2ı D 0 in D 0 .R2 /. Then, 1 1 . .r 2 ln r// D .4 ln r C 4/ 8 8 1  4  2ı C 0 D ı in D 0 .R2 /: D 8

E D

Case n D 3: @4 @4 @4 @4 @4 @4 C C C 2 C 2 C 2 ; @x14 @x24 @x34 @x12 @x22 @x22 @x32 @x32 @x12

2 D

r D .x12 C x22 C x32 /1=2 , k D 2 H) 2k  n D 4  3 > 0, but n is odd. Hence, from (3.3.50) and (3.3.53), an elementary solution E of 2 is given by: E.x/ D B2;3 r D  since from (3.3.43), H0 D

p 2. /3 .3=2/

1 r; 8

(3.3.66) 1 .43/.23/2Š4 1 .x12 C x22 C x32 / 2 ,

D 4, and from (3.3.50), B2;3 D

1 1 1 D  8 . In fact, E D Π. 8 r/ D  8 Πr. For r D 1 @ 1 1 3 r 2 Lloc .R / with its distributional derivative @x . r / D the usual partial derivative k

Œ @x@ . 1r /.x/ D  xr k3 ; r 2 D 6 in D 0 .R3 /. Since r 2 2 C 1 .R3 /; r D r 2  k L1loc .R/, applying Leibniz’s formula (2.5.2), we have

1 r

2

    3 X @r 2 1 @ 1 1 1 r D r 2   D r 2 C 2 C r 2 r r @xk @xk r r kD1

D

6 C2 r

3 X

2xk 

kD1

since . 1r / D 4ı D 0 .R3 /. Hence,

2 1 xk 6 4 C r 2 .4ı/ D   0 D  2 r r r r r

in D 0 .R3 /. Thus, r D . 2r / D 2.4ı/ D 8ı in

E D .

1 1 r/ D  r D ı 8 8

An elementary solution E of 3 D . and (3.3.54):

in D 0 .R2 /;

@2 @x12

C

E D A3;2 r 4 ln r D

@2 3 / @x22

in D 0 .R3 /:

in D 0 .R2 / is given by (3.3.51)

1 4 r ln r; 128

(3.3.67)

256

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

since k D 3, n D 2 H) 2k  n D 4 and n is even, H0 D 2, and A3;2 D

1 1 D : Œ.6  2/.4  2/231 .3  1/Š2 128

In fact, r 4 2 C 1 .R2 /; ln r 2 L1loc .R2 / with its distributional derivative the usual partial derivative formula (2.5.2), we have 3

4

2

Π@x@ k

ln r.x/ D

4

.r ln r/ D . .r ln r// D

2



xk r2

L1loc .R2 /.

2

@ @xk

ln r D

Then, applying Leibniz’s

 2 X @r 4 @ ln r 4  C r ln r ln r r C 2 @xk @xk 4

kD1

2

2

2

4

2

2

D .16r ln r C 2  4  r C r 2ı/ D 16 .r ln r/ C 2  4 2 .r 2 / C 2 .0/ in D 0 .R2 /;

D 16  8ı C 0 C 0 D 128ı

since 2 .r 2 ln r/ D 8ı in D 0 .R2 / (by (3.3.65)); 2 .r 2 / D 0 in D 0 .R2 /I hr 4 2ı; i D 2hı; r 4 i D 2.r 4 /.0/ D 0

8 2 D.R2 /

H) r 4 2ı D 0 in D 0 .R2 /. Hence,   1 4 1 1 3 3 ED r ln r D 3 .r 4 ln r/ D  128ı D ı 128 128 128 in D 0 .R2 /. An elementary solution E of 4 D .

@2 @x12

C

@2 4 / @x22

in D 0 .R2 /, which arises in elastic

cylindrical shell analysis, is given by (3.3.51) and (3.3.54): 1 r 6 ln r; 4608 since k D 4, n D 2 H) 2k  n D 6 > 0 and n is even, H0 D 2, and E D A4;2 r 6 ln r D

A4;2 D

(3.3.68)

1 1 D : 41 .8  2/.6  2/.4  2/2 .4  1/Š2 4608

Indeed,   2 X @r 6 @ ln r 4 Œr 6 ln r D 3 . .r 6 ln r// D 3 ln r .r 6 / C 2  C r 6 ln r @xk @xk kD1

3

4

4

6

3

4

D .36r ln r C 2  6  r C r 2ı/ D 36 .r ln r/ C 12 3 .r 4 / C .0/ D 36  128ı C 0 C 0 D 4608ı; since 3 .r 4 ln r/ D 128ı in D 0 .R2 / (by (3.3.67)); 3 .r 4 / D 0; hr 6 2ı; i D 2hı; r 6 i D 0 8 2 D.R2 / H) r 6 2ı D 0 in D 0 .R2 /. Hence, 4 E D 1 . 4608 r 6 ln r/ D ı in D 0 .R2 /.

Section 3.4 Applications

257

Remark 3.3.1. We will come back to this topic in a more useful setting in Chapter 8 after the introduction of Fourier transform of tempered distributions.

3.4

Applications

Construction of finite-dimensional subspaces of Sobolev space H m ./ In Section 2.15, Chapter 2, we introduced Sobolev spaces H m ./, W m;p ./, etc., and stated their elementary defining properties for arbitrary domain   Rn . Now we will construct finite-dimensional subspaces of H m ./, when   R2 is a polygonal domain in R2 for m D 1; 2. These subspaces of H m ./ are used as the finite element spaces in the finite element method of approximation of the solution of elliptic boundary value problems in . Case of two variables (n D 2) Let   R2 be a bounded polygonal domain with  D  [ ;  being the polygonal boundary of . Triangulation  h of the closed polygonal domain  [34], [35], [36] Let h denote a triangulation (i.e. subdivision) of  into closed triangles T1 ; T2 ; : : : ; TN (see Figure 3.6) such that: 



 

h D ¹T1 ; T2 ; : : : ; TN º, h D max1iN ¹dia.Ti /º; dia.Ti / D supx;y2Ti d.x; y/; S V V  D N iD1 Ti , Ti D int.Ti /, @Ti D boundary of Ti with Ti D Ti [ @Ti 8i ; (3.4.1) TVi \ TVj D ; for 1  i ¤ j  N ; For i ¤ j , Ti \ Tj D ; (empty set) or a common vertex or a common side. (3.4.2)

Remark 3.4.1. 



Triangulation h and closed triangles Ti , 1  i  N , should be understood in a generalized sense, i.e. for n D 2, the Ti ’s may be quadrilaterals, and for some closed domains , the Ti ’s may be rectangles or a combination of rectangles and triangles or quadrilaterals etc. satisfying (3.4.1) and (3.4.2). In the following discussions, T 2 h will imply closed triangles in the usual sense. For arbitrary   R2 with sufficiently smooth curved boundary , we will consider a polygonal approximation h to  and denote the polygonal domain inside h by h with h D h [ h , such that h is an approximation to . Then, closed polygonal domain h will be triangulated instead of  as shown above.

258

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

τ

τ

not

Figure 3.6 Triangulation h of  



This definition of triangulation h can be extended to the triangulation of a closed polyhedral domain   R3 into closed tetrahedrons, for example, Ti , 1  i  N , simply by replacing (3.4.2) by: For n D 3, 1  i ¤ j  N , Ti \ Tj D ; (empty set) or a common vertex or a common edge or a common face. S S For n D 2, we have N 1i¤j N iD1 @Ti D set of sides of triangles of h D .@Ti \ @Tj / [ .



@Ti \ @Tj D Lk .1  k  3/ D inter-triangular side common to Ti and Tj (i ¤ j ).



Pm .A/ D linear space of polynomials of degree  m in two variables x1 ; x2 defined in A    R2 D ¹p W p is a polynomial of degree  m in x1 ; x2 in A  º.



Pm .A/ D Span¹1I x1 ; x2 I x12 ; x1 x2 ; x22 I : : : I x1m ; x1m1 x2 ; : : : ; x2m º with dim.Pm .A// D 1 C 2 C 3 C    C .m C 1/ D .m C 1/.m C 2/=2. For example, P1 .A/ D Span¹1I x1 ; x2 º, dim.P1 .A// D 3, dim P2 .A/ D 6, etc.

Finite-dimensional vector space Xh To each triangulation h of  defined by (3.4.1) and (3.4.2), we associate a set Xh of real-valued functions vh , whose restriction to each triangle T 2 h is a polynomial of degree  m in two variables x1 ; x2 ,

259

Section 3.4 Applications

i.e. Xh D ¹vh W vh is a real-valued function with vh #T 2 Pm .T / 8T 2 h ; m 2 N0 º: (3.4.3) Properties of functions of Xh 

vh 2 Xh is a piecewise polynomial of degree  m in each T 2 h ; 8 fixed T 2 h with TV D int.T /, vh # V 2 Pm .TV / has a unique, natural polynomial T extension vT to T of vh # V , and we set vT D vh #T 2 Pm .T /. (3.4.4) T



vh 2 Xh is discontinuous, in general, across inter-triangular sides of the triangulation h .



vTi ¤ vTj for i ¤ j , i.e. in different triangles, polynomials vT are different. (3.4.5)



If vTi D vTj on their common side Ti \ Tj (i ¤ j ), then vh is continuous across the common side Ti \ Tj . (3.4.6)

Proposition 3.4.1. Xh defined by (3.4.4) is a finite-dimensional subspace of L2 ./, i.e. Xh  L2 ./:

(3.4.7)

Proof. Since Pm .T / is a vector space 8T 2 h ; 8m P 2 N0 ; Xh is obviously a vector space. By virtue of property (3.4.6), dim.Xh / D T 2h dim.Pm .T // D Card¹h º  dim.Pm .T // D N  .m C 1/.m C 2/=2 < C1, since Card¹h º D N and dim.Pm .T // D .m C 1/.m C 2/=2 H) Xh is a finite-dimensional vector space. Let vh 2 Xh . Then Z X Z 2 jvh .x/j dx1 dx2 D jvT .x/j2 dx1 dx2 

T 2h

T

 max .max jvT .x/j2 /  (area measure of ) < C1 T 2h x2T

H) vh 2 L2 ./. Hence, vh 2 Xh H) vh 2 L2 ./ H) Xh  L2 ./. But Xh 6 H 1 ./ in general. (3.4.8) To justify this, consider Example 3.1.2, in which  D 0; 1Œ  0; 1Œ; h D ¹T1 ; T2 º with N D 2;  D T1 [ T2 , T1 with vertices a2 ; a3 ; a4 ; T2 with vertices a1 ; a2 ; a4 ; 0 D T \ T2 D Œa4 ; a2  D join of a4 and a2 . Set ´ in TV1 p.x1 ; x2 / D 1 C 4x1  2x2 vh D f D q.x1 ; x2 / D 2 C 4x1 C 4x2 in TV2

260

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

such that p and q are given natural polynomial extensions to T1 and T2 respectively: vT1 D vh #T1 D 1 C 4x1  2x2 2 P1 .T1 /

8.x1 ; x2 / 2 T1 ;

vT2 D vh #T2 D 2 C 4x1 C 4x2 2 P1 .T2 /

8.x1 ; x2 / 2 T2 ;

vh being discontinuous across inter-triangular side 0 D Œa4 ; a2  with jump J D .q  p/.x1 ; x2 / D 3 C 6x2 across 0 . Then vh 2 Xh with vh #T 2 P1 .T / 8T 2 h . From (3.1.56), vh 2 L2 ./, @f h but from (3.1.58), the distributional derivative @v D @x … L2 ./. Consequently, @xi i vh … H 1 ./, although vh 2 Xh  L2 ./, i.e. Xh 6 H 1 ./. But we have the following important results. Theorem 3.4.1. Let Vh  Xh be a vector space defined by: Vh D ¹vh W vh 2 Xh ; vh 2 C 0 ./º:

(3.4.9)

Then Vh  H 1 ./ is a finite-dimensional subspace of H 1 ./. Proof. Let vh 2 Vh . Then vh 2 C 0 ./  L2 ./ H) vh 2 L2 ./. We are to prove h that the distributional derivative @v 2 L2 ./ (i D 1; 2) in order that vh 2 H 1 ./. @xi Set vT D vh #T 2 Pm .T / 8T 2 h . Then, in each TV D int.T / of T 2 h , the

T T and the usual partial derivative Π@v .x/ in the pointwise distributional derivative @v @xi @xi V sense coincide in T , both being polynomials of degree  m1. Define a new function T wi such that wi # V D Π@v .x/ 2 Pm1 .TV / (i D 1; 2) in TV 8T 2 h . Then wi @x T Ri R P 2 is defined a.e. on  and  jwi .x/j2 dx1 dx2 D T 2h TV .wi #TV .x// dx1 dx2 D R P @vT 2 2 T 2h TV Π@xi .x/ dx1 dx2 < C1 H) wi 2 L ./. Let wi #T 2 Pm1 .T / be the natural polynomial extension to T D TV [ @T of wi # , @T being the boundary of

TV

T (resp. TV ) 8T 2 h . Now we will show that the distributional derivative in

D 0 ./,

i.e.

@vh @xi

D wi 2

L2 ./,

wi .x/.x/dx1 dx2 D 

X Z T 2h

TV

wiT .x/.x/dx1 dx2

 X Z  @vT D .x/ .x/dx1 dx2 @xi V T 2h T   Z X Z  @ @ D .vT /.x/ dx1 dx2  vT dx1 dx2 : @xi TV @xi TV T 2h

D wi

i D 1; 2. In fact, 8 2 D./,

Z hwi ; i D

@vh @xi

(3.4.10)

261

Section 3.4 Applications

Applying Green’s Theorem 3.1.2 to the first double integral over each TV with boundary @T D L1 [ L2 [ L3 , ¹Lk º3kD1 being the three sides of T D TV [ @T , we have  Z  Z @ .vT / dx1 dx2 D vT  cos.nO T ; xi /ds; S TV @xi @T D 3kD1 Lk where nO T is the unit vector normal to @T , nO T  Oik D cos.nO T ; xk /, and ds is the element arc length measured along @T . Then  X Z  @ X Z .vT / dx1 dx2 D vT  cos.nO T ; xi /ds V @xi T 2h T T 2h @T Z Z X vT  cos.nO T ; xi /ds C vh  cos.nO T ; xi /ds; (3.4.11) D Lj @T;Lj 

Lj



R R where, 8 inter-triangular sides Lj , Lj .: : :/ds C L .: : :/ds D 0, since vh and  j are continuous across each Lj and the line integrals along Lj and Lj have opposite senses of orientation, and consequently Z X vT  cos.nO T ; xi /ds D 0I (3.4.12) Lj @T;Lj 

Lj

 is obtained as the union of the boundary sides Lj   of all boundary triangles, at least one side of which is a part of , and Z vh  cos.nO T ; xi /ds D 0; (3.4.13) 

since  2 D./ H) # D 0. Then, from (3.4.10)–(3.4.13), we have Z X Z @ @  vT dx1 dx2 D  vh dx1 dx2 hwi ; i D @x @x V i i T  T 2h     @ @vh D  vh ; ;  8 2 D./ D @xi @xi @vh in D 0 ./, but wi @xi 2 L2 ./ (i D 1; 2).

H) wi D H)

@vh @xi

2 L2 ./

h 2 L2 ./, i D 1; 2 H) vh 2 H 1 ./. Thus Vh  Hence, vh 2 L2 ./, @v @xi H 1 ./. But Vh  Xh ; Xh being a finite-dimensional vector space H) Vh is also a finite-dimensional vector space. Hence, Vh is a finite-dimensional subspace of H 1 ./.

262

Chapter 3 Piecewise smooth functions, Green’s formula, elementary solutions

Remark 3.4.2. For the inclusion Vh  H 1 ./ to hold, the minimal value of m to be used in defining Pm .T / in (3.4.3) and (3.4.9) is 1, i.e. Vh D ¹vh W vh 2 C 0 ./; vh #T 2 Pm .T /

8T 2 h ; m  1º  H 1 ./; (3.4.14)

since for m D 0, vh #T 2 P0 .T / H) vh #T D C ¤ 0, and in different triangles Ti 2 h , different constants C , i.e. vh #Ti D Ci ¤ 0, vh #Tk D Ck ¤ 0 with Ci ¤ Ck in general for i ¤k

H)

vh … C 0 ./:

(3.4.15)

Theorem 3.4.2. Let h be a triangulation of  defined by (3.4.1) and (3.4.2) and Vh  Xh be defined by: Vh D ¹vh W vh 2 Xh ; vh 2 C 1 ./º D ¹vh W vh 2 C 1 ./; vh #T 2 Pm .T / 8T 2 h º:

(3.4.16)

Then Vh  H 2 ./. Proof. The proof is exactly similar to that of Theorem 3.4.1, and is left as an exercise for the reader. Remark 3.4.3. 

The minimal value of m to be used in Pm .T / in defining Vh in (3.4.16) is 5 (see [35]), i.e. (3.4.16) is to be replaced by: Vh D ¹vh W vh 2 C 1 ./; vh #T 2 Pm .T / 8T 2 h ; m  5º  H 2 ./: (3.4.17)



For Vh  H k ./, Vh  Xh is defined by: Vh D ¹vh W vh 2 C k1 ./; vh #T 2 Pm .T / 8T 2 h ; m  4.k  1/ C 1º; (3.4.18) from which (3.4.14) and (3.4.17) can be retrieved as particular cases for k D 1; 2 respectively.

Chapter 4

Additional properties of D 0./

4.1

Reflexivity of D./ and density of D./ in D 0 ./

Reflexivity of D./ First of all, we will show that D./ can be identified with a subspace of D 00 ./, which is the (topological) dual space of D 0 ./, i.e. D 00 ./  .D 0 .//0 : To every  2 D./, we can associate a linear functional L on D 0 ./ defined by L .T / D T ./ D hT; i

8T 2 D 0 ./;

(4.1.1)

since L .˛1 T1 C ˛2 T2 / D h˛1 T1 C ˛2 T2 ; i D ˛1 hT1 ; i C ˛2 hT2 ; i D ˛1 L .T1 / C ˛2 L .T2 / 8Ti 2 D 0 ./; 8˛i 2 R: 8 2 D./, L defined by (4.1.1) is a continuous linear functional on D 0 ./, i.e. L 2 D 00 ./  .D 0 .//0 , which is the second (topological) dual space of D./, since Tn ! 0 in D 0 ./

H)

L .Tn / D hTn ; i ! 0 in R as n ! 1 8 2 D./: (4.1.2)

The mapping  2 D./ 7! L 2 D 00 ./ is one-to-one. In fact, L1 D L2 2 D 00 ./

(4.1.3)

8T 2 D 0 ./

H)

L1 .T / D L2 .T /

H)

hT; 1 i D hT; 2 i

8T 2 D 0 ./

H)

hT; 1  2 i D 0

8T 2 D 0 ./

H)

1  2 D 0

H)

1 D 2

in D./: (4.1.4)

Conversely, suppose that 9 D 1  2 ¤ 0 in D./ such that hT; i D 0 8T 2 D 0 ./. Then 9 at least one x0 2  such that .x0 / ¤ 0. Choose T D ıx0 D ı.xx0 / 2 D 0 ./, ıx0 being the Dirac distribution concentrated at x0 . Then hT; i D hıx0 ; i D .x0 / ¤ 0, which contradicts the fact that hT; i D 0 8T 2 D 0 ./. Hence, D./ can be identified with a subspace of D 00 ./, i.e. D./  D 00 ./. (4.1.5)

Chapter 4 Additional properties of D 0 ./

264

It remains to show that the mapping  2 D./ 7! L 2 D 00 ./ is surjective from D./ onto D 00 ./. Then the mapping will be bijective from D./ onto D 00 ./. But the proof of the surjectivity of the mapping from D./ onto D 00 ./ is quite involved, since we are to show that D./ is a Montel space, every Montel space being a reflexive one [8], [9], [25]. We have decided not to overburden the reader with these, which can be found in [8], [9], [25], and will not be required in our treatment later. Without proof, we agree to accept that the mapping  2 D./ 7! L 2 D 00 ./ is onto. Then, from (4.1.3) and (4.1.5), this mapping is bijective, and, from (4.1.2), it is continuous. Hence, D./ can be identified with D 00 ./, i.e. 8L 2 D 00 ./; 9 a unique  2 D./ such that L.T / D hT; i 8T 2 D 0 ./:

(4.1.6)

In other words, we have agreed to accept. Theorem 4.1.1. D./ is reflexive, i.e. D./  D 00 ./. Density of D./ in D 0 ./ Theorem 4.1.2. D./ is a dense subspace of D 0 ./. Proof. Here we give a direct proof. (For an alternative proof based on convolution, see Theorem 6.5.1.) D./ ,! D 0 ./ (i.e. ,!W D./ ! D 0 ./ is an imbedding operator). To every  2 D./  L1loc ./, we can associate a unique distribution T 2 D 0 ./ defined by: Z h; iD 0 ./D./ D hT ; i D .x/ .x/d x 8 2 D./: (4.1.7) 

,!W D./ ! D 0 ./ is a (linear, continuous) imbedding operator from D./ onto the subspace .,!D.//  D 0 ./: 8 2 D./, ,!  D T 2 D 0 ./, with T defined by (4.1.7). ,! is linear: ,!.˛1 1 C ˛1 2 / D T˛1 1 C˛2 2 , with Z .˛1 1 C ˛2 2 / d x D ˛1 T1 . / C ˛2 T2 . / hT˛1 1 C˛2 2 ; i D 

D h˛1 T1 C ˛2 T2 ; i 8

2 D./

H) T˛1 1 C˛2 2 D ˛1 T1 C ˛2 T2 H) ,!.˛1 1 C ˛2 2 / D ˛1 ,!1 C ˛2 ,!2 . ,! is one-to-one: ,!R1 D ,!2 in DR0 ./ H) T1 D T2 in D 0 ./ R H)  1 d x D  2 d x H)  Œ1 .x/  2 .x/ .x/d x D 0 8 2 D./. H) 1  2 D 0 in D./ by Corollary 1.2.1, since 1  2 2 D./  Lp ./, 1  p < 1 H) 1 D 2 in D./.

Section 4.2 Continuous imbedding of dual spaces of Banach spaces in D 0 ./

265

,! is continuous: n ! 0 in D./ H) 9 a compact subset K   with supp.n /  K 8n 2 N such that maxx2K jn .x/j ! 0 as n ! 1 H) .,!n / D Tn ! ,!0 D 0 in D 0 ./. In fact, for supp.n /  K   8n 2 N, ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ jh,!n ; i  0j D jhTn ; ij D ˇˇ n .x/ .x/d xˇˇ D ˇˇ n .x/ .x/d xˇˇ  K Z Z j .x/jd x ! 0 .x/d x D 0 as n ! 1:  max jn .x/j x2K

K

K

Therefore, ,!W D./ ! D 0 ./ is continuous. Hence, D./ ,! D 0 ./, ,! being a continuous injection, so D./ can be identified with a subspace of D 0 ./. In order to prove that D./ is a dense subspace of D 0 ./, it is sufficient to show that a continuous linear functional L 2 D 00 ./ D .D 0 .//0 , which vanishes on D./, is a null functional in D 00 ./, i.e. L./ D hL; i D 0 8 2 D./ H) L D 0 in D 00 ./. (This is a well-known result, which follows from the Hahn–Banach Theorem A.7.2.1 (see Corollary A.7.3.6, Appendix A)). Since D./ is reflexive by Theorem 4.1.1, 8L on D 0 ./ 9 2 D./ such that hL; T i D hT; i 8T 2 D 0 ./. But hL; i D hL; T i D hT ; i D 0 8 2 D./. We are to show that L D 0. In particular, Z Z hL; i D hL; T i D hT ; i D .x/ .x/d x D j .x/j2 d x D 0 



H) D 0 in D./ H) hL; T i D hT; 0i D 0 8T 2 D 0 ./ H) L D 0 in D 00 ./. Hence, D./ is dense in D 0 ./.

4.2

Continuous imbedding of dual spaces of Banach spaces in D 0 ./

Spaces of distributions We have already shown in Section 1.3, Chapter 1, that all important function spaces are algebraically contained in the space Lloc ./ of locally integrable functions on  and, hence, in D 0 ./. Now, we will discuss the continuous imbedding of general Banach spaces in D 0 ./. Let V be a Banach space. Then ´ I. D./  V algebraically; D./ ,! V H) (4.2.1) II. n !  in D./ H) n !  in V; (to be read as ‘D./ is continuously imbedded in V ’);

Chapter 4 Additional properties of D 0 ./

266

0

V ,! D ./

H)

´ I. V  D 0 ./ algebraically; II. vn ! v in V H) vn ! v in D 0 ./;

(4.2.2)

(to be read as ‘V is continuously imbedded in D 0 ./’). Definition 4.2.1. A Banach space V is called a subspace of distributions if and only if V ,! D 0 ./ in the sense of (4.2.2). Banach spaces C k ./, C k; ./ (0 <  < 1), k 2 N0 , Lp ./, 1  p  1, are spaces of distributions, since (4.2.2) holds with V D C k ./; C k; ./; Lp ./:

(4.2.3)

But the most important class of spaces of distributions are Sobolev spaces,which are either Hilbert spaces or Banach spaces (see Chapter 2). We now collect some important results which will be very useful in the study of duals of Sobolev spaces to be introduced in the following section. Imbedding of dual spaces of Banach spaces in D 0 ./ Let V and W be Banach spaces such that V ,! W and V is dense in W , the imbedding ,! being a continuous injection from V into W :

(4.2.4)

Let V 0 D ¹v W v is a continuous linear functional on V º

and

0

W D ¹w W w is a continuous linear functional on W º be the (topological) dual spaces of V and W respectively. Then V 0 and W 0 are also Banach spaces by Theorem A.8.1.2, and ´ I. W 0  V 0 algebraically; W 0 ,! V 0 implying II. wn ! w in W 0 H) wn ! w in V 0 ;

(4.2.5)

i.e. the imbedding ,! is continuous from W 0 into V 0 with the following identification: w 2 W 0 is identified with w#V 2 V 0 such that hw#V ; viV 0 V D hw; viW 0 W

8w 2 W 0 ; 8v 2 V ,! W satisfying (4.2.4): (4.2.6)

Section 4.2 Continuous imbedding of dual spaces of Banach spaces in D 0 ./

267

Justification of (4.2.6) This identification in (4.2.6) is possible as a consequence of the density of V in W in (4.2.4). Set wN D w#V 2 V 0 . Then, w D w" N W 2 W 0 is the unique continuous, linear extension to W of wN 2 V , since V is dense in W , and by the Hahn–Banach Theorem A.7.2.1 (see Corollary A.7.3.7), this unique extension is obtained. But this result will still be meaningful with V replaced by D./, which is no longer a Banach space, W being a Banach space, i.e. D./ ,! W and D./ is a dense subspace of W H)

W 0 ,! D 0 ./; i.e. W 0 is a subspace of distributions:

(4.2.7)

Example 4.2.1. For 1  p < 1; Lp ./ D W is a Banach space such that D./ ,! Lp ./ and D./ is dense in Lp ./ (see (1.2.25)). Hence, .Lp .//0  Lq ./ ,! D 0 ./ with 1 < q  1, p1 C q1 D 1, i.e. .Lp .//0  Lq ./ is a subspace of distributions for 1  p < 1, 1 < q  1. Remark 4.2.1. If D./ ,! W in the sense of (4.2.1), but D./ is not a dense subspace of W , then W 0 will not be a subspace of distributions and, hence, W 0 cannot be identified with a subspace of D 0 ./. In other words, W 0 6 D 0 ./. (4.2.8) Example 4.2.2. 1. For p D 1, D./ ,! L1 ./, but D./ is not dense in L1 ./. Hence, .L1 .//0 6 D 0 ./;

(4.2.9)

i.e. .L1 .//0 cannot be identified with a subspace of D 0 ./. .L1 .//0 is not a subspace of distributions. 2. D./ ,! C k; ./, 8k 2 N0 , 0 <  < 1, but D./ is not dense in C k; ./ (see Definition A.5.3.1 in Appendix A) (for  D 0, C k;0 ./  C k ./ 8k 2 N0 /. Hence, .C k; .//0 6 D 0 ./;

(4.2.10)

i.e. .C k; .//0 , 0 <  < 1, cannot be identified with a subspace of D 0 ./ and is not a subspace of distributions 8k 2 N0 . There will be important examples in the family of Sobolev spaces.

Chapter 4 Additional properties of D 0 ./

268 Case of triple spaces

Let V and W be Banach spaces with their (topological) dual spaces V 0 and W 0 (which are also Banach spaces). If the following continuous imbeddings hold: D./ ,! V ,! W with D./ dense in V and V dense in W , then W 0 ,! V 0 ,! D 0 ./.

(4.2.11)

That is, W 0 and V 0 are subspaces of distributions, and W 0 is also a subspace of V 0 with continuous imbedding from W 0 into V 0 . An important application of (4.2.11) will be for the family of Sobolev spaces in Section 4.3. Multiplication by a function If, for any fixed function  on , the mapping v 2 V 7! v 2 W is linear and continuous from V into W with D./ ,! V ,! W satisfying (4.2.11), then the mapping w 2 W 0 7! w 2 V 0 is linear and continuous from W 0 into V 0 with W 0 ,! V 0 ,! D 0 ./ and hw; viV 0 V D hw; viW 0 W

8w 2 W 0 ; 8v 2 V 0 :

(4.2.12)

Differentiation If, for any multi-index ˛, the mapping @˛ W v 2 V 7! @˛ v 2 W is linear and continuous from V into W with D./ ,! V ,! W satisfying (4.2.11), then the mapping @˛ W w 2 W 0 7! @˛ w 2 V 0 defined by: h@˛ w; viV 0 V D .1/j˛j hw; @˛ viW 0 W

8w 2 W 0 ; 8v 2 V

(4.2.13)

is linear and continuous from W 0 into V 0 , with W 0 ,! V 0 ,! D 0 ./. Restriction and null extension Let 1  2  Rn and vQ be the null extension to 2 of a function v defined on 1 , i.e. v.x/ Q D v.x/ for x 2 1

and

v.x/ Q D 0 for x 2 2 n 1 :

(4.2.14)

Let V .1 /; W .2 / be Banach spaces and D.1 /; D.2 / test spaces such that D.1 / ,! V .1 /, D.2 / ,! W .2 / with D.1 / dense in V .1 / and D.2 / dense in W .2 /. Then, if the null extension mapping v 7! vQ is continuous from V .1 / into W .2 /, the restriction mapping u 7! u#1 is continuous from W 0 .2 / into V 0 .1 / with hu#1 ; viV 0 .1 /V .1 / D hu; vi Q W 0 .2 /W .2 /

8u 2 W 0 .2 /; 8v 2 V 0 .1 /: (4.2.15)

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

4.3

269

Applications: Sobolev spaces H m ./; W m;q ./

Space H m ./, m 2 N Definition 4.3.1. Let   Rn be an open subset of Rn . Let H0m ./ D D./ in the norm k  km; of H m ./ 8m 2 N (Definition 2.15.2). Then the (topological) dual .H0m .//0 of H0m ./ is the linear space of all continuous linear functionals L defined on H0m ./ and denoted by H m ./, i.e. H m ./  .H0m .//0 D ¹L W L is a continuous linear functional on H0m ./º: (4.3.1) Theorem 4.3.1. 8m 2 N, H m ./ equipped with the usual dual norm k  km; defined, 8L 2 H m ./, by kLkm; D

sup u2H m ./¹0º

jL.u/j D sup¹jL.u/j W u 2 H0m ./; kukm;  1º kukm; (4.3.2)

is a Banach Space. H m ./ equipped with inner product h  ;  im; induced by h  ;  im; (with the help of the Riesz Representation Theorem A.13.1.1 and Proposition A.13.1.2 in Appendix A on Hilbert space H0m ./ equipped with inner product h  ;  im; ) is a Hilbert space 8m 2 N, i.e. 8L1 ; L2 2 H m ./, hL1 ; L2 im; D huL1 ; uL2 im; ;

(4.3.3)

where uL1 ; uL2 2 H0m ./ are Riesz representers of L1 and L2 respectively. Proof. The dual space H m ./ of a Banach space H0m ./ equipped with the norm k  km; in (4.3.2) is a Banach space (Theorem A.8.1.2 and Section A.13 in Appendix A). Banach space H m ./ equipped with inner product h  ;  im; becomes a Hilbert space. Imbedding results D./ ,! H0m ./ in the sense of (4.2.1) and D./ is dense in H0m ./ by definition of H0m ./ H)

H m ./  .H0m .//0 ,! D 0 ./ in the sense of (4.2.5).

(4.3.4)

H m ./ can be identified with a subspace of D 0 ./, and hence H m ./ is a subspace of distributions on :

(4.3.5)

In fact, for a distribution T 2 D 0 ./ to be in H m ./, it is necessary and sufficient that T is continuous on D./ equipped with the norm k  km; induced by H0m ./. (4.3.6)

Chapter 4 Additional properties of D 0 ./

270 Remark 4.3.1. 

For  ¤ Rn , D./ ,! H m ./, but D./ is not dense in H m ./ (H0m ./ ¤ H m ./). Hence, .H m .//0 6 D 0 ./ by Remark 4.2.1, i.e. .H m .//0 is not a subspace of distributions.



(4.3.7)

For  D Rn , D.Rn /  H0m .Rn /  H m .Rn / 8m 2 N. Then D.Rn / ,! H m .Rn /  H0m .Rn / and D.Rn / is dense in H m .Rn /. Hence, by (4.2.5), H m .Rn /  .H m .Rn //0 ,! D 0 .Rn / in the sense of (4.2.2), i.e. .H m .Rn //0  H m .Rn / is a subspace of distributions.

(4.3.8)

Hilbert triple For a Hilbert triple H0m ./; L2 ./; H m ./, we are allowed to make only one identification of a Hilbert space with its dual, i.e. L2 ./  .L2 .//0 by the Riesz representation theorem, and no more identification is possible. In other words, we cannot identify H0m ./ with its dual H m ./. Then we will have: D./ ,! H0m ./ ,! L2 ./  .L2 .//0 ,! H m ./ ,! D 0 ./: „ ƒ‚ …

(4.3.9)

Pivot space

Isometric Isomorphism P Theorem 4.3.2. The mapping 0j˛jm .1/j˛j @2˛ W H0m ./ ! H m ./ is an isometric isomorphism (i.e. canonical isomorphism) from H0m ./ onto H m ./. Proof. Let vP2 H0m ./ ,! D 0 ./. Then, 8 multi-index ˛, @2˛ v 2 D 0 ./. Set T D 0j˛jm .1/j˛j @2˛ v 2 D 0 ./. We will show that T is an element of H m ./. In fact, 8 2 D./,  X  X hT; i D .1/j˛j @2˛ v;  D h@˛ v; @˛ i 0j˛jm

D

X 0j˛jm

Z

0j˛jm

@˛ v@˛ d x D hv; im; ;



where h  ;  im; denotes the inner product (2.15.9) in H m ./ H)

jhT; ij D jhv; im; j  kvkm; kkm;

8 2 D./:

Hence, T is a linear functional, which is continuous on D./  H0m ./ in the norm k  km; induced by H0m ./. But D./ is dense in H0m ./ in the norm

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

271

k  km; . Hence, by the Hahn–Banach Theorem A.7.2.1 (Appendix A), T can be extended to a unique, continuous, linear functional L on H0m ./ such that 8 2 D./;

L./ D hT; i D hv; im; L.u/ D hv; uim;

8u 2

H0m ./;

with kT kH m ./ D kLkH m ./ D kLkm; D kvkm; for v 2 H0m ./. Hence, kLkm;

  D  X

H)

X

  @ v 

j˛j 2˛

.1/

0j˛jm

D kvkm;

8v 2 H0m ./

m;

.1/j˛j @2˛ is an isometry from H0m ./ into H m ./

0j˛jm

X

H)

.1/j˛j @2˛ is an injection from H0m ./ into H m ./:

0j˛jm

P Now, it remains to show that the mapping 0j˛jm .1/j˛j @2˛ W H0m ./ ! H m ./ is indeed a surjection. Since L 2 H m ./  .H0m .//0 and H0m ./ is a Hilbert space with inner product h  ;  im; , by the Riesz Representation Theorem A.13.1.1 in Appendix A, 9 a unique v 2 H0m ./ such that L.u/ D hu; vim; 8u 2 H0m ./ with kLkm; D kvkm; . Again, by definition, T D L#D./ H) 8 2 D./, X

L./ D hT; i D hv; im; D D

0j˛jm

X

˛

H)

h@ v; @ iD 0 ./D./ D 

hT; iD 0 ./D./ D T D

X

˛

0j˛jm

H)

h@˛ v; @˛ i0;

X

.1/j˛j h@2˛ v; iD 0 ./D./

0j˛jm

X

j˛j 2˛

.1/

0j˛jm

.1/j˛j @2˛ v



@ v; 

8 2 D./ D 0 ./D./

in D 0 ./;

0j˛jm

i.e. 8L 2 P H m ./, 9v 2 H0m ./ such that (4.3.10) holds. Hence, 0j˛jm .1/j˛j @2˛ is a surjection from H0m ./ onto H m ./.

(4.3.10)

Chapter 4 Additional properties of D 0 ./

272 Density of D./ in H m ./

Corollary 4.3.1. D./ is dense in Hilbert space H m ./. P j˛j 2˛ W H m ./ ! Proof. From Theorem 4.3.2, the mapping 0j˛jm .1/ @ 0 m m H ./ is an isometric isomorphism from H0 ./ onto H m ./, i.e. a canonical surjective isometry from H0m ./ onto H m ./, which maps dense subspaces of H0m ./ onto dense subspaces of H m ./. But D./ is a dense subspace of H0m ./, which is mapped onto D./) in H m ./ under the canonP onto itself (i.e. j˛j ical surjective isometry 0j˛jm .1/ @2˛ . Therefore, the range space D./  H m ./ is a dense subspace of H m ./. Structure of the elements of H m ./ Theorem 4.3.3. A distribution T 2 D 0 ./ on  belongs to H m ./ (i.e. T can be extended to an element of H m .// if and only if, for multi-index ˛ with j˛j  m, 9 (equivalence classes of) functions f˛ 2 L2 ./ such that X @ ˛ f˛ ; (4.3.11) T D 0j˛jm

where derivatives @˛ f˛ are in the distribution sense. P Proof. Let T D 0j˛jm @˛ f˛ with f˛ 2 L2 ./. Then T 2 D 0 ./ and, 8 2 D./,  X  X ˛ hT; i D @ f˛ ;  D .1/j˛j hf˛ ; @˛ i 0j˛jm

D

X

.1/j˛j

0j˛jm

Z

f˛ @˛ d x D



0j˛jm

X

.1/j˛j hf˛ ; @˛ i0;

0j˛jm

(where h  ;  i0; denotes the inner product in L2 ./) X X H) jhT; ij  jhf˛ ; @˛ i0; j  0j˛jm





X

kf˛ k0; k@˛ k0;

0j˛jm

 kf˛ k0; kkm;

0j˛jm

(since k@˛ k0;  kkm; for j˛j  m/, where k  k0; (resp. k  km; ) denotes the norm in L2 ./ (resp. H m ./), H) T is a continuous, linear functional on D./  H0m ./ in the norm k  km; of H0m ./  H m ./. Hence, T 2 H m ./  .H0m .//0 by (4.3.6).

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

273

m Conversely, Then, by Theorem 4.3.2 on the isometric isomorP let T 2 H j˛j ./. 2˛ phism of 0j˛jm .1/ @ from H0m ./ onto H m ./, 9v 2 H0m ./ such that

T D

X

.1/j˛j @2˛ v:

0j˛jm

8 multi-index ˛ with j˛j  m, set f˛ D .1/j˛j @˛ v. Then f˛ 2 L2 ./ 8j˛j  m, since v 2 H0m ./ H) @˛ v 2 L2 ./ 8j˛j  m. Hence, X X @˛ Œ.1/j˛j @˛ v D @ ˛ f˛ : T D 0j˛jm

0j˛jm

Example 4.3.1. Show that the Dirac distribution ı D ı0 with concentration at 0 2 R belongs to H 1 .1; 1Œ/. 2 D 0 .1; 1Œ/, where H.x/ D 1 for 0 < x < 1 Proof. From Example 2.3.2, ı D dH dx R R0 R1 1 and H.x/ D 0 for 1 < x < 0. But 1 .H.x//2 dx D 1 0dx C 0 1dx D 1 H) H 2 L2 .1; 1Œ/. Hence, ı D dH with H 2 L2 .1; 1Œ/ H) by Theorem 4.3.3, dx 1 ı 2 H .1; 1Œ/.

4.3.1 Space W m;q ./, 1 < q  1, m 2 N m;p

Definition 4.3.2. Let   Rn be an open subset of Rn and W0 ./ D D./ in the norm k  km;p; of W m;p ./ 8m 2 N, 1  p < 1 (Definition 2.15.4). Then m;p m;p the (topological) dual .W0 .//0 of W0 ./ is the linear space of all continuous, m;p linear functionals L defined on W0 ./ and denoted by W m;q ./ with 1 < q  1, p1 C q1 D 1; i.e. 8m 2 N, 1  p < 1, 1 < q  1 with p1 C q1 D 1, m;p

W m;q ./.W0

m;p

.//0 D¹L W L is a continuous linear functional on W0

./º: (4.3.12)

Theorem 4.3.4. 8m 2 N, 1 < q  1, W m;q ./ equipped with the usual dual norm k  km;q; defined by: 8L 2 W m;q ./, for 1  p < 1, 1 < q  1, with 1 1 p C q D 1, kLkm;q; D

sup m;p u2W0 ./¹0º

jL.u/j kukm;p; m;p

D sup¹jL.u/j W u 2 W0 is a Banach space.

./; kukm;p;  1º;

(4.3.13)

Chapter 4 Additional properties of D 0 ./

274 m;p

Proof. 8m 2 N, 1  p < 1, W0 ./ is a Banach space with norm k  km;p; . m;p Hence, its (topological) dual space .W0 .//0  W m;q ./ defined by (4.3.12) and equipped with the norm k  km;q; defined by (4.3.13) is also a Banach space for 1 < q  1, p1 C q1 D 1. Imbedding results 





m;p

m;p

D./ ,! W0 ./ in the sense of (4.2.1) and D./ is dense in W0 ./ m;p 8m 2 N, 1  p < 1 H) W m;q ./  .W0 .//0 ,! D 0 ./ in the sense 1 1 of (4.2.2) by (4.2.5), 1 < q  1, p C q D 1. (4.3.14) W m;q ./, m 2 N, 1 < q  1, can be identified with a subspace of D 0 ./, and hence W m;q ./ is a subspace of distributions on . (4.3.15) For a distribution T 2 D./ to be in W m;q ./, it is necessary and sufficient m;p that T is continuous on D./  W0 ./ equipped with the norm k  km;p; m;p induced by W ./, m 2 N, 1  p < 1, p1 C q1 D 1. (4.3.16) Remark 4.3.2. For  ¤ Rn , m 2 N and 1  p < 1, D./ is not dense in W m;p ./. Hence, by Remark 4.2.1, .W m;p .//0 6 D 0 ./;

(4.3.17)

i.e. .W m;p .//0 is not a subspace of distributions. 

For p D 1 and any   Rn , D./ is not dense in W m;1 ./, m 2 N. Hence, .W m;1 .//0 6 D 0 ./ by Remark 4.2.1, i.e. 8m 2 N and 8  Rn , .W m;1 .//0 is not a subspace of distributions:



(4.3.18)

m;p

For  D Rn , D.Rn / is dense in W m;p .Rn /  W0 .Rn / for m 2 N, 1  p < 1. Hence, .W m;p .Rn //0  W m;q .Rn / ,! D 0 .Rn /, i.e. for 1  p < 1, .W m;p .Rn //0  W m;q .Rn / is a subspace of distributions on Rn ; (4.3.19) ( p1 C

1 q

D 1).

Structure of elements of W m;q ./ in Theorem 4.3.3.

We have similar results as those for H m ./

Theorem 4.3.5. A distribution T 2 D 0 ./ on  belongs to W m;q ./, m 2 N; 1 < q  1 (i.e. T can be extended to an element of W m;q ./) if and only if, for

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

275

(some) multi-index ˛ with j˛j  m; 9 (equivalence classes of) functions f˛ 2 Lq ./, 1 < q  1, such that X T D @ ˛ f˛ ; (4.3.20) 0j˛jm

where derivatives @˛ f˛ are in the distributional sense. Proof. The proof is similar to that for Theorem 4.3.3. Example 4.3.2. Show that function

1 x

2 W 1;q .0; 1Œ/ with 1 < q < 1.

Proof. ln x 2 L1loc .0; 1Œ/ (see (Example 1.4.1)) H) ln x 2 D 0 .0; 1Œ/ H) .ln x/0 D 1 1 d 0 q x 2 D .0; 1Œ/. Hence, we can write x D dx .ln x/ with ln x 2 L .0; 1Œ/ for 1  1 q < 1 (see (2.3.11)) H) by Theorem 4.3.5, x 2 W 1;q .0; 1Œ/ for 1 < q < 1. Example 4.3.3. For mp > n (i.e. m  pn > 0) with m 2 N, 1  p < 1, Dirac distribution ı 2 W m;q .Rn / with p1 C q1 D 1. m;p Solution. D.Rn / is dense in W m;p .Rn /, i.e. W0 .Rn /  W m;p .Rn / for m 2 N; 1  p < 1, by Theorem 6.8.9 (see also (2.15.54b)). Moreover, for mp > n; W m;p .Rn / ,! C 0 .Rn / by Sobolev’s imbedding results (8.12.20g) (see Theorem 8.12.2 in Section 8.12). Hence, 8 2 D.Rn /, jhı; ij D j.0/j  kkC 0 .Rn /  C kkW m;p .Rn / H) Dirac distribution ı 2 D 0 .Rn / with mass/charge/force etc. concentrated at 0 is a continuous, linear functional on D.Rn / in the norm of W m;p .Rn /. But D.Rn / is dense in W m;p .Rn /. Hence, by Corollary A.7.3.7 of the Hahn–Banach Theorem A.7.2.1 in Appendix A, ı can be extended to a unique continuous, linear functional on W m;p .Rn / such that this extended unique, continuous, linear functional, which will still be denoted by ı, will belong to W m;q .Rn /  .W m;p .Rn //0 , i.e. ı 2 W m;q .Rn / with jhı; uijW m;q .Rn /W m;p .Rn /  C kukW m;p .Rn /

8u 2 W m;p .Rn /:

(4.3.20a)

Restriction and null extension in Sobolev spaces Let 1  2  Rn . For 1 < p < 1, the restriction mapping v 7! v#1 is continuous (obviously, linear) from W m;p .2 / into W m;p .1 /, with kv#1 km;p;1  kvkm;p;2

(4.3.21)

(which follows from the definition and holds also for p D 1 and 1). m;p m;p For V .1 / D W0 .1 /, W .2 / D W0 .2 /, 1 < p < 1, m 2 N, satisfying (4.2.15) and (4.2.11), the null extension mapping v 7! vQ is continum;p m;p ous from W0 .1 / into W0 .2 / (see Theorem 2.15.5 for an analogous proof). Then, by virtue of (4.2.15), the restriction mapping u 7! u#1 is continuous from

Chapter 4 Additional properties of D 0 ./

276 m;p

m;p

.W0 .2 //0 into .W0 .1 //0 , i.e. from W m;q .2 / into W m;q .1 / with 1 < p < 1, p1 C q1 D 1, 8u 2 W m;q .2 /, hu#1 ; viW m;q .1 /W m;p .1 / D hu; vi Q W m;q .2 /W m;p .2 / 0

0

m;p

8v 2 W0

.1 /: (4.3.22)

Duals of closed subspaces and quotient spaces [3] Let M be a closed subspace of Banach space V , the norm in M being that induced by V , i.e. k  kM D k  kV , such that V =M is the quotient space of V by M , which is also a Banach space equipped with the quotient norm kŒ  kV =M defined in (2.15.24) (resp. (2.15.41)) for V D H m ./ (resp. V D W m;p ./). Let M 0 and V 0 be the duals of M and V respectively. Since M and V are Banach spaces, M 0 and V 0 are also Banach spaces equipped with the dual norms k  kM 0 and k  kV 0 respectively. Annihilator M 0 of M (see Rudin [3]) Definition 4.3.3. The annihilator M ı of M is the closed subspace of V 0 defined by: M 0 D ¹l W l 2 V 0 ; hl; ui D 0 8u 2 M º  V 0 ;

(4.3.23)

which is also a Banach space equipped with the norm k  kV 0 induced by V 0 . Hence, V 0 =M 0 is the quotient space of V 0 by M 0 , which is also a Banach space equipped with the corresponding quotient norm kŒ  kV 0 =M 0 defined by [3]: Q V0 D Q 0 0 D inf klk kŒlk V =M Q l Q l2Œ

inf

m0 2M 0

Q V 0: klQ C m0 k  klk

(4.3.24)

Then the dual M 0 of the closed subspace M of V and the dual .V =M /0 of the quotient space V =M are intimately related to the annihilator M 0 . In fact, M 0 is identified with the quotient space V 0 =M 0 and .V =M /0 with M 0 by writing: M 0 D V 0 =M 0

and

.V =M /0 D M 0 ;

(4.3.25)

since these are isometrically isomorphic. Since M is a closed subspace of Banach space V , by Corollary A.7.3.1 of the Hahn–Banach Theorem A.7.2.1 in Appendix A, Q Q V 0 D klkM 0 : 8l 2 M 0 ; 9lQ 2 V 0 such that l.u/ D l.u/ 8u 2 M; klk

(4.3.26)

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

277

Then, we have the following result on isometric isomorphisms: Theorem 4.3.6. I. Let J W M 0 ! V 0 =M 0 be the mapping defined by: 8l 2 M 0 ;

Q D lQ C M 0 2 V 0 =M 0 ; J l D Œl

(4.3.27)

with lQ 2 V 0 satisfying (4.3.26). Then J is an isometric isomorphism from M 0 onto V 0 =M 0 . II. Let  W V ! V =M be the canonical surjection from V onto V =M with the same properties given by (2.15.24a)–(2.15.24d), and JQ W .V =M /0 ! M 0 be the mapping defined by: 8ƒ 2 .V =M /0 , .JQ ƒ/u D .ƒ/u D ƒ.u/ 8u 2 V such that .JQ ƒ/u D .ƒ/u D 0 8u 2 M;

i.e. JQ ƒ 2 M 0 :

(4.3.28)

Then JQ is an isometric isomorphism from .V =M /0 onto M 0 . Proof. I. J is well defined: We are to show that (4.3.27) holds for any choice of extension Q lQ1 2 V 0 be Hahn–Banach extensions of lQ of l, since lQ is not unique. Let l, l 2 M 0 . Then .lQ  lQ1 /.u/ D l.u/  l.u/ D 0 8u 2 M . Hence, lQ  lQ1 D m 2 M 0 H) lQ D lQ1 C m with m 2 M 0 H) lQ D lQ1 .mod M 0 / H) lQ C M 0 D Q 2 V 0 =M 0 , i.e. 8 choices of Hahn–Banach extension lQ 2 V 0 , lQ1 C M 0 D Œl Q 2 V 0 =M 0 and J is well defined by (4.3.27). Ql C M 0 D Œl J is linear: J.˛1 l1 C ˛2 l2 / D .˛1 lQ1 C ˛2 lQ2 / C M 0 D ˛1 .lQ1 C M 0 / C ˛2 .lQ2 C M 0 / D ˛1 J l1 C ˛2 J l2 8˛i 2 R. Q . Then, 8Œl Q 2 V 0 =M 0 , J is onto: 8lQ 2 V 0 , 9l 2 M 0 such that l D l# M 0 0 0 0 Q Q 9l 2 M such that J l D l C M D Œl 2 V =M . J is one-to-one and continuous: If J is an isometry, then J is one-to-one and continuous, since kJ lkV 0 =M 0 D klkM 0 D 0 H) l D 0 H) J is one-to-one and continuous from M 0 onto V 0 =M 0 . Hence, the proof will be complete if we can show that J is an isometry, since J will be a continuous, linear bijection from M 0 onto V 0 =M 0 and J 1 will also be linear and its continuity will follow from Corollary A.8.1.1 of the Open Mapping Theorem A.8.1.3 in Appendix A. J is an isometry: For any fixed l 2 M 0 , let lQ 2 V 0 be an extension to V 0 of l. Q Then, l.u/ D l.u/ 8u 2 M  V . Hence, klkM 0 D sup u2M

 sup u2V

Q jl.u/j jl.u/j D sup kukV u2M kukV Q jl.u/j Q V0 D klk kukV

H)

Q V 0: klkM 0  klk

(4.3.29)

Chapter 4 Additional properties of D 0 ./

278

But 8m0 2 M 0 , lQ C m0 2 V 0 is also an extension of l 2 M 0 to V , since .lQ C Q m0 /.u/ D l.u/ D l.u/ 8u 2 M . Hence, from (4.3.29), klkM 0  klQ C m0 kV 0 0 8m0 2 M , H) klkM 0 

inf

m0 2M 0

Q V0 klQ C m0 kV 0 DklQ C M 0 kV 0 =M 0 DkJ lkV 0 =M 0  klk (4.3.30)

by (4.3.24). But 8l 2 M 0 , 9 a Hahn–Banach extension lQ 2 V 0 of l such that (4.3.26) holds. Then, using (4.3.30) and (4.3.26), we get the result: Q V 0; klkM 0 D klQ C M 0 kV 0 =M 0 D kJ lkV 0 =M 0 D klk i.e. J defined by (4.3.27) is an isometric isomorphism from M 0 onto V 0 =M 0 . II. JQ is well defined: Let ƒ 2 .V =M /0 and u 2 V with u 2 V =M ,  being the canonical surjection from V onto V =M . Then the mapping u 2 V 7! ƒu is well defined and a continuous, linear functional on V , which vanishes on Ker./ D M . Hence, ƒ 2 M 0 and JQ ƒ D ƒ 2 M 0 8ƒ 2 .V =M /0 , and JQ is well defined. JQ is linear: ƒi 2 .V =M /0 H) ˛1 ƒ1 C ˛2 ƒ2 2 .V =M /0 8˛1 ; ˛2 2 R H)

JQ .˛1 ƒ1 C ˛2 ƒ2 / D .˛1 ƒ1 C ˛2 ƒ2 / D ˛1 ƒ1  C ˛2 ƒ2  D ˛1 JQ ƒ1 C ˛2 JQ ƒ2 :

JQ is onto: Let m0 2 M 0 be any fixed element, with N0 D Ker.m0 /. Then M  N0  V , since m0 .u/ D 0 8u 2 M . Hence, 9 a linear functional L on V =M such that L D m0 2 M 0 with the null space N .L/ of L D ¹u W u 2 N0 º D .N0 /. But  is the continuous (see (2.15.24d)) canonical surjection from V onto V =M . Hence, .N0 / with N0 D null space of m0 2 M 0 is a closed subspace of V =M . In fact, N0 is a closed subspace of V . Then uk ! u in N0 H) uk ! u in .N0 / by virtue of the continuity of quotient mapping . Hence, .N0 / D N .L/ is a closed subspace of V =M , L being a non-null linear functional on V =M . Then, by a well-known theorem (see Rudin [3, p. 14]), L is continuous on V =M , i.e. L 2 .V =M /0 . Hence, for any fixed m0 2 M 0 , 9L 2 .V =M /0 such that JQ L D L D m0 2 M 0 . JQ is an isometry: Let L 2 .V =M /0 be any fixed element. Let Œu 2 V =M with kŒukV =M D 1. Then 9v0 2 V with kv0 kV D 1 C " 8" > 0 such that kŒukV =M  kv0 kV (by (2.15.24d)) with v0 D Œu 2 V =M . Hence, jhL; Œuij D jLv0 j D j.JQ L/v0 j  kJQ LkV 0 kv0 kV  .1 C "/kJQ LkV 0

8" > 0:

Section 4.3 Applications: Sobolev spaces H m ./; W m;q ./

279

Then, kLk.V =M /0 D

sup

jhL; Œuij  kJQ LkV 0 D kJQ LkM 0 :

(4.3.31)

kŒukV =M D1

But kukV =M  kukV 8u 2 V (see (2.15.24d)). Then j.JQ L/.u/j D jL.u/j  kLk.V =M /0 kukV =M  kLk.V =M /0 kukV H)

kJQ LkM 0 D kJQ LkV 0  kLk.V =M /0 :

8u 2 V (4.3.32)

From (4.3.31) and (4.3.32), the isometry of JQ follows: kJQ LkM 0 D kLk.V =M /0 8L 2 .V =M /0 , from which it also follows immediately that JQ is continuous and one-to-one (the proof is similar to that of the injectivity and the continuity of J given earlier), i.e. JQ is a continuous, linear bijection from .V =M /0 onto M 0 . Consequently, JQ 1 W M 0 ! .V =M /0 is also linear and its continuity follows from Corollary A.8.1.1 of the Open Mapping Theorem A.8.1.3 in Appendix A. Hence, JQ is an isometric isomorphism from .V =M /0 onto M 0 .

Chapter 5

Local properties, restrictions, unification principle, space E 0.Rn/ of distributions with compact support

5.1

Null distribution in an open set

Although a distribution T 2 D 0 ./ with   Rn does not have point values, i.e. a value at a given point x 2 , we can say that T is zero in an open subset 0  . Definition 5.1.1. A distribution T 2 D 0 ./ is a zero or null distribution in an open set 0   if and only if T ./ D 0

8 2 D./ with supp./  0 .i.e. 8 2 D.0 //:

(5.1.1)

A distribution T 2 D 0 ./ is a null distribution in a neighbourhood of a point x0 2  if and only if T is zero in an open set U containing the point x0 , i.e. T D 0 in a neighbourhood U of x0 2  ”

hT; i D 0 8 2 D./ with supp./  U;

(5.1.2)

(i.e. 8 2 D.U /). T D 0 in D 0 ./ ” 8x0 2 , 9 an open set U   with x0 2 U such that T D 0 in U , i.e. 8 open sets U  , hT; i D 0 8 2 D./ with supp./  U .i.e. 8 2 D.U //:

5.2

(5.1.3)

Equality of distributions in an open set

Definition 5.2.1. Two distributions T1 ; T2 2 D 0 ./ are said to be equal in an open set 0   if and only if T1  T2 D 0 in 0 , i.e. hT1 ; i D hT2 ; i

5.3

8 2 D./ with supp./  0 .i.e. 8 2 D.0 //: (5.2.1)

Restriction of a distribution to an open set

Definition 5.3.1. Let 0   be an open subset of  and T 2 D 0 ./ be a distribution on . Then the restriction to 0 of the distribution T is the distribution

281

Section 5.3 Restriction of a distribution to an open set

T #0 D T0 2 D 0 .0 / on 0 defined by: hT0 ; iDhT #0 ; i D hT; i 8 2 D./ with supp./  0 .i.e. 8 2 D.0 //: (5.3.1) A distribution T 2 D 0 ./ with   Rn can not necessarily be extended to a distribution on Rn . In fact, we have: 2

Example 5.3.1. For e 1=x 2 D 0 .R n ¹0º/, there does not exist any distribution T 2 2 D 0 .R/ on R such that T #Rn¹0º D e 1=x 2 D 0 .R n ¹0º/, i.e. hT; i D he

1=x 2

Z

1

; i D

2

e 1=x .x/dx

8 2 D.R n ¹0º/:

(5.3.2)

1 2

In other words, the distribution e 1=x 2 D 0 .R n ¹0º/ on R n ¹0º can not be extended to a distribution T 2 D 0 .R/ on R. Proof. The scheme of the proof is as follows: we choose a sequence .n / in D.R/ R1 2 such that n ! 0 in D.R/. Then we show that hT; n i D 1 e 1=x n .x/dx ! 1 as n ! 1, establishing that T is not a distribution on R. Let  2 D.R/ such that supp./  1; 2Œ, 0  .x/  1 8x 2 R and .x/ D 1 for a  x  b with 1 < a < b < 2. Define n D e n .nx/ 8n 2 N. Then n 2 D.R/ with supp.n /  ¹x W 1 < nx < 2º D ¹x W n1 < x < n2 º 8n 2 N. Hence, 8n 2 N, supp.n /  Œ0; 2, Œ0; 2 .m/

being a compact subset of R. Moreover, 8m 2 N0 , n .x/ D e n nm  .m/ .nx/, .m/ and supx2R jn .x/j  .nm  e n  sup1y2 j .m/ .y/j/ ! 0 as n ! 1, since 8m 2 N0 , .nm  e n / ! 0 as n ! 1 H) n ! 0 in D.R/ as n ! 1. 2 Suppose that the contrary holds, i.e. 9T 2 D 0 .R/ which extends e 1=x to R satis2 fying (5.3.2). In other words, 9T 2 D 0 .R/ which coincides with e 1=x on R n ¹0º. Then, n ! 0 in D.R/

H)

hT; n i ! 0 as n ! 1:

(5.3.3)

But supp.n /  1=n; 2=nŒ  R n ¹0º 8n 2 N. Since n .x/  0 and .nx/ D 1 for 1 < a < nx < b < 2, i.e. for a=n < x < b=n, n .x/ D e n .nx/ D e n for 1=n < a=n < x < b=n < 2=n. Hence, Z

2=n

hT; n i D

2

e 1=x n .x/dx

1=n

Z

b=n

e

 a=n

1=x 2

Z

b=n

n .x/dx D a=n

2

e 1=x  e n dx

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

282

(since the integrand is positive) H) hT; n i  e n 2

2

2

R b=n a=n

2

e 1=x dx. But for 0
0 such that kvk Q V2 .2 /  C kvkV1 .1 /

8v 2 V1 .1 /;

(5.3.9)

then the restriction operator defined by (5.3.7)–(5.3.8) is linear and continuous from V20 .2 / into V10 .1 /. (5.3.10) In fact, jh w; vij D jhw; vij Q  kwkV20 .2 / kvk Q V2 .2 /  C kwkV20 .2 / kvkV1 .1 / H)

5.4

k wkV10 .1 /  C kwkV20 .2 /

(by (5.3.9))

8w 2 V20 .2 /:

(5.3.11)

Unification principle

From the local knowledge of a distribution on a family of open sets, the following theorem on the Unification Principle (called principe du recollement des morceaux in French, and also known as the Principle of Localization) allows us to have a global knowledge of the distribution on the union of these open sets. Theorem 5.4.1 (Unification Principle Theorem of Schwartz). Let   Rn be an open subset of Rn , ¹i ºi2I be a finite or infinite family of open sets in Rn such that S  D i2I i is an open set, and ¹Ti ºi2I be a family of distributions Ti 2 D 0 .i / on i 8i 2 I , I  N being a set of indices. Suppose that 8i; j 2 I (i 6D j ), if i ; j have nonempty intersection i \ j ¤ ; and Ti D Tj on i \ j , then 9 a unique distribution T 2 D 0 ./ on  such that T #j D Tj on each j , j 2 I , i.e. 8j 2 I , hTj ; i D hT; i

8 2 D./ with supp./  j .i.e. 8 2 D.j //:

(5.4.1)

Proof. Assume that I D N and Ti 2 D 0 .i / is a distribution on i 8i 2 N. 1 Let ¹‚i º1 iD1 Sbe a partition of unity subordinate to the family P1of open sets ¹i ºiD1 of  with i2N i D , i.e. ‚i 2 D.i / 8i 2 N, iD1 ‚i .x/ D 1 8x 2  (see Appendix C). Let  2 D./ with compact support, i.e. supp./  . Then define with i 2 D.i /, supp.i /  i 8i 2 N such that P i D ‚i  P 1 .x/ D 1  .x/ D iD1 i iD1 ‚i .x/.x/ 8x 2 . But supp./ is compact in . Hence, supp./ will intersect a finite number of the (compact) supports of ‚i such

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

284

P P that the series i ‚i  will contain only a finite number of terms, i.e.  D 1 iD1 i D PN PN kD1 ik D kD1 ‚ik . Since ‚i  D i 2 D.i /, hTi ; i i is well defined 8i 2 N. But i 2 D.i / H) i 2 D./ H) hT; i i is also well defined. Hence, 8 2 D.i /, we set hT; i i D hTi ; i i D hTi ; ‚i i with  2 D./. Then we define  X  X X hT; i D T; ‚i  D hT; ‚i i D hTi ; ‚i i 8 2 D./; (5.4.2) i

i

which uniquely defines T 2

D 0 ./

i

in terms of Ti 2 D 0 .i /.

Existence of T 2 D 0 ./ In the proof given above, we assumed the existence of T 2 D 0 ./. Now, we establish its existence. For this we define T by the right-hand P side of (5.4.2): hT; i D i hTi ; ‚i i 8 2 D./. Then T is a linear functional on D./ by virtue of the linearity of Ti 2 D 0 .i / 8i 2 N. Now, for the continuity of T on D./, by Proposition 1.3.1, it is sufficient to show that 8 compactP K  , T is continuous in DK ./, i.e. 8 2 D./ with supp./  K, the series 1 iD1 hTi ; ‚i i consists only of the finite numberPof terms for which the support of ‚i intersects 0 K, i.e. 8 2 DK ./, hT; i D N hT ; ‚lk i, where each Tlk 2 D 0 .lk / is kD1 lk 0 ./ 8K  . Therefore, T is continuous. Hence, it is easily shown that T 2 DK 0 continuous and belongs to D ./. T #j D Tj To complete the proof, we are to show that T #j D Tj . Let  2 D./ with supp./  j , i.e.  2 D.j /, and i D ‚i  with i ¤ j . Then i 2 D.i / with supp.i / D supp.‚i / \ supp./  .i \ j /, for i D ‚i  ¤ 0 with supp.i / ¤ ; and supp.i /  supp.j /. Hence, i \ j ¤ ; H) Ti D Tj in i \ Pj (by hypothesis) H) hTj ; ‚i i D hTi ; ‚i i 8 2PD.j /. Then, 8 D i ‚i  2 D.j / (finite number of terms in the summation ),  X  X X ‚i  D hTj ; ‚i i D hTi ; ‚i i D hT; i hTj ; i D Tj ; i

i

i

H) T #j D Tj . Consequences of the unification principle 

A distribution T D 0 in a family of open sets ¹i ºi2I0 with I0  I [ H) T D 0 on the their union i (see also (5.1.3)):

(5.4.3)

i2I0 

The union of all open sets i in which T D 0 is the largest open set in which T D 0.

285

Section 5.5 Support of a distribution

5.5

Support of a distribution

Definition 5.5.1. The support of a distribution T on  is the smallest closed subset of  outside which T D 0 or, equivalently, the support of a distribution T on  is the complement of the largest open set in which T D 0.

supp.T / D A  

”

hT; i D 0 8 2 D./ with supp./  A{ (5.5.1)

(A{ D complement of A in /. A point x0 2 supp.T / ” T ¤ 0 in an open set U containing x0 (i.e. in a neighbourhood of x0 /. (5.5.2) supp.T / \ supp./ D ;

H)

hT; i D 0:

(5.5.3)

Examples of supports of distributions 

f is a continuous function with supp.f / D K   and Tf 2 D 0 ./ is the distribution defined by f H)



supp.Tf / D supp.f / D K:

(5.5.4)

The support of Dirac distribution ıa , hıa ; i D .a/ 8 2 D.Rn /, is ¹aº, a being the point at which the mass/force/charge etc. is concentrated: supp.ıa / D ¹aº;

(5.5.5)

since hıa ; i D 0 8 2 D.Rn / with supp./  .Rn n ¹aº/ (i.e. 8 2 D.Rn n ¹aº/), Rn n ¹aº being the largest open set in which ıa D 0. 

For Dirac distribution ıS with mass/charge/force concentrated on the hypersurface S  Rn defined by the equation xn D 0, Z hıS ; i D .x1 ; x2 : : : ; xn1 ; 0/dx1 ; dx2 : : : dxn1 8 2 D.Rn /; Rn1

the support of ıS is the surface S  Rn : supp.ıS / D S;

(5.5.6)

since hıS ; i D 0 8 2 D.Rn / with supp./  .Rn n S/, .Rn n S/ being the largest open set in which ıS D 0. 

The support of c:p:v:. x1 / is 1; 1Œ D R, since c:p:v: x1 is the locally integrable function x1 in the open set R n ¹0º, (i.e. 1=x 2 L1loc .R n ¹0º/) and the closure of this set is R. (5.5.7)

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

286

5.6

Distributions with compact support

Definition 5.6.1. A distribution T 2 D 0 .Rn / is said to have compact support K in Rn if and only if K  Rn is a compact subset of Rn and K D supp.T /. A distribution T has compact support K  Rn if and only if hT; i D 0 8 2 D.Rn / with supp./ \ K D ;:

(5.6.1)

Definition of hT; i for 2 C 1 .Rn / with arbitrary support and T 2 D 0 .Rn / with compact support Let T 2 D 0 .Rn / be a distribution with compact support K0  Rn , and  2 C 1 .Rn / be an infinitely differentiable function with arbitrary support, i.e.  may not belong to D.Rn /. Let ˛ be a function of D.Rn / such that ˛.x/ D 1 8x 2 U with K0  U; U being an open set containing K0 (i.e. U is a neighbourhood of K0 ). Then ˛ 2 D.Rn / with supp.˛/  Rn compact in Rn and hT; ˛i is well defined, since T 2 D 0 .Rn /. Now we show that the value hT; ˛i is independent of the choice of the function ˛. Let ˛; ˇ 2 D.Rn / be any two functions such that ˛.x/ D ˇ.x/ D 1 8x 2 U with K0  U . Then .˛  ˇ/ D 0 in K0 H) supp..˛  ˇ// is contained in the complement K0{ of K0 H)

supp.T / \ supp..˛  ˇ// D ;

H)

H)

hT; ˛  ˇi D hT; ˛i  hT; ˇi D 0

H)

hT; ˛i D hT; ˇi

hT; .˛  ˇ/i D 0

8 2 C 1 .Rn /:

(5.6.2)

hT; ˛i coincides with hT; i if  2 C 1 .Rn / and T 2 D 0 .Rn / has compact support K0  Rn and ˛ 2 D.Rn / with ˛.x/ D 1 8x 2 U; K0  U . Hence, we can set hT; i D hT; ˛i

for all  2 C 1 .Rn /:

(5.6.3)

Thus, for a distribution T 2 D 0 .Rn / with compact support, hT; i is well defined when  2 C 1 .Rn / is an infinitely differentiable function with arbitrary support in Rn . Important properties of distribution T with compact support Let T be a distribution with compact support K0  Rn . Then the following properties hold: 

hT; i is well defined 8 2 C 1 .Rn / with arbitrary support, and hT; i D hT; ˛i;

(5.6.4)

where ˛ 2 D.Rn /, ˛.x/ D 1 in a neighbourhood U of K0 with K0  U . 

hT; i depends only on the values of  in the neighbourhood U of K0  U . (5.6.5)

Section 5.7 Space E 0 .Rn / of distributions with compact support

287

In particular, if .x/ D 1 8x 2 Rn ,  2 C 1 .Rn / with supp./ D Rn (i.e.  D 1 … D.Rn //, 

hT; 1i is well defined and usually called the total mass/charge/force etc. or the integral of T . (5.6.6)

Space E 0 .Rn / of distributions with compact support

5.7

5.7.1 Space E.Rn / Definition 5.7.1. E.Rn /  C 1 .Rn / is the linear space of infinitely differentiable functions having arbitrary support in Rn , equipped with the following notion of conn vergence: a sequence .m /1 mD1 converges to 0 in E.R / if and only if, 8˛ with j˛j 2 N0 , @˛ m ! 0 uniformly on every compact set as m ! 1: D.Rn /  E.Rn /; m ! 0 in D.Rn /

H)

m ! 0 in E.Rn /:

(5.7.1)

(5.7.2)

But m ! 0 in E.Rn / does not imply m ! 0 in D.Rn /. Definition 5.7.2. A linear functional L W E.Rn / ! R on E.Rn / is continuous on E.Rn / if and only if m ! 0 in E.Rn /

H)

L.m / ! 0 in R as m ! 1:

(5.7.3)

A distribution T 2 D 0 .Rn / with compact support K  Rn is a continuous, linear functional L on E.Rn /.



In fact, L./ D hT; i D hT; ˛i

8 2 E.Rn /;

(5.7.4)

with ˛ 2 D.Rn ), ˛.x/ D 1 8x 2 U , and supp.T /  U defines a continuous, linear functional on E.Rn /. Conversely, a continuous, linear functional L on E.Rn / defines a distribution T 2 D 0 .Rn / with compact support. (5.7.5)



Indeed, D.Rn /  E.Rn /, m ! 0 in D.Rn / H) m ! 0 in E.Rn /. Hence, L is a linear functional on E.Rn / H) L is a linear functional on D.Rn / and m ! 0 in D.Rn / H) L.m / ! 0 in R as m ! 1 H) L is a distribution in D 0 .Rn / with L./ D hT; i 8 2 D.Rn /.

288

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

supp.T / is a compact subset of Rn . Suppose that the contrary holds, i.e. supp.T / is unbounded in Rn . Then it is possible to choose a sequence .m / in D.Rn / such that m D 0 in ¹x W kxk < mº 8m 2 N, i.e. 9 a sequence .m / with m 2 D.Rn / and supp.m /  ¹x W kxk  mº such that supp.T / \ supp.m / ¤ ; (since supp.T / \ supp.m / D ; H) hT; m i D 0) and hT; m i D 1 8m 2 N. Consequently, @˛ m ! 0 uniformly on every compact set as m ! 1, 8j˛j 2 N0 . Thus, m ! 0 in E.Rn / as m ! 1 H) L.m / D hT; m i ! 0 as m ! 1, since L is continuous on E.Rn / by hypothesis. But L.m / D hT; m i D 1 8m 2 N H) L.m / ! 1 as m ! 1. Thus, we meet with a contradiction and our assumption that supp.T / is not bounded is wrong. Hence, supp.T / is a compact subset of Rn . 

Now, we state this result as: Theorem 5.7.1. Every distribution T 2 D 0 .Rn / with compact support can be extended to a unique, continuous, linear functional L on E.Rn /, i.e. L./ D hT; i

8 2 D.Rn /  E.Rn /:

(5.7.6)

Consequently, L can be identified with T 2 D 0 .Rn / having compact support.

5.7.2 Space E 0 .Rn / Definition 5.7.3. The distributions T 2 D 0 .Rn / with compact support form a vector space denoted by E 0 .Rn /, which is the dual space of E.Rn /  C 1 .Rn / by virtue of Theorem 5.7.1. Now we collect the important properties of E 0 .Rn /: Property 1 T 2 E 0 .Rn / if and only if T is a distribution with compact support in Rn , i.e. T 2 D 0 .Rn / and supp.T /  Rn . E 0 .Rn /  D 0 .Rn /:

(5.7.7)

Property 2 T 2 E 0 .Rn / H) T ./ is well defined for every  2 C 1 .Rn /  E.Rn / with arbitrary support, and is defined by (5.6.4). (5.7.8) In particular, if .x/ D 1 8x 2 Rn ,  2 C 1 .Rn /  E.Rn / with supp./ D Rn (i.e.  D 1 … D.Rn /). Hence, 8T 2 E 0 .Rn /, hT; 1i is well defined and usually called the integral of T or total mass/charge/force etc. (see (5.6.6) also). (5.7.9) n n For a continuous function f 2 C0 .R / with compact support in R , T D Tf 2 E 0 .Rn / (since Tf 2 D 0 .Rn / and supp.Tf / D supp.f /  Rn /, Z Z hTf ; 1i D hf; 1i D f .x/d x D f .x/d x (5.7.10) Rn

supp.f /

Section 5.7 Space E 0 .Rn / of distributions with compact support

289

denotes the total mass/force/charge etc. corresponding to the volume density distribution of mass/force/charge etc. defined by f 2 C0 .Rn /. For T D ıa 2 E 0 .Rn / with compact support supp.ıa / D ¹aº (the Dirac distribution ıa with mass/force/charge etc. concentrated at ¹aº), hıa ; 1i D C1 denotes the total mass/charge/force etc. (5.7.11) Pn 2 0 n 2 2 T 2 E .R /, hT; r i with r D iD1 xi denotes the total moment of inertia: (5.7.12) Z hTf ; r 2 i D hf; r 2 i D f .x/r 2 d x for f 2 C0 .Rn / (5.7.13) Rn

is the total moment ofP inertia. For T D ıa 2 E 0 .Rn / with supp.ıa / D ¹aº, 2 2 hıa ; r i D kak D niD1 ai2 is the total moment of inertia. (5.7.14) For a continuous function f 2 C0 .R3 / with compact support, T D Tf 2 E 0 .R3 / 1 1 i with kxbk 2 C 1 .R3 n ¹bº/ defines with supp.Tf / D supp.f /  R3 , hTf ; kxbk Newtonian potential at b 2 R3 with  Z  1 f .x/ D d x; (5.7.15) Tf ; 2 2 2 1=2 3 kx  bk R Œ.x1  b1 / C .x2  b2 / C .x3  b3 /  where kx  bk D Œ.x1  b1 /2 C .x2  b2 /2 C .x3  b3 /2 1=2 . For T D ıa 2 E 0 .R3 / with supp.ıa / D ¹aº and b 6D a,   1 1 1 D D P3 : ıa ; kx  bk ka  bk Œ iD1 .ai  bi /2 1=2

(5.7.16)

Property 3 Every distribution T 2 E 0 .Rn / is of finite order m0 2 N0 . Property 4 Local structures of distributions with compact support [8, p. 91]. X T 2 E 0 .Rn / H) T D @ ˛ f˛ (5.7.17) j˛jm0

(the representation being a non-unique one), where f˛ 2 C0 .Rn / with j˛j  m0 with supp.f˛ /  U , supp.T /  U , U being an arbitrary neighbourhood of supp.T /, m0 being the order of T . P Remark 5.7.1. 8T 2 D 0 .Rn /, 9f˛ 2 C0 .Rn / such that T D ˛ @˛ f˛ in D 0 .Rn / in the sense that 8K  Rn , 8 2 D.Rn / with supp./  K (i.e. 8 2 DK .Rn /), 9m0 D m0 .K/ 2 N such that  X  Z X ˛ j˛j hT; i D @ f˛ ;  D .1/ f˛ @˛ d x; (5.7.18) j˛jm0

j˛jm0

Rn

the proof of which involves a partition of unity and convolution to be introduced in Chapter 6.

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

290

We then have the following interesting result. Example 5.7.1. Show that Dirac distribution ı 2 E 0 .R/ on R can not be equal to the derivative of certain order m 2 N of a single continuous function f 2 C0 .R/ m with compact support in R, i.e. we cannot write ddx mf D ı 2 D 0 .R/ for a single f 2 C0 .R/, m 2 N. Proof. Suppose that the contrary holds, i.e. 9f 2 C0 .R/ with compact support in R m such that ddx mf D ı in D 0 .R/ for some m 2 N. Since ı#Rn¹0º D 0 2 D 0 .R n ¹0º/, f m must satisfy ddx mf D 0 in R n ¹0º, whose general solution is a polynomial of degree  m  1 (see Proposition 2.7.1 and Remark 2.7.1): f .x/ D ˛0 x m1 C ˛1 x m2 C    C ˛m1 . But f 2 C0 .R/ H) f must have compact support H)

f D 0 in R n ¹0º; i.e. ˛0 D ˛1 D    D ˛m1 D 0;

(5.7.19)

since every non-null polynomial has support R. Hence, f D 0 on R n ¹0º. But f m is continuous on R H) f D 0 on R H) ddx mf D 0 in D 0 .R/, which contradicts m the hypothesis that ddx mf D ı 6D 0 in D 0 .R/. Thus, our assumption is wrong and the result follows. Independent proof of (5.7.19). .x/ limjxj!1 xfm1

f .x/ x m1

D ˛0 C

˛1 x

C  C

˛m1 x m1

in R n ¹0º

H) D 0, since f has compact support in R H) for sufficiently large jxj, f .x/ D 0. f .x/ But limjxj!1 .˛0 C ˛x1 C  C x˛m1 m1 / D ˛0 D limjxj!1 x m1 D 0. Thus, ˛0 D 0. Hence, f .x/ D ˛1 x m2 C    C ˛m1 in R n ¹0º. Similarly, we can show sequentially that ˛1 D ˛2 D    D ˛m1 D 0. Property 5 8T 2 E 0 .Rn /  D 0 .Rn /, 8 multi-index ˛, @˛ T 2 E 0 .Rn / is defined by: h@˛ T; i D .1/j˛j hT; @˛ i

8 2 C 1 .Rn / D E.Rn /:

(5.7.20)

Example 5.7.2. For T D ıa and .x/ D 1 8x 2 R, hıa0 ; 1i D hıa ;

d1 i D hıa ; 0i D 0: dx

(5.7.21)

Example 5.7.3. For T D @˛ S 2 E 0 .Rn /, hT; 1i D 0. In fact, hT; 1i D h@˛ S; 1i D .1/j˛j hS; @˛ 1i D .1/j˛j hS; 0i D 0:

(5.7.22)

Section 5.7 Space E 0 .Rn / of distributions with compact support

291

Property 6 Let T 2 E 0 .Rn /  D 0 .Rn / and f 2 C 1 .Rn / such that f .x/ D 0 8x 2 supp.T /. Then, f T does not vanish in D 0 .Rn / in general. Example 5.7.4. Let T D ı 0 2 E 0 .R/  D 0 .R/ with supp.ı 0 / D ¹0º and f .x/ D x 8x 2 R. Then f 2 C 1 .R/ with f .x/ D 0 for x 2 supp.ı 0 / D ¹0º. But 8 2 D.R/, hf ı 0 ; i D hxı 0 ; i D hı 0 ; xi D hı; .x/0 i D Œx 0 .x/ C .x/xD0 D .0/ D hı; i H)

xı 0 D ı 6D 0 in D 0 .R/:

Example 5.7.5. Show that the following hold for T 2 E 0 .R/  D 0 .R/,  2 D.R/: 1. T D 0 in D 0 .R/ H) hT; i D 0; 2. hT; i D 0 does not imply T D 0 in D 0 .R/. Proof. 1. Let  2 D.R/. Choose 2 D.R/ such that .x/ D 1 for x 2 supp./. Then .x/ D .x/ 8x 2 R, and 0 D hT; i D hT;  i D hT; i 8 2 D.R/. 2. For n D 1, T D ı 0 2 E 0 .R/, choose  2 D.R/ with  D 1 in U , ¹0º D supp.ı 0 /  U . Then .0/ D 1 and  0 .0/ D 0. Hence, hT; i D hı 0 ; i D hı;  0 i D  0 .0/ D 0. Now we show that ı 0 6D 0 in D 0 .R/. For this, choose 2 D.R/ with 0 .0/ 6D 0: hT; i D hı 0 ; i D hı 0 ;  i D hı; . /0 i D Œ 0 .x/ .x/ C .x/ D 0  H)

.0/  .0/

0

0

.x/xD0

.0/ D 

0

.0/ 6D 0

ı 0 6D 0 in D 0 .R/:

Property 7 For T 2 E 0 .Rn / with supp.T / D K  Rn , T ./ 6D 0 in general for functions  2 D.Rn / with .x/ D 0 8x 2 K. Example 5.7.6. Let T D ı 0 2 E 0 .R/ with K D supp.ı 0 / D ¹0º. Let 2 D.R/ with .0/ 6D 0. Define  D x 2 D.R/ with .0/ D 0, i.e. .x/ D 0 on supp.ı 0 /. But hı 0 ; i D hı 0 ; x i D hı; .x /0 i D Œx 0 .x/ C .x/xD0 D  .0/ 6D 0. But we have: Proposition 5.7.1. Let T 2 E 0 .Rn / with supp.T / D K  Rn , m0 being the order of T (by Property 3). If  2 D.Rn / such that .x/ D @˛ .x/ D 0 8x 2 K, 8j˛j  m0 , then T ./ D 0.

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

292

Example 5.7.7. Let T D @˛ ı 2 E 0 .Rn /  D 0 .Rn / be a distribution of order j˛j D m0 with supp.@˛ ı/ D ¹0º. Then, 8 2 D.Rn / with .0/ D @˛ .0/ D 0 8j˛j  m0 , we have h@˛ ı; i D .1/j˛j hı; @˛ i D .1/j˛j hı; @˛ i D .1/j˛j @˛ .0/ D 0: Property 8 Theorem 5.7.2. A distribution T 2 E 0 .Rn / with supp.T / D ¹0º is a finite, linear combination of Dirac distribution ı and its derivatives @˛ ı (with mass/charge/force etc. concentrated at 0). Proof. Let T 2 E 0 .Rn / with supp.T / D ¹0º. Then, by Property 3, T is of finite order m0 2 N0 . 8 2 D.Rn /. Using Maclaurin’s Theorem, we have .x/ D .x/  .0/ 

X

xk

k ˇ

H)

.0/ D @

H)

T. / D 0

H)

.0/ D 0

X 1 @ x˛ @˛ .0/ .0/      @xk ˛Š j˛jDm0

8jˇj  m0

by Proposition 5.7.1   n X 1 X @ x˛ @˛ .0/ D 0 xk .0/C  C T . / D T ./  T .0/ C @xk ˛Š j˛jDm0

kD1

H)

T ./ D .0/hT; 1i C

n X kD1

  X @ x˛ .0/hT; xk i C    C @˛ .0/ T; ; @xk ˛Š j˛jDm0

˛

˛

where 1; xk ; : : : ; x˛Š 2 C 1 .Rn / and hence hT; 1i, hT; xk i; : : : ; hT; x˛Š i are well defined for T 2 E 0 .Rn /. ˛ (Then, b0 D hT; 1i, b˛ D hT; x˛Š i 8j˛j  m0 ) X X H) 8 2 D.Rn /; T ./ D b0 .0/ C b˛ @˛ .0/ C    C b˛ @˛ .0/ j˛jD1

D

m0 X j˛jD0

D

m0 X

b˛ @˛ .0/ D

j˛jDm0 m0 X

b˛ hı; @˛ i

j˛jD0

.1/j˛j b˛ h@˛ ı; i

j˛jD0

H)

T D

m0 X j˛jD0

a˛ @˛ ı in D 0 .Rn / with a˛ D .1/j˛j b˛ 8j˛j  m0 :

(5.7.23)

Section 5.7 Space E 0 .Rn / of distributions with compact support

293

P Example 5.7.8. Show that T D j˛jm0 a˛ @˛ ı D 0 in E 0 .Rn / with a˛ 2 R 8j˛j  m0 if and only if a˛ D 0 8j˛j  m0 . Proof. For multi-index ˇ 0 with jˇ 0 j  m0 , let xˇ0 2 C 1 .Rn /. Hence, hT; xˇ0 i D

 X

 X a˛ @˛ ı; xˇ0 D a˛ h@˛ ı; xˇ0 i

j˛jm0

D

X

j˛jm0

a˛ .1/j˛j hı; @˛ .xˇ0 /i D

j˛jm0 ˇ0

X

a˛ .1/j˛j Œ@˛ .xˇ0 /.0/;

j˛jm0

ˇ0

hT; x i D h0; x i D 0: But ´ 0 Œ@ .x /.0/ D ˇ0Š ˛

ˇ0

for ˛ 6D ˇ 0 for ˛ D ˇ 0

H) hT; xˇ0 i D aˇ0 .1/jˇ0 j ˇ 0 Š D 0 ” aˇ0 D 0 8jˇ 0 j  m0 . Example 5.7.9. Find all the distributions T 2 E 0 .Rn / with supp.T / D ¹0º which are invariant under the transformations Fi W .x1 ; : : : ; xi ; : : : ; xn / 7! .x1 ; : : : ; xi ; : : : ; xn / from Rn into Rn , 1  i  n (using (1.10.21)–(1.10.23)). Solution. Fi W Rn ! Rn is defined by the diagonal matrix Ai of order n: Ai D d1; 1; : : : ; 1; : : : ; 1c with ai i D 1, ajj D C1 for j 6D i , aj k D 0 for 1  j 6D k  n, j det.Ai /j D 1, A1 D Ai 8i D 1; 2; : : : ; n, Ai x D  with j D xj for i 1  j D 6 i  n,  D x . But T 2 E 0 .Rn / with supp.T / D ¹0º H) T D i i P ˛ 0 n j˛jm0 a˛ @ ı by Theorem 5.7.1, m0 being the orderPof T 2 E .R /. Now, for ˛ invariance (1.10.42), we are to show that Fi T D T D j˛jm0 a˛ @ ı, 1  i  n, 1 where Fi T D T .F1 i / D T .Ai / (see (1.10.21)–(1.10.23)), hFi T; i D hT .A1 i /; ./i D hT .x/; .Ai x/ij det.Ai /j; .j det.Ai /j D 1/  X  X ˛ D a˛ @ ı; .Ai x/ D .1/j˛j a˛ hı; @˛ . .Ai x//i j˛jm0

D

X

j˛jm0

.1/j˛j a˛ Œ@˛ .Ai x/.0/

8

2 D.Rn /:

j˛jm0

But @˛ . .Ai x//.0/ D .1/˛i @˛ .0/ D .1/˛i hı; @˛ i D .1/˛i .1/j˛j h@˛ ı; i:

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

294 Hence,

hFi T; i D

X

a˛ .1/j˛j .1/˛i .1/j˛j h@˛ ı; i

j˛jm0

D

X

a˛ .1/˛i h@˛ ı; i

8

2 D.Rn /

j˛jm0

P

i @˛ ı. H) Fi T D j˛jm0 a˛ .1/˛P P SinceP T is invariant under Fi , j˛jm0 a˛ .1/˛i @˛ ı D j˛jm0 a˛ @˛ ı H) j˛jm0 Œ.1/˛i  1a˛ @˛ ı D 0 in E 0 .Rn /, 1  i  n H) Œ.1/˛i  1a˛ D 0 8j˛j  m0 (from Example 5.7.9), 1  i  n. H) 8˛ with a˛ 6D 0, Œ.1/˛i 1 D 0, 1  i  n H) ˛i is even 8i D 1; 2; : : : ; n P 2ˇ 2ˇ 2ˇ H) T D 2jˇjm0 a2ˇ1 ;2ˇ2 ;:::;2ˇn @1 1 @2 2 : : : @n n ı. (5.7.24) Hence, every distribution of the form (5.7.24) will be invariant under the given transformation Fi defined above.

Property 9 Convergence in E 0 .Rn /. Convergence of sequences of distributions with compact support in E 0 .Rn / Definition 5.7.4. A sequence .Tk / in E 0 .Rn / is said to converge to T 2 E 0 .Rn / if and only if lim hTk ; i D hT; i

k!1

8 2 E.Rn / D C 1 .Rn /:

(5.7.25)

Convergence of series of distributions with compact support in E 0 .Rn / P 0 n Definition 5.7.5.PLet 1 kD1 ak Tk with ak 2 R (resp. C) and Tk 2 E .R / 8k 2 N. 1 0 n Then the series kD1 ak Tk is said to converge in E .R / if and only if the sequence .SN / of thePpartial sums of the first N terms of the series converges to S 2 E 0 .Rn /, i.e. SN D N kD1 ak Tk 8N 2 N and 8 2 E.Rn /  C 1 .Rn /I

lim hSN ; i D hS; i

N !1

S 2 E 0 .Rn / is called the sum of the series SD

P1

1 X kD1

kD1 ak Tk ,

a k Tk :

(5.7.26)

and we write (5.7.27)

Section 5.7 Space E 0 .Rn / of distributions with compact support

295

Example 5.7.10. Find a sequence .Tn / of distributions Tn 2 E 0 .R/ with P supp.Tn / D ¹0º 8n 2 N such that the sequence .Sn / defined by hSn ; i D hTn ; i  jnD1 . j1 /



 1 8 2 D.R/ converges in D 0 .R/, using  j1 D .0/ C j1  0 .0/ C j12 by j Proposition 1.2.1 for  with supp./ D ŒA; A; A > 0. Solution. By Proposition 1.2.1, 8 2 D.R/ with supp./ D K  ŒA; A, A > 0, x m1 .m1/  .0/Cx m .x/, with 2 C 0 .R/ we have .x/ D .0/Cx 0 .0/C  C .m1/Š and sup j .x/j  C sup j .m/ .x/j. 8 choices of m  2, we will get a sequence .Tn /. Case m D 2: . j1 / D .0/ C j1  0 .0/ C j12 . j1 /, and   n  X 1 1 1 0 .0/ C  .0/ C 2 hSn ; i D hTn ; i  j j j j D1  X   n n X 1 1 0 1 D hTn ; i  n.0/   .0/  : (5.7.28) 2 j j j j D1

But

j D1

1 j j2

. j1 /j  j12 sup j .x/j  jC2 sup j 00 .x/j  jM2 , (where M P P n). j jnD1 j12 . j1 /j  M jnD1 j12 ! a finite limit in R as

j and Hence, if we set

hTn ; i D n.0/ C

X n j D1

jhSn ; ij  M

n P j D1

1 j2

 1 0  .0/ j

8n 2 N;

is independent of n ! 1.

(5.7.29)

! a finite limit as n ! 1 implies.

hSn ; i converges in R as n ! 1 8 2 D.R/. H) .Sn / converges in D 0 .R/ for the sequence .Tn / in D 0 .R/ with   X X n n 1 0 1  .0/ D hnı; i C hı;  0 i hTn ; i D n.0/ C j j j D1 j D1  X   n 1 0 D nı  ı ;  8 2 D.R/ j j D1 P H) for m D 2; Tn D nı  . jnD1 j1 /ı 0 in D 0 .R/ with supp.Tn / D ¹0º8n 2 N H) Tn 2 E 0 .R/ 8n 2 N. For m  3, Tn can be found in a similar way. P n Example 5.7.11. Show that the series 1 nD0 a ın with a > 0 and hın ; i D .n/ 1 8 2 E.R/  C .R/ 8n 2 N (which converges in D 0 .R/ for arbitrary a > 0 (see Example 1.9.1)), does not converge in E 0 .R/ for any a > 0.

Chapter 5 Local properties, restrictions, unification principle, E 0 .Rn /

296

P n 0 0 Proof. Set SN D N nD0 a ın . Since ın 2 E .R/ 8n 2 N0 , SN 2 E .R/ 8N 2 N. Then, 8 2 C 1 .R/  E.R/, hSN ; i is well defined and X  N n a ın ;  hSN ; i D D

nD0 N X

N X

nD0

nD0

an hın ; i D

an .n/

8N 2 N; 8 2 C 1 .R/:

P n For a  1, let .x/ D 1 8x 2 R. Then  2 C 1 .R/ and hSN ; 1i D N nD0 a ! 1 as N ! 1,Psince a  1. Hence, the sequence .SN / does not converge in E 0 .R/ n 0 for a  1 H) 1 nD0 a ın does not converge in E .R/ for a  1. 1 x 1 1 x For 0 < a < 1, let .x/ D . a / D e lnŒ. a /  D e x ln. a / 8x 2 R. Then  2 C 1 .R/ PN P n 1 n and hSN ; i D nD0 an .n/ D N nD0 a . a / D 1 C    C 1 D N C 1 ! 1 as N ! 1. H) the .SN / does not converge in E 0 .R/ for 0 < a < 1. P1sequence n H) nD0 P a ın does not converge in E 0 .R/ for 0 < a < 1. Thus, we have proved n 0 the result that 1 nD0 a ın does not converge in E .R/ for a > 0.

5.8

Definition of hT; i for 2 C 1 .Rn / and T 2 D 0 .Rn / with non-compact support

The procedure of definition of hT; i for distributions T with compact support and  2 C 1 .Rn / with arbitrary support can be extended to distributions T with noncompact support in the following situation. For a distribution T 2 D 0 .Rn / with non-compact support and  2 C 1 .Rn / with arbitrary support, if the intersection of the supports of T and  is compact, i.e. if K0 D supp.T / \ supp./ is a compact subset of Rn , then hT; i can always be defined. (5.8.1) Let ˛ 2 D.Rn / be a function with ˛.x/ D 1 8x 2 a compact neighbourhood U of K0 , i.e. K0  U . Then ˛ 2 D.Rn / and we set hT; i D hT; ˛i

8 2 C 1 .Rn /;

(5.8.2)

the right-hand side of which does not depend on the choice of ˛. Let ˛; ˇ be two functions of D.Rn / such that ˛.x/ D ˇ.x/ D 1 8x 2 a compact neighbourhood U of K0 , K0  U . Then .˛  ˇ/ 2 D.Rn / with supp..˛  ˇ// contained in supp./\supp.˛ˇ/. But ˛.x/ˇ.x/ D 0 8x 2 K0 H) supp.˛ˇ/  K0{ D complement of K0 . Hence, supp..˛  ˇ// is contained in supp./ and also in K0{ , i.e. outside K0 .

Section 5.8 Definition for hh; T i for T 2 D 0 .Rn / with non-compact support

297

x 2 supp./, x 2 K0 H) x 2 supp.T /. But x 2 supp./, x … K0 H) x … supp.T / H) supp..˛  ˇ// is outside the support of T H) hT; .˛  ˇ/i D 0, since supp.T / \ supp..˛  ˇ// D ; H) hT; ˛i D hT; ˇi. Thus, 8 2 C 1 .Rn / with compact intersection supp.T / \ supp./, hT; i D hT; ˛i:

(5.8.3)

Chapter 6

Convolution of distributions

6.1

Tensor product

Let x  Rn ; y  Rm be open subsets of Rn and Rm whose generic points are x D .x1 ; x2 ; : : : ; xn / 2 x ; y D .y1 ; y2 ; : : : ; ym / 2 y : Then, x  y D ¹.x; y/ W x 2 x ; y 2 y º  Rn  Rm D RnCm

(6.1.1)

is an open subset of Rn  Rm with .x; y/ D .x1 ; x2 ; : : : ; xn I y1 ; y2 ; : : : ; ym /. By abuse of notations, functions x 2 x 7! f .x/, y 2 y 7! g.y/, and .x; y/ 2 x  y 7! h.x; y/ will be denoted by f .x/; g.y/ and h.x; y/ respectively. Let D.x /; D.y / and D.x  y / be the test spaces of infinitely differentiable real- or complex-valued functions with compact support in Rn ; Rm and Rn  Rm respectively, and let their dual spaces D 0 .x /; D 0 .y / and D 0 .x y / be the spaces of distributions Tx ; Sy and Wx;y (or equivalently T .x/; S.y), W .x; y/) on x ; y and x  y respectively. Remark 6.1.1. Tx or T .x/; Sy or S.y/; Wx;y or W .x; y/ do not indicate the values of T; S; W at x; y and .x; y/ respectively, since a distribution cannot have a value at a point. The notation Tx or T .x/, Sy or S.y/, Wx;y or W .x; y/ is used to indicate that the generic points of x ; y ; x  y are denoted by x; y; .x; y/ respectively. Tensor product of functions Definition 6.1.1. Let f .x/; g.y/ be functions on x and y respectively. Then the tensor product f .x/ ˝ g.y/ of f .x/ and g.y/ is the function h.x; y/ on x  y defined by: h.x; y/ D f .x/ ˝ g.y/ D f .x/  g.y/

8.x; y/ 2 x  y :

(6.1.2)

Example 6.1.1. 

For m D n D 1, x D y D R, x  y D R  R D R2 . 1. f .x/ D sin mx, g.y/ D cos ny, h.x; y/ D sin mx ˝ cos ny D sin mx: cos ny 8.x; y/ 2 R2 ; 2. f .x/ D e i2x , g.y/ D e i2y , h.x; y/ D e i2x ˝ e i2y D e i2x  e i2y D e i2.xCy/ 8.x; y/ 2 R2 .

299

Section 6.1 Tensor product 

For m D 2, n D 1, x D R2 , y D R, f .x1 ; x2 / D sin mx1 cos nx2 , g.y/ D 2 2 2 e y , h.x1 ; x2 I y/ D .sin mx1 cos nx2 / ˝ e y D e y sin mx1 cos nx2 8.x1 ; x2 I y/ 2 R2  R.

Let 1x W x 7! 1x .x/ D 1 8x 2 Rn , and 1y W y 7! 1y .y/ D 1 8y 2 Rn . (6.1.3) A function g (resp. f ) is said to be independent of x (resp. y) if it is of the form: 8.x; y/ 2 x  y ;

g.y/ D 1x ˝ g.y/

.resp. f .x/ D f .x/ ˝ 1y /:

(6.1.4)

Generalized integrals dependent on a parameter The classical theorems of calculus on the continuityR and differentiability of integrals dependent on a parameter (under the integral sign ) has been extended to distributions by [8]; since it will be used later to define the tensor product of distributions. Let .xI / with  D .1 ; 2 ; : : : ; m / 2 ƒ  Rm be a function not only of x D .x1 ; x2 ; : : : ; xn / 2 x , but also of parameter  such that for every fixed value  2 ƒ, .xI / can be considered to be a function of x only, and .xI / 2 D.x / 8 fixed  2 ƒ. Let Tx 2 D 0 .x / be a fixed distribution on x , in which x is a generic point. Then, 8 fixed  2 ƒ with .xI / 2 D.x /, I./ D Tx Œ.xI / D hTx ; .xI /i;

(6.1.5)

well defined for each value of  2 ƒ, will be called a generalized integral dependent on the parameter . In fact, if Tx D f .x/ 2 L1loc .x /, then we can rewrite (6.1.5) in integral form: 8 fixed  2 ƒ with .xI / 2 D.x /, Z I./ D Tx Œ.xI / D f .x/.xI /d x: (6.1.6) x 0 Otherwise (i.e. if Tx 2 R D .x / is a singular distribution), we can not write I./ using the integral sign as in (6.1.6). Hence, we have called I./ in (6.1.5) the generalized integral. We agree to accept the result (see Schwartz [8]):

Lemma 6.1.1. I./ defined by (6.1.5) is continuous in  and infinitely differentiable (in the usual pointwise sense) with respect to i ; 1  i  m; i.e. 8 multi-index @j˛j ˛ D .˛1 ; ˛2 ; : : : ; ˛m / and @˛ ˛m ,  D ˛1 ˛2 @ @ :::@m 1

2

˛ ˛ @˛  I./ D @ Tx Œ.xI / D Tx Œ@ .xI / D Tx

  @j˛j .xI / D Tx ; ˛ 1 ˛ 2 @1 @2    @˛m m



@j˛j .xI / ˛1 ˛2 @1 @2 : : : @˛m m



8 2 ƒ with .xI / 2 D.x /:

(6.1.7)

300

Chapter 6 Convolution of distributions

Corollary 6.1.1. Let .x; y/ be a function of x 2 x which depends on parameter y 2 y and is infinitely differentiable with respect to variables x; y in x  y such that, for fixed y 2 y , .x; y/ 2 D.x /. Then the generalized integral I.y/ D Tx Œ.x; y/ D hTx ; .x; y/i;

(6.1.8)

well defined for each value of y 2 y , is an infinitely differentiable function of y in y . Moreover, if .x; y/ 2 D.x  y / with supp..x; y//  y for fixed x 2 x , then supp.I.y//  y , I.y/ 2 D.y / and Sy .I.y// D hSy ; I.y/i D hSy ; hTx ; .x; y/ii:

(6.1.9)

Linear functional defined by tensor product f ˝ g Let f .x/ 2 L1 .x /; g.y/ 2 L1 .y / and .x; y/ D .x/ .y/, with .x/ 2 D.x /; .y/ 2 D.y /. Then define a linear functional Lf ˝g by: 8 .x; y/ D .x/ .y/ with .x/ 2 D.x /; .y/ 2 D.y /, Z Lf ˝g . / D hf .x/ ˝ g.y/; .x/ .y/i D Z

Z f .x/.x/d x

D x

H)

y

f .x/g.y/.x/ .y/d xd y x y

g.y/ .y/d y D Lf ./  Lg . / D hf; i  hg; i

hf .x/ ˝ g.y/; .x/ .y/i D hf; i  hg; i

8 2 D.x /; 8

2 D.y /: (6.1.10)

If .x; y/ 2 D.x  y / is not of the tensor product form (i.e. .x; y/ ¤ .x/ .y//, then applying Fubini’s Theorem (see Theorem 7.1.2C), we have, 8 2 D.x  y /: Z f .x/g.y/ .x; y/d xd y

hf .x/ ˝ g.y/; .x; y/i D x y

´R

R f .x/. y g.y/ .x; y/d y/d x D hf .x/; hg.y/; .x; y/ii R D R y g.y/. x f .x/ .x; y/d x/d y D hg.y/; hf .x/; .x; y/ii x

H) for f 2 L1 .x /, g 2 L1 .y / and 8 2 D.x  y /, hf .x/ ˝ g.y/; .x; y/i D hf .x/; hg.y/; .x; y/ii

(6.1.11)

D hg.y/; hf .x/; .x; y/ii:

(6.1.12)

301

Section 6.1 Tensor product

Tensor product of distributions Following formulae (6.1.10)–(6.1.12), Lemma 6.1.1 and Corollary 6.1.1, we can define the tensor product of distributions as follows: Definition 6.1.2. Let Tx 2 D 0 .x /; Sy 2 D 0 .y / be any two distributions on x and y respectively. Then the tensor product Tx ˝ Sy of two distributions Tx and Sy is the unique distribution Wx;y 2 D 0 .x  y / defined by: hWx;y ; .x/ .y/i D hTx ˝ Sy ; .x/ .y/i D hTx ; .x/i  hSy ; .y/i

(6.1.13)

8 .x; y/ D .x/ .y/ 2 D.x /  D.y /; hWx;y ; .x; y/i D hTx ˝ Sy ; .x; y/i D hSy ; hTx ; .x; y/ii D hTx ; hSy ; .x; y/ii (6.1.14) 8 .x; y/ 2 D.x  y / not in the tensor product form. For every .x; y/ D .x/ .y/ with .x/ 2 D.x /; .y/ 2 D.y /, we get (6.1.13) from (6.1.14), i.e. 8.x/ .y/ 2 D.x /  D.y /  D.x  y /, hTx ; hSy ; .x/ .y/ii D hSy ; hTx ; .x/ .y/ii D hTx ; .x/i  hSy ; .y/i:

(6.1.15)

Support of tensor product Tx ˝ Sy supp.Tx ˝ Ty / D supp.Tx /  supp.Sy / D ¹.x; y/ W x 2 supp.Tx /; y 2 supp.Sy /º: (6.1.16) Example 6.1.2. Let x D Rl , y D Rm , Tx D ı.x/ 2 D 0 .Rn / and Sy D ı.y/ 2 D 0 .Rm /. Then their tensor product ı.x/ ˝ ı.y/ is defined, 8.x; y/ 2 D.Rl  Rm /, by: hı.x/ ˝ ı.y/; .x; y/i D hı.x/; hı.y/; .x; y/ii D hı.x/; .x; 0/i D .0; 0/; i.e. ı.x/ ˝ ı.y/ D ıx;y with .x; y/ 2 Rl  Rm , supp.ı.x/ ˝ ı.y// D .0; 0/ 2 Rl  Rm :

(6.1.17)

A distribution is said to be independent of x (resp. y) if it is of the form 1x ˝ Sy (resp. Tx ˝ 1y /, where 1x (resp. 1y ) is defined by (6.1.3). This definition is suggested

302

Chapter 6 Convolution of distributions

by (6.1.4). Then, 8 2 D.x  y /, Z

Z

h1x ˝ Sy ; i D h1x ; hSy ; .x; y/ii D

1  hSy ; .x; y/id x D x

Z

D hSy ; h1x ; .x; y/ii D hSy ;

.x; y/d xiI Z

hSy ; .x; y/id x x

(6.1.18)

x

hTx ˝ 1y ; i D hTx ; h1y ; .x; y/ii D hTx ;

.x; y/d yi y

Z D h1y ; hTx ; .x; y/ii D

hTx ; .x; y/id y;

(6.1.19)

y

since 1x .x/ D 1 8x 2 x , 1y .y/ D 1 8y 2 y .

Derivatives of tensor product Let @˛ x D

@j˛j ˛ ˛ ˛n , @x11 @x22 :::@xn

@jˇj

ˇ

@y D

ˇ ˇ ˇm @y11 @y22 @ym

. Then

ˇ ˛ ˇ @˛ x @y ŒTx ˝ Sy  D @x Tx ˝ @y Sy :

(6.1.20)

In fact, ˇ ˛ ˇ ˛ jˇj ˇ h@˛ x Tx ˝ @y Sy ; i D h@x Tx ; h@y Sy ; ii D h@x Tx ; .1/ hSy ; @y .x; y/ii ˇ D .1/jˇj h@˛ x Tx ; hSy ; @y .x; y/ii ˇ D .1/jˇj hSy ; h@˛ x Tx ; @y .x; y/ii ˇ D .1/jˇj .1/j˛j hSy ; hTx ; @˛ x @y .x; y/ii ˇ D .1/jˇj .1/j˛j hTx ˝ Sy ; @˛ x @y .x; y/i ˇ D h@˛ x @y .Tx ˝ Sy /; .x; y/i

.by definition of Tx ˝ Sy /

8 2 D.x  y /;

since Tx ˝ Sy is a distribution.

Tensor product of several distributions Let x  Rl , y  Rm , z  Rn and Tx 2 D 0 .x /, Sy 2 D 0 .y /, Rz 2 D 0 .z /. Then the tensor product Tx ˝ Sy ˝ Rz of the three distributions Tx ; Sy ; Rz is a distribution on x  y  z defined, 8.x/ 2 D.x /, .y/ 2 D.y /, .z/ 2

303

Section 6.2 Convolution of functions

D.z /, by: hTx ˝ Sy ˝ Rz ; .x/ .y/ .z/i D hTx ; .x/ihSy ; .y/i  hRz ; .z/iI hTx ˝ Sy ˝ Rz ; .x; y; z/i D hTx ; hSy ˝ Rz ; .x; y; z/ii

(6.1.21) (6.1.22)

8.x; y; z/ 2 D.x  y  z / D hTx ˝ Sy ; hRz ; .x; y; z/ii:

(6.1.23)

Example 6.1.3. Let H.xi / be the Heaviside function in variable xi ; 1  i  3, defined by H.xi / D 1 for xi > 0 and H.xi / D 0 for xi < 0. Then H.x1 ; x2 ; x3 / D H.x1 / ˝ H.x2 / ˝ H.x3 / is a Heaviside function in 3 variables, which equals 1 in the octant x1 > 0, x2 > 0, x3 > 0 and equals 0 otherwise. i/ Now dH.x D ı D ıxi , 1  i  3 (xi is a generic point of R). Hence, dx i

@3 @3 H D .H.x1 / ˝ H.x2 / ˝ H.x3 // @x1 @x2 @x3 @x1 @x2 @x3 dH.x1 / dH.x2 / dH.x3 / D ˝ ˝ D ıx1 ˝ ıx2 ˝ ıx3 D ıx1 ;x2 ;x3 : dx1 dx2 dx3 (6.1.24)

6.2

Convolution of functions

Notations

Let A; B  Rn be any two non empty sets in Rn . Then

A ˙ B D ¹z W z 2 Rn

such that z D x ˙ y with x 2 A; y 2 Bº:

(6.2.1)

Some properties of A C B are summarized here: 

If A or B is open, then A C B is open.

(6.2.2)



If A and B are compact, then A C B is compact.

(6.2.3)



If one of the two sets is closed and the other one is compact, then A C B is closed. (6.2.4)

Proof of (6.2.4). Assume that A is compact and B is closed in Rn . Let .xm /1 mD1 be a Cauchy sequence in A C B with xm D am C bm , am 2 A, bm 2 B 8m 2 N. Then .xm / is a Cauchy sequence in Rn , which is complete. Hence, 9x 2 Rn such that xm ! x in Rn as m ! 1. We are to show that x 2 A C B with x D a C b, a 2 A, b 2 B. Since A is compact in Rn and .am /1 mD1 is a bounded sequence in A, 9 a 1 subsequence .amk /1 of .a / in A such that amk ! a 2 A as k ! 1. Then m mD1 kD1 xmk D amk C bmk 8k 2 N, with xmk ! x in Rn , amk ! a in A as k ! 1 and bmk D xmk  amk 2 B 8k 2 N. Hence, limk!1 bmk D limk!1 .xmk  amk / D x  a 2 Rn with x 2 Rn , a 2 A. But B is closed in Rn and .bmk /1 converges in kD1

304

Chapter 6 Convolution of distributions

Rn H) 9b 2 B such that limk!1 bmk D b D xa 2 B by virtue of the uniqueness of the limit. Hence, b D x  a H) x D a C b with a 2 A, b 2 B H) x 2 A C B. Thus, A C B is closed. Convolution of functions integrable on Rn Let f; g 2 L1 .Rn / be functions integrable on Rn . Then the tensor product f ./ ˝ g. / is integrable on Rn  Rn  R2n , since Z Z Z Z jf ./ ˝ g. /jd d D jf ./g. /jd d Rn Rn Rn Rn Z Z D jf ./jd  jg. /jd < C1: Rn

Rn

By change of variables x D  C , t D with their Jacobian D 1 and d d D d xd t, we get Z Z Z Z f ./g. /d d D f .x  t/g.t/d xd t: Rn

Rn

Rn

Rn

Then, by Fubini’s Theorem, the function h.x/ D in Rn and h 2 L1 .Rn / H)

h D f g 2 L1 .Rn /

R

Rn

f .x  t/g.t/d t is defined a.e.

8f; g 2 L1 .Rn /:

(6.2.5)

In general, we have the following results (see Schwartz [8, p. 151] and also Vladimirov [6, pp. 60–61]): Theorem 6.2.1. If f 2 Lp .Rn /, g 2 Lq .Rn / with 1  p; q  1 and p1 C q1  1, then R I. h.x/ D .f g/.x/ D Rn f .x  /g./d  is well defined for almost all x 2 Rn such that: II. h D f g 2 Lr .Rn / with

1 r

D

1 p

C

1 q

 1, 1  r  1;

(6.2.6)

III. khkLr .Rn / D kf gkLr .Rn /  kf kLp .Rn / kgkLq .Rn / ,

(6.2.7)

i.e. 1=r  Z jh.x/j d x 

Z

r

Rn

Rn

1=p  Z jf .x/j d x p

1=q jg.x/j d x : q

(6.2.8)

Rn

Proposition 6.2.1. LetR f 2 L1 .Rn / and g 2 Lp .Rn / with 1  p  1. Then h.x/ D .f g/.x/ D Rn f .x  /g./d  is well defined for almost all x 2 Rn such that h D f g 2 Lp .Rn /, with khkLp .Rn / D kf gkLp .Rn /  kf kL1 .Rn / kgkLp .Rn / :

(6.2.9)

305

Section 6.2 Convolution of functions

Proof. Replacing ‘q’ by ‘p’ and ‘p’ by ‘1’ in Theorem 6.2.1, we get p1 C q1 D 1 1 1 1 1 1 1 C p  1 and r D 1 C p  1 D p , i.e. r D p. Then, by Theorem 6.2.1, the results follow. Remark 6.2.1. For f  0, g  0 and p D q D r D 1, the inequality (6.2.8) becomes an equality: khkL1 .Rn / D kf gkL1 .Rn / D kf kL1 .Rn / kgkL1 .Rn / :

(6.2.10)

Proposition 6.2.2. If f; g 2 L2 .Rn /, then h D f g 2 L1 .Rn /.

(6.2.11)

Proof. p D q D 2 H) r D 1 and khkL1 .Rn / D ess sup jh.x/j  kf kL2 .Rn / kgkL2 .Rn / : x2Rn

For r D 1, h exists everywhere and is continuous on Rn . Moreover, for r D 1, h.x/ ! 0 for kxk ! 1 except when p D 1 or q D 1. (6.2.11a) Theorem 6.2.2. Let f 2 L1 .Rn / and g 2 Lp .Rn / with 1  p  1. Then supp.f g/  supp.f / C supp.g/:

(6.2.12) R

Proof. By Proposition 6.2.1, .f g/.x/ is well defined and .f g/.x/ D Rn f .x  /g./d . But f .x  / D 0 for x   D y … supp.f / H) f .x  / D 0 for  D x  y … x  supp.f /, where x  supp.f / D ¹z W z D x  y with y 2 supp.f /º. Hence, Z .f g/.x/ D f .x  /g./d : .xsupp.f //\supp.g/

But x 2 .supp.f / C supp.g// H) x D y C  with y 2 supp.f /,  2 supp.g/ H) x  y D  2 supp.g/ with y 2 supp.f / H) .x  supp.f // \ supp.g/ ¤ ;. Hence, xR … .supp.f / C supp.g// H) .x  supp.f // \ supp.g/ D ; H) .f g/.x/ D .xsupp.f //\supp.g/ f .x  /g./d  D 0 for almost all x … .supp.f / C supp.g// H)

.f g/.x/ D 0

a.e. in the complement of .supp.f / C supp.g//

H)

.f g/.x/ D 0

a.e. in the interior of Œ.supp.f / C supp.g//{ 

H)

supp.f g/  supp.f / C supp.g/ D supp.f / C supp.g/:

Remark 6.2.2. 

If supp.f / and supp.g/ are both compact, then supp.f / C supp.g/ is compact by (6.2.3), and supp.f g/ is also compact. (6.2.13)



If only one of the supports, i.e. supp.f / or supp.g/ is compact, then supp.f g/ is not compact in general. (6.2.14)

306

Chapter 6 Convolution of distributions

Regularization with the help of convolution Regularizing functions n D 1) be defined by:

Let  2 C01 .Rn / (see Example 1.2.1 and Figure 1.1 for ´

.x/ D

exp.1=.1  kxk2 // 0

for kxk < 1 for kxk  1

R N 1/, n .x/d x > 0. with the properties .x/  0, supp./ D B.0I R Define .x/ D ˛.x/ with ˛ > 0 such that Z Z ı

.x/d x D 1; i.e. ˛ D 1 .x/d x: Rn

(6.2.15)

(6.2.16)

Rn

Definition 6.2.1. For " > 0, the functions

" .x/ D "n .x="/

(6.2.17)

with .x="/ D ˛.x="/ are called regularizing functions or mollifiers, with the following properties 8" > 0: Z 1 n N

" 2 C0 .R /I supp. " / D closed ball B.0I "/I

" .x/d x D 1: (6.2.18) Rn

In fact, Z Z

" .x/d x D Rn

N B.0I"/

"n .x="/d x D "n

Z N B.0I1/

.y/"n d y D 1;

(6.2.19)

which is obtained by the change of variables x D "y with jJ j D "n . An alternative notation used for a regularizing function is J" . Definition 6.2.2. 8" > 0, the convolution " u defined by Z

" .x  /u./d  . " u/.x/ D

(6.2.20)

Rn

for functions u, for which the integral over Rn on the right-hand side of (6.2.20) is well defined, is called a regularization or mollification of u with the help of regularizing functions " with " > 0. Remark 6.2.3. 

The importance of " u is due to the fact that that " u behaves much like u, but " u is extremely smooth or regular – hence the name regularization or mollification. This fact will be established in a series of theorems below.

307

Section 6.2 Convolution of functions 

In [30, p. 329], " u is called Sobolev’s regularization of u with " , " > 0. But [26, p. 71] states that the technique of regularization by convolution was introduced by Leray and Friedrichs. Finally, according to [10, p. 1075], in 1926 Norbert Wiener used regularization, i.e. convolution with compactly supported C 1 -functions, to approximate continuous functions f by smooth ones. Choosing " D 1=m in the right-hand side expression of (6.2.17), we get  n 1

.mx/ D mn .mx/ 8m 2 N: m We set

m .x/ D mn .mx/ D ˛mn .mx/;

(6.2.21)

where  and ˛ are defined by (6.2.15) and (6.2.16) respectively. Definition 6.2.3. A sequence . m /m2N with m defined by (6.2.21) and having the properties, 8m 2 N, Z 1 n N

m .x/  0; m 2 C0 .R /I supp. m / D B.0I 1=m/I

m .x/d x D 1; Rn

(6.2.22) is called the sequence of regularizing functions or sequence of mollifiers and, 8m 2 N, Z . m u/.x/ D

m .x  /u./d  (6.2.23) Rn

is called the regularization of u, for which the integral in (6.2.23) is well defined, with the help of the sequence . m /. By abuse of terminology, . " /">0 is also called a sequence of regularizing functions. (6.2.24) Elementary properties of convolution By change of variables, 8" > 0, Z Z . " u/.x/ D

" .x  /u./d  D Rn

Rn

" ./u.x  /d  D .u " /.x/; (6.2.25)

˛1 " u1 C ˛2 " u2 D " .˛1 u1 C ˛2 u2 /:

(6.2.26)

In (6.2.25) and (6.2.26) it is assumed that the defining integral of the convolution is well defined.

308

Chapter 6 Convolution of distributions

Proposition 6.2.3. Let u 2 C 0 .Rn /. Then " u ! u uniformly on every compact subset K of Rn as " ! 0C . Proof. Let K  Rn be any fixed compact subset of Rn . Since u 2 C 0 .Rn /, 8 > 0, 9ı D ı.K; / > 0 such that ju.x  /  u.x/j < 

8x 2 K; 8

(6.2.27)

with kx  .x  /k D kk < ı, i.e. 8 2 B.0I ı/. But . " u/.x/  u.x/ D .u " /.x/  u.x/ Z Z R D u.x  / " ./d   u.x/

" ./d  .by (6.2.18), Rn " ./d  D 1/ Rn Rn Z Z D Œu.x  /  u.x/ " ./d  D Œu.x  /  u.x/ " ./d : Rn

B.0I"/

Hence, for any fixed compact K  Rn , 8 > 0, 9ı D ı.K; / > 0 such that 8x 2 K, Z j. " u/.x/  u.x/j  ju.x  /  u.x/j " ./d j B.0I"/

Z

" ./d  D 



8" < ı

.by (6.2.27), (6.2.18)/

B.0I"/

H) " u ! u uniformly on K  Rn as " ! 0C . But K is any compact of subset of Rn . Hence, the result holds for every compact subset of Rn . Lemma 6.2.1. Let f 2 C0 .Rn /  D 0 .Rn / be a continuous function with compact support in Rn . Let functions f" be defined, 8" > 0, by: Z

" .x  /f ./d : (6.2.28) f" .x/ D . " f /.x/ D Rn

Then I. f" 2 C01 .Rn /  D.Rn /; II. lim"!0C f" D lim"!0C . " f / D f in D 0 .Rn /  C0 .Rn /, i.e. C01 .Rn / is dense in C0 .Rn /. (6.2.29) Proof. I. Compact support of f" 8" > 0: Let f 2 C0 .Rn /. Then, 9 a compact set K  Rn such that supp.f / D K. Since " , f 2 L1 .Rn /, using (6.2.12), N "/ C K D K" supp.f" / D supp. " f /  supp. " / C supp.f / D B.0I (6.2.30)

309

Section 6.2 Convolution of functions

8 fixed " > 0, K"  Rn being a compact subset of Rn by (6.2.13). Hence, 8" < "0 with "0 > 0, 9 a fixed compact set K0  Rn such that supp.f" /  K0 :

(6.2.31)

Continuity and infinite differentiability of f" 8" > 0: The continuity of f" (resp. the infinite differentiability of f" ) follows from the classicalR theorems on integrals depending on parameter x 2 K" , since in the integral Rn " .x  R /f ./d  D KDsupp.f / " .x/f ./d  on a compact set K (precisely speaking on .x  supp. " // \ K), the integrand function F .x  / D " .x  /f ./ is continuously dependent on parameter x (resp. infinitely differentiable with respect to the variables x1 ; x2 ; : : : ; xn ) on the compact set K" 8 fixed " > 0. Hence, f" 2 C 1 .Rn / and supp.f" /  K"  Rn 8 fixed " > 0, i.e. f" 2 C01 .Rn /. (6.2.32) II. Uniform convergence of f" ! f in D 0 .Rn /  C0 .Rn /: Since f 2 C0 .Rn / H) f 2 C 0 .Rn /, by Proposition 6.2.3, f" D " f ! f uniformly as " ! 0C on every compact subset of Rn . Hence, 8" < "0 with "0 > 0, 9 a fixed compact set K0 such that supp.f" /  K0 and f" ! f uniformly as " ! 0C , i.e. f" ! f in D 0 .Rn /  C0 .Rn / as " ! 0C . Thus, C01 .Rn / is dense in C0 .Rn /.

Corollary 6.2.1. 8m 2 N, let f 2 C0m .Rn /  D m .Rn / with compact support in Rn . Let f" D " f be defined by (6.2.28). Then I. f" 2 C01 .Rn / 8 fixed " > 0; ˛ II. 8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n / with 0  j˛j  m, @˛ x f" .x/ D @x . " ˛ m n C f /.x/ ! @x f .x/ in D .R / as " ! 0 (derivatives being in the usual pointwise sense), i.e. D.Rn / is dense in D m .Rn /. (6.2.33)

Proof. I. f 2 C0m .Rn / H) f 2 C0 .Rn / H) f" 2 C01 .Rn / 8 fixed " > 0 by Lemma 6.2.1. R II. By change of variables z D x   with jJ j D 1, we have f" .x/ D Rn " .z/ f .x  z/d z. Then, 8 multi-index ˛ with j˛j  m and @˛ x D ˛ @˛ x f" .x/ D @x

Z

Z Rn

" .z/f .x  z/d z D

Rn

@j˛j ˛ ˛ , @x1 1 :::@xn n

" .z/@˛ x f .x  z/d z;

i.e. the derivative @˛ by differentiating in the usual pointx f" .x/ can be obtained R n wise sense under the integral sign . But f 2 C0m .Rn / H) @˛ x f 2 C0 .R / 8j˛j  m. Then, by Lemma 6.2.1 the result is obtained, i.e. 8" > "0 with

310

Chapter 6 Convolution of distributions

"0 > 0, 9 a fixed compact set K0  Rn such that supp.f" /  K0 , and, ˛ C 8˛ with 0  j˛j  m, @˛ x f" ! @x f uniformly as " ! 0 , i.e. f" ! f m 1 m n n C n in D .R /  C0 .R / as " ! 0 . Hence, C0 .R /  D.Rn / is dense in D m .Rn /. Some important results of regularization in Lp -spaces Proposition 6.2.4. Let  2 C0 .Rn /. Then "  2 Lp .Rn / and

"  !  in Lp .Rn /

as " ! 0C

for 1  p < 1;

(6.2.34)

i.e. C0 .Rn / is dense in Lp .Rn /, 1  p < 1. Proof. Let  2 C0 .Rn /  Lp .Rn /, 1  p < 1, with compact supp./ D K  N "/ C K D K" , K" being a compact Rn . By Theorem 6.2.2, supp. " /  B.0I set by (6.2.13) 8 fixed " > 0. Hence, 9 a fixed compact set K0 with K, K"  K0 8" < "0 with some "0 > 0. Then . " /.x/ D 0, .x/ D 0 8x … K0 . By Lemma 6.2.1 and Proposition 6.2.3, "  2 C01 .Rn /  Lp .Rn / for 1  p < 1 and . " /.x/ ! .x/ uniformly on compact set K0 , i.e. 8 > 0, 9ı D ı.K0 ; / > 0 such that 8x 2 K0 , j. " /.x/  .x/j  Œ.K /1=p 8" < ı. Then 0 Z p j. " /.x/  .x/jp d x k "   kLp .Rn / D Rn Z j. " /.x/  .x/jp d x D 

K0 p

.K0 /

.K0 / D p

8" < "0 D min¹"0 ; ıº

H) 8 > 0, 9"0 > 0 such that k "   kLp .Rn /   8" < "0 , i.e. lim"!0C . " / D  in Lp .Rn /, 1  p < 1. Density result Proposition 6.2.5. C01 .Rn /  D.Rn / is dense in Lp .Rn /, 1  p < 1. Proof. Let u 2 Lp .Rn /, 1  p < 1. Let  > 0. Then, by virtue of the density of C0 .Rn / in Lp .Rn /, 1  p < 1, 9 2 C0 .Rn / with compact supp./ D K  Rn such that ku  kLp .Rn / < =2:

(6.2.35)

By Lemma 6.2.1, "  2 C01 .Rn / 8 fixed " > 0, and by Proposition 6.2.4, "  !  in Lp .Rn /, 1  p < 1, as " ! 0C , i.e. 8 > 0, 9"0 > 0 such that k "   kLp .Rn / < =2

8" < "0 :

(6.2.36)

311

Section 6.2 Convolution of functions

Then, combining (6.2.35) and (6.2.36), we have: 8 > 0, 9"0 > 0 such that ku  " kLp .Rn /  ku  kLp .Rn / C k  " kLp .Rn / < =2 C =2 D 

8" < "0 ;

i.e. for u 2 Lp .Rn /, 1  p < 1, 8 > 0, 9. " / 2 C01 .Rn / with " < "0 such that ku  " kLp .Rn / < . Thus, C01 .Rn / is dense in Lp .Rn /, 1  p < 1. Theorem 6.2.3. Let " 2 C01 .Rn / be defined by (6.2.17) 8" > 0. Then: I. If u 2 L1loc .Rn /; then

" u 2 C 1 .Rn /

8" > 0:

(6.2.37)

1 Moreover, if u 2 LLoc .Rn / has compact support, i.e. supp.u/ D K  Rn is n a compact subset of R ,

" u 2 C01 .Rn /

8" > 0:

(6.2.38)

II. If u 2 Lp .Rn / with 1  p < 1, then

" u 2 Lp .Rn /;

k " ukLp .Rn /  kukLp .Rn / lim ku  " ukLp .Rn / D 0:

"!0C

8" > 0I

(6.2.39) (6.2.40)

Proof. I. 8 fixed x 2 Rn , the function  2 Rn 7! " .x  /u./ is integrable on Rn , since " 2 C01 .Rn / has compact support in Rn and u 2 L1loc .Rn / is integrable on compact subsets of Rn . Hence, . " u/.x/ is well defined 8x 2 Rn , 8" > 0. Continuity of " u: Let x 2 Rn be any fixed point and .xm / be a sequence in Rn such that xm ! x in Rn as m ! 1. Then, 8 fixed  2 Rn , " .xm  / !

" .x  / as m ! 1 since " .x  / is continuous in x. Now we will show that . " u/.xm / ! . " u/.x/ 8" > 0 with the help of Lebesgue’s Dominated Convergence Theorem B.3.2.2 (Appendix B), as follows. 8" > 0 define fm ./ D " .xm  /u./ 8m 2 N and f ./ D " .x  /u./ such that fm ./ D Œ " .xm  /u./ ! f ./ as m ! 1 a.e. in Rn . Moreover, xm ! x as m ! 1 H) 9M > 0 such that kxm k  M 8m 2 N (since .xm / is then a bounded sequence in Rn ). Z . " u/.xm / D

" .xm  /u./d  8fixed " > 0: .xm supp." //\supp.u/

But  2 xm  supp. " / H) kk  kxm k C kzk  M C " D M" with z 2 supp. " / H) xm  supp. " / is a bounded set 8 fixed " > 0. Let K0 be a

312

Chapter 6 Convolution of distributions

fixed compact set such that xm  supp. " /  K0 8m 2 N. Then  … K0 H)  … xm  supp. " / H) " .xm  / D 0 for  … K0 . Hence, 8m 2 N, jfm ./j D j " .xm  / K0 ./u./j  sup " .z/ K0 ./ju./j z2Rn

D k " kL1 .Rn / K0 ./ju./j D g./ a.e. in Rn ; where

´ K0 ./ D

1 for  2 K0 0 for  … K0

is the characteristic function of K0 and g./  0 is the majoring integrable function in Rn : Z Z g./d  D k " kL1 .Rn / ju./jd  < C1; Rn

K0

since u 2 L1loc .Rn /. Hence, fm ./ ! f ./ a.e. in Rn as m ! 1, and 8m 2 N, jfm ./j  g./ a.e. in Rn with g 2 L1 .Rn /. Then, by Lebesgue’s Dominated Convergence Theorem, 8 fixed " > 0, Z Z . " u/.xm / D

" .xm  /u./d  !

" .x  /u./d  Rn

D . " u/.x/

Rn

as m ! 1:

Thus, xm ! x H) . " u/.xm / ! . " u/.x/ as m ! 1. Hence, " u 2 C 0 .Rn / 8 fixed " > 0. Infinite differentiability of . " u/: Following the proof of Lemma 6.2.1, the differentiation can Rbe carried out under the integral sign, and, 8j˛j 2 N, ˛ n @˛ x . " u/.x/ D K0 @x Œ " .x  /u./d  exists 8x 2 R . Hence,

" u 2 C 1 .Rn /

8 fixed " > 0:

(6.2.41)

Compact support of . " u/ for u with compact support: Let u 2 L1loc .Rn / such that supp.u/ D K  Rn . Then u 2 L1 .Rn / with supp.u/ D K and, N "/ C K D K" , K" being a compact by Theorem 6.2.2, supp. " u/  B.0I set by (6.2.13) 8 fixed " > 0. Hence, from (6.2.41) " u 2 C 1 .Rn / with supp. " u/  Rn 8 fixed " > 0 H) " u 2 C01 .Rn / 8 fixed " > 0. II. " 2 C01 .Rn / 8" > 0 H) " 2 L1 .Rn / 8" > 0. Then, by Proposition 6.2.1, 8u 2 Lp .Rn /, 1  p < 1, " u 2 Lp .Rn / 8" > 0 and k " ukLp .Rn /  k " kL1 .Rn / kukLp .Rn / D 1  kukLp .Rn / D kukLp .Rn / : (6.2.42)

313

Section 6.2 Convolution of functions

In fact, we can prove (6.2.42) without using Proposition 6.2.1 as follows: Case 1 < p < 1: For 1 < p < 1 with p1 C q1 D 1, using Hölder’s inequality: Z j. " u/.x/j  j Œ " .x  /1=q .Œ " .x  /1=p u.//d j Rn

Z 

Rn

Z H)

1=q  Z

" .x  /d 

Rn

p

1=p

" .x  /ju./j d 

j. " u/.x/jp d x  Z Z R p 

" .x  /ju./j d  d x .since Rn " .x  /d  D 1/ Rn Rn Z Z

" .x  /d x ju./jp d  .by Fubini’s Theorem 7.1.2C/ D Rn

Rn

D1

Rn

p kukLp .Rn /

< C1:

Case p D 1: Z Z Z j. " u/.x/jd x 

" .x  /ju./jd d x Rn Rn Rn Z Z . " .x  /d x/ju./jd  .by Fubini’s Theorem 7.1.2C/ D Rn

Rn

D kukL1 .Rn / : Now we will prove (6.2.40). Let  > 0. Then, by virtue of the density of C01 .Rn / in Lp .Rn /, 1  p < 1 (see Proposition 6.2.5), 9 2 C01 .Rn / such that ku  kLp .Rn /  =3:

(6.2.43)

Using (6.2.42), k " u  " kLp .Rn / D k " .u  /kLp .Rn /  ku  kLp .Rn /  =3: (6.2.44) Moreover,  2 C01 .Rn /, " 2 C01 .Rn / H) "  !  uniformly on every compact subset of Rn by Proposition 6.2.4, and, by Theorem 6.2.2, supp. " N "/ C supp./  K" , K" being a compact set by (6.2.13). Hence, /  B.0I for all sufficiently small " < "0 with "0 > 0, 9 a fixed compact set K0 such that K"  K0 . Then "  !  uniformly on K0 as " ! 0C H) 8x 2 K0 , 8 > 0, 9ı D ı.K0 ; / > 0 such that  8" < "0 j. " /.x/  .xj  3Œ.K0 /1=p

314

Chapter 6 Convolution of distributions

with "0 D min¹"0 ; ıº > 0, where .K0 / D n-dimensional Lebesgue volume measure of K0 . Then, 8 > 0, 9"0 > 0 such that, 8" < "0 , k "  

p kLp .Rn /

Z D

j. " /.x/  .x/jp d x

K0

p  p .K0 / D 3 Œ.K0 /

 p  : 3

(6.2.45)

Hence, from (6.2.43)–(6.2.45), 8 > 0, 9"0 > 0 such that k " u  ukLp .Rn /  k " u  " kLp .Rn / C k "   kLp .Rn / C k  ukLp .Rn /  =3 C =3 C =3 D  8" < "0 H) lim"!0C k " u  ukLp .Rn / D 0. Construction of cut-off functions 2 C01 ./ with .x/ D 1 8x 2 K   In many situations we are to replace a given continuous function by another one with compact support such that the two functions are identical on a large compact set. This is achieved by multiplying the given function by a suitable cut-off function (see also other forms of cut-off function in (6.8.55), (7.5.3)). For example, for a given function 2 C./, continuous on  with non-compact support in , and a given compact set K  , it is required to construct a continuous function 0 2 C0 ./ with compact support in  such that 0 .x/ D .x/ 8x 2 K  . Let  2 C01 ./ D D./ be a test function with .x/ D 1 8x 2 K0 with K  K0  supp./   and 0  .x/  1 8x 2 . Then  is the required cut-off function, since 0 D  has the desired properties: 0 2 C0 ./, 0 .x/ D .x/ .x/ D 1  .x/ D .x/ 8x 2 K  , supp.0 / D supp./ \ supp. /  supp./  .1 The existence of such a cut-off function  2 D./ is given by: Theorem 6.2.4. Let   Rn be an open subset of Rn and K   be a compact subset of . Then, 9 2 C01 ./ with 0  .x/  1 8x 2  such that  D 1 in a compact neighbourhood K" D ¹y W y 2 , d.y; K/ D infx2K kx  yk  "º of K in  for a sufficiently small " > 0 (see (1.2.19) and Figure 1.6). Consequently, .x/ D 1 8x 2 K with K  K" . 1 See Section 5.6, Chapter 5, where such a cut-off function ˛ 2 D.Rn /, with ˛.x/ D 1 8x 2 K , 0 K0 being a (compact) neighbourhood of supp.T / D K  Rn , and 0  ˛.x/  1 8x 2 Rn , has been used to define hT; i D hT; ˛i for  2 E.Rn /  C 1 .Rn / with non-compact support and distribution T 2 E 0 .Rn / with compact support.

Section 6.3 Convolution of two distributions

315

Proof. By sufficiently small " > 0, we mean that kx  yk  4" for x 2 K and y 2 { D complement of  in Rn . For sufficiently small " > 0, define compact neighbourhoods K" , K2" , K3" of K in  by Kı D ¹y W y 2 , d.y; K/ D infx2K kx yk  ı, ı > 0º with K  Kı and ı D "; 2"; 3" (see (1.2.19) and Figure 1.6). (6.2.46) Let K2" be the characteristic function of K2" , i.e. K2" .x/ D 1 for x 2 K2" and D 0 for x 2 Rn n K2" . For " > 0 satisfying (6.2.46), let " be the regularizing functions defined in (6.2.18)–(6.2.19). Then, define  D K2" " 2 C 1 .Rn / by Theorem 6.2.3, since K2" 2 L2 .Rn /, and supp./  supp. K2" / C supp. " / D N "/ H) supp./  K3"   (by Theorem 6.2.2) H)  2 C 1 ./ K2" C B.0I 0 with .x/ D 1 in K" , K  K" . In fact, 1   D .1  K2" / " , since 1 " D 1. N "/ D K { C B.0I N "/ H) 1  .x/ D 0 Then supp.1  /  supp.1  K2" / C B.0I 2" for x 2 K" H) .x/ D 1 for x 2 K" with K  K" .

6.3

Convolution of two distributions

Convolution T  of a distribution T and a test function

Let f 2 L1 .Rn / and  2 D.Rn /  L1 .Rn /. Then f  is well defined by: Z Z f .x  /./d  D f ./.x  /d  8 2 D.Rn /; .f /.x/ D Rn Rn Z L L D hTf ; x i; L D D hf; x i (6.3.1) f ./.x /./ Rn

L x L are defined by: where ; L .x/ D .x/

8x 2 Rn ;

.x /./ D .  x/

8 2 Rn ; 8 fixed x 2 Rn ;

L L  x/ D ..  x// D .x  /I D . .x /./ Tf 2 D 0 .Rn / is the distribution defined by f 2 L1 .Rn /. Equation (6.3.1) suggests the definition of the convolution T  of an arbitrary distribution T 2 D 0 .Rn / and  2 D.Rn / by: Definition 6.3.1A. The convolution T  of T 2 D 0 .Rn / and  2 D.Rn / is defined by: L  x/i D hT ; .x  /i: L D hT ; . .T /.x/ D hT; x i

(6.3.2)

Remark 6.3.1. T  is a C 1 -function on Rn (see Lemma 6.1.1, Corollary 6.1.1 and L also Section 6.4 later). In particular, .T /.0/ D hT; i.

316

Chapter 6 Convolution of distributions

Let T D Tf and TL D TfL be the regular distributions in D 0 .Rn / defined by f and fL respectively. Then hTL ; i D hfL; i D

Z

Z f .x/.x/d x D Rn

f .x/.x/d x Rn

L D hT; i L D hf; i

8 2 D.Rn /

L 8 2 D.Rn /, which suggests the following definition: H) hTL ; i D hT; i Definition 6.3.1B. For every distribution T 2 D 0 .Rn /, the distribution TL is defined by: L 8 2 D.Rn /: hTL ; i D hT; i

(6.3.3)

Since the mapping  2 D.Rn / 7! hTL ; i is continuous on D.Rn /, TL 2 D 0 .Rn / 8T 2 D 0 .Rn /. Hence, LL D hT; i L D hT; i hTL ; i

8 2 D.Rn /:

TL  is a C 1 -function 8 2 D.Rn / and TL  … D.Rn /.

(6.3.4) (6.3.5)

Case when both distributions are functions Let f; g 2 L1 .Rn / such that T D Tf and S D Sg are regular distributions defined by f and g respectively and SL D SgL . Then T S is defined and a regular distribution defined by f g. In fact, 8 2 D.Rn /, Z Z f .x  t/g.t/.x/d td x (6.3.6) hT S; i D hf g; i D Rn Rn Z Z D f ./g. /.  /d d (6.3.7) Rn

Rn

(by change of variables:  D x  t, D t with jJ j D 1) Z  Z D f ./ g. /. L  /d d  Rn Rn Z f ./.gL /./d  D Rn

D hf; gL i D hTf ; SgL i D hT; SL i H)

hT S; i D hT; SL i

8 2 D.Rn /:

(6.3.8)

Since SL  is a C 1 -function and does not belong to D.Rn /, hT; SL i is well defined 8T 2 E 0 .Rn /  D 0 .Rn / i.e. 8 distributions T with compact support. Thus, (6.3.6)

317

Section 6.3 Convolution of two distributions

can be accepted as the definition of the convolution T S, when T 2 E 0 .Rn /  D 0 .Rn / is a distribution with compact support and S 2 D 0 .Rn / is an arbitrary distribution. Then, T S is a distribution, since the mapping  2 D.Rn / 7! hT; SL i is continuous, and we have: Definition 6.3.2. The convolution T S of T 2 E 0 .Rn /  D 0 .Rn / and S 2 D 0 .Rn / is a distribution defined by: hT S; i D hT; SL i

8 2 D.Rn /:

(6.3.9)

Alternative definition of the convolution T  S in terms of the tensor product T ˝S Again, we consider f; g 2 L1 .Rn / with T D Tf and S D Tg . Then h D f g 2 L1 .Rn / is a distribution defined, 8 2 D.Rn /, by:  Z Z f .x  t/g.t/d t .x/d x: hTh ; i D hh; i D hf g; i D Rn

Rn

By changing the variables,  D x  t, D t with jJ j D 1, we have, 8 2 D.Rn /, Z  Z f ./ g. /. C /d d  hf g; i D Rn

Rn

D hf ./; hg. /; . C /ii D hf ./ ˝ g. /; . C /i (6.3.10) H)

hTf Tg ; i D hTf g ; i D hf g; i D hf ./ ˝ g. /; . C /i

8 2 D.Rn /:

Thus, 8f; g; 2 L1 .Rn /, the convolution Tf Tg D Tf g D f g is a distribution in D 0 .Rn / defined by (6.3.10) in terms of the tensor product f ./ ˝ g. /. Hence, it suggests the possibility of defining the convolution T S , if it exists, of two distributions T 2 D 0 .Rn /; S 2 D 0 .Rn / by: 8 2 D.Rn /, hT S; i D hT ˝S ; . C /i or, equivalently, hT S; i D hT ./ ˝ S. /; . C /i:

(6.3.11)

Remark 6.3.2. An immediate remark is in order: for T 2 E 0 .Rn / and S 2 D 0 .Rn /, the two definitions (6.3.9) and(6.3.11) coincide. In fact, using the definitions of tensor L  / D L  . / D  . / D . C / 8 fixed  2 Rn , product T ˝ S , SL and . we get L C /ii hT S; i D hT ˝ S ; . C /i D hT ; hS ; . L D hT; SL i D hT ; .S .//i

8 2 D.Rn /:

(6.3.12)

318

Chapter 6 Convolution of distributions

For the moment we ignore this most important case of T 2 E 0 .Rn /  D 0 .Rn / with compact support and S 2 D 0 .Rn /; we will start with the probable definition (6.3.9), which can be accepted as the definition of the convolution T S of two distributions T and S if the right-hand side of (6.3.9) is well defined for every .x/ 2 D.Rn /. Since T ˝ S 2 D 0 .Rn  Rn / is well defined as a distribution for arbitrary T and S, the right-hand side of (6.3.9) will make sense if . C / 2 D.R2n /, for .x/ 2 D.Rn /. But for .x/ 2 D.Rn /, . C / 2 C 1 .Rn  Rn / (as the function .x.; // D . C / with x D  C of  and ), supp . C / is not compact in R2n , and consequently . C / … D.R2n /. For example, for n D 1, .x/ 2 D.R/ is defined by: ´  1 e 1x2 for jxj  1 .x/ D 0 for jxj > 1 with K D supp./ D Œ1; 1. Set x D  C , with ;  2 R. Then supp.. C // D ¹.; / W .; / 2 R2 with  C  2 Œ1; 1º, which is the infinite strip 1   C   C1 parallel to  C  D 0 (see Figure 6.1) and not bounded in R2 , and hence not compact in R2 .

supp

0

Figure 6.1 Infinite strip bounded by 1   C   C1, parallel to  C  D 0, defining the unbounded support . C / in R2

Thus, we arrive at the important conclusion: For arbitrary distributions T ,S 2 D 0 .Rn /, the right-hand side of (6.3.9) will not make sense, since . C / … D.R2n / for .x/ 2 D.Rn /, i.e. for arbitrary distributions T ,S 2 D 0 .Rn /, T S is not defined. Now we identify the situations in which the right-hand side of (6.3.9) will have a meaning. Let A D supp.T /; B D supp.S /.

319

Section 6.3 Convolution of two distributions

Then, from (6.1.16), supp.T ˝ S / D A  B  Rn  Rn

with A  B D ¹.; / W  2 A; 2 Bº:

We state the result as follows: Theorem 6.3.1. If the set supp.T ˝ S / \ supp.. C // D ¹.; / W  2 A; 2 B;  C 2 supp./;  2 D.Rn /º

(6.3.13)

is bounded in Rn  Rn (i.e. for  2 A, 2 B,  C will only remain bounded if both  and remain bounded), the right-hand side of (6.3.9) is well defined for .x/ 2 D.Rn /, and the convolution T S D S T is a distribution defined by (6.3.9). Remark 6.3.3. For n D 1, the condition relating the supports of distributions T and S in Theorem 6.3.1 can be replaced by the following: if both the distributions T and S on R have supports bounded from the left, i.e. their supports are contained in a; 1Œ, or bounded from the right, i.e. their supports are contained in 1; bŒ, then T S exists.

• supp(T ) ]a, [ , supp(S ) ]a, [ • supp( (x)) = [ c, d ]

d

• supp(T

S )

supp( ( + ))

+ c =

a d 0

a

+

c

d = + =

c

0 Figure 6.2 Boundedness of supp.T ˝ S / \ supp.. C // in R2 with T and S having supports bounded from the left, i.e. contained in a; 1Œ

In fact, suppose that T and S have supports bounded from the left, i.e. contained in a; C1Œ.

320

Chapter 6 Convolution of distributions

Hence,   a;   a. Then, for supp..x// D Œc; d , supp.T ˝ S / \ supp.. C // is bounded in R  R (see Figure 6.2). Hence, T S exists. Now we will study the important case in which the existence of the convolution depends on the property of compactness of the support of T or (and) S . (We have already given the definition of T S in (6.3.7) (resp. (6.3.9)) when T has compact support, and shown that both the definitions coincide). Case I. At least one of the distributions T or S has compact support Let A D supp.T / and B D supp.S / be the supports of T and S such that at least one of them, say A, is compact. Let  2 D.Rn / with supp./ D K compact in Rn . Then the intersection I D supp.T ˝ S / \ .. C // is a bounded subset of Rn  Rn D R2n . In fact, supp.T ˝ S / D A  B and supp.. C // D ¹.; / W ; 2 Rn with  C 2 Kº. Then I D ¹.; / W  2 A; 2 B;  C 2 Kº. But  2 A,  C 2 K H) D . C /   2 .K  A/ H) I  A  .K  A/, which is a compact subset of R2n H) I is bounded in R2n . Now we introduce a function ˛./ 2 D.Rn / such that ˛./ D 1 8 2 U with A  U , U being a neighbourhood of the compact support A of T , and supp.˛.// is compact. Then the function ˛./. C / has compact support in R2n , and ˛./. C / D . C / 8.; / 2 V with AB  V; V being a neighbourhood of supp.T ˝ S /. Then hT S; i is well defined as a linear functional 8 2 D.Rn /: hT S; i D hT ˝ S ; . C /i D hT ˝ S ; ˛./. C /i

8 2 D.Rn /:

Continuity of T  S on D (Rn ) Let .m / be a sequence in D.Rn / such that 8m 2 N, supp.m /  K0 , K0 being a fixed compact subset of Rn and m ! 0 in D.Rn / as m ! 1. Then, 8m 2 N, ˛./m . C / has support contained in a fixed compact subset of Rn  Rn and converges, along with all derivatives, uniformly to 0, i.e. ˛./m . C / ! 0 in D.Rn  Rn / as m ! 1. But T ˝ S 2 D 0 .Rn  Rn / and, consequently, hT ˝ S ; ˛./m .. C //i ! 0 as m ! 1. Hence, m ! 0 in D.Rn / H) hT S; m i D hT ˝S ; m . C /i D hT ˝S ; ˛./m . C /i ! 0 in R as m ! 1 H) the linear functional T S is continuous on D.Rn / and, consequently, a distribution on Rn . Hence, we have: Theorem 6.3.2. Let T and S be arbitrary distributions on Rn such that at least one of the two has compact support. Then the convolution of T and S , denoted by T S or S T , is well defined as a distribution on Rn such that hT S; i D hT ˝ S ; .. C //i

8 2 D.Rn /:

(6.3.14)

321

Section 6.3 Convolution of two distributions

Moreover, using the definition of the tensor product of two distributions (see (6.1.14)), we can write: 8 2 D.Rn /, hT S; i D hT ˝ S ; .. C //i D hT ; hS ; .. C //ii D hS ; hT ; .. C //ii:

(6.3.15)

In fact, for T 2 D 0 .Rn / with compact support A and  2 D.Rn /, hT ; .. C //i D h. / is a well defined function of which is infinitely differentiable on Rn (see Remark 6.2.1) and has compact support in Rn , i.e. h 2 D.Rn /. Then, for S 2 D 0 .Rn / with arbitrary support, hS ; h. /i is well defined 8h. / 2 D.Rn /. Thus, hS ; h. /i D hS ; hT ; . C /ii:

(6.3.16)

Again, for S 2 D 0 .Rn / with arbitrary support, g./ D hS ; . C /i is a well defined function of  which is infinitely differentiable on Rn and may have arbitrary support in Rn (see Remark 6.2.1), i.e. g 2 E.Rn /. But T 2 E 0 .Rn /, i.e. T 2 D 0 .Rn / is a distribution with compact support A. Hence, hT ; g./i is well defined. Thus, hT ; g./i D hT ; hS ; . C /ii

(6.3.17)

is well defined. Then (6.3.16) and (6.3.17) must give the same result such that (6.3.15) holds, i.e. hT S; i D hT ; hS ; . C /ii D hS ; hT ; . C /ii 8 2 D.Rn /. Support of the convolution of two distributions Theorem 6.3.3. Let T and S be two distributions with A D supp.T / and B D supp.S / such that either T or S has compact support, i.e. either A or B is compact in Rn . Then supp.T S/  A C B:

(6.3.18)

In particular, if A and B are compact, then T S has compact support, i.e. T 2 E 0 .Rn /; S 2 E 0 .Rn /

H)

T S 2 E 0 .Rn /:

(6.3.19)

For n D 1, supp.T /  a; 1Œ;

supp.S /  b; 1Œ;

supp.T S /  a C b; 1Œ:

(6.3.20)

Proof of Theorem 6.3.3. Since either A or B is compact and A and B are closed sets, A C B is closed (see (6.2.4)). Let  D .A C B/{ D the complement of the closed set A C B. Then  is open and we are to show that hT S; i D 0 8 2 D.Rn / with supp./  . For such a , the support of . C / in Rn  Rn is contained in the open set defined by  C 2  (i.e.  C … .A C B/). But supp.T ˝

322

Chapter 6 Convolution of distributions

S / D A  B D ¹.; / W  2 A, 2 Bº  Rn  Rn . Hence, .; / 2 A  B H)  C 2 A C B H) supp.T ˝ S / D A  B  A C B. Consequently, supp.. C // \ supp.T ˝ S / D ; H) hT ˝ S ; . C /i D 0 8 2 D.Rn / with supp./   H) hT S; i D hT ˝ S ; . C /i D 0 8 2 D.Rn / with supp./   H) supp.T S/  C D A C B. In particular, if A and B are compact, then A C B is compact (6.2.3). Hence, supp.T S / is a closed subset of the compact set A C B, i.e. supp.T S / is compact. Example 6.3.1. 1. Let ı D ı.0/ be the Dirac distribution with mass/force/charge concentrated at 0 2 Rn such that hı; i D .0/ 8 2 D.Rn /. Hence, ı is a distribution with compact support ¹0º, and ı T exists 8T 2 D 0 .Rn / by Theorem 6.3.2 and is given by ı T D T 8T 2 D 0 .Rn /, since hı T; i D hı ˝ T ; . C /i D hT ; hı ; . C /ii D hT ; .0 C /i D hT ; . /i D hT; i 8 2 D.Rn / H) ı T D T , i.e. in convolution, ı is the unity. (6.3.21) Remark 6.3.4. In physics and mechanics, for T D Tf , ı Tf D ı f is usually written incorrectly in the form: Z Z ı.x  /f ./d  D f .x  /ı./d  D f .x/; (6.3.21a) Rn

Rn

which is, in fact, the definition of ı function given by Dirac, although the integral sign has no meaning. 2. ı 0 T D T 0 8 T 2 D 0 .Rn /. (6.3.22) 0 0 0 In fact, hı T; i D hı ˝ T ; . C /i D hT ; hı ; . C /ii D hT ;  0 . /i D hT;  0 i D hT 0 ; i 8 2 D.Rn / H) ı 0 T D T 0 in D 0 .Rn /. @ı @ı @T H) @x T D @x 8T 2 D 0 .Rn /. In particular, ı 0 D @x k

k

In physics and mechanics, for T D Tf , @f @xk

k

@ .ı @xk

Tf / D

@ı @xk

Tf D

is usually written in the form: Z @ı @f .x  /f ./d  D ; n @x @x R k k

@ı @xk

f D

(6.3.23)

although the integral sign has no meaning here. Equation (6.3.23) can be formally obtained from the (incorrect) formula (6.3.21a) by ‘differentiating under the integral sign’! 3. 8 multi-index ˛ D .˛1 ; ˛2 : : : ; ˛n / with @˛ D

@j˛j ˛ ˛n , @x11 :::@xn

@˛ ı T D @˛ T

8T 2 D 0 .Rn /, since h@˛ ı T; i D h@˛ ı ˝T ; .C /i D hT ; h@˛ ı ; .C /ii D hT ; .1/j˛j @˛ . /i D .1/j˛j hT; @˛ i D h@˛ T; i 8 2 D.Rn / H)

@˛ ı T D @ ˛ T

in D 0 .Rn /:

(6.3.24)

323

Section 6.3 Convolution of two distributions

Translation operator a For a 2 Rn , let a be the translation operator defined, for any function f on Rn , by: .a f /.x/ D f .x  a/:

(6.3.25)

. fL/. / D fL.  / D f ..  // D f .  /:

(6.3.26)

Then

Let f 2 L1 .Rn /. Then fL and a f belong to L1 .Rn / such that TL D TfL and a T D Ta f are distributions defined by fL and a f respectively: Z Z .a f /.x/.x/d x D f .x  a/.x/d x: ha T; i D hTa f ; i D Rn

Rn

Changing variables defined by  D x  a with the absolute value of the Jacobian of the transformation equal to 1, we get Z Z f .x  a/.x/d x D f ./. C a/d  Rn

Rn

D hTf ; . C a/i D hT ; . C a/i D hTx ; .x C a/i: Hence, ha T; i D hTx ; .x C a/i

8 2 D.Rn /;

(6.3.27)

which suggests we define the translated or shifted distribution a T for arbitrary T 2 D 0 .Rn / by: 8T 2 D 0 .Rn /, ha T; i D hTx ; .x C a/i

8 2 D.Rn /:

(6.3.28)

Since .a f /.x/ D f .x  a/, a Tx is denoted by Txa , or T .x  a/ (the latter is more frequently used in physics and mechanics). For example, for Tx D ı.x/ (the Dirac distribution associated with variable x with charge/mass/force concentrated at x D 0), a ıx D ıxa or ı.x  a/ is the translated Dirac distribution associated with x with charge/force/mass concentrated at a. Remark 6.3.5. In T .x  a/ (resp. ı.x  a/), no point values of T (resp. ı) at x  a are to be understood, since T .x  a/ (resp. ı.x  a/) is a distribution, not a function. Hence, we have written Txa , but .x C a/ is the point value of the function  at the point x C a. The mapping  2 D.Rn / 7! ha T; i is continuous. Hence, T 2 D 0 .Rn / H) a T 2 D 0 .Rn / is a distribution.

324

Chapter 6 Convolution of distributions

Example 6.3.2. Let ıa be the Dirac distribution with mass/charge/force concentrated at a 2 Rn . Then ıa T D a T

8T 2 D 0 .Rn /;

(6.3.29)

where a T is defined by (6.3.28), since hıa T; i D hıa; ˝ T ; . C /i D hT ; hıa; ; . C /ii D hT ; .aC /i D ha T; i 8 2 D.Rn / H) ıa T D a T . Remark 6.3.6. Again, in physics and mechanics, for T D Tf ; ıa Tf D ıa f D a f is usually written in the form: Z Z ıa ./f .x  /d  D ı.  a/f .x  /d  Rn Rn Z D ı..x  a/  /f ./d  D f .x  a/ D a f .x/; Rn

(6.3.30) which, in fact, follows from the definition of ı.x  a/ given by Dirac. 4. ıa ıb D ıaCb , since T D ıb H) ıa ıb D a ıb H)

ha ıb ; i D hıb ; .x C a/i D .b C a/ D hıaCb ; i

H)

a ıb D ıa ıb D ıaCb ;

8 2 D.Rn / (6.3.31)

where ıaCb is the Dirac distribution with mass/force/charge concentrated at the point a C b 2 Rn . Case II. For  2 A D supp.T /, 2 B D supp.S /,  C remains bounded only if both  and remain bounded Case I is a particular situation of Case II: A is compact H)  is bounded. If  C is bounded, then D . C /   is also bounded H) Case II holds. For f; g 2 L1loc .Rn /, Tf Sg D f g is not defined in general, since supp.. C n n n n // ’ is not bounded in R  R and the double integral on R  R in (6.3.8), Rn Rn f ./g. /. C /d d has no meaning. But we have: Theorem 6.3.4. Let Tf and Sg be the distributions defined by functions f; g 2 L1loc .Rn /. Let A =supp.Tf / D supp.f / and B D supp.Sg / D supp.g/ satisfy the following condition: supp.f ./ ˝ g. // \ supp.. C // D ¹.; / W  2 A; 2 B; C  2 K; K D supp..x//º

325

Section 6.3 Convolution of two distributions

is bounded in Rn  Rn . Then the convolution Tf Sg is well defined by (6.3.9) and a distribution Wf g defined by f g D h 2 L1loc .Rn /: 8 2 D.Rn /, Z h.x/.x/d x; hTf Sg ; i D hWf g ; i D hf g; i D Rn

where Z

Z

h.x/ D .f g/.x/ D

f .x  t/g.t/d t D

f .t/g.x  t/d t

Rn

a.e. in Rn :

Rn

(6.3.32) Proof. From definition (6.3.9), 8 2 D.Rn /, “ hTf Tg ; i D hf ./ ˝ g. /; . C /i D

f ./g. /. C /d d ; Rn Rn

(6.3.33) since the right-hand side double integral on Rn Rn exists by virtue of the assumption on A and B. In fact, f and g 2 L1loc .Rn / H) f ./g. /. C / is locally summable on Rn Rn , but supp.f ./g. /. C // D ¹.; / W  2 A, 2 B, C 2 K, K D supp..x//º is a bounded subset of Rn  Rn . Hence, f ./g. /. C / is integrable on Rn Rn , i.e. f ./g. /.C / 2 L1 .Rn /. Then, by Fubini’s Theorem 7.1.2C, we can interchange the order of integration. In order to show (6.3.32), we are to change the variables: x D  C , t D , the Jacobian of this transformation being C1, i.e. from (6.3.33), “ Z Z hTf Tg ; i D f .  t/g.t/.x/d xd t D dx .x/f .x  t/g.t/d tI Rn

Rn Rn

Rn

R since f .x  t/g.t/.x/ 2 L1 .Rn  Rn /, by Fubini’s Theorem 7.1.2C, RRn f .  t/g.t/.x/d t exists for almost all values of x 2 Rn . Hence, the R integral Rn f .x  t/g.t/d t has a meaning for almost all values of x, and h.x/ D Rn f .x  t/g.t/d t D .f g/.x/ is a function wellR defined for almost all values of x 2 Rn . Again, by Fubini’s Theorem 7.1.2C, Rn .x/h.x/d x exists and .x/h.x/ 2 L1 .Rn / H) h.x/.x/ 2 L1loc .Rn / with  2 D.Rn / H) h 2 L1loc .Rn /. Hence, Z Z h.x/.x/d x D .f g/.x/.x/d x hTf Sg ; i D Rn

Rn

D hWf g ; i

8 2 D.Rn /

H) Tf Sg D Wf g with f g D h.RSubstituting  D t,  C D x in (6.3.33) and proceeding similarly, we get h.x/ D Rn f .t/g.x  t/d t.

326

Chapter 6 Convolution of distributions

Remark 6.3.7. For functions f and g, their convolution f g may exist, even if their supports do not satisfy the condition in Theorem 6.3.4. For example, 



f; g 2 L1 .Rn /, f g always exists and f g 2 L1 .Rn / with kf gkL1 .Rn /  kf kL1 .Rn / kgkL1 .Rn / (see (6.2.5)). For f 2 L1 .Rn / and g bounded in Rn ; f g exists and is a continuous function bounded on Rn with the properties: h.x/ D .f g/.x/ is defined 8x 2 Rn ; Z  jh.x/j  jf .t/jd t sup jg.t/j Rn

H)

t2Rn

khkL1 .Rn /  kf kL1 .Rn / kgkL1 .Rn / :

(6.3.34)

In particular, for f; g 2 C.R/ with supp.f / and supp.g/ contained in 0; 1Œ, i.e. f .x/ D 0, g.x/ D 0 for x  0, Z 1 Z 1 h.x/ D .f g/.x/ D f .x  t /g.t /dt D f .x  t /g.t /dt Z

1 x

0 1

Z

f .x  t /g.t /dt C

D 0

f .x  t /g.t /dt: x

– For x  0, x  t  0 8t  0 H) f .x  t / D 0 H) H) h.x/ D 0 for x  0.

Rx 0

and

– For x  0, xR  t  0 8t  x H) f .x  t / D 0 H) x H) h.x/ D 0 f .x  t /g.t /dt 8x  0.

R1 x

R1 x

vanish

vanishes

Thus, ´

0 h.x/ D f g.x/ D R x 0

f .x  t /g.t /dt

for x  0 for x  0:

(6.3.35)

Example 6.3.3. For any complex number a 2 C, let f and g be the functions defined by: f .x/ D H.x/e ax

x ˛1 ; .˛/

g.x/ D H.x/e ax

x ˇ 1 ; .ˇ/

where ˛ > 0, ˇ > 0 and H.x/ denotes the Heaviside function. Then H.x/e ax

x ˛1 x ˇ 1 x ˛Cˇ 1 H.x/e ax D H.x/e ax : .˛/ .ˇ/ .˛ C ˇ/

Solution. f and g are continuous for x  0, since f .x/ D 0, g.x/ D 0 8x  0, and f and g are also continuous for x > 0. Hence, f and g are continuous on R.

327

Section 6.4 Regularization of distributions by convolution

Then f g is given by (6.3.35): 8x  0, f g.x/ D 0, 8x  0, Z x .x  t /˛1 t ˇ 1 a.xt/ at  e  e dt f g.x/ D .˛/ .ˇ/ 0 Z e ax  x ˛1  x ˇ 1  x 1 D .1  /˛1  ˇ 1 d  .˛/.ˇ/ 0 (by change of variables: t D x, dt D xd ) e ax  x ˛Cˇ 1 e ax x ˛Cˇ 1 .˛/.ˇ/  D .˛/.ˇ/ .˛ C ˇ/ .˛ C ˇ/ R1 / ). (since the Beta function B.˛; ˇ/ D 0 .1  /˛1  ˇ 1 d  D .˛/.ˇ .˛Cˇ / Hence, ´ ˛Cˇ 1 0 for x  0 ax x f g.x/ D H.x/e D ax x ˛Cˇ1 .˛ C ˇ/ e .˛Cˇ / for x  0: D

6.4

Regularization of distributions by convolution

Let T 2 D 0 .Rn / be a distribution and 2 C 1 .Rn / be a function infinitely differentiable in the usual pointwise sense such that either T has compact support, i.e. T 2 E 0 .Rn /, or has compact support, i.e. 2 D.Rn /. Then, for fixed x; .x  t/ is a function of t only and, as a function of t, .x  t/ 2 C 1 .Rn / is infinitely differentiable. For T 2 E 0 .Rn /  D 0 .Rn / with compact support, and for

.x  t/ 2 E.Rn / D C 1 .Rn / 8 fixed x 2 Rn ; h.x/ D hTt ; .x  t/i is a well-defined function of x. Similarly, for T 2 D 0 .Rn / and for .x  t/ 2 D.Rn / for fixed x 2 Rn , h.x/ D hT t ; .x  t/i is a well-defined function of x. In both the situations, h.x/ D hT t ; .x  t/i has the following properties: 

h is infinitely differentiable in the usual pointwise sense;



8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n / with @˛ x D

@j˛j ˛ ˛ ˛n , @x11 @x22 :::@xn

˛ ˛ @˛ x h.x/ D @x hTt ; .x  t/i D hTt ; @x .x  t/i (see Remark 6.2.1); 

h defines a distribution on Rn which coincides with T , T being a distribution since either T or has compact support, and defines a regular distribution T in D 0 .Rn /.

In fact, 8 2 D.Rn /, hT ; i D hT ˝ ; . C /i D< T ; h . /; . C /i > Z Z D hT ;

. /. C /d i D hT ;

.x  /.x/d xi Rn

Rn

328

Chapter 6 Convolution of distributions

(putting x D  C , jJ j D 1 and d x D d ) D hT ; h.x/; .x  /ii D hT ˝ .x/; .x  /i D h.x/; hT ; .x  /ii D h.x/; h.x/i D hh; i H) T D h D hT ; .x  /i 2 C 1 .Rn / (but the support of h is not compact in general and hence h … D.Rn / in general). We state this result in the following form: Theorem 6.4.1. The convolution T of a distribution T and an infinitely differentiable function , at least one of which has compact support (i.e. either T 2 E 0 .Rn /,

2 E.Rn / or T 2 D 0 .Rn /, 2 D.Rn /) is defined by: T .x/ D hT ; .x  /i 2 C 1 .Rn /;

(6.4.1)

and the infinitely differentiable (in the usual pointwise sense) function T is called regularization or mollification of the distribution T by with ˛ @˛ x .T /.x/ D T @x .x/;

where @˛ x D or mollifier.

@j˛j  ˛ ˛n @x11 :::::@xn

(6.4.2)

is in the usual pointwise sense. Then, is called the regulator

Example 6.4.1. 1. For .x/ D 1 8x 2 Rn , T 1 D hT; 1i D constant.  

(6.4.3)

T D ı, ı 1 D hı; 1i D 1 D constant. R T D Tf , Tf 1 D hTf ; 1i D Rn f .x/d x D constant.

2. For polynomial p 2 Pm of degree  m and T 2

(6.4.4) (6.4.5)

E 0 .Rn /,

T p D hT t ; p.x  t/i

(6.4.6)

is a polynomial of degree  m. For the sake of simplicity, we consider the case of polynomial p 2 Pm in a single variable. Then, .T p/.x/ D hT t ; p.x  t /i D hT t ; p..t / C x/i  X  m m m X X x k .k/ xk ˛k k p .t / D hT t ; p .k/ .t /i D x 2 Pm ; D Tt ; kŠ kŠ kŠ kD0

kD0

kD0

(6.4.7) where ˛k D hT t ; p .k/ .t /i 2 R 8k D 0; 1; 2 : : : m, p .k/ .t / D 8k D 0; 1; 2 : : : m.

dkp .t / dt k

Section 6.5 Approximation of distributions by C 1 -functions

329

Continuity of the convolution operation We agree to accept the following result without proof (see Schwartz [8, p. 170]). Theorem 6.4.2. Let .Tk / and .Sk / be any two sequences of distributions in D 0 .Rn / such that Tk ! T and Sk ! S in D 0 .Rn / as k ! 1. Let A; B  Rn be fixed closed sets in Rn with supp.Tk /  A, supp.Sk /  B 8k 2 N such that for  2 A, 2 B,  C remains bounded only if both  and remain bounded, i.e. Tk Sk is defined 8k 2 N and T S is also defined. Then, Tk Sk ! T S in D 0 .Rn / as k ! 1.

Approximation of distributions by C 1 -functions

6.5

Proposition 6.5.1. Every distribution T 2 D 0 .Rn / is the limit of C 1 -functions (i.e. functions belonging to C 1 .Rn /) in D 0 .Rn /. In other words, E.Rn /  C 1 .Rn / is dense in D 0 .Rn /:

(6.5.1)

Proof. 8" > 0, let " be defined by (6.2.15). Then " 2R C01 .Rn / with supp. " / D B.0I "/ D ¹x W kxk  "º  Rn . lim"!0C Rn " .x/.x/d x D .0/ 8 2 D.Rn /. In fact, Z Z

" .x/.x/d x  .0/ D Π" .x/.x/  .0/ " .x/d x Rn Rn Z D

" .x/..x/  .0//d x Rn Z D

" .x/..x/  .0//d x H)

ˇZ ˇ ˇ ˇ

B.0I"/

ˇ Z ˇ

" .x/.x/d x  .0/ˇˇ  max j.x/  .0/j n kxk"

" .x/d x

B.0;"/

R

D max j.x/  .0/j; kxk"

R

since B.0;"/ " .x/d x D 1. Since  2 D.Rn / is continuous in Rn , 8 > 0, 9"0 D "0 ./ > 0 such that j.x/  .0/j <  8kxk  " with " < "0 . Hence, 8 > 0, 9"0 D "0 ./ > 0 such that ˇ ˇZ ˇ ˇ ˇ   8" < "0 ; ˇ

.x/.x/d x  .0/ " ˇ ˇ Rn

i.e.

Z lim

"!0C

Rn

" .x/.x/d x D .0/:

(6.5.2)

330

Chapter 6 Convolution of distributions

" ! ı in D 0 .Rn / as " ! 0C : 8 2 D.Rn /, Z

" .x/.x/d x D .0/ D hı; i lim h " ; i D lim "!0C

H)

"!0C

Rn

lim " D ı 2 D 0 .Rn /:

(6.5.3)

"!0C

T " ! T as " ! 0C : Since " 2 D.Rn / 8" > 0, T " is a C 1 -function 8 distributions T 2 D.Rn / by Theorem 6.4.1. But " ! ı in D 0 .Rn / as " ! 0C H) T " ! T ı D T in D 0 .Rn / as " ! 0C by Theorem 6.4.2, i.e. lim"!0C hT " ; i D hT ı; i D hT; i 8 2 D.Rn /. Thus, 8T 2 D 0 .Rn /, 9 a sequence of C 1 -functions T " 2 C 1 .Rn / such that lim .T " / D T 2 D 0 .Rn /:

"!0C

(6.5.4)

Density of D.Rn / in D 0 .Rn / Theorem 6.5.1. D.Rn / is dense in D 0 .Rn /. Proof. D.Rn / is dense in E.Rn / which is dense in D 0 .Rn / by (6.5.1), and D.Rn / ,! E.Rn / ,! D 0 .Rn /, the imbeddings being continuous ones. Hence, D.Rn / is dense in D 0 .Rn /. For an alternative proof, see Theorem 4.1.2. We agree to accept the following result without proof. Proposition 6.5.2. Let ı 2 D 0 .Rn / be the Dirac distribution (measure) with concentration at 0 2 Rn . Then 9 a sequence .pk / of polynomials pk such that pk ! ı D ı0 in D 0 .Rn / as k ! 1, i.e. hpk ; i ! hı; i D .0/

as k ! 1 8 2 D.Rn /:

(6.5.5)

Weierstrass’s Approximation Theorem Let T 2 E 0 .Rn /  D 0 .Rn / be a distribution with compact support. Let .pk / be a sequence of polynomials such that pk ! ı in D 0 .Rn /, i.e. limk!1 hpk ; i D hı; i D .0/. Then, analogously to Weierstrass’s theorem on the approximation of continuous functions on compact sets by polynomials, we have: Theorem 6.5.2. 8T 2 E 0 .Rn / with compact support in Rn , 9 a sequence .pk / of polynomials pk with pk ! ı (the Dirac distribution) in D 0 .Rn / as k ! 1 such that lim .T pk / D T

k!1

in D 0 .Rn /:

(6.5.6)

331

Section 6.6 Convolution of several distributions

Proof. From Proposition 6.5.2, 9 a sequence .pk / of polynomials such that pk ! ı in D 0 .Rn / as k ! 1. T pk is well defined 8T 2 E 0 .Rn /, and 8 polynomials pk with k 2 N (see (6.4.6)–(6.4.7)). Moreover, T ı D T is defined 8T . From Theorem 6.4.2, T pk ! T ı D T as k ! 1.

6.6

Convolution of several distributions

Let T , S , R in D 0 .Rn / be distributions such that, 8 2 D.Rn /, supp.T ˝ S ˝ R / \ supp . C C / is a bounded set in Rn  Rn  Rn D R3n . Then the convolution T S R of T , S and R is well defined and a distribution defined, 8 2 D.Rn /, by: hT S R; i D hT ˝ S ˝ R ; . C C /i:

(6.6.1)

Remark 6.6.1.  Even if T S R is not defined, .T S/ R and T .S R/ may be defined and may not be equal. For example, for the Heaviside function H with its derivative dH D ı (in the sense of distribution), 1 ı 0 H does not exist, but .1 ı 0 / H dx dı /. In fact, and 1 .ı 0 H / both exist, .ı 0 D dx 0 0 .1 ı / H D .ı 1/ H D 0 H D 0, 1 .ı 0 H / D 1 H 0 D 1 ı D ı 1 D 1, i.e. .1 ı 0 / H and 1 .ı 0 H / exist, but are not equal.  T S R defined by (6.6.1) exists H) it is associative, i.e. T S R D T .S R/ D .T S / R:

(6.6.2)

The following result is of importance in applications: Theorem 6.6.1. If at least two of the three distributions T , S , R have compact support in Rn , then the convolution T S R exists and is associative and commutative. Moreover, for n D 1, if T , S and R have supports bounded from the left, i.e. contained in a; 1Œ (resp. bounded from the right, i.e. contained in 1; bŒ), then T S R exists and is associative and commutative. Example 6.6.1. Let H.x/ be the Heaviside function with H.x/ D 1 for x  0 and H.x/ D 0 for x < 0, and let .x/ be the Gamma function. Then, for ˛; ˇ > 0 and  2 C, prove that   ˛1   ˇ 1  ˛Cˇ 1  1. x x x x x x H.x/e  H.x/e  D H.x/e : (6.6.3) .˛/ .ˇ/ .˛ C ˇ/ In particular, for  D 0, ˛; ˇ > 0, H˛ .x/ D

H.x/x ˛1 , .˛/

.H˛ Hˇ /.x/ D H˛Cˇ .x/:

Hˇ .x/ D

H.x/x ˇ1 , .ˇ /

332

Chapter 6 Convolution of distributions

x n1 H.x/e x H.x/e x    H.x/e x D H.x/e x : „ ƒ‚ … .n  1/Š

2.

n distributions

Solution.

 ˛1   ˇ 1  x x x x H.x/e  H.x/e  .˛/ .ˇ/ Z 1 ˛1 .x  / H./e   ˇ 1 D d H.x  /e .x / .˛/ .ˇ/ 1 8 for x < 0 ˆ x  0; for x    0; i.e. for x    0: R . / d x p fp d  (see Vladimirov [6, pp. 88– Hence, D 1=2 f D D.H1=2 f / D dx 0  x 89] for more detail).

6.7

Derivatives of convolution, convolution of distributions on a circle  and their Fourier series representations on 

Since the Dirac distribution ı and its derivatives have compact support in Rn , by Theorem 6.6.1 @˛ ı T S will be well defined and given by (6.6.1) 8 multi-index j˛j ˛, @˛ D ˛1@ ˛n , if either T or S has compact support, and it will be associative and @x1 :::@xn

commutative (6.6.2). Proposition 6.7.1. Let T 2 E 0 .Rn /  D 0 .Rn /, S 2 D 0 .Rn /. Then T S 2 D 0 .Rn /, and the distributional derivatives of T S are given by: @ @T @S .T S/ D S DT I @xk @xk @xk @2 @2 T @2 S .T S/ D S DT @xi @xj @xi @xj @xi @xj D

@T @S @T @S D : @xi @xj @xj @xi

(6.7.1) (6.7.2) (6.7.3)

In general, 8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n /, @˛ .T S/ D @˛ T S D T @˛ S:

(6.7.4)

334

Chapter 6 Convolution of distributions

@ı @S Proof. From (6.3.21)–(6.3.24), we know that ı S D S , @x S D @x , @˛ ı S D k k @˛ S 8 distributions S 2 D 0 .Rn /. Now, applying the associative and commutative properties, we have:   @ @ı @ı @T .T S/ D .T S/ D T S D S @xk @xk @xk @xk   @ı @S @S @ı .S T / D S T D T DT : D @xk @xk @xk @xk

Applying this formula, we get     @ @2 @ @ @T .T S/ D .T S/ D S @xi @xj @xi @xj @xi @xj   @S @2 T @T @ @T D SI S D D @xi @xj @xj @xi @xi @xj     @ @ @ @S @2 .T S/ D .T S/ D T @xi @xj @xi @xj @xi @xj   @ @S @S @2 S @T DT DT : D @xi @xj @xi @xj @xi @xj Similarly, (6.7.4) can be proved. Remark 6.7.1. 

For arbitrary distributions T and S, T S is not defined. Even if T S is not defined, @˛ T S and T @˛ S may be defined and may not be equal. For example, for the Heaviside function H with dH D ı; dH 1 D ı 1 D 1, but dx dx d1 d1 D H 0 D 0, i.e. dH 1 ¤ H , since H 1 is not defined, and H dx dx dx consequently (6.7.1) does not hold. Thus, from the existence of @˛ f g and f @˛ g, we cannot conclude that their equality in formula (6.7.4) holds.



The formulae (6.7.1)–(6.7.4) do not hold for functions in general, if the derivatives of f are in the usual pointwise sense. For example, for f 2 C0 .R/ with compact support and Heaviside function H with dH D ı in the distributional dx sense, d dH .H f / D f D ı f D f: dx dx .x/ D But if we consider the derivative of H in the usual pointwise sense, Œ dH dx 0 8x ¤ 0, then     Z 1 dH dH f .t / f D .x  t /dt D 0; dx dx 1

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

335

d i.e. dx .H f / ¤ Œ dH  f . Hence, in the formula for functions, the derivatives dx are in the sense of distributions.

Example 6.7.1. Let P ./ D 

m

C a1 

m1

C    C am1  C am D

m Y

.  i /

(6.7.5)

iD1

be a polynomial of degree m with real roots 1 ; 2 ; : : : m such that the i s may or may not be distinct numbers, and the corresponding ordinary differential operator P .D/ be defined by     dm d d d m1 P .D/      C a C    C a D    (6.7.6) 1 m 1 m dx m dx m1 dx dx d with D D dx . Then, show that       dı dı dı P .D/ı D  1 ı  2 ı     m ı ; dx dx dx

(6.7.7)

ı being the Dirac distribution. Solution. Using (6.3.21),      d d d P .D/ı D P .D/ı ı D  1  2     m ı ı dx dx dx      d d d  1  2     m ı ı D dx dx dx       d d d  2     m ı  1 ı (by (6.7.4)) D dx dx dx        d d d dı  2  3     m ı  1 ı ı (by (6.3.21)) D dx dx dx dx         d d d dı D  3     m ı  1 ı  2 ı D    dx dx dx dx         dı dı d dı  m ı  1 ı  2 ı     m1 ı D dx dx dx dx       dı dı dı D  1 ı  2 ı     m ı : dx dx dx d  /m and In particular, for 1 D 2 D    D m D , P .D/ D . dx      m dı d dı P .D/ı D  ı     ı D  ı : dx dx dx

(6.7.8)

336

Chapter 6 Convolution of distributions

Convolution of distributions of D 0 . / defined on a circle  Consider the test space D./ (see Definition 1.10.2) of infinitely differentiable functions on a circle  D .0I r/, and the dual D 0 ./ of distributions on the circle  (see Definition 1.10.3). Since arcs of different lengths on  can be added, we can define the convolution of distributions on . For this we begin with functions on . Let f; g 2 L1loc ./. Then their convolution f g on  is defined by: 8 arc length s, Z Z f .s  /g./d  D f ./g.s  /d : (6.7.9) h.s/ D f g.s/ D 



D 0 ./,

their convolution S1 S2 2 Similarly, for any pair of distributions S1 ; S2 2 D 0 ./ is well defined, since S1 ; S2 2 D 0 ./ have bounded support, and is given, 8 2 D./, by: hS1 S2 ; iD 0 ./D./ D hS1 ./ ˝ S2 ./; . C /i D hS1 ./; hS2 ./; . C /ii: (6.7.10) S1 S2 D S2 S1 in D 0 ./ 8S1 ; S2 2 D 0 ./. Example 6.7.2. For ı.s/ 2 D 0 ./ and S 2 D 0 ./, ı.s/ S D S in D 0 ./, since hı S; i D hS ı; i D hS./; hı./; . C /ii D hS./; . C 0/i D hS; i ı S D S 2 D 0 ./

”

8 2 D./

(see also (6.3.21)).

(6.7.11) (6.7.12)

For ı 0 .s/ 2 D 0 ./ and S 2 D 0 ./, ı 0 S D ı S 0 D S 0 in D 0 ./, where D ddsı (see also (6.3.22)). In fact,    @ hı 0 S; i D hS./; hı 0 ./; . C /ii D S./; .1/ ı./; . C / @

ı 0 .s/

D hS./;  0 . C 0/i D hS 0 ; i ”

0

0

ı S DS Dı S

0

0

in D ./:

8 2 D./ (6.7.13)

S1 ; S2 2 D 0 ./ dS1 dS2 d.S1 S2 / D S2 D S1 in D 0 ./: (6.7.14) ds ds ds Indeed, from (6.7.12)–(6.7.13), ı Si D Si , ı 0 Si D Si0 in D 0 ./. Then, applying the commutative and associative properties, we have   d.S1 S2 / dı dı dS1 D .S1 S2 / D S1 S2 D S2 ds ds ds ds H)

D S10 S2 D S1 S20 : The last equality in (6.7.14) is similarly established.

337

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

Remark 6.7.2. In (6.7.9), f g has been defined for f; g 2 L1loc ./. Let fQ; gQ be the associated periodic functions on R with R 1 period T > 0 defined in (1.10.55). Then, in general, fQ gQ is not defined, since 1 f .x  /g./d  does not exist for arbitrary Q are not bounded periodic fQ; gQ 2 L1loc .R/ owing to the fact that supp.fQ/ and supp.g/ Q on R. Hence, for periodic f , gQ on R with period T , we define Z aCT Z aCT Q f .x  /g./d Q D fQ./g.x Q  /d : (6.7.15) h.x/ D a

a

Fourier series of distributions in D 0 . / Let f 2 L1loc ./ be a locally summable function on . Thus, from Property 4 of Fourier coefficients in (2.11.7e), we can define ck .f / by: for T D 2 r, Z 1 1 ck .f / D f .s/e ik!s ds D hf; e ik!s iD 0 ./D./ 8k 2 Z; (6.7.16) T  T which suggests we define Fourier coefficients of distributions S 2 D 0 ./ by: ck .S / D

1 hS; e ik!s iD 0 ./D./ T

8k 2 Z:

(6.7.17)

Fourier series of distributions on  P ik!s with The trigonometric series 1 kD1 ck .S /e ck .S / D

1 hS; e ik!s iD 0 ./D./ T

(6.7.18)

is called the Fourier series of S 2 D 0 ./. P ik!s converges to It is interesting to note that if P a trigonometric series 1 kD1 dk e 1 0 ik!s a distribution S 2 D ./, then kD1 dk e is the Fourier series of S 2 D 0 ./ with dk DP ck .S / 8k 2 Z. (6.7.19) ik!s converges to S 2 D 0 ./ d e In fact, 1 kD1 k  X  n ik!s ” dk e ; ! hS; iD 0 ./D./ D 0 ./D./

kDn

in C 8 2 D./ as n ! 1. Since Z Z ik!s il!s ik!s il!s ;e iD e e ds D he 

T 0

e ik!x e il!x dx D T ıkl

(by (2.11.7d)), for  D e il!s 2 D./,  X  n n X ik!s il!s dk e ;e dk T ıkl D T dl ; D kDn

kDn

8k; l 2 Z

n  l  n 8n 2 N

338

Chapter 6 Convolution of distributions

H) T dl P D hS; e il!s i D T cl .S / H) dl D cl .S / 8l 2 Z H) P1 8l 2 Z, 1 ik!s D ik!s with d D c .S / 8k 2 Z. k k kD1 dk e kD1 ck .S /e Example 6.7.3. Let ı 2 D 0 ./ be the Dirac distribution with unit mass/charge/force concentrated at s D 0 on . Then the Fourier coefficients ck .ı/ of ı 2 D 0 ./ are given by ck .ı/ D T1 hı; e ik!s i D T1  e ik!s jsD0 D T1 8k 2 Z, and the Fourier series of ı 2 D 0 ./ is given by: 1 X

ck .ı/e ik!s D

kD1

(6.7.20)

kD1

Theorem 6.7.1. The Fourier series verges to ı 2 D 0 ./: ıD

1 X 1 ik!s e : T

P1

1 ik!s kD1 T e

of ı 2 D 0 ./ in (6.7.20) con-

1 X 1 ik!s e 2 D 0 ./: T

(6.7.21)

kD1

Proof. For the convergence of Fourier series (6.7.20), we can apply Theorem 2.11.2 on the convergence of trigonometric series. Since ck .ı/ D T1  jkjp for some p 2 N 8jkj 2 N, Fourier series converges in D 0 ./. Let the distribution S 2 D 0 ./ P1 (6.7.20) 1 ik!s be its sum, i.e. S D kD1 T e in D 0 ./. Multiplying term by term by e i!s , we P1 P 1 i.kC1/!s 1 ik 0 !s i!s D 1 D S H) .e i!s  1/S D 0 get e S D kD1 T e k 0 D1 T e in D 0 ./ H) ˛.s/S D 0 with ˛.s/ D .e i!s  1/. ˛ 2 C 1 ./ vanishes only at s D 0, i.e. ˛.0/ D 0, but ˛ 0 .0/ D i !e ik!0 D i ! ¤ 0. Hence, ˛.s/ has a simple zero only at s D 0 and, by Proposition 1.6.1, ˛.s/S D 0 in D 0 ./ H) S D C ı, P where ı is a Dirac distribution concentrated at s D 0 and C 1 ik!s is a constant. Since S D 1 in D 0 ./ with T1 D the Fourier coefficient kD1 T e of S D ck .S / D ck .C ı/ D T1 hC ı; e ik!s i D T1  C  hı; e k!s i D T1  C  1. Hence, 1 1 0 T D T  C ” C D 1. Thus, S D ı in D ./. P 1 ik!s Alternative method. S D C ı in D 0 ./ H) C ı D 1 kD1 T e P1 P 1 H) hC ı; .s/i D h kD1 T1 e ik!s ; .s/i D kD1 T1 he ik!s ; i P1 1 ik!s H) C .0/ D ; i 8 2 D./. In particular, for  D 1 2 kD1 T he D./, .0/ D 1 and ´ Z meas./ D 2 r D T for k D 0; he ik!s ; 1i D e ik!s ds D 0 for k ¤ 0; k 2 Z:  Hence, for  D 1, .0/ D 1 and C .0/ D C D

1 T

 T C 0 D 1.

339

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

Remark 6.7.3. Since e ik!s 2 D./, the associated periodic function e ik!x on R with T belongs to DT .R/ such that e ik!s D e ik!x D e ik!.xClT / , and P1 period 1 ik!x converges to the periodic distribution ıQ 2 D 0 .R/ on R with pekD1 T e riod T (see also Examples 2.11.2 and 2.11.3). In fact, using (1.10.57),  X   X  n n 1 ik!x 1 ik!s e e ; .x/ D ; ˆ.s/ T T D 0 .R/D.R/ D 0 ./D./ kDn

kDn

! hı; ˆ.s/iD 0 ./D./ Q D ˆ.0/ D ˆ.0/ D

1 X

(as n ! 1)

(see (1.10.53))

.x C lT /jxD0

(see (1.10.59))

lD1

D

1 X

.lT / D

lD1

D

1 X

hı.x  lT /; .x/iD 0 .R/D.R/

lD1

 X 1

 ı.x  lT /; .x/ 8 2 D.R/ as n ! 1

lD1

H)

H)

lim

n!1

1 n X X 1 ik!x e D ı.x  lT / T

kDn

1 X kD1

1 ik!x e D T

in D 0 .R/

lD1

1 X

Q ı.x  lT / D ı:

(6.7.22)

lD1

Hence, 1 1 X X 1 ik!x Q e D ı.x  lT / D ı; T

kD1

(6.7.23)

lD1

where ıQ is the periodic Dirac distribution with concentrated unit mass/charge/force at the points x D lT on the real line 8l 2 Z. (For another alternative proof see Theorem 6.7.5, and see also the proof of (2.11.9) in Example 2.11.2.) Convergence of Fourier series (6.7.18) of distributions on  In Theorem 6.7.1, we have already proved the convergence of the Fourier series (6.7.20) of Dirac distribution ı. Now, with the help of convolutions of distributions of D 0 ./, the properties of which are similar to those of convolutions of distributions on R, we will prove the convergence of the Fourier series.

340

Chapter 6 Convolution of distributions

P ik!s be the Fourier series of a distribution S 2 Theorem 6.7.2. Let 1 kD1 ck .S /e D 0 ./ on . Then the series converges to S 2 D 0 ./, i.e. 1 X

SD

ck .S /e ik!s :

(6.7.24)

kD1

P 1 ik!s . By virtue of the conProof. Consider Fourier series (6.7.21), ı D 1 kD1 T e tinuity of convolution (Theorem 6.4.2), which also holds for distributions on , termby-term convolution of Fourier series (6.7.21) with any S 2 D 0 ./ can be performed, and we get, from (6.7.12): S DS ı DS

1 1 X X 1 ik!s 1 e .S e ik!s /; D T T

kD1

(6.7.25)

kD1

which converges in D 0 ./. We will show that the series in (6.7.24) and (6.7.25) are the same. Since e ik!s is a C 1 -function on  and Theorem 6.4.1 is applicable for distributions S 2 D 0 ./, i.e. the convolution product of a distribution on  with a C 1 -function on  is a C 1 -function, we have S e ik!s D hS./; e ik!.s / i D hS./; e ik! e ik!s i De

ik!s

hS./; e

ik!

iDe

ik!s

(6.7.26)

T ck .S /:

(6.7.27)

Then, from (6.7.24)–(6.7.26), we have 1 1 X X 1 T ck .S /e ik!s D ck .S /e ik!s D S 2 D 0 ./: T

kD1

kD1

Term-by-term differentiation of Fourier series (6.7.18) Theorem 6.7.3. Fourier series (6.7.18) of the distribution S 2 D 0 ./ can be differentiated term by term, and the sum of the resultant series obtained by m-times term-by-term differentiation of the series is the Fourier series of S

.m/

d mS D 8m 2 N; ds m

Proof. By Theorem 6.7.2, S D Sn D

n X kDn

i.e. S

.m/

D

1 X

ck .S /.i k!/m e ik!s :

(6.7.28)

kD1

P1

kD1 ck .S /e

ck .S /e ik!s ! S D

1 X kD1

ik!s

in D 0 ./, i.e.

ck .S /e ik!s

in D 0 ./

as n ! 1:

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

341

m

d 0 0 But 8m 2 N, ds m W D ./ ! D ./ is continuous (see Theorem 2.9.1, which also dm dm holds in D 0 ./). Hence, Sn ! S in D 0 ./ as n ! 1 H) ds m Sn ! ds m S D Pn m ik!s ! S .m/ in D 0 ./ as N H) S .m/ in D 0 ./ 8m 2 P kDn ck .S /.i k!/ e 1 .m/ m ik!s n ! 1 H) S D kD1 ck .S /.i k!/ e in D 0 ./ 8m 2 N. The righthand side series is a trigonometric series, which converges to the distribution S .m/ 2 D 0 ./ H) the trigonometric series is the Fourier series of S .m/ (see (6.7.19)), i.e. Fourier coefficients ck .S .m/ / D ck .S /.i k!/m 8k 2 Z, 8m 2 N.

Fourier series of convolutions of distributions in D 0 . / Let S1 ; S2 2 D 0 ./. Then S1 S2 2 D 0 ./ is defined by (6.7.10) and the Fourier coefficients ck .S1 S2 / of S1 S2 2 D 0 ./ are defined, 8k 2 Z, by: 1 1 hS1 S2 ; e ik!s i D hS1 ./ ˝ S2 ./; e ik!. C / i (by (6.7.10)) T T 1 1 P 2 ./; e ik! i D hS1 ./; hS2 ./; e ik!  e ik! ii D hS1 ./; e ik! ihS T T   1 ik! hS2 ./; e i D T ck .S1 /ck .S2 /: (6.7.29) D ck .S1 /  T  T

ck .S1 S2 / D

The Fourier series of S1 S2 2 D 0 ./ is defined by: 1 X

1 X

ck .S1 S2 /e ik!s D T

kD1

ck .S1 /  ck .S2 /e ik!s :

(6.7.30)

kD1

Theorem 6.7.4. Fourier P series (6.7.30) of S1 S2 2 D 0 ./ converges to S1 S2 2 ik!s . D 0 ./, i.e. S1 S2 D T 1 kD1 ck .S1 /  ck .S2 /e P ik!s Proof. Let S1 ; S2 2 D 0 ./ with S1 S2 2 D 0 ./ and Si D P1 kD1 ck .Si /e n n n n 0 0 ik!s in D ./ S1 ; S2 2 D ./ be defined by S1 D kDn ck .S1 /e , Pn(i D 1; 2). Letik!s n n n 0 S2 D kDn ck .S2 /e such that S1 S2 ! S1 S2 in D ./ as n ! 1 by virtue of the continuity of the convolution operation (see Theorem 6.4.2, which also holds in D 0 ./). In fact, S1n S2n D

n n X X

ck .S1 /cl .S2 /e ik!s e il!s

kDn lDn

D

n n X X kDn lDn

D

n n X X kDn lDn

Z ck .S1 /cl .S2 /

e ik!.s / e il! d 

(by (6.7.9))



ck .S1 /cl .S2 /  e ik!s 

Z

e ik! e il! d  

342

Chapter 6 Convolution of distributions

D

n n X X

ck .S1 /cl .S2 /  e ik!s  T ıkl

(using (2.11.7b))

kDn lDn

D

n X

1 X

T ck .S1 /ck .S2 /e ik!s ! S1 S2 D

kDn

ck .S1 S2 /e ik!s ;

kD1

(6.7.31) which is the Fourier series of S1 S2 2 D 0 ./ with Fourier coefficients ck .S1 S2 / D T ck .S1 /ck .S2 /. Example 6.7.4. With the help of Fourier series (6.7.18) of distributions S 2 D 0 ./, and using Theorem 6.7.3 on term-by-term differentiation of Fourier series, prove the following: 1. ı S D S ; 2. ı 0 S D S 0 in D 0 ./. Solution.

P1 1 ik!s 1. From (6.7.21), ı D . By kD1 T e P 1 0 ik!s . Then, from TheoTheorem 6.7.2, 8S 2 D ./, S D kD1 ck .S /e rem 6.7.4, 1 X

ı S D

ck .ı S/e

kD1 1 X

D

kD1

ik!s

1 X

D

T  ck .ı/  ck .S /e ik!s

kD1

T 

1 X 1  ck .S /e ik!s D ck .S /e ik!s D S: T kD1

P 1 ik!s with c .ı 0 / D .ik!/ . 2. By Theorem 6.7.3, ı 0 D ddsı D 1 k kD1 T .i k!/e T P 1 0 ik!s , by Theorem 6.7.3 on termFor S 2 D ./ with S D kD1 ck .S /e P1 ik!s . By TheoD by-term differentiation, S 0 D dS kD1 Œck .S /.i k!/e ds rem 6.7.4, 0

ı S D

1 X

T  ck .ı 0 /  ck .S /e ik!s

kD1

D

1 X kD1

T 

1 X i k! :ck .S /e ik!s D Œck .S /.i k!/e ik!s D S 0 : T kD1

Remark 6.7.4. We have followed Schwartz [7] to study Fourier series of distributions in D 0 ./, which are in one-to-one correspondence with periodic distributions on R with period T D circumference of . Since DT .R/ ª D.R/ and D.R/ ª DT .R/,

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

343

i.e. DT .R/ is not related to D.R/ (see (1.10.48)), we did not consider DT0 .R/, although DT0 .R/ and D 0 .R/ are related by (1.10.57) (see (6.7.32) below), DT0 .R/ being the dual of DT .R/. But an alternative treatment of Fourier series of periodic distributions on R with period T is also interesting and can be given as follows (see [30]). For periodic fQ 2 L1loc .R/ with period T , we have shown in (1.10.54)–(1.10.57): 8 2 D.R/, Z

T

0

Q fQ.x/ˆ.x/dx D

Z

fQ.x/.x/dx D hfQ; iD 0 .R/D.R/ Z D hf .s/; ˆ.s/iD 0 ./D./ D f .s/ˆ.s/ds; (6.7.32)

Q D 0 .R/D .R/ D hfQ; ˆi T T

R



P1

P Q Q 2 DT .R/ and f .s/ where ˆ.x/ D lD1 .x C lT / (finite summation ) with ˆ Q Q (resp. ˆ.s/) is a function on  associated with the periodic f .x/ (resp. ˆ.x/) on R. 1 Q Q Then, for periodic f1 ; f2 2 Lloc .R/ with period T , from (6.7.32), we have Q D 0 .R/D .R/ D hfQ2 ; ˆi Q D 0 .R/D .R/ hfQ1 ; ˆi T T T T H)

Q 2 D 0 .R/ 8ˆ T

hfQ1 ; iD 0 .R/D.R/ D hfQ2 ; iD 0 .R/D.R/

H) fQ1 D fQ2 a.e. in R. Hence, distinct periodic functions of L1loc .R/ with period T define distinct linear functionals in DT0 .R/. We need the following results to introduce the notion of convergence in DT .R/ and the continuity of linear functionals in DT0 .R/. P Q Q 2 DT .R/ such that its Fourier series is given by 1 Lemma 6.7.1. Let ˆ kD1 ck .ˆ/ R 1 T Q ik!x ik!x Q , with ck .ˆ/ D T 0 ˆ.x/e dx 8k 2 Z. Then, e Pn Q D Q ik!x ! ˆ Q uniformly on R; I. Sn .ˆ/ kDn ck .ˆ/e P1 dm m Q Q .m/ .x/ D Q ik!x uniII. 8m 2 N, dx m ŒSn .ˆ/.x/ ! ˆ kD1 .i k!/ ck .ˆ/e formly on R as n ! 1. (6.7.33) Proof. The results follow from Theorem 2.11.4, since DT .R/ D CT1 .R/. Now, using the results of Lemma 6.7.1, we can introduce the notion of convergence in DT .R/ and DT0 .R/ as follows: Q n / be a sequence in DT .R/. Then Convergence in DT .R/ Let .ˆ Qn !ˆ Q in DT .R/ ˆ ´ Qn !ˆ Q uniformly as n ! 1I I. ˆ ” .m/ Q .m/ uniformly as n ! 1 8m 2 N: Qn !ˆ II. ˆ

(6.7.34)

344

Chapter 6 Convolution of distributions

Q 2 DT .R/, Convergence in DT0 .R/ Let .SQn / be a sequence in DT0 .R/. Then, 8ˆ SQn ! SQ in DT0 .R/ ”

Q D 0 .R/D .R/ ! hS; Q ˆi Q D 0 .R/D .R/ as n ! 1: (6.7.35) hSQn ; ˆi T T T T

Convergence of SQn in DT0 .R/ H) convergence of SQn in D 0 .R/ i.e. SQn ! SQ in DT0 .R/

SQn ! SQ in D 0 .R/ as n ! 1:

H)

(6.7.36)

Q D 0 .R/D .R/ ! 0 In fact, 8 2 D.R/, hSQ  SQn ; iD 0 .R/D.R/ D hSQ  SQn ; ˆi T T 0 Q 2 DT .R/ is defined by: Q 2 D .R/, since 8 2 D.R/, ˆ for ˆ T Q ˆ.x/ D

1 X

.x C lT / (finite summation

P ):

(6.7.37)

lD1

Continuity of functionals on DT .R/ SQ 2 DT0 .R/ is continuous on DT .R/ if and Q in DT .R/ in the sense of (6.7.34), Qn !ˆ only if ˆ H)

Q ˆ Q n iD 0 .R/D .R/ ! hSQ ; ˆi Q D 0 .R/D .R/ as n ! 1: hS; T T T T

(6.7.38)

Definition 6.7.1. DT0 .R/ is the linear space of continuous (in the sense of (6.7.38)) linear functionals defined on DT .R/. SQ 2 DT0 .R/

”

SQ is a periodic distribution on R with period T :

(6.7.39)

Fourier series of periodic distributions on R Definition 6.7.2. A trigonometric series 1 X

Q ik!x ck .S/e

(6.7.40)

kD1

is called the Fourier series of the periodic distribution SQ 2 DT0 .R/  D 0 .R/ on R with period T , ck .SQ / being the Fourier coefficients of SQ 2 DT0 .R/  D 0 .R/ defined, 8k 2 Z, by: Q e ik!x iD 0 .R/D .R/ ; Q D 1 hS; ck .S/ T T T where e ik!x 2 DT .R/ 8k 2 Z.

(6.7.41)

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

345

Example 6.7.5. Let ıQ be the periodic Dirac distribution (see (1.10.59)) on R with Q are given period T . Then, for ıQ 2 DT0 .R/  D 0 .R/, the Fourier coefficients ck .ı/ by: Q D ck .ı/

1 Q ik!x 1 1 hı; e i D e ik!x jxD0 D T T T

8k 2 Z;

and the Fourier series of the periodic Dirac distribution ıQ is given by (see (6.7.23)).

(6.7.42)

P1

1 ik!x kD1 T e

Convergence and term-by-term differentiation of Fourier series (6.7.39) Theorem 6.7.5. The Fourier series D 0 .R/ to ıQ 2 D 0 .R/:

P1

1 ik!x kD1 T e

of ıQ in (6.7.42) converges in

1 1 X X 1 ik!x e D ı.x  lT / D ıQ T

kD1

in D 0 .R/;

(6.7.43)

kD1

where ı.x  lT / is the Dirac P distribution with unit mass/force/charge concentrated at x D lT , l 2 Z; ıQ D 1 kD1 ı.x  lT / is the periodic Dirac distribution (see (1.10.59)). Proof. For the particular case T D 2; ! D

2 T

D 1, see the proof of (2.11.9) in ExP ik!x ample 2.11.2. We give here an alternative independent proof. Consider nkDn e T in D 0 .R/. Then, 8 2 D.R/,   X Z n n X e ik!x 1 1 ik!x ; D e .x/dx T T 1 D 0 .R/D.R/ kDn

D

n X kDn

D

kDn

1 T

1 X lD1

Z

.lC1/T

e ik!x .x/dx

lT

n 1 Z T X 1 X e ik! . C lT /d  T 0

kDn

(by change of variables x D  C lT )

lD1

Z n 1 X 1 T ik!x X D e ..x C lT //dx T 0 kDn

D

lD1

Z n X 1 T ik!x Q ˆ.x/dx e T 0

kDn

Z n X 1 T Q D ˆ.x/e ik!x dx T 0 kDn

(by (1.10.51))

(finite summation with respect to l)

346

Chapter 6 Convolution of distributions

Q 2 DT .R/ is a C 1 (both sides of the last equality contain the same terms), where ˆ function which is periodic on R with period T such that its uniformly and absolutely convergent Fourier series is given by Theorem 2.11.4/Lemma 6.7.1: 1 X

Q D ˆ

1 X

Q ik!x D ck .ˆ/e

kD1

Q e ik!x iT e ik!x hˆ;

(see (2.11.7d))

kD1

 Z 1  X 1 T Q ik!x D ˆ.x/e dx e ik!x ; T 0 kD1 RT P ik!x dx/ Q Q Q Q with Sn .ˆ/.x/ D nkDn . T1 0 ˆ.x/e i.e. ˆ.x/ D limn!1 Sn .ˆ/.x/ ik!x and e  Z n  X 1 T Q Q ˆ.x/e ik!x dx e ik!0 Sn .ˆ/.0/ D T 0 kDn

 Z n  X 1 T Q ik!x ˆ.x/e D dx T 0 kDn

such that Q lim Sn .ˆ/.0/ D

n!1

 Z 1  X 1 T Q Q ˆ.x/e ik!x dx D ˆ.0/: T 0

kD1

Hence,   X n e ik!x Q Q ; D Sn .ˆ/.0/ ! ˆ.0/ T D 0 .R/D.R/

as n ! 1

kDn

H)

  X n e ik!x Q ;  D lim Sn .ˆ/.0/ n!1 n!1 T lim

kDn

Q D ˆ.0/ D

1 X

.0 C lT / D

lD1

H)

lD1

 X  1 1 X e ik!x ; D .lT / T D 0 .R/D.R/ kD1

D

 X 1

lD1

 ı.x  lT /; 

8 2 D.R/ D 0 .R/D.R/

lD1

H)

1 X

1 1 X X e ik!x D ı.x  lT / D ıQ T

kD1

in D 0 .R/,

lD1

which is the periodic Dirac distribution ıQ with period T (see (1.10.59).

.lT /

347

Section 6.7 Derivatives of convolutions, convolution of distributions on a circle 

Theorem 6.7.6.

P1 Q ik!x of the periodic distribution SQ 2 I. The Fourier series kD1 ck .S/e 0 0 0 DT .R/  D .R/ converges in D .R/ to SQ .

II. 8m 2 N, 1 X d m SQ SQ .m/ D D ck .SQ /.i k!/m e ik!x ; dx m

(6.7.44)

lD1

i.e. m-times term-by-term differentiation of the Fourier series of SQ is admissible 8m 2 N. Proof. I. Let SQ 2 DT0 .R/ be a periodic distribution in D 0 .R/ with period T . We are Pn Q ik!x ; i D hSQ ; i 8 2 D.R/. Set to show that limn!1 .S/e Pn h kDn ckik!x Q SQn D Sn .SQ / D 8n 2 N. Then SQn 2 L1loc .R/ and is kDn ck .S /e periodic on R with period T . Hence, from (6.7.32) we have, 8 2 D.R/,     Z T Q Q Q Q Sn ;  D Sn ; ˆ D Sn .SQ /.x/ˆ.x/dx 0 DT .R/DT .R/

D 0 .R/D.R/

Z

n X

T

D 0

Q Q ik!x ˆ.x/dx D ck .S/e

kDn

n X

0

ck .SQ /

kDn

Z

T

ik!x Q ˆ.x/e dx

0

 Z T n X 1 Q ik!x ik!x Q D ˆ.x/e hS ; e iDT0 .R/DT .R/ dx (using (6.7.41)) T 0 kDn

D

  Z T  n  X ik!x Q Q 1 ˆ.x/e dx e ik!x S; T 0 D0

T .R/DT .R/

kDn

   Z n  X 1 T Q ik!x ik!x Q ˆ.x/e D S; dx e T 0 D0

T .R/DT .R/

kDn

  n X Q ik!x D SQ ; ck .ˆ/e kDn

0 DT .R/DT .R/

Q D 0 .R/D .R/ ; D hSQ ; Sn .ˆ/i T T

P P Q D n Q ik!x D n Q ik!x 2 ˆ/e where 8n 2 N, Sn .ˆ/ kDn ck .ˆ/e P1 kDn ck .ik!x Q Q 2 DT .R/ is the nth partial sum of the Fourier series kD1 ck .ˆ/e of ˆ DT .R/. Hence, Q D 0 .R/D .R/ hSQn ; iD 0 .R/D.R/ D hSQn ; ˆi T T Q Sn .ˆ/i Q D 0 .R/D .R/ D hS; T T

8 2 D.R/:

(6.7.45)

348

Chapter 6 Convolution of distributions

Q ! ˆ Q uniformly as n ! 1 and Sn.m/ .ˆ/ Q ! ˆ Q .m/ By Lemma 6.7.1, Sn .ˆ/ Q Q uniformly 8m 2 N as n ! 1. Hence, Sn .ˆ/ ! ˆ in DT .R/ as n ! 1 by Q D 0 .R/D .R/ ! hSQ ; ˆi Q D 0 .R/D .R/ as n ! 1 by (6.7.34) H) hSQ ; Sn .ˆ/i T T T T (6.7.35). From (6.7.32), Q D 0 .R/D .R/ D hSQ ; iD 0 .R/D.R/ : hSQ ; ˆi T T

(6.7.46)

Hence, from (6.7.45) and (6.7.46), 8 2 D.R/, Q D 0 .R/D .R/ lim hSQn ; iD 0 .R/D.R/ D lim hSQ ; Sn .ˆ/i T T

n!1

n!1

Q ˆi Q D 0 .R/D .R/ D hSQ ; iD 0 .R/D.R/ D hS; T T P Q ik!x D SQ in D 0 .R/. ” limn!1 SQn D 1 kD1 ck .S /e P1 Q ik!x in D 0 .R/, SQ .m/ 2 D 0 .R/8m 2 N. In II. Since SQ D kD1 ck .S /e dm fact, 8m 2 N, dx m W D 0 .R/ ! D 0 .R/ is continuous on D 0 .R/. Hence, P m Q SQn D nkDn ck .SQ /e ik!x ! SQ in D 0 .R/ (proved above in (I)) H) ddxSmn D Pn m Q ck .SQ /.i k!/m e ik!x ! ddx mS in D 0 .R/ as n ! 1 with SQ .m/ D PkDn n m ik!x 2 D 0 .R/, which is the Fourier series of SQ .m/ . In Q kDn ck .S /.i k!/ e fact, from (6.7.41), 8m 2 N, 1 Q .m/ ik!x hS ; e iDT0 .R/DT .R/ T  m ik!x /  1 m Q d .e D .1/ S; T dx m D 0 .R/DT .R/

ck .SQ .m/ / D

T

1 Q .1/m .i k!/m e ik!x iD 0 .R/D .R/ D .1/m hS; T T T 1 Q e ik!x iD 0 .R/D .R/ D .i k!/m ck .SQ / 8k 2 Z; D .i k!/m hS; T T T Q D since ck .S/

1 Q ik!x iDT0 .R/DT .R/ . T hS ; e

Example 6.7.6. 1 Q  a/ is represented by the complex Fourier series ı.x Q  a/ D P1 1. ı.x kD1 T Q  a// D 1 hı.x Q  a/; e ik!.xa/ i D e ik!.xa/ with Fourier coefficients ck .ı.x

2.

T 1 ik!.xa/ jxDa D T1 e ik!0 D T1 8k 2 Z, and by the real Fourier series Te Q  a/ D 1 C 2 P1 cos.k!.x  a//. ı.x kD1 T T dm Q Q.m/ .x  a/, we have: 8m 2 N, for the derivatives dx m ı.x  a/ D ı 1 m ik!.xa/ with c .ıQ .m/ .x  a// D Q.m/ .x  a/ D P1  ı k kD1 T .i k!/ e 1 m; .i k!/ T

349

Section 6.8 Applications  

P m2 2m cos.k!.x  a//; ıQ.2m/ .x  a/ D 1 kD1 .1/ T .k!/ P m2 2m sin.k!.x  a//; ıQ.2m1/ .x  a/ D 1 kD1 .1/ T .k!/

Q  a/ is the periodic Dirac distribution with unit mass/force/charge where ı.x etc. concentrated at x D a C lT with l 2 Z. Remark 6.7.5. In Example 6.7.6, the real Fourier series is obtained by writing the nth partial sum: Sn D

n X

Q  a//e ik!.xa/ D ck .ı.x

kDn

n X 1 ik!.xa/ e T

kDn

n n X X 1 1 1 ik!.xa/ 2 Œe cos.k!.x  a// D C C e ik!.xa/  D C T T T T kD1

kD1

Q  a/ D ! ı.x

1 C T

1 X kD1

2 cos.k!.x  a// in D 0 .R/ as n ! 1. T

The other results are obtained by m or 2m or 2m  1 times, as the case may be, term-by-term differentiation of the Fourier series.

6.8

Applications

Application to partial differential equations P P Let P .@/ D j˛jm a˛ @˛ D j˛jm a.˛1 ;:::;˛n /

@j˛j ˛ ˛ ˛n @x11 @x22 :::@xn

be a partial differential

operator with constant coefficients a˛ 2 R 8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n /. Then P .@/.ı T / D P .@/ı T D P .@/T D

X

a ˛ @˛ T

8T 2 D 0 .Rn /:

j˛jm

Proof. P .@/.ı T / D

 X

 X a˛ @˛ .ı T / D a˛ .@˛ .ı T //

j˛jm

D

X

j˛jm ˛

a˛ .@ T ı/

(by (6.7.4))

j˛jm

D

X

j˛jm

a˛ @˛ T D P .@/T:

(6.8.1)

350

Chapter 6 Convolution of distributions

Example 6.8.1. P 1. D niD1 2. D 3.  D

@4 @x14

@2 @t 2

@2 @xi2

C2

H) .ı T / D T ı D T ; @4 @x12 @x22

C

@4 @x24

H) .ı T / D T ı D T ;

 H) .ı T / D T ı D T .

(6.8.2) (6.8.3)

4. Consider the partial differential equation P .@/. For a given distribution S 2 E 0 .Rn / with compact support in Rn , find T 2 D 0 .Rn / such that P .@/T D S in D 0 .Rn /. (6.8.4) Theorem 6.8.1 (Malgrange and Ehrenpreis [38], [5]). Let S 2 E 0 .Rn /  D 0 .Rn / be a distribution with compact support in Rn , and E 2 D 0 .Rn / be an elementary solution of P .@/, i.e. P .@/E D ı in D 0 .Rn /. Then T D S E is a solution of the equation (6.8.4). Proof. Since S 2 E 0 .Rn / is a distribution with compact support in Rn , S E is well defined. Then, using (6.7.4), P .@/.S E/ D S P .@/E D S ı D ı S D S . Example 6.8.2. For given f 2 E 0 .Rn / (i.e. a distribution with compact support in Rn ), n D 2 or 3, find u 2 D 0 .Rn / such that 1.  u D f in D 0 .Rn /; 2. for n D 2, u D f in D 0 .R2 /. Solution. 1. From Theorem an elementary solution E 2 D 0 .Rn / of  , for r D Pn 3.3.2, 1 2 ln 1r for n D 2. kxkRn D iD1 xi , is given by E D 41 r for n D 3; E D 2 Hence, by Theorem 6.8.1, u D f E D E f with 8 R f ./ 0; II. u" D . " u/# p u" D . " u/# Q  ! u in L ./ as " ! 0C .

(6.8.21) (6.8.22) (6.8.23)

III. Moreover, if u 2 Lp ./, 1  p < 1, has support contained in a compact set K  , and if d.K; @/ D distance between K and @ > ", @ being the boundary of , then u" 2 C01 ./. (6.8.24) IV. If u 2 Lp ./, 1  p < 1, is continuous at a point x 2 , then u" .x/ D . " u/# Q  .x/ ! u.x/ as " ! 0C , the convergence being uniform on every compact set of points at which u is continuous. Proof. I. Since Lp ./  L1loc ./ for 1  p < 1 and uQ 2 L1loc .Rn /, from (6.2.34),

" uQ 2 C 1 .Rn /. Then u" D . " u/# Q  2 C 1 ./ 8" > 0.

355

Section 6.8 Applications

II. For u 2 Lp ./, 1  p < 1, uQ 2 Lp .Rn /. Then, by Theorem 6.2.3, . " u/ Q 2 p n p n L .R / with k " uk Q Lp .Rn /  kuk Q Lp .Rn / and " uQ ! uQ in L .R / as " ! 0C . (6.8.25) Q  2 Lp ./, 1  p < 1, and, by (6.8.25), Hence, u" D . " u/# p ku" kLp ./

Z D Z



j. " u/# Q  .x/jp d x p

 Rn

H)

p

p j. " u/.x/j Q d x D k " uk Q Lp .Rn /  kuk Q Lp .Rn /

ku" kLp ./  kuk Q Lp .Rn / D kukLp ./

8" > 0; 1  p < 1:

Then, ku"  H)

p ukLp ./

Z

p

 Rn

p j. " u/.x/ Q  u.x/j Q d x D k " uQ  uk Q Lp .Rn /

ku"  ukLp ./  k " uQ  uk Q Lp .Rn / ! 0 as " ! 0C (by (6.8.25)):

Q  2 C 1 ./ 8" > 0. III. For u 2 Lp ./, " uQ 2 C 1 .Rn / and u" D . " u/# N "/ C supp.u/  B.0I N "/ C K D K" , From Theorem 6.2.2, supp. " u/ Q  B.0I n K" being a compact subset of R by (6.2.3). But d.K; @/ > " H) K"   is a compact subset of  for all sufficiently small " > 0 satisfying d.K; @/ > ". Hence, 8" > 0 with d.K; @/ > ", supp.u" /  supp. " u/ Q  K"  . Thus, u" 2 C01 ./ 8" > 0 with d.K; @/ > ". (6.8.26) IV. Let x 2  be a point of continuity of u 2 LpR./, 1  p < 1. RThen uQ is also continuous atR the point x, and u.x/ R D u.x/ Rn " .x  /d  D Rn " .x  /u.x/d , since Rn " .x  /d  D Rn " .y/d y D 1 8" > 0 Z H) ju" .x/  u.x/jDj. " u/# Q  .x/  u.x/j

" .x  /ju./ Q  u.x/jd Q  Rn Z

" .x  /ju./ Q  u.x/jd Q  D kxk"



sup kxk"

D sup

Z

ju./ Q  u.x/j Q Rn

" .x  /d 

ju./ Q  u.x/j Q !0

(6.8.27)

kxk"

R R as " ! 0C (since Rn " .x  /d  D Rn " .y/d y D 1 and uQ is continuous at x 2 ), the convergence being uniform on all compact subsets of points of continuity of u in  (see Proposition 6.2.4 for more detail).

356

Chapter 6 Convolution of distributions

Density result Theorem 6.8.3. Let   Rn be any open subset of Rn . Then C01 ./  D./ is dense in Lp ./, 1  p < 1. Proof. Let u 2 Lp ./, 1  p < 1. Let  > 0. Then, by virtue of the density of C0 ./ in Lp ./, 1  p < 1 (see Appendix B, Theorem B.3.3.4 and Theorem 1.2.3 of Chapter 1), 9 2 C0 ./ with compact supp./ D K   such that ku  kLp ./ < 2 . (6.8.28) Q D .x/ for x 2  and Let Q be the null extension to Rn of  2 C0 ./: .x/ Q Q D supp./ D K  . Then Q 2 .x/ D 0 for x 2 Rn n  with supp./ C0 .Rn /  Lp .Rn / for 1  p < 1. Hence, by Proposition 6.2.5, Q Lp .Rn / ! 0 k " Q  k

as " ! 0C :

(6.8.29)

Q  Moreover, by Lemma 6.2.1, 9"0 > 0 such that " Q 2 C01 .Rn / with supp. " / Q K0   8" < "0 , K0 being a fixed compact set in . Define u" D . " /# 8" > 0. Then u" 2 C01 ./ with supp.u" /  supp. " Q " /  K0   8" < "0 . (6.8.30) Then, Z p Q  .x/  .x/jp d x ku"  kLp ./ D j. " /#  Z p Q Q Q pp n ! 0  j. " /.x/  .x/j d x D k " Q  k L .R / Rn

as " ! 0C (by (6.8.29)) H) 8 > 0, 9"1 > 0 such that  ku"  kLp .Rn / < 8" < "1 : (6.8.31) 2 Hence, combining (6.8.28) and (6.8.31), we get: 8 > 0, 9"0 D min¹"0 ; "1 º > 0 such that, for 1  p < 1,   ku  u" kLp ./  ku  kLp ./ C k  u" kLp ./ < C D  8" < "0 ; 2 2 i.e. for u 2 Lp ./, 1  p < 1, 8 > 0, 9u" 2 C01 ./ with " < "0 such that ku  u" kLp ./ < . Thus, C01 ./  D./ is dense in Lp ./, 1  p < 1. Now we can deal with the results in Sobolev spaces H m ./ defined by (2.15.1)– (2.15.3). Let u 2 H m ./ with m 2 N. Then its distributional derivatives @˛ u 2 L2 ./ 8j˛j  m are defined by (2.15.2). Let uQ and @˛ u be the null extensions to Rn of u and @˛ u 8j˛j  m, respectively: ´ ´ u.x/ for x 2  .@˛ u/.x/ for x 2  ˛ and u.x/ Q D @ u.x/ D 0 for x 2 Rn n  0 for x 2 Rn n  (6.8.32)

e

e

357

Section 6.8 Applications

e

Then uQ 2 L2 .Rn / with kuk Q L2 .Rn / D kukL2 ./ and, 8j˛j  m, @˛ u 2 L2 .Rn /, although uQ … H m .Rn / and @˛e u … L2 .Rn / for 1  j˛j  m in general.

(6.8.33)

For example, for n D 1 and  D 0; 1Œ, u.x/ D 1 8x 2 0; 1Œ, uQ … H 1 .R/, since     Z 1 d uQ d d ; dx D.x/jxD1 D u; Q 1 D xD0 D.1/ C .0/ dx dx dx 0 D 0 .R/D.R/ Dhı1 ; i C hı0 ; i D hı0 C ı1 ; i

8 2 D.R/;

where ı0 , ı1 are Dirac distributions with concentration of mass/force/charge etc. at 0 and 1 respectively H)

d uQ D ı0 C ı1 … L2 .R/ dx

H)

uQ … H 1 .R/:

(6.8.34)

e

Q 2 L2 .Rn / and . " @˛ u/ 2 L2 .Rn / 8j˛j  m, By Theorem 6.2.3, 8" > 0, . " u/ and consequently . " u/# Q  2 L2 ./, . " @˛ u/# 2 L2 ./ 8j˛j  m. Then we have the following result:

e

Theorem 6.8.4. Let u 2 H m ./. Then, for x 2  with d.x; @/ > ", @ being the boundary of ,

e

@˛ Œ. " u/# Q  .x/ D . " @˛ u/# .x/ 8j˛j  m: (6.8.35) R R Proof. 8x 2 , . " u/# Q  .x/ D Rn " .x  /u./d Q  D  " .x  /u./d  and Z Z ˛ ˛ . " @ u/# .x/D

" .x  /@ u./d  D " .x  /.@˛ u/./d  8j˛j  m:

e

Rn

e



(6.8.36) Then, from Theorem 6.8.2, . " u/# Q  2 C 1 ./. Hence, 8j˛j  m, ˛ Q  .x/ is well defined 8x 2 , 8" > 0 and can be obtained by dif@ Œ. " u/# ferentiating (in the usual pointwise sense) under the integral sign: 8j˛j  m, Z Z ˛ ˛ @ Œ. " u/# Q  .x/ D @x

" .x  /u./d  D @˛ x Œ " .x  /u./d    Z j˛j D @˛  u./d   Œ " .x  /.1/  Z j˛j u./@˛ (6.8.37) D .1/  Œ " .x  /d : 

For fixed x 2 , set " ./ D " .x  / 8 2 , 8" > 0. Then, for fixed x 2 , d.x; @/ > " H) " 2 C01 ./  D./. In fact, for fixed x 2 , the function  2

358

Chapter 6 Convolution of distributions

Rn 7! " .x  / belongs to C01 .Rn / by definition of " . Then, for all sufficiently small " > 0, " 2 C 1 ./ with " ./ D 0 8 2  with kx  k > ". Thus, for  2 @ and x 2 , kx  k  inf 2@ kx  k D d.x; @/ > " H) " .x  / D 0 8x 2  with d.x; @/ > " H) supp." /   is a compact subset of . Hence, " 2 C01 ./  D./ for all sufficiently small " > 0 with d.x; @/ > ". Then, from (6.8.37), 8j˛j  m, ˛ @˛ Œ. " u/# Q  .x/ D .1/j˛j hu; @˛  " iD 0 ./D./ D h@ u; " iD 0 ./D./ Z Z D @˛ u./" ./d  D

" .x  /@˛ u./d  (6.8.38)

e

Rn



for x 2  with d.x; @/ > ", since for u 2 H m ./, its distributional derivatives @˛ u 2 L2 ./ 8j˛j  m. From (6.8.36)–(6.8.38), the result follows: @˛ Œ. " u/# Q  .x/ D . " @˛ u/# .x/ for x 2  with d.x; @/ > ", i.e. for all sufficiently small " > 0.

e

Theorem 6.8.5. Let ; 1  Rn be open subsets of Rn such that 1   is a compact subset of  (i.e. 1 is relatively compact in ). Let u 2 H m ./ and uQ and @˛ u be the null extensions to Rn of u and @˛ u (8j˛j  m) respectively, as defined by (6.8.32). If u" D . " u/# Q  8" > 0, then

e

u" #1 ! u#1 in H m .1 / as " ! 0C :

e

(6.8.39)

Proof. Let u 2 H m ./. Then u; Q @˛ u 2 L2 .Rn / 8j˛j  m. So, " uQ 2 L2 .Rn / H) 2 u" D . " u/# Q  2 L ./ H)

u" #1 2 L2 .1 / 8" > 0 and u" #1 ! u#1 in L2 .1 / as " ! 0C (6.8.40)

e

by Theorem 6.8.2 with p D 2. Again, from Theorem 6.8.4, for all sufficiently small " > 0 with d.1 ; @/ > ", 8j˛j  m, .@˛ u" /.x/ D . " @˛ u/.x/ 8x 2 1 H)

e

. " @˛ u/#1 D .@˛ u" /#1 D @˛ .u" #1 /;

(6.8.41)

since u" 2 C 1 ./ \ L2 ./ by (6.8.21) and (6.8.22). But u 2 H m ./ H) .@˛ u/#1 D @˛ .u#1 / 8j˛j  m: R R In fact,  .@˛ u/d x D .1/j˛j  u@˛  8 2 D./. Hence, 8j˛j  m, Z Z ˛ j˛j .@ u/#1 d x D .1/ .u#1 /@˛ d x 8 2 D.1 / 1

(6.8.42)

1

˛

D h@ .u#1 /; iD 0 .1 /D.1 /

8 2 D.1 /

H) .@˛ u/#1 D @˛ .u#1 / in D 0 .1 /, but .@˛ u/#1 2 L2 .1 / 8j˛j  m.

359

Section 6.8 Applications

Hence, 8j˛j  m, @˛ .u#1 / D .@˛ u/#1 in L2 .1 /. Again, applying Theorem 6.8.2, 8j˛j  m,

e

in L2 .1 / as " ! 0C ,

. " @˛ u/#1 ! .@˛ u/#1

(6.8.43)

since Z 1

e

j. " @˛ u/#1 .x/  .@˛ u/#1 .x/j2 d x Z  

e

j. " @˛ u/# .x/  @˛ u.x/j2 d x ! 0

as " ! 0C . Hence, from (6.8.41),(6.8.42) and (6.8.43), 8j˛j  m,

e

. " @˛ u/#1 D @˛ .u" #1 / ! .@˛ u/#1 D @˛ .u#1 / in L2 .1 / as " ! 0C : (6.8.44) Finally, from (6.8.40) and (6.8.44), we have u" #1 ! u#1 in L2 .1 / and @˛ .u" #1 / ! @˛ .u#1 / in L2 .1 / as " ! 0C 8j˛j  m H) u" #1 ! u#1 in H m .1 / as " ! 0C . The following discussion needs two auxiliary spaces, C m ./ and H m ./, which will be introduced now. Space C m ./ Definition 6.8.2. Let   Rn be any open subset of Rn , and C m ./ be the space of m times continuously differentiable (in the usual sense) functions on . Then C m ./ is the subspace of C m ./ defined by C m ./ D ¹u W u 2 C m ./ such that, 8j˛j  m, @˛ u 2 L2 ./º, which is equipped with the inner product h  ;  im; and the norm k  km; defined by:  hu; viC m ./ D hu; vim; D

Z

X 0j˛jm

 @ u@ vd x ˛

˛

8u; v 2 C m ./I



(6.8.45) kukC m ./ D kukm; D hu; ui1=2 D



X 0j˛jm

Z

1=2 j@˛ uj2 d x 8u 2 C m ./:



For arbitrary   Rn , C m ./  C m ./, C m ./  L2 ./.

(6.8.46)

360

Chapter 6 Convolution of distributions

Space H m ./ Definition 6.8.3. H m ./  C m ./ in the norm k  km; defined by (6.8.45), i.e. H m ./ is the completion of C m ./ in the norm k  km; . Following Agmon [39, p. 2], we give another useful characterization as follows: u 2 H m ./ ” u 2 L2 ./ and 9 a sequence .uk /1 in C m ./ such that kD1 1 2 ˛ uk ! u in L ./ and .@ uk /kD1 is a Cauchy sequence in L2 ./ 8j˛j  m as k ! 1. (6.8.47) Now we state some important results without proof (see, for example, Agmon [39, pp. 2–4]). Theorem 6.8.6. I. H m ./ equipped with the inner product h  ;  im; in (6.8.45) is a Hilbert space. II. H m ./  H m ./, where H m ./ is defined by (2.15.1)–(2.15.3). III. For  satisfying the segment property,2 H m ./  H m ./. Theorem 6.8.7 (Leibniz Rule). Let u 2 H m ./ and v 2 C m ./ such that v and its usual partial derivatives (in the pointwise sense) @˛ v of order j˛j  m are bounded in . Then, I. uv 2 H m ./I ˛ ˇ ˛ˇ P II. @˛ .uv/ D v 8j˛j  m, where ˇ  ˛ ” ˇi  ˛i ˇ˛ ˇ @ u@

˛1 ˛2

˛ 8i D 1; 2; : : : ; n and ˇ D ˇ1 ˇ2    ˛ˇnn . Proof. First we will consider the case of u 2 H m ./  H m ./. Let u 2 H m ./. Then, 9 a sequence .uk / in C m ./ such that uk ! u

and

@˛ uk ! u˛

in L2 ./ 8j˛j  m:

(6.8.48)

But v and @˛ v are continuous and bounded on  8j˛j  m. Hence, 8j˛j  m, ! ! X ˛ X ˛ @˛ .uk v/ D @ˇ uk @˛ˇ v ! @ˇ u@˛ˇ v in L2 ./ as k ! 1: ˇ ˇ ˇ˛

ˇ˛

(6.8.49) So the sequence .wk / with wk D uk v 8k 2 N is a Cauchy sequence in C m ./. Hence, 9w 2 H m ./ such that wk D uk v ! w D uv and @˛ wk D @˛ .uk v/ ! w˛ 2 Roughly

in L2 ./ as k ! 1 8j˛j  m:

(6.8.50)

speaking,  has the segment property if at each point x on the boundary  of , there exists a linear segment L with origin at x such that L n ¹xº  . For more details, see Appendix D.  has the segment property H)  is not locally a two-sided domain (see Appendix D).

361

Section 6.8 Applications

But L2 ./  D 0 ./ H) wk D uk v ! w D uv in D 0 ./ and @˛ .uk v/ ! w˛ in D 0 ./ as k ! 1:

(6.8.51)

Since @˛ W D 0 ./ ! D 0 ./ is continuous, uk v ! uv in D 0 ./ H) @˛ .uk v/ ! @˛ .uv/ in D 0 ./ as k ! 1. From the uniqueness of the limit, @˛ .uv/ D w˛ 2 L2 ./ 8j˛j  m. Hence, w D uv 2 L2 ./; @˛ w D @˛ .uv/ 2 L2 ./ 8j˛j  m

H)

w D uv 2 H m ./: (6.8.52)

Then, for H m ./, 8j˛j  m, @˛ .uv/ D @˛ w D limk!˛ @˛ .uk v/ D ˛ uˇ 2 ˛ˇ P v in L2 ./ (by (6.8.52) and (6.8.49)). ˇ˛ ˇ @ u@ Now assume that u 2 H m ./. Then, 8 relatively compact 1   (i.e. 1  ), by Theorem 6.8.5, u" #1 ! u#1 in H m .1 / as " ! 0C , where u" D . " u/# Q  2 C 1 ./. Hence, u" #1 2 C m .1 / ! u#1 in H m .1 / H) m u#1 2 H .1 /, since u" #1 ! u#1 in L2 .1 /, and 8j˛j  m, @˛ .u" #1 / D .@˛ u" /#1 ! @˛ .u#1 / D .@˛ u/#1 in L2 .1 /. Then, using the first part of this proof, u#1 2 H m .1 / H) .uv/#1 D u#1  v#1 2 H m .1 / and ˛ ˇ P @ .u#1 /@˛ˇ .v#1 /. But @ˇ .u#1 / D .@ˇ u/#1 and @˛ Œ.uv/#1  D ˇ˛ ˇ ˛ ˛ @ Œ.uv/#1  D Œ@ .uv/#1 and v 2 C m ./ ! X ˛ @ˇ u@˛ˇ v @ .uv/ D ˇ ˛

H)

in 1 ;

(6.8.53)

ˇ˛

which holds 81 with 1  . But the right-hand side of (6.8.53) belongs to L2 ./, since @˛ˇ v is bounded in , and @ˇ u 2 L2 ./ 8jˇj  m. Thus, 81  m ./ D ¹w W w# m , uv 2 H m .1 /, i.e. uv 2 Hloc 1 2 H .1 / 81  º. Since the relation (6.8.53) holds 81 with 1  , it will also hold in , i.e. 8j˛j  m ! X ˛ @ˇ u@˛ˇ v @ .uv/ D ˇ ˛

in :

(6.8.54)

ˇ˛

Hence, @˛ .uv/ 2 L2 ./ 8j˛j  m and uv 2 H m ./ with (6.8.54) (see also Agmon for more details [39, p. 10]).

362

Chapter 6 Convolution of distributions

Density result Theorem 6.8.8. The set of functions u 2 H m ./ with compact support in  is dense in H m ./.

Proof. Let u 2 H m ./ with supp.u/  . Let with the properties: 0  .x/  1 8x 2 Rn and ´ .x/ D

2 D.Rn / be a cut-off function

1 for kxk  1 0 for kxk  2;

(6.8.55)

as shown in Figure 6.3 for n D 1.

ψ x

Figure 6.3 Cut-off function

on R

Define a sequence .uk /1 with uk .x/ D . kx /u.x/ 8k 2 N. Then, by Leibniz’s kD1 rule, uk 2 H m ./ and uk has compact support in  8 fixed k 2 N. But 8j˛j  m, !    X ˛ x @ uk D @ˇ @˛ˇ u ˇ k ˇ˛ !   X ˛ 1 x ˛ˇ ˇ @ .@ / uC D ˇ k jˇj k ˛

ˇ˛ ˇ¤0

H)

  x ˛ @ u k

!       X ˛ 1 x ˛ˇ x ˇ .@ / u C @ uk  @ u D @  1 @˛ u: ˇ k jˇj k k ˛

˛

ˇ˛ ˇ¤0

363

Section 6.8 Applications

Applying the triangular inequality, j˛j  m, ˛

˛

k@ uk  @ ukL2 ./ 

! ˛ 1 max j@ˇ .x/jk@˛ˇ ukL2 ./ ˇ k x2Rn

X ˇ˛ ˇ¤0

Z C

1=2  j@ u.x/j d x !0 ˛

2

as k ! 1;

kxk>k

(6.8.56) i.e. the right-hand side tends to 0 as k ! 1, since . kx / D 1 for kxk  k H) . kx /  1 D 0 for kxk  k and j . kx /  1j  1 for kxk > k, k 1jˇj  k1 8jˇj, 8k 2 N, supp.u/  R H) for all sufficiently large k 2 N, @˛ u.x/ D 0 for kxk > k 8j˛j  m H) kxk>k j@˛ u.x/j2 d x D 0 for all sufficiently large k 2 N. Hence, uk ! u in H m ./ as k ! 1, i.e. 8u 2 H m ./, 9uk D k u 2 H m ./ with supp.uk /  , such that uk ! u in L2 ./ and @˛ uk ! @˛ u in L2 ./ 8j˛j  m H) uk ! u in H m ./ as k ! 1. Theorem 6.8.9. D.Rn / is dense in W m;p .Rn / 8m 2 N, 1  p < 1. Proof. The subspace W0 D ¹u W u 2 W m;p .Rn /, supp.u/  Rn º is dense in W m;p .Rn / 8m 2 N, 1  p < 1: Let u 2 W m;p .Rn /, and 2 D.Rn / be the cutoff function in (6.8.55). Define uk .x/ D .x=k/u.x/ 8k 2 N. Then .uk /k2N is a sequence in W0 , supp.uk /  Rn 8k 2 N. Following the steps of the proof of Theorem 6.8.8, replacing ‘’ by ‘Rn ’, ‘L2 ./’ by ‘Lp .Rn /’, 1  p < 1, in (6.8.56), and introducing necessary minor modifications, we get uk ! u in W m;p .Rn / with m 2 N, 1  p < 1, i.e. 8u 2 W m;p .Rn /, 9.uk /k2N in W0 such that uk ! u in W m;p .Rn /. Hence, W0 is dense in W m;p .Rn / with m 2 N, 1  p < 1. D.Rn / is dense in W0 : Let v 2 W0 , i.e. v 2 W m;p .Rn / with supp.v/  Rn , and . k /k2N be a sequence of regularizing functions as defined in (6.2.2) with supp. k / D 1 N B.0I / 8k 2 N. Define k D k v with v 2 W0 8k 2 N. But v 2 W m;p .Rn / H) k v 2 L1loc .Rn / and v 2 W0 H) supp.v/ is compact in Rn . Then k 2 D.Rn / 8k 2 N by (6.2.38). v 2 W0 H) v 2 Lp .Rn /, @˛ v 2 Lp .Rn / for 1  j˛j  m H) k D k v ! v in Lp .Rn / and @˛ k D @˛ . k v/ D k @˛ v ! @˛ v in Lp .Rn / for 1  j˛j  m as k ! 1 (by (6.2.40) with " D k1 ! 0); the second equality n @˛ . k v/ D k @˛ v follows from (6.7.4). Hence, 0 , 9.k /k2N in D.R / P for v 2 W p p p ˛ ˛ such that kvk km;p;Rn D Œkvk kLp .Rn / C 1j˛jm k@ v@ k kLp .Rn /  ! 0 as k ! 1 H) D.Rn / is dense in W0 , and W0 is dense in W m;p .Rn / (proved earlier). Hence, D.Rn / is dense in W m;p .Rn /, since 8u 2 W m;p .Rn /, 8" > 0, 9v" 2 W0 and 9" 2 D.Rn / such that ku  v" km;p;Rn < "=2 and kv"  " km;p;Rn < "=2 H) ku  " km;p;Rn < ".

364

6.9

Chapter 6 Convolution of distributions

Convolution equations (see also Section 8.7, Chapter 8)

A convolution equation is an equation of the form A T D B;

(6.9.1)

where A and B are given distributions, A being the coefficient distribution and B being the right-hand-side distribution, and T is the unknown distribution. Since the convolution of any two distributions is not necessarily defined, we make an assumption that A 2 E 0 .Rn /;

(6.9.2)

(i.e. A is a distribution with compact support in Rn ) such that A T is well defined for arbitrary T 2 D 0 .Rn / by Theorem 6.3.2. Homogeneous (convolution) equation For B D 0, A T D 0 is called homogeneous:

(6.9.3)

Particular cases of convolution equations are: 1. Linear partial differential equations with constant coefficients P ˛ 0 n n For A D j˛jm a˛ @ ı 2 E .R / with supp.A/  R , a˛ 2 R, @˛ D

@j˛j ˛ ˛ @x1 1 @xn n

8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n /, X

A T D B H)

a ˛ @˛ ı T D

j˛jm

X

a˛ @˛ T D B;

(6.9.4)

j˛jm

which is a partial differential equation with constant coefficients a˛ for the unknown distribution T . 2. Linear difference equations with constant coefficients PN PN For A D iD1 ahi ıhi D iD1 ahi ı.x  hi / with supp.A/ D ¹h1 ; h2 ; : : : ; hN º  Rn , ahi 2 R, A T DB

H)

N X

ahi ıhi T D

iD1

where hi T D T .x  hi / by (6.3.28).

N X iD1

ahi hi T D B;

(6.9.5)

Section 6.9 Convolution equations (see also Section 8.7, Chapter 8)

365

3. Volterra’s integral equations of the first and second kinds (a) First kind: For x  0, Z

x

K.x  t /f .t /dt D g.x/I

(6.9.6)

0

(b) Second kind: For x  0, Z

x

f .x/ C

K.x  t /f .t /dt D g.x/I

(6.9.7)

0

where the kernel K and the right-hand-side function g are given and locally summable on Œ0; 1Œ, and f is the unknown function. We extend f; g; K by 0 for x < 0. Then, setting  A D K.x/ (resp. A D ı C K.x// with K 2 L1 .R/, supp.K/  Œ0; 1Œ; loc  T D f .x/ with supp.f /  Œ0; 1Œ; (6.9.8) 1  B D g.x/ with g 2 L loc .R/, supp.g/  Œ0; 1Œ; we get Volterra’s integral equations in convolution form: Rx (a0 ) First kind: .A f /.x/ D .K f /.x/ D 0 K.x  t /f .t /dt D g.x/; (6.9.9) (b0 ) Second kind: .A f /.x/ D ..ı C K.x// f /.x/ D g.x/ Z x K.x  t /f .t /dt D g.x/: H) ı f C K f D g H) f .x/ C 0

(6.9.10) Remark 6.9.1. There R x are Volterra’s equations, which are not convolution equations. For example, 0 K.x; /f ./d  D g.x/ is Volterra’s equation of the first kind, where the kernel K.x; / is a function of two variables, with x  0, 0    x. 4. Integro-differential equations These can be obtained by combining diverse types of linear differential and integral equations. See (6.10.3) for an example. Systems of convolution equations A system of m convolution equations for m unknown distributions T1 ; T2 ; : : : ; Tm is given, for 1  i  m, by: Ai1 T1 C Ai2 T2 C    C Aim Tm D Bi ;

(6.9.11)

where Aij 2 E 0 .Rn /, 1  i; j  m, are the given m2 coefficient distributions such that Aij Tj is well defined 8i; j D 1; 2; : : : ; m, and B1 ; B2 ; : : : ; Bm are the given right-hand-side distributions.

366

Chapter 6 Convolution of distributions

Elementary solution (see also Section 3.3 and Section 8.7) Definition 6.9.1. A distribution E 2 D 0 .Rn / satisfying the convolution equation (6.9.1) with B D ı, i.e. A E D ı; is called an elementary or fundamental solution of the convolution equation. Then E is denoted by A1 or simply by A1 , i.e. A E D A A1 D A A1 D A1 A D ı:

(6.9.12)

If A1 (i.e. E) exists, the coefficient distribution A is called invertible. Non-uniqueness of A 1 In general, elementary solution E 2D 0 .Rn / is not unique. Hence, A1 is not unique in general. In fact, E is determined up to an additive distribution E0 2 D 0 .Rn / satisfying A E0 D 0 (i.e. the corresponding homogeneous equation). Indeed, A .E C E0 / D A E C A E0 D ı C 0 D ı:

(6.9.13)

Non-invertibility of A 9 distributions A which are not invertible, i.e. for which A1 does not exist. For example, let A D  2 D.Rn /. Then A 2 D 0 .Rn / and A T D  T D T  D is well defined 8T 2 D 0 .Rn /. But is a C 1 -function by Definition 6.3.1A and Remark 6.3.1 (see also Theorem 6.4.1). Hence, for A D , A T can never be Dirac distribution ı, i.e. A D  2 D.Rn /  D 0 .Rn / is not invertible. dı Example 6.9.1. For n D 1, A D dx D ı 0 2 E 0 .R/ has inverse A1 2 D 0 .R/, but A1 … E 0 .R/. In fact, A E D ı 0 E D ı dE D dE D ı (applying first (6.3.24) dx dx 1 and then (6.3.21)) H) E D H.x/ H) A D H.x/ (the Heaviside function) with supp.A1 / D Œ0; 1Œ, which is not compact in R H) A1 … E 0 .R/.

Example 6.9.2. For n D 3, A D ı 2 E 0 .R3 / with D

@2 @x12

C

@2 @x22

C

@2 @x32

(the

Laplace operator), A E D ı E D ı E D E D ı (applying first (6.3.24) and then (6.3.21)) H) A1 D E D  41 r 2 D 0 .R3 / by Theorem 3.3.2. But 1 3 1 … E 0 .R3 /, although A 2 E 0 .R3 /. supp. 4 r /  R H) A Example 6.9.3. Every Pcorresponding to Ppolynomial in derivatives is invertible, i.e. polynomial P ./ D j˛jm a˛  ˛ , A D P .@/ı with P .@/ D j˛jm a˛ @˛ , ı 2 E 0 .Rn /, is invertible. In fact, P .@/ı E D ı H) P .@/E D ı has a solution by Theorem 6.8.1. Methods for constructing elementary solutions can be found in Chapter 8, Section 8.7 and Chapter 3, Section 3.3.

Section 6.9 Convolution equations (see also Section 8.7, Chapter 8)

367

Convolution algebra A The discussions on convolution equations and their solutions, elementary solutions and the inverses A1 D A1 suggest the study of convolution equations and their solutions in a special subspace A of D 0 .Rn /, called a convolution algebra, which we will now define, following [7]. Definition 6.9.2. A subspace A  D 0 .Rn /, i.e. T1 ; T2 2 A H) T1 ; T2 2 D 0 .Rn / and ˛1 T1 C ˛2 T2 2 A 8˛1 ; ˛2 2 C, is called a convolution algebra if and only if 

ı 2 A (i.e. the Dirac distribution ı is an element of A/;



T1 ; T2 ; : : : ; Tm 2 A H) T1 T2    Tm is defined and belongs to A with Ti Tj D Tj Ti

(6.9.14a)

(6.9.14b)

in A, 1  i; j  m; Ti Tj Tk D Ti .Tj Tk / D .Ti Tj / Tk

(6.9.14c)

in A, 1  i; j; k  m, and so on. Examples of convolution algebras are: 1. A D E 0 .Rn / (see Chapter 5, Section 5.6); 2. A D D

0C

(6.9.15a)

D ¹T 2 D 0 .R/ W supp.T /  Œ0; 1Œº (see Definition 8.8.2); (6.9.15b)

3. A D D 0 ./ D the space of distributions on a circle  (see Section 1.10 and Section 6.7). (6.9.15c) D 0 .Rn / is not a convolution algebra, since (6.9.15b)–(6.9.15c) do not hold in general. Consider the convolution equation (6.9.1) in A, i.e. for given A; B 2 A, find T 2 A such that A T D B:

(6.9.16)

Theorem 6.9.1. For given A 2 A and arbitrary B 2 A, the convolution equation (6.9.16) in A has at least one solution T 2 A if and only if A has an inverse A1 D A1 2 A, i.e. A E D A A1 D A1 A D ı 2 A. Then A1 2 A is unique, and the unique solution T 2 A of (6.9.16) is given by T D A1 B. Proof. Existence of A1 2 A: Suppose that there exists at least one solution T 2 A of (6.9.16) for arbitrary B 2 A. Then, for B D ı 2 A, 9 a solution T D E 2 A such that A E D ı H) A has an inverse A1 2 A. Uniqueness of A1 2 A: Suppose that A E D ı and A E1 D ı. Then A .E  E1 / D 0 H) A E0 D 0 with E0 D E  E1 2 A. But E0 D ı E0 D

368

Chapter 6 Convolution of distributions

.A1 A/ E0 D A1 .A E0 / D A1 0 D 0 H) E D E1 H) A1 D E 2 A is unique. Conversely, suppose that A1 2 A exists with A A1 D A1 A D ı 2 A. Then, by taking the convolution of both sides of (6.9.16) with A1 2 A, we get A1 A T D A1 B in A H) ı T D A1 B H) T D A1 B 2 A. Again, T D A1 B H) A T D A A1 B D ı B D B H) T D A1 B is the solution of (6.9.16). Hence, if, for A 2 A, A1 2 A exists, A T D B ” T D A1 B 2 A, i.e. equation (6.9.16) has a unique solution T D A1 B 2 A. For solving convolution equation (6.9.16) in A, the essential problem is to find an elementary solution E 2 A. (6.9.17) Application Let E.x/ D H.x/y.x/ be an elementary solution of the differential operator P .D/  Lm derived in (3.3.25)–(3.3.30), H being the Heaviside function and y being the unique solution of the Cauchy problem P .D/y D 0 with the initial conditions y .k/ .0/ D 0, 0  k  m  2, y .m1/ .0/ D 1. Then the solution z D z.x/ of the corresponding non-homogeneous equation P .D/z D f and satisfying the initial conditions z .k/ .0/ D zk;0 , 0  k  m  1, where f is a given sufficiently regular function, ¹zk0 ºm1 are arbitrary given real kD0 numbers, is given by:   m1 X .k/ k ı ; H z D Hy Hf C kD0

with k D zmk1;0 C a1 zmk2;0 C    C amk1 z00 ;

ı .k/ D

d kı ; dx k

or, for x  0, Z

x

y.x  /f ./d  C

z.x/ D 0

m1 X

k y .k/ :

(6.9.18)

kD0

Proof. E D Hy 2 D 0 .R/ is an elementary solution of P .D/  Lm in (3.3.25)– (3.3.30) with supp.E/ D supp.Hy/  Œ0; 1Œ H) E D Hy 2 D 0C and P .D/E D ı H) P .D/ı E D ı H) ŒP .D/ı1 D E D Hy 2 D 0C . Moreover, following the steps of the proof of (3.3.29) and using initial values zk;0 D z .k/ .0/, we get .H z/.1/ D H z .1/ C ız.0/ D H z .1/ C ız0;0 I .H z/.2/ D H z (2) C ız1;0 C ı .1/ z0;0 I :: : .H z/.k/ D H z .k/ C ızk1;0 C ı .1/ zk2;0 C    C ı .k1/ z0;0 ;

1  k  m:

369

Section 6.9 Convolution equations (see also Section 8.7, Chapter 8)

Then P .D/.H z/ D H Œz .m/ C a1 z .m1/ C    C am z C ıŒzm1;0 C a1 zm2;0 C    C am1 z0;0  C    C ı .k/ Œzmk1;0 C a1 zmk2;0 C    C amk1 z0;0  C    C ı .m1/ Œz0;0  with a0 D 1, H)

P .D/.H z/ D H ŒP .D/z „ ƒ‚ … f

m1 X

C

kD0

Œzmk1;0 C a1 zmk2;0 C    C amk1 z0;0  ı .k/ ; ƒ‚ … „ k

Pm1

with ı .0/ D ı D Hf C k0 k ı .k/ with k 2 R. Hence, in the notations of Theorem 6.9.1, the convolution algebra A D D 0C , A P D P .D/ı 2 D 0C , A1 D 1 0C .k/ 2 D 0C . Then, ŒP .D/ı D E D Hy 2 D , T D H z, B D Hf C m1 kD0 k ı P m1 1 by Theorem 6.9.1, T D H z D A B D Hy ŒHf C kD0 k ı .k/  2 D 0C (see P .k/ . Then, for x  0, Proposition 8.8.2) H) H z D Hy Hf C m1 kD0 k Hy ı Rx Pm1 .k/ z.x/ D 0 y.x/f ./d  C kD0 k y , since H./ D 0 for  < 0, H.x/ D 0 for  > x  0 Z 1 H) Hy Hf D H.x  /  y.x  /H./f ./d  1 1

Z

Z

H.x  /y.x  /f ./d  D

D 0

x

y.x  /f ./d ; 0

and Hy ı .k/ D ı .k/ Hy D ı .Hy/.k/ D Hy .k/ D y .k/ for x  0 and for 0  k  m  1 (by (3.3.29), with y .0/ D y, (y .k/ .0/ D 0 for 0  k  m  2 by virtue of the initial conditions). Remark 6.9.2. Consider Example 6.9.2 with A  E 0 .R3 /, A D ı 2 A. But A1 D  41 r … A. Hence, Theorem 6.9.1 is not applicable and we cannot deduce T D A1 B from A T D B. In fact, A1 A T is not meaningful if T does not have compact support (see Theorem 6.6.1). Now we consider the following particular cases: Case I: supp.T / is compact. For A D ı, if T has compact support, then B D A T has compact support (see (6.3.19)). Now, if T has compact support and satisfies A T D B, then T D A1 B, which is well defined. But Theorem 6.9.1 is not applicable (since A1 … A/, and consequently we cannot prove that the solution

370

Chapter 6 Convolution of distributions

is unique. In fact, it is not unique, since the corresponding homogeneous equation ı T D T D 0 has an infinite number of solutions, for example harmonic functions with non-compact support. Case II: supp.B/ is compact. Then T D A1 B is well defined, from which the equation A T D B can be deduced. Hence, ı T D T D B has a particular solution T D  41 r B, where B has compact support. Then the general solution is given by T D  41 r B C harmonic distributions. Case III: supp.B/ is not compact. In this case, we can not prove the existence of a solution by this method. Remark 6.9.3. If A1 does not exist, then for B D ı, A T D ı has no solution. But for B 6D ı, there may exist only one or more than one solution (see Schwartz [7, p. 125]). Proposition 6.9.1. Let A be a convolution algebra. If A1 ; A2 2 A have inverses 1 1 2 A and .A A /1 D A1 A1 D A1 A1 . A1 1 2 1 ; A2 2 A, then .A1 A2 / 1 2 2 1 1 Proof. A1 ; A2 2 A have inverses in A H) A1 A1 1 D ı, A2 A2 D ı. Then 1 1 1 1 .A1 A2 / .A1 A2 / D .A1 A1 / .A2 A2 / D ı ı D ı 2 A H) 1 1 1 .A1 A2 /1 D A1 1 A2 D A2 A1 .

Example 6.9.4. Let P .D/ be the differential operator defined by (6.7.6) and P .D/ı be defined by (6.7.7). Then ŒP .D/ı1 D H.x/e 1 x H.x/e 2 x    H.x/e m x , where H is the Heaviside function. x m1 In particular, for 1 D 2 D    D m D , ŒP .D/ı1 D H.x/e x  .m1/Š . Proof. ı 2 D 0 .R/ and supp.ı/  Œ0; 1Œ H) ı 2 D 0C , which is a convolution algebra A (see (6.9.15b) H) P .D/ı 2 D 0C . But, from (6.7.7),       dı dı dı  1 ı  2 ı     m ı 2 D 0C ; P .D/ı D dx dx dx dı  k ı/ being an element of D 0C (see also Section 8.8). From (3.3.24)– each . dx (3.3.30), E.x/ D H.x/y.x/ 2 D 0 .R/, where H is the Heaviside function and y is a C 1 -function which is the unique solution of Cauchy problem P .D/y D 0 with y.0/ D y 0 .0/ D    D y .m2/ .0/ D 0, y .m1/ .0/ D 1, is an elementary solution of P .D/, i.e.

P .D/E D ı

H)

P .D/ı E D ı

H)

ŒP .D/ı1 D E D Hy 2 D 0 .R/;

with supp.ŒP .D/ı1 / D supp.Hy/  Œ0; 1Œ H) ŒP .D/ı1 2 D 0C . Moreover, d d from (3.3.31), the elementary solution of . dx  k / is H.x/e k x , i.e. . dx  k /E D dı d dı ı H) . dx  k ı/ E D . dx  k /E D ı H) . dx  k ı/1 D H.x/e k x 2 D 0C .

371

Section 6.9 Convolution equations (see also Section 8.7, Chapter 8)

Now, using (6.7.7) and repeatedly applying Proposition 6.9.1, we have 1

ŒP .D/ı



    1 dı dı dı  1 ı  2 ı     m ı D dx dx dx  1  1  1 dı dı dı D   1 ı  2 ı  m ı dx dx dx D H.x/e 1 x H.x/e 2 x    H.x/e m x :

In particular, for 1 D 2 D    D m D , from (6.6.3) and (6.7.8), we have ŒP .D/ı1 D

 

D „

dı  ı dx dı  ı dx

m 1



1 1  dı dı  ı  ı  dx dx ƒ‚ … m distributions

D H.x/e x

x m1 .m  1/Š

:

Systems of convolution equations in A For the system of convolution equations with Aij 2 A, Bi 2 A, 1  i; j  m, it will be convenient to use matrix calculus and determinants of distributions in A, with necessary modifications, as follows. Convolution matrix product ŒA  ŒB in A Let ŒA D .Aij /1i;j m , ŒB D .Bij /1i;j m , with Aij ; Bij 2 A such that Aik Bkj 2 A 8i; j; k D 1; 2; : : : ; n (by virtue of (6.9.15b)). Then we define ŒA ŒB as the matrix ŒC  D .Cij /1i;j m D ŒA ŒB with Cij D

m X

Aik Bkj 2 A 8i; j D 1; 2; : : : ; m;

(6.9.19)

kD1

where the multiplication operation ‘  ’ in the usual matrix multiplication formula is replaced by convolution operation ‘ ’ [7]. ŒA ŒB ¤ ŒB ŒA

in general.

(6.9.20)

372

Chapter 6 Convolution of distributions

 B11 B12  11 A12 For example, for m D 2, ŒA D A A21 A22 , ŒB D B21 B22 ,   C11 C12 ŒC  D ŒA ŒB D C21 C22   A11 B11 C A12 B21 A11 B12 C A12 B22 D ; (6.9.21) A21 B11 C A22 B21 A21 B12 C A22 B22 where Aij ; Bij ; Aik Bkj 2 A, i; j; k D 1; 2. Convolution determinant  .A/ of ŒA in A The convolution determinant det .A/ D  .A/ of ŒA D .Aij /1i;j m with Aij 2 A is a distribution in A which is obtained by opening the determinant  .A/ in the usual way and replacing multiplication ‘  ’ by convolution ‘ ’, so that  .A/ 2 A by (6.9.15b). For example, for ŒA in (6.9.21), ˇ ˇ ˇA11 A12 ˇ  ˇ ˇ D A11 A22  A12 A21 2 A; .A/ D ˇ (6.9.22) A21 A22 ˇ where multiplication ‘  ’ is replaced by convolution ‘ ’. Then ŒA ŒB D ŒC 

H)

 .A/  .B/ D  .C / 2 A:

(6.9.23)

For the example in (6.9.21), using (6.9.15b)–(6.9.15c), we have  .A/  .B/ D .A11 A22  A12 A21 / .B11 B22  B12 B21 / D A11 A22 B11 B22  A11 A22 B12 B21 „ ƒ‚ … „ ƒ‚ … .1/

.2/

 A12 A21 B11 B22 C A12 A21 B12 B21 „ ƒ‚ … „ ƒ‚ … .3/

.4/

D .1/  .2/  .3/ C .4/   A11 B11 C A12 B21 A11 B12 C A12 B22  .ŒA ŒB/ D  A21 B11 C A22 B21 A21 B12 C A22 B22 D .1/  .2/  .3/ C .4/ C .5/  .5/ C .6/  .6/ D .1/  .2/  .3/ C .4/; where .5/ D A11 B11 A21 B12 , .6/ D A12 B21 A22 B22 . H)  .A/  .B/ D  .ŒA ŒB/ in A. Convolution cofactors Cij of Aij in  .A/ These are defined in the usual way with multiplication ‘  ’ replaced by convolution ‘ ’, i.e. convolution cofactor Cij of Aij in  .A/ is given by: Cij D .1/iCj  .ŒA.i jj // 2 A;

(6.9.24)

Section 6.9 Convolution equations (see also Section 8.7, Chapter 8)

373

where ŒA.i jj / is the matrix of order m  1 obtained from ŒA by deleting its i th row and j th column,  .ŒA.i jj // being its convolution determinant. For example, for m D 3, ŒA D .Aij /1i;j;3 , ˇ ˇ ˇ A12 ˇˇ ˇA31 A32 ˇ

2C3 ˇA11

C23 D convolution cofactor of A23 D .1/ D .A11 A32  A12 A31 / 2 A:

Inverse 1 .A/ 2 A of the determinant  .A/ in A In convolution algebra A, Dirac distribution ı 2 A plays the rôle of unity (1), since ı T D T ı D T 8T 2 A (see (6.3.21)). Hence,  .A/ 2 A has an inverse 1 .A/ 2 A ”

 .A/ 1 .A/ D ı:

(6.9.25)

Convolution inverse ŒA 1 of ŒA in A In the convolution matrix product (6.9.19), the diagonal matrix ŒıI  D .ııij /1i;j m D dı; ı; : : : ; ıc „ ƒ‚ …

m diagonal elements

plays the rôle of the identity matrix ŒI  in usual matrix multiplication, Kronecker delta, elements ı being the Dirac distribution. ıij D 1 for i D j and 0 for i ¤ j , diagonal For example, for m D 2, ŒıI  D 0ı 0ı D dı; ıc.  .ŒıI / D ı„ ı ƒ‚    …ı D ı

(by (6.3.21)):

(6.9.25a)

.n1/ convolutions

For ŒA D .Aij /1i;j m with Aij 2 A, if there exists ŒE D .Eij /1i;j m with Eij 2 A such that ŒA ŒE D ŒıI , then ŒE D ŒA1  is a convolution inverse of ŒA. Definition 6.9.3. A matrix ŒE D .Eij /1i;j m with distributions Eij 2 A is called an elementary solution of the convolution matrix equation ŒA ŒT  D ŒB in A if ŒA ŒE D ŒıI . Then ŒE is called a convolution inverse of ŒA and will be denoted by ŒA1 , i.e. ŒA ŒE D ŒA ŒA1  D ŒıI . (6.9.26) If 1 .A/ 2 A exists, then ŒE D ŒA1  exists and is given by ŒE D 1 .A/ ŒC t , with ŒC  D .Cij /1i;j m , Cij D cofactor of Aij in  .A/ defined by (6.9.26) H)

Eij D 1 .A/ Cj i ;

1  i; j  m:

(6.9.27)

374

Chapter 6 Convolution of distributions

For example, for ŒA with  .A/; 1 .A/ 2 A such that  .A/ 1 .A/ D A22 A12 ı. Then ŒC t D A and ŒE D .Eij /1i;j 2 D ŒA1  is given by 21 A11  A A ŒE D 1 .A/ A2221 A1112 such that Eij D 1 .A/ Cj i , 1  i; j  2, i.e. E11 D 1 .A/ A22 ;

E12 D 1 .A/ .A12 /;

E21 D 1 .A/ .A21 /; E22 D 1 .A/ A11 :

(6.9.28)

Then 

  1  A11 A12 .A/ A22  1 .A/ A12 ŒA ŒE D A21 A22  1 .A/ A21 1 .A/ A11  1  .A/ .A11 A22  A12 A21 / 1 .A/ .A12 A11 C A11 A12 / D 1 .A/ .A21 A22  A22 A21 / 1 .A/ .A21 A12 C A22 A11 /  1    .A/  .A/ 0 ı 0 D D D ŒıI  0 1 .A/  .A/ 0 ı H) ŒE D ŒA1  is defined by (6.9.27). System of convolution equations in A Now we consider the system (6.9.11) in A. For given Aij 2 A and P the right-hand side Bi 2 A, 1  i; j  m, find Tj 2 A, 1  j  m such that jmD1 Aij Tj D Bi , 1  i  m or, equivalently, ŒAmm ŒT m1 D ŒBm1 ;

(6.9.29)

where ŒAmm D .Aij /1i;j m , ŒT m1 D .T1 ; T2 ; : : : ; Tm /t D ŒBm1 D .B1 ; B2 ; : : : ; Bm /t : Theorem 6.9.2. For arbitrary Bi 2 A, 1  i  m, the system (6.9.29) has at least one solution ŒT m1 D .T1 ; T2 ; : : : ; Tm /t , with Ti 2 A, if and only if the convolution determinant  .A/ 2 A has an inverse 1 .A/ 2 A, i.e.  .A/ 1 .A/ D ı. Then, the inverse 1 .A/ 2 A is unique and the system (6.9.29) has a unique solution ŒT m1 for arbitrary ŒBm1 defined by: ŒT m1 D ŒEmm ŒBm1 ;

(6.9.30)

ŒEmm D ŒA1 mm being the convolution inverse of ŒAmm satisfying (6.9.26) and given by (6.9.27). Note: In the following, the sizes m  m, m  1 of square matrices and column vectors respectively will not be shown.

Section 6.10 Application in electrical circuit analysis and heat flow problems

375

Proof. Assume that for arbitrary Bi 2 A, 1  i  m, the system (6.9.29) has a solution. Then, for ŒB D ŒıIk  D .0; : : : ; 0; ı; 0; : : : ; 0/t , let ŒT  D ŒEk , 1  k  m, be a solution of the system (6.9.29), i.e. ŒA ŒEk  D ŒıIk  8k D 1; 2; : : : ; m. Hence, ŒA ŒE D ŒıI  with ŒE D ŒE1 ; E2 ; : : : ; Em , i.e. elementary solution ŒE D ŒA1 . Then ŒE is defined by (6.9.27). Moreover, using (6.9.23),  .A/  .E/ D  .ıI / D ı„ ı ƒ‚    …ı D ı 2 A .n1/ convolutions

H) 1 .A/ D  .E/ 2 A exists. Q 1 .A/ be another inverse such that  .A/ Q 1 Uniqueness of 1 .A/: Let   1 1 1 Q .A/ D ı. Then, .A/ Œ .A/  .A/ D 0 2 A. Set 0 D .A/  Q 1 .A/ 2 A with  .A/  D 0. But 0 0 D 0 ı D ı 0 D 1 .A/  .A/ 0 D 1 .A/ 0 D 0: Conversely, suppose that 1 .A/ 2 A exists. Define ŒE by (6.9.27): ŒE D 1 .A/ ŒC t . Then ŒA ŒE D ŒE ŒA D ŒıI  by (6.9.26), and ŒT  D ŒE ŒB is a solution of (6.9.29) for arbitrary ŒB with Bi 2 A. In fact, ŒA ŒT  D ŒA ŒE ŒB D ŒıI  ŒB D ŒB, i.e. ŒT  D ŒE ŒB is a solution of (6.9.29). In other words, there exists a solution ŒT  D ŒE ŒB of (6.9.29). (6.9.31) Now suppose that (6.9.29) has a solution, i.e. 9T such that (6.9.29) holds. Then, taking convolution of both sides with ŒE, we have ŒE ŒB D ŒE ŒA ŒT  D ŒıI  ŒT  D ŒT  H) ŒT  D ŒE ŒB, i.e. every solution of (6.9.29) is given by (6.9.30). (6.9.32) Hence, from (6.9.31) and (6.9.32), the uniqueness of the solution (6.9.30) follows, i.e. if 1 .A/ 2 A exists, ŒA ŒT  D ŒB ” ŒT  D ŒE ŒB.

6.10

Application of convolutions in electrical circuit analysis and heat flow problems

6.10.1 Electric circuit analysis problem [7] Let us consider an R-L-C electrical circuit (see Figure 6.4), i.e. a circuit consisting of resistor R, inductor L and capacitor C and a source of electromotive force (EMF) e.t / (see Example 6.10.1 for more details) which is switched on at t D t0 such that a current i.t / begins to flow in this circuit for t  t0 . Then the EMF e.t / defines the excitation of the circuit for t  t0 , and the current i.t / flowing in the circuit is the response of the circuit corresponding to this excitation. Both e.t / and i.t / are zero for t < 0. Excitations e.t / and their responses i.t / may not just be functions on R with e.t / D 0, i.t / D 0 for t < 0, but also distributions in D 0 C (see (6.9.15b) and Chapter 8, Section 8.8) with their supports in Œ0; 1Œ. For example, e.t / may be a transient

376

Chapter 6 Convolution of distributions

R

L

e(t)

C Figure 6.4 R-L-C electrical circuit

instantaneous excitation defined by the Dirac distribution ı, i.e. e.t / D ı 2 D 0 C , which is not a function. Thus, distributions in D 0 C are used to define excitations of the circuit in practical situations. Then the question arises as to whether distributions are necessary to also define the current in the circuit. The answer is an affirmative one. In fact, current is the derivative (in the distributional sense) of the quantity of electricity in the circuit. Suppose that a single constant charge q is passed through an ammeter at t D  . Then the quantity of electricity passed equals ´ q qH.t   / D 0

for t   for t < ;

H being the Heaviside function, and the current i is the distributional derivative (not d the usual derivative in the pointwise sense), i.e. i D dx ŒqH.t   / D qı.t   /, 0C which is a distribution in D and not a function of t . Hence, both excitations and responses of a circuit may be distributions in D 0C , rather than functions. Thus, in general, to each excitation distribution e 2 D 0 C there corresponds a response distribution i 2 D 0 C . This defines the excitation–response relationship in the circuit and, mathematically, an operator with the following properties (see Schwartz [7, pp. 134– 135]): 1. Linearity (the ‘principle of superposition’). If ik is the response to the excitation ek (k D 1; 2), then ˛1 i1 C ˛2 i2 is the response to ˛1 e1 C ˛2 e2 8˛k . 2. Translational invariance (the ‘principle of invariance’ in time). If i.t / is the response to e.t /, then i.t   / is the response to e.t   /, i.e. both undergo the same shift  in time. 3. For t < 0, a null excitation yields a null response. e.t / D 0 for t < 0 H) i.t / D 0 for t < 0. But e.t / D 0 for t >   0 H) 6 i.t / D 0 8t >  , since due to self-induction and capacitance, the response will not vanish immediately after the disappearance of excitation e.t / at t D  . Moreover, by virtue of Property 2, e.t   / D 0 8t <  H) i.t   / D 0 8t <  .

377

Section 6.10 Application in electrical circuit analysis and heat flow problems

4. Continuity. Let .ek /1 be a sequence of excitations in D 0 C and .ik /1 be kD1 kD1 C 0 0 the sequence of corresponding responses in D . Then, if ek ! e in D C as k ! 1, then ik ! i in D 0 C , where i is the response to e (for D 0 C , see Section 8.8, Chapter 8). Unit impulse and impulse response The excitation e.t / D ı.t / 2 D 0 C , where ı D ı.t / is the Dirac distribution (not a function) representing an instantaneous excitation at t D 0, is called the unit impulse, with hı; 1i D C1 (see (1.11.2) and Sections 1.1 and 1.11, Chapter 1). Then the response i.t / D E.t / of the circuit due to the unit impulse excitation ı.t / is called the impulse response. The impulse response E.t / is a function of t with E.t / D 0 for t < 0 and E.t / ¤ 0 for 0  t <  , ( > 0 being a small number), even when e.t / D 0 8t > 0 by virtue of Property 3. If a response function or distribution i corresponds to an excitation function or distribution e satisfying Properties 1–4, and E is the impulse response, then the response i is given by: i D E e: In fact, for e D ı, i D E ı D E is the impulse response.

(6.10.1) (6.10.1a)

Remark 6.10.1. Instead of the EMF e, the current i in the circuit can be taken as the excitation. Then e is the response to i and the role of the impulse response is replaced by that of the impedance ‘Z’ [7] (see Example 6.10.1 later) of the circuit such that the response e is given by: e D Z i:

(6.10.2)

Example 6.10.1. We consider the R-L-C circuit consisting of three elements: 1. Resistor R, which resists the flow of current i D i.t / in the circuit resulting in a voltage drop eR D Ri , with current i.t / in amperes and resistance R in ohms; 2. Inductor L, which opposes a change in the current i , having an inertia effect in electricity similar to that of mass in mechanics, causing a voltage drop eL D L ddti , i.e. the voltage drop eL is proportional to the instantaneous time rate of change of current i.t /, L being the constant of proportionality called inductance, measured in henrys; 3. Capacitor C , which stores energy causing a voltage drop eC proportional to the instantaneous charge Q.t / (in coulombs) on the capacitor, i.e. eC D Q=C , where C is the capacitance, measured in farads. Then dQ D i.t / H) Q.t / D dt Rt R 1 t i. /d  H) e D i. /d  . C t0 C t0

378

Chapter 6 Convolution of distributions

By Kirchhoff’s second law, the algebraic sum of all the instantaneous voltage drops around any closed loop is zero, and the total voltage drop on a closed loop (or circuit) is equal to the sum of the voltage drops in the rest of the loop. Hence, for the R-L-C circuit with resistor R, inductor L and capacitor R t C , we have the total voltage drop e.t / D eR C eL C eC D Ri C L ddti C C1 0 i. /d  . Then we have the integrodifferential equation for the excitation i.t /: for t  0, Z di 1 t Ri C L C i. /d  D e.t /; (6.10.3) dt C 0 which can be rewritten as: e.t / D .Z i /.t / (see (6.10.2));

(6.10.4)

where e.t / is the response to the excitation i.t /, with i.t / D 0 and e.t / D 0 for t < 0; Z D Rı C Lı 0 C

1 C H 2 D0 C

(6.10.5)

is a distribution called the impedance [7] of the R-L-C circuit. ı D ı.t / is the unit impulse, ı 0 D ddtı , H D H.t / is the Heaviside function with H.t / D 1 for t  0 and D 0 for t > 0. In fact, 1 1 H / i D R.ı i / C L.ı 0 i / C H i C C Z 1 1 di /C H. /i.t   /d  D Ri C L.ı dt C 1 Z Z di 1 1 1 t di D Ri C L C H.t   /i. /d  D Ri C L C i. /d ; dt C 1 dt C 0

.Z i /.t / D .Rı C Lı 0 C

since by (6.3.21) ı i D ı, and ı 0 i D ddti by (6.3.22), i.t / D 0 for t < 0 and Z 1 Z 1 H.t   /i. /d  D H.t   /i. /d  1

Z

0

Z

t

1

H.t   /i. /d  C

D Z

0

Z

t

D

1

1:i. /d  C 0

H.t   /i. /d  t

Z

t

0:i. /d  D t

i. /d  0

(H.t   / D 0 for t <  < 1). Then Z 2 D 0 C gives the response e in terms of the excitation i by convolution (6.10.4). A 2 D 0 C , called the admittance [7], is the inverse of Z in the convolution algebra 0 D C (see (6.9.26)), i.e. A D Z 1 with A Z D ı. (6.10.6) Then formula (6.10.5) corresponds to equation (6.10.4) with e D ı and i D A, i.e. ı D Z A D A Z H) A D Z 1 .

Section 6.10 Application in electrical circuit analysis and heat flow problems

379

Alternative relation Differentiating (see (6.7.1)–(6.7.3)) both sides of (6.10.6) and (6.10.5) with respect to t , we get: d .A Z/ D ı 0 dt

H) H) H)

2

0 since ı 00 A D ddt A 2 , ı A D Schwartz [7, p. 137]).

dA , dt

dZ d .Z A/ D A D ı0 dt dt   1 Lı 00 C Rı 0 C ı A D ı 0 C L

1 d 2A dA C A D ı0; CR 2 dt dt C

ı A D A,

dH dt

D ı (for more details, see

Physical interpretation of (6.10.1) Let e.t / D ı.t / be the unit impulse excitation defined by Dirac’s ı function, i.e. heuristically speaking, the excitation e.t / will be very large, of the order 1" , for a very, very short period, 0  t  " (see also page 2, Footnote 2 and Section 1.11), and e.t / D 0 for t < 0. Then the impulse response is i.t / D E.t / with E.t / D 0 for t < 0 by Property 3. By virtue of Property 2, the unit impulse excitation ı.t   / will generate the impulse response i.t / D E.t   /:

(6.10.7)

We will take recourse to the fact that any excitation e.t / is composed of impulse point excitations e. /ı.t   /, as any material body is composed of point masses (see also Section 1.11) or any current is composed of a stream of charged particles, etc. (see Schwartz [7, p. 135]). Then, any excitation e.t / can be expressed heuristically as the linear combination in the integral form of impulsive excitations e. /ı.t   / such that Z Z t e.t / D e. /ı.t   /d  D e. /ı.t   /d  (6.10.8) R

0

H) e D e ı (see Example 6.3.1). In (6.10.8) the integral has no mathematical meaning, it has a heuristic interpretation only! Then, by virtue of Property 1 and (6.10.7), the response i.t / is, again heuristically, the corresponding linear combination in the integral form of the impulse responses E.t   / such that Z Z t i.t / D e.t /E.t   /d  D e. /E.t   /d  (6.10.9) R

H)

i De E

0

(6.10.10)

is the formula giving the excitation–response relation, i.e. the impulse response E gives the response i to any excitation e by (6.10.1)/(6.10.10).

380

Chapter 6 Convolution of distributions

6.10.2 Excitations and responses defined by several functions or distributions [7] The problem of excitations and responses defined by several functions or distributions leads to a system of convolution equations (6.9.11). As an example, we consider an electrical network involving several EMFs as excitations and unknown currents as responses. For n D 3, let .ek .t //3kD1 be the functions or distributions defining EMFs which together represent the excitation of a network. Let the corresponding response of the network be defined by the functions or distributions .ik .t //3kD1 , which are the unknown currents in the network. Then, instead of (6.10.10), the excitation–response relation will now be defined by a system of three convolution equations (see (6.9.11)/ (6.9.29)) for the unknown currents i1 .t /, i2 .t / and i3 .t /: i1 D E11 e1 C E12 e2 C E13 e3 ; i2 D E21 e1 C E22 e2 C E23 e3 ; i3 D E31 e1 C E32 e2 C E33 e3 ;

(6.10.11)

which can be rewritten in matrix form: Œi D ŒE Œe

(6.10.12)

with ŒE D .Eij /1i;j 3 , Œi D .i1 ; i2 ; i3 /t , Œe D .e1 ; e2 ; e3 /t . Then ŒE ŒıI  D ŒE

(6.10.13)

is the impulse response matrix (see also (6.10.1a) and (6.9.26)), ŒI  is the identity matrix, 3 ı 0 0 ŒıI  D 40 ı 05 0 0 ı 2

is a diagonal matrix of order 3 with Dirac distributions ı as diagonal elements, all other elements being zero. For example, the response to the excitation .e1 ; e2 ; e3 /t D .0; ı; 0/t is .E12 ; E22 ; E32 /t . Then the element Ej k of the impulse response matrix ŒE is the value of the current ij for ek D ı, el D 0 for 1  l ¤ k  3, i.e. E32 is the value of i3 for e1 D 0, e2 D ı, e3 D 0, or E13 is the value of i1 for e1 D 0, e2 D 0, e3 D ı.

Section 6.10 Application in electrical circuit analysis and heat flow problems

381

Application of convolution in the problem of heat flow in a rod [7] Case of several excitations causing a single response Consider a one-dimensional heat-flow problem in a rod AB of length l with end points A and B defined by x D 0 and x D l respectively. In the rod, heat is transmitted only by conduction along the axis of the rod from x D 0 to x D l, i.e. there is no heat exchange by radiation or convection with the exterior of the rod. For this, the rod is assumed to be completely insulated everywhere except at the ends A and B, where the sources of heat are placed. A time-varying quantity of heat is being transmitted by these sources and received by the rod without any heat loss from the initial instant t D 0. Let q1 .t / (resp. q2 .t /) be the quantity of heat transmitted by the heat source at the end A (resp. B) and received by the rod per unit time at the instant t . Then the total amount of heat received during Œ0; t  from the heat source at A (resp. B) is given by: Z t Z t q1 . /d  .resp. q2 . /d  /: (6.10.14) 0

0

The temperature of the rod will change due to the flow of heat along the rod by conduction, since there will be no loss of heat due to radiation or convection. Let u D u.x; t / denote the temperature at the point x 2 Œ0; l and at the instant t  0. It is assumed that the temperature in the rod is zero for t < 0, i.e. u.x; t / D 0 for t < 0, 8x 2 Œ0; l. Hence, u.x; t / will denote the change in temperature, and it is required to find u.x; t / in the rod for t  0 and x 2 Œ0; l. Excitation by an instantaneous source of heat and the resultant impulse response Let q1 .t / D ı.t / 2 D 0C (resp. q2 .t / D ı.t / 2 D 0C ) denote the instantaneous excitation heat source placed at A (resp. B) at t D 0. Such a heat source at A or B will also be called the unit (excitation) impulse ı.t /, with hı; 1i D 1. Due to the unit excitation impulse ı.t /, there will be a change in temperature of the rod, which will be called the impulse response and denoted by E1 .x; t / (resp. E2 .x; t /) corresponding to q1 .t / D ı.t / at A (resp. q2 .t / D ı.t / at B), since the change in temperature will also depend on x. For fixed x 2 Œ0; l, E1 .x; t / (resp. E2 .x; t /) is a function of t , with Ei .x; t / D 0 8t < 0, but Ei .x; t / ¤ 0 for 0  t <  ,  being a very small number, even when qi .t / D 0 8t > 0, (i D 1; 2) (Properties 1–4 of the R-L-C circuit analysis also hold in this case). Moreover, the unit excitation impulse ı.t   / will yield the impulse response Ei .x; t   / by Property 2. Let q1 .t / (resp. q2 .t /) denote an arbitrary excitation heat source at A (resp. B) which is not necessarily a unit excitation impulse. Then the response u1 (resp. u2 ) due to q1 at A (resp. q2 at B) is given by: u1 D E1 q1

.resp. u2 D E1 q2 /:

(6.10.15)

In fact, for qi .t / D ı.t /, ui D Ei ı D Ei (i D 1; 2). Hence, the total response u due to two arbitrary excitation heat sources q1 at A and q2 at B is given by the

382

Chapter 6 Convolution of distributions

algebraic sum u1 C u2 of the responses u1 ; u2 in (6.10.15), i.e. u.x; t / D E1 .x; / q1 .  / C E2 .x;  / q2 .  / Z t Z t D E1 .x; t   /e1 . /d  C E2 .x; t   /e2 . /d : 0

0

Here we meet with the case of two excitations causing one response.

Chapter 7

Fourier transforms of functions of L1.Rn/ and S.Rn/

7.1

Fourier transforms of integrable functions in L1 .Rn /

Let x D .x1 ; x2 ; : : : ; xn / and  D .1 ; 2 ; : : : ; n / 2 Rn be any two points in Rn with their inner product hx; i D x   D x1 1 C x2 2 C    C xn n . Then, for functions f 2 L1 .Rn /, i.e. Z Z jf .x/jd x D jf .x1 ; x2 ; : : : ; xn /jdx1 dx2 : : : dxn < C1; (7.1.1) Rn

Rn

the Fourier transform fO D F Œf  of f is defined by: Z fO./ D F Œf .x/ D f .x/e i2hx;i d x:

(7.1.2)

Rn

Similarly, we can define the Fourier co-transform (also called Fourier transform) FN Œf  of f 2 L1 .Rn / by: Z FN Œf ./ D f .x/e i2hx;i d x: (7.1.3) Rn

R

R In fact, for f 2 L1 .Rn /, j Rn f .x/e i2hx;i d xj  Rn jf .x/jd x < C1, since je i2hx;i j D je i2x1 1 j  je i2x2 2 j    je i2xn n j D 1. Hence, both definitions (7.1.2) and (7.1.3) are well defined for f 2 L1 .Rn / and we have: Theorem 7.1.1. Every function f 2 L1 .Rn / has a Fourier transform fO D F f and Fourier co-transform FN f defined by (7.1.2) and (7.1.3) respectively.  fO W Rn ! C is a complex-valued function of n real variables  ;  ; : : : ;  1 2 n with



 D .1 ; 2 ; : : : ; n / 2 Rn ; fO./ 2 C: Z f .x/d x: fO.0/ D

(7.1.4) (7.1.5)

Rn 

bk1 D kfOkL1 .Rn /  kf k 1 n . fO is bounded in Rn with kf L .R /

(7.1.6)

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

384 In fact, 8 2 Rn , ˇZ ˇ O jf ./j D ˇˇ

f .x/e

i2hx;i

Rn

ˇ Z ˇ d xˇˇ 

Rn

jf .x/jd x D kf kL1 .Rn / :

Hence, kfOk1 D kfOkL1 .Rn / D sup jfO./j  kf kL1 .Rn / :

(7.1.7)

2Rn

Remark 7.1.1. We will show later that for certain classes of functions f (for example, for functions f 2 S.Rn /  L1 .Rn /) (see Definition 7.2.3), the Fourier cotransform FN D F 1 (i.e. the inverse of F ) and F D .FN /1 (i.e. the inverse of FN ) are such that FN F f D F FN f D f 8f 2 S.Rn / (see Theorem 7.7.1). FN .F f / D FN fO is not defined 8f 2 L1 .Rn /, since fO … L1 .Rn / for f 2 L1 .Rn / in general (see Example 7.1.1). In particular, for n D 1 and f 2 L1 .1; 1Œ/, Z 1 fO./ D F Œf .x/ D f .x/e i2x dx; (7.1.8) FN Œf .x/ D

Z

1 1

f .x/e i2x dx:

(7.1.9)

1

Example 7.1.1. Let f be defined by: ´ f .x/ D

1 0

for jxj < 1 otherwise.

R1 R1 Then 1 jf .x/jdx D 1 1dx D 2 < C1 H) f 2 L1 .1; 1Œ/ and its Fourier transform  ˇxDC1 Z 1 ˇ 1 i2x i2x ˇ O f ./ D F Œf .x/ D e 1e dx D ˇ i 2 1 xD1 D

1 e i2  e i2 sin.2/ D  2i 

. ¤ 0/

and, for  D 0, fO.0/ D

Z

Z

1

f .x/dx D 1

1

2 sin.2/ 1dx D 2 D lim fO./ D lim 2 !0 !0 1

H) fO is continuous at  D 0. is continuous for all . Hence, fO is continuous on 1; 1Œ. But fO./ D si n.2 /  But fO is not summable on R D1; 1Œ, since ˇ ˇ Z 1ˇ Z 1 Z 1ˇ ˇ sin 2 ˇ ˇ sin.2/ ˇ O ˇ ˇ ˇ ˇ jf ./j D ˇ  ˇd  D 2 ˇ 2 ˇd  ! 1 1 1 1

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

385

H) fO … L1 .1; 1Œ/ H) Fourier co-transform FN fO of fO (resp. Fourier transform F fO of fO) does not exist, i.e. FN F f (resp. F FN f ) of this function f is not defined.

-a + i A

a+i

i

D

B

C 0

-a

a

Figure 7.1 Path of integration along ABCD from A D a C i  to D D a C i 

Example 7.1.2 (Fourier transform fO is the function f itself, i.e. fO D f ). For 2 2 f .x/ D e x 8x 2 R, F f D f , i.e. fO./ D f ./ D e  . Proof. Here, we give a proof based on the theory of analytic functions of a complex R1 2 variable. For an alternative proof, see page 421 later in this chapter. 1 e x dx D 2 1 (the Gauss integral) H) e x 2 L1 .1; 1Œ/ H) its Fourier transform fO exists and is given by: Z 1 Z 1 2 2 2 2 fO./ D e x  e i2x dx D e   e Œ.x  /Ci2x  dx 1

De

 2

Z

1 1

e

.xCi /2

dx:

1

For the evaluation of the last integral for fixed , we introduce complex variable z D x C iy such that for fixed y D , z D x C i  with dz D dx and fO./ D R aCi 2 2 e  Œlima!1 aCi e z dz. 2

Since e z is analytic everywhere, the contour integration along fixed  from a C i  to a C i  with a > 0 (see Figure 7.1) can be replaced by: Z lim

aCi

a!1 aCi

e

z 2

Z

a

dz D lim

a!1

Z

e aCi aCi

C a

z 2

Z

a

dz C a

2

e z dz



2

e x dx

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

386

Z

a

D lim

a!1 aCi

Z

e

z 2

aCi

C lim

a!1 a

Z

2

e z dz;

0

2

2

e x dx

1

where the first and third integrals vanish. For example, for the third integral, ˇ Z aCi ˇ ˇZ ˇ ˇ Z ˇ ˇ ˇ ˇ ˇ z 2 .aCiy/2 ˇ ˇ ˇ e dz ˇ D ˇ e idy ˇˇ  ˇˇi ˇ a

1

dz C



e

.a2 y 2 /i2ay

0

ˇ ˇ dy ˇˇ

2

 .e a e  jj/ ! 0 as a ! 1 for fixed  ( may be positive or negative, although  is positive in Figure 7.1). R aCi R1 2 2 2 Hence, lima!1 aCi e z dz D 1 e x dx D 1, and fO./ D e   1 D 2 e  D f ./ 8 2 R H) fO D f . Counterexample 7.1.3. For n D 1, f .x/ D 1 8x R2 R, Fourier transform fO D 1 F f (resp. co-transform FN f ) does not exist, since 1 1  dx D 1 H) 1 … 1 L .1; 1Œ/. Similarly, f .x/ D x n or e x or sin x etc. does not possess a Fourier transform fO D F f (resp. co-transform gO D FN f ), since f … L1 .1; 1Œ/. In other words, many elementary functions do not possess a Fourier transform and co-transform as defined by (7.1.2) and (7.1.3) respectively. This suggests that we should develop new definitions of Fourier transform to overcome such annoyances; these will be introduced in Chapter 8. Counterexample 7.1.4. For n D 1, consider f1 .x/ D 1 ; 8x 1Cx 2

f2 .x/ D .f1 .x//2 D

2 1; 1Œ. Z

1

p 1

H)

p 1 , 1Cx 2

p 1 1Cx 2

Z

dx 1 C x2

D

 2  2

sec d D 1 .setting x D tan /

… L1 .R/. Z

1 1

1 dx D 1 C x2

Z

 2  2

d D 

.setting x D tan /

1 1 H) 1Cx 2 2 L .R/. Hence,

p

1 1C

x2

… L1 .R/;

1 but p 2 L2 .R/: 1 C x2

(7.1.10)

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

387

1 Therefore, the Fourier transform of p 1 2 does not exist, whereas f2 .x/ D 1Cx 2 1Cx has a Fourier transform. (See Remark 8.3.1 for a definition of the Fourier transform of L2 -functions.)

Alternative definitions of Fourier transforms Although we have accepted and followed Schwartz’s definition of Fourier transform fO./ D F Œf .x/ of f 2 L1 .Rn / in (7.1.2), various forms of definition of Fourier transforms are found in mathematical literature. For example, for f 2 L1 .Rn /, the Fourier transform of f can also be defined by any one of the following alternative formulae: Z 1 I. gO 1 ./ D p f .x/e ihx;i d x; or (7.1.11) . 2/n Rn Z 1 gO 2 ./ D p f .x/e ihx;i d xI . 2/n Rn Z II. gO 3 ./ D f .x/e ihx;i d x; or (7.1.12) Rn Z f .x/e ihx;i d x; or gO 4 ./ D n R Z 1 gO 5 ./ D f .x/e ihx;i d x; etc. .2/n Rn Other choices of definition are also possible. Relations between fO and gO i Let fO D F f be defined by (7.1.2). Then the Fourier transforms gO i (i D 1; 2; 3; 4; 5) are related to fO by the following formulae:       1 1 1 1 1 O O O  I gO 2 ./ D p  I gO 3 ./ D f  I f f gO 1 ./ D p 2 2 2 . 2/n . 2/n     1  1 1 O O gO 4 ./ D f  I gO 5 ./ D Dp f gO 1 ./: (7.1.13) n 2 .2/ 2 2/n For example,

H)

  Z ˙ fO D f .x/e ihx;i d x 2 Rn   Z 1 1 1 O  D p f f .x/e ihx;i d x D gO 1 ./ p 2 . 2n/n . 2/n Rn   Z 1 1 1 O  D p f f .x/e ihx;i d x D gO 2 ./: p 2 . 2n/n . 2n/n Rn

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

388 Remark 7.1.2. 



The multiplying factor p 1 n is introduced for the convenience of some com. 2/ putations in applications such as in multi-dimensional Fourier series, and the associated definition is followed in many applied books. We have accepted Schwartz’s definition for the sake of elegance in many of the results and formulae to be obtained later. For example, the Fourier transform of Dirac distribution ı will be 1 and the Fourier co-transform of 1 will be ı as tempered distributions (see Chapter 8).

Properties of Fourier transform F and co-transform FN All the properties of Fourier transform F will also hold for the co-transform FN , since by replacing ‘i ’ by ‘i ’ in the proofs for the case of F defined by (7.1.2), the corresponding results for the case of the co-transform FN defined by (7.1.3) can be obtained. Hence, we will state and prove the results only for Fourier transform F . Property 1 Fourier transform F is linear. F Œ˛1 f1 C ˛2 f2  D ˛1 F f1 C ˛2 F f2

8f1 ; f2 2 L1 .Rn /; 8˛1 ; ˛2 2 R: (7.1.14)

In fact, Z

.˛1 f1 C ˛2 f2 /.x/e i2hx;i d x Z Z D ˛1 f1 .x/e i2hx;i d x C ˛2 f2 .x/e i2hx;i d x

F Œ˛1 f1 C ˛2 f2 ./ D

Rn

Rn

Rn n

D .˛1 F f1 C ˛2 F f2 /./ 8 2 R ; 8˛1 ; ˛2 2 R: Property 2 Let ¹fi ºniD1 be a set of n functions such that fi D fi .xi / 8xi 2 R, R 1 .R/ and fO . / D .F f /. / D 1 f .x /e i2xi i dx . 1  i  n, fi 2 LN i i i i i 1 i i Then, for f D niD1 fi D f1 ˝ f2 ˝    ˝ fn defined by f .x/ D f .x1 ; x2 ; : : : ; xn / D f1 .x1 /f2 .x2 / : : : fn .xn / 8x D .x1 ; x2 ; : : : ; xn /; Z F .f /./ D f .x/e i2hx;i d x Rn Z D f1.x1 /f2.x2 /  fn .xn /e i2.x1 1 Cx2 2 CCxn n / dx1 dx2 : : : dxn Rn DRRR

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

Z

f1 .x1 /e i2x1 1 dx1

D R

Z

f2 .x2 /e i2x2 2 dx2   

R

fn .xn /e i2xn n dxn R

D .F f1 /.1 /  .F f2 /.2 /    .F fn /.n / H)

Z

389

8 D .1 ; 2 ; : : : n /

F f D .F f1 / ˝ .F f2 / ˝    ˝ .F fn /:

(7.1.15) 2

2

2

2

Example 7.1.5. f .x/ D f .x1 ; x2 ; : : : ; xn / D e kxk D e x1 x2 xn 2 H) f .x/ D f1 .x1 /  f2 .x2 /    fn .xn / with fi .xi / D e xi 8xi 2 R, fOi D F fi D fi (see Example 7.1.2) H)

fO./ D .F f /./ D .F f1 /.1 /  .F f2 /.2 /    .F fn /.n / 2

2

2

D fO1 .1 /fO2 .2 /    fOn .n / D e  1  e  2    e  n 2

2

n

2

D e . 1 C 2 CC n / D e kk D f ./ 8 2 Rn H) F f D .F f1 / ˝ .F f2 / ˝ : : : .F fn / D f . 2 Similarly, for f .x/ D e kxk , FN f D f;

(7.1.16)

2

since FN f .i / D F f .i / D e  i for 1  i  n 2 H) FN f ./ D F f ./ D e kk D f ./ 8  2 Rn . Property 3 Let fL be the function defined by fL.x/ D f .x/ for x 2 Rn . Then, I. FN .fL/ D F f ; II. F .fL/ D FN f ; III. FN .f / D F f for complex-valued f ; IV. .F f /_ D FN f , where .F f /_ ./ D .F f /./; V. .FN f /_ D F f .

(7.1.17)

Proof. I. From the definition (7.1.3) of FN , we have Z Z Ci2hx;i L L N .F .f //./ D f .x/e dx D Rn

f .x/e Ci2hx;i dx: Rn

By change of variables x D y with Jacobian J , jJ j D 1, we have 8 2 Rn , Z Z Ci2hy;i L N .F .f //./D f .y/e jJ jd yD f .y/e i2hy;i d yD.F f /./ Rn

H) FN .fL/ D F f .

Rn

390

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

II. F .fL/ D FN .fL/_ D FN f , since .fL/_ .x/ D fL.x/ D f .x/ for x 2 Rn . III. For complex-valued f .x/, Z Z Ci2hx;i N F .f / D f .x/e dx D Rn

Z D

f .x/e i2hx;i d x Rn

f .x/e i2hx;i d x D F f :

Rn

IV.

Z .F f /_ ./ D .F f /./ D f .x/e i2hx;i d x n R Z D f .x/e i2hx;i d x D FN f: Rn

V. .FN f /_ D ..F f /_ /_ D F f . Property 4 Let A W Rn ! Rn be an invertible linear mapping defined by a square matrix of order n, and f  A W Rn ! R be defined, 8f 2 L1 .Rn /, by .f ı A/.x/ D f .Ax/ a.e. in Rn . Then F Œf ı A./ D

1 .F f /.At /; j det Aj

i.e. F Œf ı A D

1 Œ.F f / ı At ; j det Aj

(7.1.18)

where At D .A1 /t is the transpose of A1 . 1 for this Proof. Set y D Ax. Then, x D A1 y with Jacobian J D det.A1 / D det.A/ 1 t change of variables, and hx; i D hA y; i D hy; A i. Hence, Z Z i2hx;i .f ı A/.x/e dx D f .Ax/e i2hx;i d x F Œf ı A./ D n n R R Z t f .y/e i2hy;A i jJ jd y D Rn Z 1 t D f .y/e i2hy;A i d y j det.A/j Rn 1 D .F f /.At / 8 2 Rn j det Aj 1 .F f ı At /: H) F Œf ı A D j det Aj

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

391

Particular cases (a) y D kx with k 2 R; k ¤ 0 H) A D k I , I being the identity matrix of order n H) A1 D k1 I and det.A/ D k n . Then      1 2 1 n 1 F Œf .kx/./ D ; ;:::; .F f / .F f / D : (7.1.19) jkjn k jkjn k k k For n D 1, f is even (resp. odd) H) fO is even (resp. odd). Indeed, f is even H) f .x/ D f .x/ for x 2 R H) fO./ D F Œf .x/ D F Œf .x/ D fO./ with k D 1 (from (7.1.19)) H) fO./ D fO./ H) fO is even. (b) y D S x, where A D S is the rotation matrix of order n. Then S S t D I , det.S / D 1, S 1 D S t H) S t D S . Hence, F Œ.f ı S/.x/./ D .F f /.S/. (c) If f is a radial function, then its Fourier transform will also be a radial function. 1 For r D .x12 C x22 C    C xn2 / 2 , let f .x/ D f .x1 ; x2 ; : : : ; xn / be a radial function ˆ of r only: f .x1 ; x2 ; : : : :; xn / D ˆ.r/. Then its Fourier transform 1 .F f / D fO./ will be a radial function ‰ of D .12 C 22 C    C n2 / 2 only, i.e. .F f /./ D ‰. /:

(7.1.20)

Proof. It is sufficient to show that fO./ is invariant under rotation about the origin. Let S be such a rotation matrix. Then, 8x 2 Rn with kxk D r, kS xk2 D hS x; S xi D hx; S t S xi D hx; xi D kxk2 D r 2 (since S t S D I ). f is a radial function of x H) f .x/ D f .S x/, since f .x/ is invariant under the rotation about the origin defined by S . Hence, f .S x/ D f .x/ D ˆ.r/. Now we will show that fO./ is also invariant under the rotation about the origin defined by S , i.e. fO.S / D fO./. Set D S , i.e. the point R  is carried to the point by the rotation defined by S . Hence, fO. / D fO.S / D Rn f .x/e i2hx;Si d x. But the inner product h  ;  i is invariant under S 1 , and hx; Si D hS S 1 x; S i D R 1 hS 1 x; S t S i D hS 1 x; i. Hence, fO.S / D Rn f .x/e i2hS x;i d x. By change Rof variables y D S 1 x suchRthat x D Sy with Jacobian J D det.S / D 1, fO.S/ D Rn f .S y/e i2hy;i d y D Rn f .y/e i2hy;i d y D fO./, since f is invariant under rotation about the origin, i.e. f .y/ D f .S y/. So, fO.S/ D fO./ H) fO is invariant under rotation about the origin. Hence, 1 we can write fO./ D ‰. / with D .12 C 22 C    C n2 / 2 . We refer to [7] for finding the explicit formula of ‰. / as the Fourier transform of ˆ.r/ in terms of Bessel functions.

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

392

Property 5 Let .a f /.x/ be the translation of f .x/ by a 2 Rn , a f .x/ D f .x  a/;

(7.1.21)

and e ˙i2ha;  i be denoted by ˙a .  /, such that ˙a .x/ D e ˙i2ha;xi ;

˙a ./ D e ˙i2ha;i :

(7.1.22)

Then

b f D  fO. II. b

I. a f D a fO; a

(7.1.23)

a

Proof.

b

I. a f ./ D

R

Rn .a f

/.x/e i2hx;i d x D

R

Rn

f .x  a/e i2hx;i d x.

Set y D x  a. Then x D y C a with Jacobian J D jI j D 1, and Z f .x  a/e

i2hx;i

Z dx D

Rn

f .y/e i2hyCa;i jJ jd y

Rn

Z

f .y/e i2Œhy;iCha;i d y Z i2ha;i De f .y/e i2hy;i d y D a ./fO./

D

Rn

Rn

b b

b

H) a f ./ D a ./fO./ 8 2 Rn H) a f D a fO. Z Z II. a f ./ D . a f /.x/e i2hx;i d x D e i2ha;xi f .x/e i2hx;i d x n n R R Z D f .x/e i2hx;ai d x D fO.  a/ D a fO./ 8 2 Rn Rn

b

H) a f D a fO. In order to study the deeper properties of Fourier transforms of functions f 2 L1 .Rn /, which are continuous or differentiable, we need: Some auxiliary results on the continuity and differentiability of integrals Definition 7.1.1A. A functionf D f .x; y/ with x; y 2 R is called separately (or partially) continuous with respect to each variable x and y if and only if f is continuous with respect to one variable when the other variable is fixed.

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

393

The separate (or partial) continuity of f does not imply its continuity, but the continuity of f in .x; y/ implies its separate continuity with respect to x and y. For example, f defined by: ´ f .x; y/ D

xy x 2 Cy 2

for .x; y/ 6D .0; 0/

0

for .x; y/ D .0; 0/

is separately continuous at .0; 0/, but f is not continuous at .0; 0/. In fact, for fixed y, ´ fy .  / W x 7! fy .x/ D f .x; y/ D

xy x 2 Cy 2

for x 6D 0

0

for x D 0

H) limx!0 fy .x/ D 0 D fy .0/ H) fy .  / is continuous at x D 0 for fixed y. Similarly, fx .  / is continuous at y D 0 for fixed x. Hence, f is separately continm uous at .0; 0/. But f .x; mx/ D 1Cm 2 6D 0 for m 6D 0 m H) limx!0 .m6D0/ f .x; mx/ D 1Cm 2 6D f .0; 0/ H) f is not continuous at .0; 0/. Definition 7.1.1A holds when f D f .x; y/ with x 2 Rn ; y 2 Rm . Rb Define function F W I  R ! R by F .x/ D a f .x; t /dt 8x 2 I  R such that the integral exists 8x 2 I , the interval a; bΠ R (resp. I  R) being a finite or infinite one. The continuity, differentiability and integrability of F will be dependent on the analogous properties of f , which will be stated now. We agree to accept the following theorems without proof. Continuity of the integral

F .x/ D

Rb a

f .x; t /dt

Theorem 7.1.2A (Lebesgue). Let f D f .x; t / be separately continuous with respect to x at x D x0 2 I  R for almost all t 2 a; bŒ, a; bŒ  R being an arbitrary finite or infinite interval of integration. If g  0 is a positively valued function of Rb t such that jf .x; t /j  g.t / for almost all t 2 a; bŒ and a g.t /dt < C1, then Rb F .x/ D a f .x; t /dt is continuous at x D x0 2 I . Differentiability of the integral

F .x/ D

Rb a

f .x; t /dt

Theorem 7.1.2B (Lebesgue). Let I D Œ˛; ˇ  R be a finite interval with ˛  x  ˇ. D fx0 .x; t / be the partial derivative of f for almost all t 2 a; bŒ such Let @f @x 0 that fx .x; t / is separately continuous with respect to x for almost all t 2 a; bŒ and Rb jfx0 .x; t /j  g.t / for almost all t 2 a; bŒ, g.t /  0 with a g.t /dt < C1. Then, if

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

394 F .x/ D

Rb a

f .x; t /dt exists for a particular value of x D x0 , the following hold:

I. F .x/ exists for any x 2 Œ˛; ˇ; II. F is continuous and differentiable and the derivative F 0 .x/ is obtained by difR Rb ferentiating under the integral sign , i.e. F 0 .x/ D a fx0 .x; t /dt . F .x/ D

Integrability of the integral

Rb a

f .x; t /dt

Theorem 7.1.2C (Fubini). Let f be summable on rectangle ˛; ˇŒ  a; bŒ, i.e. ’ ˛;ˇ Œa;bŒ jf .x; t /jdxdt < C1. Then Rˇ I. F is summable on ˛; ˇŒ, i.e. ˛ jF .x/jdx < C1; Z ˇ Z ˇ Z b Z b Z ˇ II. F .x/dx D dx f .x; t /dt D dt f .x; t /dx ˛

˛

a

“

a

˛

f .x; t /dxdt

D

(7.1.23a)

˛;ˇ Œa;bŒ

(i.e. interchange of the order of integration in the iterated integral is possible). Instead of giving the proof of Fubini’s Theorem 7.1.2C (see, for example, [20], [30], [33] for proof), we give some interesting counterexamples to highlight the different situations which may arise. For example, if f .x; y/ is not summable on ˛; ˇŒa; bŒ, R i.e. ˛;ˇ Œ  a;bŒ jf .x; y/jdxdy D 1, it may happen that one of the iterated integrals Rˇ Rb Rb Rˇ in (7.1.23a), ˛ dx a f .x; y/dy or a dy ˛ f .x; y/dx, may exist and the other one may not exist, or both the iterated integrals may exist with equal or unequal values, i.e. (7.1.23a) does not hold in all these cases. Example 7.1.6. Let  D 0; 1Œ  0; 1Œ  R2 and f .x; y/ D Then R 2 y 2 j 1.  .xjx2 Cy 2 /2 dxdy does not exist.

x 2 y 2 .x 2 Cy 2 /2

2. The iterated integrals have different values, i.e. Z

1Z 1

I1 D 0

Z

0 1Z 1

I2 D 0

0

 x2  y2 dx dy D =4I .x 2 C y 2 /2  x2  y2 dy dx D =4: .x 2 C y 2 /2

8.x; y/ 2 .

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

395

Proof. 1. Consider the first quadrant Q1 of the unit circle or ball B.0I 1/ with radius 1 such that Q1  , and introduce polar coordinates with x D r cos , y D r sin , jJ j D r, 0 < r < 1, 0 < < =2. Then Z

Z Z 1 Z =2 2 jx 2  y 2 j jx 2  y 2 j r j cos j dxdy > dxdy D rdrd

2 C y 2 /2 2 C y 2 /2 .x .x r4  Q1 0 0  Z =2 Z 1 Z 1  Z =4 1 dr D1 cos 2 d C . cos 2 /d dr D D r 0 r 0 =4 0

H) Œ.x 2  y 2 /=.x 2 C y 2 /2  is not summable/integrable on . R1 2. I1 D 0 I1 .y/dy, with Z

1

I1 .y/ D Z

0 1

D 0

Set u D

1 x 2 Cy 2

Z 1 2 x2  y2 x C y 2  2y 2 dx D dx .x 2 C y 2 /2 .x 2 C y 2 /2 0 Z 1 dx dx 2  2y : 2 C y 2 /2 x2 C y2 .x 0

with du D

2x dx, .x 2 Cy 2 /2

vD

R

dx D x. Then

Z

Z dx x 2x 2 D C dx x2 C y2 x2 C y2 .x 2 C y 2 /2 Z Z x dx dx 2 D 2 C2  2y x C y2 x2 C y2 .x 2 C y 2 /2 Z Z dx dx x 2 H)  2y D 2 2 2 2 2 2 x Cy .x C y / x C y2 ˇ1 ˇ x ˇ D 1 H) I1 .y/ D  2 2 x C y ˇ0 1 C y2 Z 1 1 H) I1 D  dy D  tan1 .y/j10 D =4: 2 1 C y 0

Similarly, I2 D Z I2 .x/ D 0

1

R1 0

I2 .x/ with

x2  y2 dy D  .x 2 C y 2 /2

Z 0

1

y2  x2 dy D .=4/ D =4; .y 2 C x 2 /2

which is obtained by replacing ‘x’ by ‘y’ and vice versa.

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

396

Example 7.1.7. Let  D 1; 1Π 1; 1Π R2 be a square domain and xy f .x; y/ D .x 2 Cy 2 /2 8.x; y/ 2 . Then R 1.  jf .x; y/jdxdy does not exist; 2. but the corresponding iterated integrals exist and are equal, i.e. Z

1

Z

1

 Z f .x; y/dx dy D

yD1

yD1

1

Z



xD1

f .x; y/dy dx D 0:

1

xD1

Proof. 1. Consider the unit circle or ball B.0; 1/  . Introducing polar coordinates, x D r cos , y D r sin , jJ j D r, we get Z

Z

Z

jf .x; y/jdxdy > 

1 Z 2

r 2 j cos sin j rdrd

r4 0 0  Z 1 j sin 2 j 1 d dr D 2 dr D 1; 2 0 r

jf .x; y/jdxdy D B.0;1/

Z D

0

1

1 r

Z 0

2

R =2 R  R 3=2 R 2 R 2 C 3=2 D 1 C 1 C 1 C 1 D 4, since 0 j sin 2 jd D 0 C =2 C  R H)  jf .x; y/jdxdy D 1. Hence, f .x; y/ is not summable/integrable on  in the sense of Lebesgue.   Z 1  Z xD1 Z 1  Z xD1 xy y 2x 2. dx dy D dx dy 2 2 2 2 2 2 1 xD1 .x C y / 1 2 xD1 .x C y /  ˇxD1 Z Z ˇ 1 1 1 1 1 ˇ y dy D y  0 D 0: D 2 1 x 2 C y 2 ˇxD1 2 1 Similarly,

R 1 R yD1 1 . yD1

xy dy/dx .x 2 Cy 2 /2

D 0.

Example 7.1.8. Let  D 0;R1Œ  0; 1Œ  R2 and f .x; y/ D e xy sin x 8.x; y/ 2 . Then the double integral  jf .x; y/jdxdy does not exist, but the corresponding iterated integrals exist and are equal. R R Proof.  jf .x; y/jdxdy does not exist, since if  jf .x; y/jdxdy < C1, then by Fubini’s Theorem 7.1.2C the corresponding iterated integrals in (7.1.23a) exist and are equal, i.e.   Z 1  Z xD1 Z 1  Z yD1 jf .x; y/jdx dy D jf .x; y/jdy dx < C1: 0

xD0

0

yD0

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

In fact, Z

1  Z y!1 0

 Z j sin xje xy dy dx D

yD0

Z

1

D 0

y!1

j sin xj

0

Z

397

 e xy dy dx

yD0 1

j sin xj dx D 1: x

But Z

1 0

1 Z .nC1/ 1 Z  X X j sin xj j sin xj sin t dx D dx D dt; x x n Ct n 0 nD0

nD0

which is obtained by the change of variable x D n C t . Since .n C t /  .n C 1/ 8t 2 Œ0; , we have, 8n 2 N0 : ˇ Z  Z  ˇ sin t sin t 1 2 dt  dt D . cos t /ˇˇ D .n C 1/ .n C 1/ 0 n C t 0 .n C 1/ 0 Z Z 1 1 1 X  sin t 2 X 1 j sin xj dx D dt  D 1; H) x  nC1 0 0 n C t nD0

nD0

R1

j sin xj x dx

since 1 C 12 C    C n1 C    D 1. Hence, 0 D 1. So the iterated R 1 R y!1 integral 0 . yD0 j sin xje xy dy/dx does not exist. Consequently, the double inR1R1 xy jdxdy does not exist by Fubini’s Theorem 7.1.2C. Hence, tegral 0 0 j sin xe R1R1 the double integral 0 0 sin xe xy dxdy does not exist in the sense of Lebesgue. But as an improper Riemann integral, we have Z yD1   xy ˇyD1 Z 1 Z 1 Z 1 ˇ e sin x xy ˇ dx D=2: sin x e dy dx D sin x dx D ˇ x x 0 yD0 0 0 yD0   ˇ Z 1  Z xD1 Z 1 y2 sin xe xy cos xe xy ˇˇxD1 xy  e sin xdx dy D dy  ˇ 2 y y2 0 xD0 0 1Cy xD0 ˇ1 Z 1 ˇ dy 1 D tan .y/ˇˇ D =2: D 2 1Cy 0

0

Now we state the additional properties of Fourier transform fO D F f of f 2 L1 .Rn /. Property 6 fO is continuous in Rn , i.e. fO 2 C 0 .Rn /.

(7.1.24)

Proof. f .x/e i2hx;i is separately continuous with respect to  forRalmost all x 2 n i2hx;i j D jf .x/j. Choose g.x/ D jf .x/j with R Rn g.x/d x D R and jf .x/e Rn jf .x/jd x 0 (Suppose that the contrary holds, i.e. limx!1 R1 1 L j 8x > M H) such that jf .x/j > j L M jf .x/jdx  M j 2 jdx, but 2 R1 L R1 M j 2 jdx D 1 for L 6D 0 H) R M jf .x/jdx D 1, which contradicts 1 that f is summable on 1; 1Œ: 1 jf .x/jdx < C1. Hence, our assumption is wrong, i.e. limx!1 f .x/ D L D 0. Similarly, we can prove that limx!1 f .x/ D 0.)

Property 11 The Riemann–Lebesgue Property. Let f 2 L1 .Rn / and fO./ D .F f /./ be its Fourier transform. Then fO./ ! 0 as kk ! 1. (7.1.36) Proof. Let  2 D.Rn /  L1 .Rn /, D.Rn / being a dense subspace of L1 .Rn /. Then, from (7.1.33) with ˛i D 1, 1  i  n, we have O i 2i ./ D

b

Z Rn

@ @ .x/e i2hx;i d x D ./: @xi @xi

Hence, for i 6D 0,

b

ˇ ˇ ˇ @ ˇ 1 ˇ O j./j D ./ˇˇ: ˇ 2ji j @xi From (7.1.6),

b

b

 ˇ  ˇ    ˇ  ˇ   @  D sup ˇ @  ./ˇ   @   ./  ˇ ˇ  @x  @xi L1 .Rn / i 2Rn @xi 1 O H) for i 6D 0, j./j  the ji j ! 1

1 k @ k . 2j i j @xi L1 .Rn /

But kk ! 1 H) at least one of

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

402 H) at least one of Thus,

1 j i j

O ! 0 H) j./j 

O j./j !0

as kk ! 1

1 k @ k 2j i j @xi L1 .Rn /

! 0 as kk ! 1.

for  2 D.Rn /:

(7.1.37)

Then the result jfO./j ! 0 as kk ! 1 8f 2 L1 .Rn / will follow from the density of D.Rn / in L1 .Rn /. In fact, let f 2 L1 .Rn /. Then 9 a sequence .k / in D.Rn / such that k ! f in L1 .Rn / as k ! 1, i.e. kf  k kL1 .Rn / ! 0 as k ! 1. But k 2 D.Rn /, f 2 L1 .Rn / H) .fO  O k / 2 C 0 .Rn / 8k 2 N (by (7.1.24) of Property 6) with kfO  O k k1 D sup2Rn jfO./  O k ./j  kf  k kL1 .Rn / ! 0 as k ! 1 H) .O k / converges uniformly to fO 2 C 0 .Rn / in Rn H) 8" > 0, 9n0 D n0 ."/ 2 N such that " jfO./  O k ./j  kfO  O k k1 < 2

8k  n0 ; 8 2 Rn :

(7.1.38)

From (7.1.38), 8k 2 N, jO k ./j ! 0 as kk ! 1. We fix k D n0 such that (7.1.38) holds with k D n0 . Then, 8" > 0, 9M > 0 such that jO n0 ./j < "=2

for kk  M:

(7.1.39)

Hence, for k D n0 satisfying (7.1.39), we have, from (7.1.38) and (7.1.39), 8" > 0; 9M > 0 such that jfO./j  jfO./  O n0 ./j C jO n0 ./j < "=2 C "=2 D " 8kk  M H)

lim fO./ D 0;

i.e. fO./ ! 0 as kk ! 1:

kk!1

˛1

Example 7.1.9. Let H.x/e ax x.˛/ 2 L1 .R/ with unbounded support. Prove that ˛1

1. F ŒH.x/e ax x.˛/  D 2. F ŒH.x/ e

ajxj jxj˛1

3. F Œe ajxj  D Proof.

.˛/

1 .aCi2 /˛

D

D fO./ for a > 0, ˛ > 0;

1 ai2 /˛

for x < 0;

2a . a2 C4 2 2

  x ˛1 i2x dx H.x/e ax e .˛/ 1 Z 1 Z R x ˛1 x ˛1 D dx D lim dx: e .aCi2 /x  e .aCi2 /x  .˛/ .˛/ "!0C " 0

1. F Œf ./fO./ D

Z

1

R!1

We will follow [7] to evaluate this improper integral with the help of complex integration along a half-line in the complex plane. For this, we introduce the

Section 7.1 Fourier transforms of integrable functions in L1 .Rn /

403

complex variable z defined by z D .a C i 2/x 2 C, from which dx D dz , such that z varies from 0 to 1 (i.e. .a C i 2/1) along the half-line aCi2 0; .a C i 2/1Œ in the complex plane of z as x varies from 0 to 1 along the real axis. Then, Z x /˛1 . aCi2 dz fO./ D  e z  .˛/ a C i 2 0;.aCi2 /1Œ Z 1 e z  z ˛1 dz: D .a C i 2/˛ .˛/ 0;.aCi2 /1Œ Since z ˛1 is a multiple-valued function, we will choose the branch of z ˛1 for Re.z/ > 0 such that e z z ˛1 is analytic (or holomorphic) for Re.z/ 2 0; 1Œ. Then, by virtue of the analyticity of e z z ˛1 in the right-hand side half-plane of z, the complex integral along 0; .a C i 2/1Œ is evaluated by: Z Z .aCi2 /R z ˛1 e z dz D lim e z  z ˛1 dz "!0C .aCi2 /" R!1

0;.aCi2 /1Œ

D lim I."I R/: "!0C R!1

For the evaluation of complex integral I."I R/ along the complex half-line 0; .a C i 2/1Πfrom the point .a C i 2/" to .a C i 2/R, we can replace this path of integration along the complex half-line from .a C i 2/" to .aCi 2/R by any suitable path in the right-hand side half-plane of z in which the integrand is analytic. Hence we choose a contour with 4 vertices: A W .a C i 2/"I

B W ."; 0/I

C W .R; 0/I

D W .a C i 2/R;

 is a curve joining A and B, BC is a segment of the real axis such that AB  is a curve joining C and D. joining B and C and CD Z e z  z ˛1 dz I."I R/ D

Z D

ŒAD

 AB

e

z

z

˛1

Z dz C

e

z

z

˛1

R

"!0C

R!1

"

Z dx C

 CD

e z  z ˛1 dz:

Z dz D lim I."I R/ D lim

0;.aCi2 /1Œ

Z

x

˛1

ŒBC 

Then Z

D lim

e

x

e x  x ˛1 dx D

"!0C R!1

Z

0

1

"!0C ŒBC  R!1

e x  x ˛1 dx D .˛/;

e x  x ˛1 dx

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

404 since Z lim

"!0C

e

 AB

z

„

z ƒ‚

˛1

I1 ."/

Z dz D 0; …

lim

 R!1 CD

„

e z  z ˛1 dz D 0; ƒ‚ … I2 .R/

with ´  ABW

´ x D a" with y from 2" to 0;  y D 0 with x from R to aR; CDW y D 0 with x from a" to "I x D aR with y from 0 to 2RI Z e z  z ˛1 dz I1 ."/ D Z D

 AB 0

Z " e a"iy .a" C iy/˛1 idy C e x x ˛1 dx ; 2 " a" ƒ‚ … ƒ‚ … „ „

I2 .R/ D Z

I1;2 ."/

I1;1 ."/

Z  CD

aR

D „R

e z  z ˛1 dz Z 2 R e x x ˛1 dx C e aRiy .aR C iy/˛1 idy : ƒ‚ … „0 ƒ‚ … I2;1 .R/

I2;2 .R/

We consider the case 0 < a < 1 with a" < ", aR < R. For a > 1, the same proof will hold with modifications owing to the sign change. lim"!0C I1 ."/ D 0: jI1;1 ."/j  e

a"

Z

2 "

ja" C iyj˛1 dy

0

q  e a" . a2 C 4 2  2 /˛1  "˛1  2" q D 2. a2 C 4 2  2 /˛1  e a"  "˛ ! 0 as " ! 0, since e a" ! e 0 D 1, "˛ ! 0 as " ! 0 8˛ > 0. For a D 1, a" < " (similarly, for a" > ") I1;2 ."/ ! 0. I1;2 D 0. Hence, for R a ¤z1 with ˛1 dzj  .jI C Finally, jI1 ."/j D j A e z 1;1 ."/jCjI1;2 ."/j/ ! 0 as " ! 0 . c B RR ˛ limR!1 I2 .R/ D 0: jI2;1 .R/j D aR e x x ˛1 dx D e  .R/  x˛ jR aR (by the generalized Mean Value Theorem) ! 0 as R ! 1, since .R/ ! 1 as R ! 1, e  .R/ R˛ , e  .R/ .aR/˛ ! 0 as R ! 1 (by L’Hospital’s rule)

405

Section 7.2 Space of infinitely differentiable functions with rapid decay at infinity

8a > 0. jI2;2 .R/j  e aR

Z

2 R

q . a2 R2 C 4 2  2 R2 /˛1 dy

0

q  2. a2 C 4 2  2 /˛1  e aR R˛ ! 0 as R ! 1 8˛ > 0 (using L’Hospital’s rule). R z  z ˛1 dzj  .jI Hence, jI2 .R/j D j CD 2;1 .R/j C jI2;2 .R/j/ ! 0 as e R ! 1. R Thus, 0;.aCi2 /1Œ e z  z ˛1 dz D lim"!0C ;R!1 I."I R/ D .˛/. .˛/ 1 Therefore, fO./ D D ˛ ˛. .aCi2 / .˛/

.aCi2 /

2. For x < 0, x D jxj and F Œf .x/ D fO./, we have for x < 0,   ˛1  ˛1  a.x/ .x/ ajxj jxj F H.x/e D F H.x/e .˛/ .˛/ 1 1 D : D ˛ .a C i 2.// .a  i 2/˛ ˛1

3. Adding (1) and (2), we have F Œe ajxj jxj D .˛/ for ˛ D 1, we have .1/ D 1 and jxj˛1 D 1

7.2

1 1 C ai2 / ˛ . Finally, .aCi2 /˛ 2a ajxj and F Œe  D a2 C4 2 2 .

Space S.Rn / of infinitely differentiable functions with rapid decay at infinity

Infinitely differentiable functions with rapid decay at infinity For the sake of simplicity we consider the case of a single variable (n D 1). Definition 7.2.1. An infinitely differentiable complex-valued function  2 C 1 .1; 1Œ/ of the real variable x, and its derivatives  .l/ .x/ of all orders l 2 N, are said to decrease or decay rapidly at infinity if, 8l 2 N0 , j .l/ .x/j with  .0/ .x/ D .x/ 1 decreases more rapidly than any power of jxj as jxj ! 1, i.e. 8k; l 2 N0 , j .l/ .x/j D jx k  .l/ .x/j ! 0 as jxj ! 1; 1=jxjk or, in other words, lim jx k  .l/ .x/j D 0 8k; l 2 N0 :

jxj!1

(7.2.1)

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

406

2

For example, .x/ D e x 2 C 1 .1; 1Œ/ and its derivatives of all orders 1 decrease faster than any power of jxj , since ˇ ˇ ˇ k d l x 2 ˇ ˇ lim ˇx .e /ˇˇ D 0 8k; l 2 N0 jxj!1 dx l

(7.2.2)

by L’Hospital’s rule. Functions  2 D.1; 1Œ/ and their derivatives of all orders have compact support in R. Consequently,  and all its derivatives  .l/ .x/ of order l 2 N are a fortiori functions with rapid decay at infinity. (7.2.3) k .l/ In fact, 8k; l 2 N0 , x  .x/ D 0 8x lying outside the compact support of . Hence, limjxj!1 jx k  .l/ .x/j D 0 8k; l 2 N0 . Now we state the following lemma which will be used frequently. Lemma 7.2.1 ([40]). 8 fixed k 2 N; 9 a strictly positive constant C > 0, dependent on k; n, such that the following inequalities hold: 8 multi-index ˇ D .ˇ1 ; ˇ2 ; : : : ; ˇn / with jˇj D ˇ1 C ˇ2 C    C ˇn , 8 D .1 ; 2 ; : : : ; n / 2 C n with kk D .j1 j2 C 1 j2 j2 C    C jn j2 / 2 , C.1 C kk2 /k  sup j ˇ j2  sup j ˇ j  .1 C kk2 /k ; jˇjk

(7.2.4)

jˇj2k

where the second inequality is actually an equality. ˇ

ˇ

ˇ

Proof.  ˇ D  11  22 : : :  nn H) j ˇ j2 D j1 ˇ1 2 ˇ2 : : : n ˇn j2 D j 2ˇ j with 2ˇ D .2ˇ1 ; 2ˇ2 ; : : : ; 2ˇn / H)

sup j ˇ j2 D sup j 2ˇ j D sup j ˇ j: jˇjk

jˇjk

(7.2.5)

jˇj2k

But j 2ˇ j D j1 j2ˇ1 j2 j2ˇ2    jn j2ˇn  kk2ˇ1 kk2ˇ2    kk2ˇn D kk2.ˇ1 Cˇ2 CCˇn / D kk2jˇj : Hence, for jˇj  2k, j 2ˇ j  kk4k  .1 C kk2 /2k H) For jˇj  2k, j ˇ j2 D j 2ˇ j  .1 C kk2 /2k H) j ˇ j  .1 C kk2 /k H) supjˇj2k j ˇ j  .1 C kk2 /k H)

sup j ˇ j2 D sup j ˇ j  .1 C kk2 /k jˇjk

jˇj2k

(using (7.2.5)):

(7.2.6)

407

Section 7.2 Space of infinitely differentiable functions with rapid decay at infinity

Now, we prove the first inequality using the multinomial of Newton. Introducing the multi-index notation ˇŠ D ˇ1 Šˇ2 Š : : : ˇn Š, we have .1 C kk2 /k D .1 C j1 j2 C j2 j2 C    C jn j2 /k X X kŠ kŠ j1 j2ˇ1 j2 j2ˇ2 : : : jn j2ˇn D j 2ˇ j D ˇŠ.k  jˇj/Š ˇŠ.k  jˇj/Š jˇjk

jˇjk

 C0 .k; n/ sup j

2ˇ

ˇ 2

j D C0 .k; n/ sup j j ;

jˇjk

with C0 .k; n/ D

jˇjk

P

kŠ jˇjk ˇŠ.kjˇj/Š

H)

>0

C.1 C kk2 /k  sup j ˇ j2

(7.2.7)

jˇjk

with C D

1 C0

> 0. Hence, from (7.2.5)–(7.2.7), we get the result (7.2.4).

Definition 7.2.2. An infinitely differentiable complex-valued function  2 C 1 .Rn / of n real variables x1 ; x2 ; : : : ; xn 2 R and its partial derivatives @ˇ .x/ D @jˇj .x/ of all orders jˇj 2 N are said to decrease or decay rapidly at inˇ1 ˇ2 ˇn @x1 @x2 :::@xn

finity if @ˇ .x/ of all orders jˇj 2 N0 (with @.0/ .x/ D .x/) decrease more rapidly than jx1˛ j 8 multi-index ˛ with x˛ D x1˛1 x2˛2 : : : xn˛n as kxk ! 1; i.e. if, 8˛; ˇ, j@ˇ .x/j D lim jx˛ @ˇ .x/j D 0 kxk!1 1=jx˛ j kxk!1 lim

(7.2.8)

or, equivalently, 8k 2 N0 , 8jˇj 2 N0 , limkxk!1 j.1 C kxk2 /k @ˇ .x/j D 0, which follows from (7.2.8) and Lemma 7.2.1. 2

For example, for kxk2 D x1 2 C x2 2 C    C xn 2 ; .x/ D e kxk 2 C 1 .Rn / and the partial derivatives @ˇ .x/ of all orders jˇj 2 N are functions with rapid decay at infinity, since 2

lim jx˛ @ˇ Œe kxk j D 0

kxk!1

8˛; ˇ 2 N0n

.see (7.2.2)/:

(7.2.9)

Functions  2 D.Rn / and their partial derivatives @ˇ .x/ of all orders jˇj 2 N are a fortiori functions with rapid decay at infinity (see (7.2.3)).

7.2.1 Space S.Rn / Definition 7.2.3. S.Rn / is the linear space of all complex-valued functions  2 C 1 .Rn / which, together with all partial derivatives @ˇ .x/, decay rapidly at infinity,

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

408 i.e.

S.Rn / D ¹ W  2 C 1 .Rn /I for all multi-index ˛; ˇ;

lim jx˛ @ˇ .x/j D 0º;

kxk!1

(7.2.10) or, equivalently, S.Rn /D¹ W  2 C 1 .Rn /I 8k 2 N0 ; 8jˇj 2 N0 ; lim j.1 C kxk2 /k @ˇ .x/jD0º; kxk!1

(7.2.11) which follows from (7.2.10) by virtue of Lemma 7.2.1. Important properties of functions of S.Rn / Proposition 7.2.1. Let  2 S.Rn /. Then the following properties hold: I. For all multi-index ˛; ˇ, i. x˛ @ˇ .x/, ii. @ˇ .x˛ .x//, iii. p.x/@ˇ .x/ for any polynomial p.x/ in n variables, are all bounded in Rn . Consequently, 

8˛; ˇ; q˛;ˇ ./ D sup jx˛ @ˇ .x/j < C1I

(7.2.12)

x2Rn



8l; m 2 N0 ; ql;m ./ D sup q˛;ˇ ./ D sup sup jx˛ @ˇ .x/j < C1I





(7.2.13)

j˛jl x2Rn jˇjm

j˛jl jˇjm

8l 2 N0 , 8ˇ, .1 C kxk2 /l @ˇ .x/ is bounded in Rn I

(7.2.14)

 ql;ˇ ./ D sup j.1 C kxk2 /l @ˇ .x/j < C1I

(7.2.15)

8l 2 N0 , 8ˇ, x2Rn



8l; m 2 N0 ,   ql;m ./ D sup ql;ˇ ./ D sup sup j.1 C kxk2 /l @ˇ .x/j < C1I jˇjm

jˇjm x2Rn

(7.2.16)

Section 7.2 Space of infinitely differentiable functions with rapid decay at infinity

II.

 

409

8˛; ˇ, x˛ @ˇ , @ˇ .x˛ / 2 S.Rn /;

(7.2.17)

8l 2 N0 , 8ˇ, .1 C kxk2 /l @ˇ  2 S.Rn /, i.e.  2 S.Rn / H)

(7.2.18)

i. all derivatives @˛  2 S.Rn / 8˛; ii. the product x˛  2 S.Rn / 8x˛ ; iii. the product .1 C kxk2 /l  2 S.Rn / 8l 2 N0 . Remark 7.2.1. For arbitrary f 2 C 1 .Rn / and  2 S.Rn /, the product f  … 2 2 S.Rn /. For example, for n D 1, f .x/ D e x 2 C 1 .R/, .x/ D e x 2 S.R/, the product f  D 1 … S.R/. But for  2 S.Rn / and a function f 2 C 1 .Rn / which has, along with all its derivatives, polynomial or slow growth at infinity: 8 multi-index ˛, 9 a constant C > 0 and an integer (see also (3)) l D l.˛/ 2 N0 such that j@˛ f .x/j  C.1 C kxk2 /l 8x 2 Rn , the product f  2 S.Rn /. (7.2.19) For example, for n D 1; .ix/k with k 2 N is a C 1 -function of polynomial growth (by Lemma 7.2.1). Proof of Proposition 7.2.1. I. It is sufficient to show the boundedness of x˛ @ˇ .x/ in Rn , because other cases can be shown similarly with minor modifications. In fact, since  2 S.Rn /, by the definition of S.Rn / in (7.2.10), 8 fixed ˛; ˇ, limkxk!1 jx˛ @ˇ .x/j D 0 H) 8" 2 0; 1Œ, 9R > 0 such that jx˛ @ˇ .x/j < 2" 8kxk > R. But x˛ @ˇ .x/ is continuous in the compact set K D ¹x W x 2 Rn , kxk  Rº H) 9M > 0 such that jx˛ @ˇ .x/j  M 8x 2 K. Then, 8 fixed ˛; ˇ, 9M0 > 0 such that 8x 2 Rn , jx˛ @ˇ .x/j  M0 D max¹ 2" ; M º H) 8 fixed ˛; ˇ, x˛ @ˇ .x/ is bounded in Rn . Then the results (7.2.12) and (7.2.13) follow from the property of the existence of the supremum of bounded functions in Rn . Applying Lemma 7.2.1 and using the boundedness of x˛ @ˇ .x/ and the property of supremum, the boundedness of .1 C kxk2 /l @ˇ .x/ in (7.2.14), (7.2.15) and (7.2.16) is proved. II. Using Leibniz’s theorem on the derivatives of products of functions: @ˇ .uv/ D

X

ˇ

ˇŠ @ˇ u@ v; .ˇ  /ŠŠ

(7.2.20)

where ˇŠ D ˇ1 Šˇ2 Š : : : ; ˇn Š, ˇ   D .ˇ1  1 ; ˇ2  2 ; : : : ; ˇn  n / with ˇi  i  0, .ˇ  /Š D .ˇ1  1 /Š.ˇ2  2 /Š    .ˇn  n /Š, Lemma 7.2.1 and the definition of S.Rn / in (7.2.10) (resp. (7.2.11)), the result (7.2.17) (resp. (7.2.18)) is proved.

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

410

Alternative definition of S.Rn / As a consequence of Properties I and II in (7.2.12)–(7.2.18), instead of Definition 7.2.3 ((7.2.10)/(7.2.11)), S.Rn / can equivalently be defined by: Definition 7.2.4. S.Rn / D ¹ W  2 C 1 .Rn /I 8k; m 2 N0 ;  sup sup j.1 C kxk2 /k @ˇ .x/j D qk;m ./ < C1º (7.2.21)

jˇjm x2Rn

or, equivalently, S.Rn / D ¹ W  2 C 1 .Rn /I 8˛; ˇ; q˛;ˇ ./ D sup jx˛ @ˇ .x/j < C1º: x2Rn

Proposition 7.2.2. Let  2 S.Rn /. Then, 8˛; ˇ, I.



, x˛ @ˇ , @ˇ .x˛ .x// 2 L1 .Rn /; kxk2 /l @ˇ 

.1 C 2 n 1 n i.e. S.R /  L .R /; 

L1 .Rn /

(7.2.22)

8l 2 N0 ,

II. supx2Rn j.x/j  supjˇjn k@ˇ kL1 .Rn / .

(7.2.23) (7.2.24) (7.2.25)

Proof. I. It is sufficient to prove that  2 S.Rn / H)  2 L1 .Rn / and, 8˛; ˇ, x˛ @ˇ  .x/ 2 L1 .Rn /. From (7.2.15), supx2Rn j.1 C kxk2 /n .x/j D qn;0 ./ < C1. 1 Then .x/ D .1 C kxk2 /n .x/ .1Ckxk 2 /n .

But 1 C kxk2 D 1 C x12 C    C xn2  1 C xi2 8i D 1; 2; : : : ; n 1 1 1  1 2 H) .1Ckxk 2 /n  1Cx12 1Cx22 1Cxn R R 1 dx1 R 1 dx2 R 1 dxn dx n H) Rn .1Ckxk2 /n  1 2 1 2    1 2 D      D  , 1Cx1 1Cx2 1Cxn  R1 R since, setting xi D tan , 1 dxi 2 D 2 d D . 1Cxi

Hence, Z Z j.x/jd x D Rn

H)

2

1 dx .1 C kxk2 /n Rn Z dx 2 n   sup j.1Ckxk / .x/j qn;0 ./ n 0, independent of , and a semi-norm qm;ı with m 2 N0 , multi-index ı such that, 8l 2 N0 , 8ˇ,   ql;ˇ .A/  C1 qm;ı ./

8 2 S.Rn /;

(7.3.2)

i.e. 8l 2 N0 , 8ˇ, supx2Rn j.1Ckxk2 /l @ˇ .A/.x/j  C1 supx2Rn j.1Ckxk2 /m @ı .x/j 8 2 S.Rn /. Consequences

If A W S.Rn / ! S.Rn / is a continuous linear mapping, then

k !  in S.Rn / n

k ! 0 in S.R /

H) H)

Ak ! A in S.Rn / as k ! 1; n

Ak ! 0 in S.R / as k ! 1:

(7.3.3) (7.3.4)

413

Section 7.4 Imbedding results

Applications jj

8 multi-index ; @ : S.Rn / 3  7! @  D @x1 1 @x@2 2:::@xn n 2 S.Rn / is a continuous linear operator from S.Rn / into S.Rn /. (7.3.5) Proof. From the linearity of the differential operator, the linearity of @ follows. It remains to show that (7.3.1) holds. In fact, 8˛; ˇ, q˛;ˇ .@ / D sup jx˛ @ˇ .@ .x//j D sup jx˛ @ˇC .x/j D q˛;ı ./; x2Rn

x2Rn

with C D 1, ı D ˇ C  and 8 2 S.Rn /. Hence, @ W S.Rn / ! S.Rn / is continuous. Alternatively, k ! 0 in S.Rn / H) q˛;ˇ .k / ! 0 8˛; ˇ as k ! 1 by (7.2.31) H) q˛;ˇ .@ k / D q˛;ı .k / ! 0 as k ! 18˛; ˇ H) @ k ! 0 in S.Rn / H) @ W S.Rn / ! S.Rn / is continuous. The multiplication of  2 S.Rn / by an arbitrary monomial x (resp. by arbitrary polynomial .1 C kxk2 /l with l 2 N),  2 S.Rn / 7! x  2 S.Rn / (resp. .1 C kxk2 /l  2 S.Rn /) defines a continuous linear mapping from S.Rn / into S.Rn /. Proof. The linearity is obvious. For continuity to show that (7.3.1) holds. In P we are ˇŠ fact, using Leibniz’s theorem @ˇ .x / D ıˇ .ˇı/ŠıŠ @ˇı .x /@ı , we find that 9 a constant C > 0, independent of , such that 8l; m 2 N0 with j˛j  l; jˇj  m, ql;m .x /  C qlCj j;m ./ 8 2 S.Rn /, where ql;m .x / D supj˛jl;jˇjm jx˛ @ˇ .x /j, qlCj j;m ./ D supjjlCj j;jˇjm jx @ˇ j (multi-index  D .1 ; 2 ; : : : ; n /) H) the mapping  2 S.Rn / 7! x  2 S.Rn / is continuous by (7.3.1). Then, applying Lemma 7.2.1, the continuity of the linear mapping  2 S.Rn / 7! .1 C kxk2 /l  2 S.Rn / can be proved by (7.3.2).

7.4

Imbedding results

Proposition 7.4.1. I. S.Rn / ,! Lp .Rn /, 1  p  1, with continuous imbedding ,!, i.e. 9 a constant C > 0, independent of , such that  k,!kLp .Rn / D kkLp .Rn /  C qn;0 ./

8 2 S.Rn /:

(7.4.1)

II. D.Rn / ,! S.Rn / with continuous imbedding ,!, (7.4.2) n where D.R / is the space of complex-valued test functions (see (1.3.46)– (1.3.49)).

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

414 Proof.

I. Case p D 1: The result follows from: kkL1 .Rn / D supx2Rn j.x/j D  q0;0 ./.  Case p D 1: The result follows from (7.2.26) with kkL1 .Rn /   n qn;0 ./. Case p D 2: Following the proof of Proposition 7.2.2, we have, 8 2 S.Rn /, 1 1   qn;0 ./ .1 C kxk2 /n .1 C kxk2 /n Z Z 1 j.x/j2 d x  Œq n;0 ./2 dx 2 2n Rn Rn .1 C kxk /

j.x/j  . sup j.1 C kxk2 /n .x/j/ x2Rn

H)

 ./2 < C1;   n Œqn;0

since .1 C kxk2 /2n  .1 C kxk2 /n  .1 C x12 /.1 C x22 /    .1 C xn2 / 8x 2 Rn R R 1 1 n H) Rn .1Ckxk 2 /2n d x  Rn Π.1Ckxk2 /n d x   (see proof of Proposition 7.2.2). Hence,  2 L2 .Rn / H) S.Rn /  L2 .Rn / with Z k kL2 .Rn / D

 12  j.x/j d x  C qn;0 ./ 2

Rn

8 2 S.Rn /;

n

with C D  2 > 0. Cases p 2 2; 1Œ, p 2 1; 2Œ: These can be proved as in the case of p D 2 with n C D p. II. Let K be any compact subset of Rn and DK .Rn / be defined by DK .Rn / D ¹ W  2 D.Rn /; supp./  Kº; with semi-norm pK;m defined, 8m 2 N0 , by pK;m ./ D sup sup j@ˇ .x/j

8 2 DK .Rn /:

jˇjm x2K

Then it is sufficient to show the continuity of the imbedding DK .Rn / ,! S.Rn / 8 compact K  Rn , from which the continuous imbedding (7.4.2) will follow. In fact, 8 2 DK .Rn /, 8l; m 2 N0 , ql;m ./ D

sup

sup jx˛ @ˇ .x/j D

j˛jl;jˇjm x2Rn

sup

sup jx˛ @ˇ .x/j

j˛jl;jˇjm x2K

(since .x/ D 0 outside K)  . sup sup jx˛ j/ sup sup j@ˇ .x/j < C1 j˛jl x2K

jˇjm x2K

415

Section 7.5 Density results

H) 8l; m 2 N0 , ql;m ./  CK;l pK;m ./ 8 2 DK .Rn / 8K  Rn with CK;l D supj˛jl supx2K jx˛ j H) 8 compact K  Rn , the imbedding DK .Rn / ,! S.Rn / is continuous, H) the imbedding D.Rn / ,! S.Rn / is continuous.

7.5

Density results

Proposition 7.5.1. I. D.Rn / is a dense subspace of S.Rn /;

(7.5.1)

II. S.Rn / is a dense subspace of L2 .Rn /.

(7.5.2)

Proof. I. We will prove the density of D.Rn / in S.Rn / with the help of cut-off functions. Let 2 D.Rn / such that .x/ D 1 for kxk  1. Now, 8k 2 N, define x n k .x/ D . k / 8x 2 R . Then . k / is a sequence of cut-off functions with the following properties: 2 D.Rn / 8k 2 N;



k



k .x/



k .x/



. kx / D 1 for kxk  k 8k 2 N;

D

(7.5.3)

 1 D 0 for kxk  k 8k 2 N; ˇ  ˇ ˇ ˇ x ˇˇ ˇ ˇ sup sup j@ k .x/j D sup sup ˇ@  sup j@ˇ .x/j k ˇ x2Rn k2N x2Rn k2N x2Rn

8ˇ: (7.5.4)

Let  2 S.Rn /. Define k D  k 8k 2 N. Hence, 8k 2 N, k 2 D.Rn / (since  2 C 1 .Rn /, k 2 D.Rn / 8k 2 N). Then the sequence .k / converges to  in S.Rn /, for which we are to show that the sequence of semi-norms q˛;ˇ .k / D q˛;ˇ . k / (with k D k  D . k 1/) tends to 0 as k ! 1 P ˇŠ @ˇ . k  1/@  with 8˛; ˇ (see (7.2.31)). In fact, @ˇ k D ˇ .ˇ /Š Š @ˇ . k  1/ D 0 for kxk  k (by virtue of (7.5.3)). P ˇŠ j@ˇ . Hence, 8˛; ˇ, jx˛ @ˇ k .x/j  ˇ .ˇ /Š Š H)

k .x/

 1/jjx˛ @ .x/j

sup jx˛ @ˇ k .x/j

x2Rn



X

ˇ



ˇŠ sup j@ˇ . .ˇ  /ŠŠ kxk>k

X

ˇ

k .x/

ˇŠ sup j@ˇ . .ˇ  /ŠŠ kxk>k

 1/j sup jx˛ @ .x/j kxk>k



k .x/

 1/j sup sup jx˛ @ .x/j:

ˇ kxk>k

(7.5.5)

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

416 But 8k 2 N, sup j@ˇ .

k .x/

 1/j  sup sup j@ˇ

k2N x2Rn

kxk>k

k .x/j

C1

 sup j@ˇ .x/j C 1

8  ˇ

(by (7.5.4))

x2Rn

H) 8k 2 N, sup sup j@ˇ . .x/  1/j  sup sup j@ˇ .x/j C 1 D Mˇ C 1;

ˇ x2Rn

ˇ kxk>k

(7.5.6) since for

2 D.Rn /, 9Mˇ > 0 such that sup sup j@ˇ .x/j  Mˇ :

(7.5.7)

ˇ x2Rn

Hence, from (7.5.5)–(7.5.7), we have, 8k 2 N, 8˛; ˇ, q˛;ˇ . k / D sup jx˛ @ˇ k .x/j 

X

ˇ



X

ˇ

x2Rn

ˇŠ .ˇ  /ŠŠ ˇŠ .ˇ  /ŠŠ

Set C.ˇ/ D

P



sup sup j@ˇ . .x/  1/j  sup sup jx˛ @ .x/j

ˇ kxk>k

ˇ kxk>k

 .Mˇ C 1/ sup sup jx˛ @ .x/j:

ˇŠ

ˇ .ˇ /Š Š .Mˇ

ˇ kxk>k

C 1/ > 0.

Then 8˛; ˇ, q˛;ˇ . k /  C.ˇ/ sup ˇ supkxk>k jx˛ @ .x/j ! 0 as k ! 1, since  2 S.Rn / H) 8˛; , x˛ @ .x/ 2 S.Rn / by Proposition 7.2.1 H) 8˛; , x˛ @ .x/ ! 0 as kxk ! 1 (by (7.2.10)) H) supkxk>k jx˛ @ .x/j ! 0 as k ! 1 8˛;  H) 8˛; ˇ, q˛;ˇ . k / ! 0 as k ! 1. Hence, D.Rn / is dense in S.Rn /, since k !  in S.Rn / as k ! 1. II. Let f 2 L2 .Rn /. Since D.Rn / is dense in L2 .Rn /, 9 a sequence .k / in D.Rn / such that kf  k kL2 .Rn / ! 0 as k ! 1. (7.5.8) But D.Rn / ,! S.Rn / by (7.4.2) H) k 2 S.Rn / 8k 2 N such that (7.5.8) holds. Hence, for f 2 L2 .Rn /, 9 a sequence .k / in S.Rn / such that k ! f as k ! 1. Thus, S.Rn / is dense in L2 .Rn /.

Section 7.6 Fourier transform of functions of S.Rn /

7.6

417

Fourier transform of functions of S.Rn /

By virtue of the imbedding result S.Rn / ,! L1 .Rn / (7.4.1), all the Properties 1–11 of Fourier transform fO D F f (resp. co-transform F f ) stated in Theorems 7.1.1– 7.1.3, in Proposition 7.1.1 and Corollary 7.1.1 for functions f 2 L1 .Rn / will also hold for functions  2 S.Rn /. For f 2 L1 .Rn /, its Fourier transform fO … L1 .Rn / in general (see Example 7.1.1), but for every  2 S.Rn / with its Fourier transform O D F  2 S.Rn / ,! L1 .Rn /:

Theorem 7.6.1. I. 8 2 S.Rn /, Fourier transform O D F  2 S.Rn / and co-transform F  2 S.Rn / are defined by (7.1.2) and (7.1.3): O ./ D .F /./ D

Z Z

.x/e i2hx;i d x;

(7.6.1)

.x/e i2hx;i d x:

(7.6.2)

Rn

.F /./ D Rn

II. F W S.Rn / ! S.Rn / (resp. F W S.Rn / ! S.Rn /) is continuous from S.Rn / into S.Rn /.

Proof. We give the proof for F . Then, replacing ‘i ’ by ‘i ’ in the proof, the results for F are obtained. I.  2 S.Rn / H) O D F  2 C 1 .Rn /: Let  2 S.Rn /. Then, by Proposition 7.2.2, xˇ  2 L1 .Rn / 8 multi-index ˇ with jˇj 2 N0 , and, from Theorem 7.1.3, jˇj O ˇ O D F  2 C jˇj .Rn / 8jˇj 2 N0 , i.e. O 2 C 1 .Rn / and @ O D ˇ@1  ˇn D @ 1 :::@ n

F Œ.i 2x/ˇ  8jˇj 2 N0 , where .i 2x/ˇ D .i 2x1 /ˇ1 .i 2x2 /ˇ2    .i 2xn /ˇn D .i 2/jˇj xˇ with jˇj D ˇ1 C ˇ2 C    C ˇn . O < C1 for O 2 S.Rn /: For this it remains to show that q˛;ˇ ./ all multi- index ˛; ˇ (by Definition 7.2.4). Applying Proposition 7.2.2, ˇ ˇ 1 n ˛ ˇ x˛ @x .x/ and @˛ x .x .x// 2 L .R / 8˛; ˇ with kx @x kL1 .Rn / < C1, ˇ k@˛ x .x /kL1 .Rn / < C1. Then, from Theorem 7.1.3, we have, 8˛; ˇ, ˇ

ˇ O .i 2/˛ @ ./ D .i 2/˛ F Œ.i 2x/ˇ .x/ D F Œ@˛ x ¹.i 2x/ º:

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

418 Applying (7.1.7):

ˇ

ˇ O 1 D kF Œ@˛ 8˛; ˇ; k.i 2/˛ @ k x ¹.i 2x/ ºkL1 .Rn / ˇ  k@˛ x Œ.i 2x/ kL1 .Rn /

H)

ˇO j˛j ˛ ˇO 8˛; ˇ; ji 2jj˛j k ˛ @ k 1 D .2/ k @ k L1 .Rn / ˇ  ji 2jjˇj k@˛ x .x /kL1 .Rn /

H)

ˇO O O D sup .j ˛ @ˇ ./j/ D k ˛ @ ./k 8˛; ˇ; q˛;ˇ ./ L1 .Rn /  2Rn

ˇ  .2/jˇjj˛j k@˛ x .x /kL1 .Rn / < C1

H)

(7.6.3)

O 2 S.Rn /:

Hence,  2 S.Rn / H) O D F  2 S.Rn /. ˇ n ˛ n II. k ! 0 in S.Rn / H) @˛ x .x k / ! 0 in S.R / 8˛; ˇ, since @x W S.R / ! S.Rn / and k 2 S.Rn / 7! xˇ k 2 S.Rn / are continuous and hence their ˇ n n composition k 2 S.Rn / 7! @˛ x .x k / 2 S.R / is continuous from S.R / n n ˛ ˇ n into S.R /, i.e. k ! 0 in S.R / H) @x .x k / ! 0 in S.R / 8˛; ˇ. As a consequence of the imbedding result S.Rn / ,! L1 .Rn / (Proposition 7.4.1), ˇ 1 n k ! 0 in S.Rn / H) @˛ x .x k / ! 0 in L .R / 8˛; ˇ,

H)

ˇ 8˛; ˇ; k@˛ x .x k /kL1 .Rn / ! 0 as k ! 1:

(7.6.4)

But from (7.6.3) and (7.6.4), 8˛; ˇ, ˇ q˛;ˇ .F k /  .2/jˇjj˛j k@˛ x .x k /kL1 .Rn / ! 0

as k ! 1 H) F k D O k ! 0 in S.Rn / as k ! 1, i.e. k ! 0 in S.Rn / H) F k ! 0 in S.Rn / H) F W S.Rn / ! S.Rn / is continuous from S.Rn / into S.Rn /.

7.7

Fourier inversion theorem in S.Rn /

For f 2 L1 .Rn R/; fO D F f … L1 .Rn / in general (fO is bounded in Rn ), and consequently F fO D Rn fO./e i2hx;i d  is not defined in general. But for functions of S.Rn /, we have: Theorem 7.7.1 (Fourier Inversion Theorem). 8f 2 S.Rn /, F F f D F F f D f:

(7.7.1)

Section 7.7 Fourier inversion theorem in S.Rn /

419

In other words, F D F 1 is the inverse to Fourier transform F , and F D F the inverse to Fourier co-transform F : 8f 2 S.Rn /, F f D fO H) F fO D f I

F f D g H) F g D f:

1

is

(7.7.2)

Proof. It is sufficient to show that 8a 2 Rn , .F F f /.a/ D f .a/ 8f 2 S.Rn /, i.e. Z .F F f /.a/ D ŒF .F f /.a/ D .F f /./e Ci2ha;i d  n R  Z Z i2hx;i D f .x/e d x e i2ha;i d  D f .a/; (7.7.3) Rn

Rn

since following the steps of this proof and replacing ‘i ’ with ‘i ’, we will get .F F f / .a/ D f .a/ 8a 2 Rn , 8f 2 S.Rn /. R But for the function F .x; / D f .x/e i2hx;i e i2ha;i , Rn Rn jF .x; /jd xd  D 1, i.e. F is not summable (integrable) on Rn  Rn . Hence, we cannot apply Fubini’s Theorem 7.1.2C and consequently cannot interchange the order of integration in (7.7.3). Therefore we will apply Riesz’s Formula (Corollary 7.1.1): 8f .x/ 2 L1 .Rn /, 8g./ 2 L1 .Rn /, Z Z i2ha;i O f ./g./e d D f .a C x/g.x/d O x; Rn

Rn

where fO./ D .F f /./ D

Z

f .x/e i2hx;i d xI

Rn

Z g.x/ O D .F g/.x/ D

g./e i2h;xi d ;

Rn

and Lebesgue’s Dominated Convergence Theorem B.3.2.2 (Appendix B) for a sequence .gk / of auxiliary functions such that gk ! 1 as k ! 1 in order to establish the result. Let  2 S.Rn /. Define gk .x/ D . kx / 8k 2 N such that gk .x/ ! .0/ as k ! 1.  will finally be chosen such that .0/ D 1. Then, 8f , gk 2 S.Rn /  L1 .Rn /, we can apply: Z Z i2ha;i O f ./gk ./e d D f .a C x/gO k .x/d x; Rn

Rn

R R where gk ./ D . k /, gO k .x/ D Rn gk ./e i2h;xi d  D Rn . k /e i2h;xi d . Set y D k with k 2 N. Then  D ky H) the Jacobian of this transformation D k n > 0 8k 2 N. Hence, Z Z O gO k .x/ D .y/e i2hky;xi .k n /d y D k n .y/e i2hy;kxi d y D k n .kx/; Rn

Rn

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

420 and

Z Rn

fO./gk ./e i2ha;i d  D k n

Z

O f .a C x/.kx/d x

8k 2 N:

Rn

Set D kx with k 2 N. Then x D k H) the Jacobian of this transformation is R 1 O > 0 8k 2 N. Hence, f 2 S.Rn /;  2 S.Rn / H) k n Rn f .a C x/.kx/d xD kn R O 1 n k Rn f .a C k /. / k n d Z Z O fO./gk ./e i2ha;i d  D 8k 2 N: (7.7.4) f .a C /. /d H) k Rn Rn Now we let k ! 1. Then the sequence gk ./ D . k / ! .0/ and the sequence fO./gk ./e i2ha;i ! fO./.0/e i2ha;i

as k ! 1:

(7.7.5)

On the other hand, 8 2 Rn , 8k 2 N,  jfO./gk ./e i2ha;i j D jfO./kgk ./j D jfO./j  j. /j  kk1 jfO./j; k with kk1 D sup2Rn j./j. Moreover, Z Z kk1 jfO./jd  D kk1 jfO./jd  < C1; (7.7.6) Rn

Rn

since f 2 S.Rn /  L1 .Rn / H) fO 2 S.Rn /  L1 .Rn /. Hence, by virtue of (7.7.5) and (7.7.6), Lebesgue’s Dominated Convergence Theorem B.3.2.2 (Appendix B) can be applied and we get: Z Z lim fO./gk ./e i2ha;i d  D lim .fO./gk ./e i2ha;i /d  n k!1 Rn k!1 R Z fO./.0/e i2ha;i d  D Rn Z Z fO./gk ./e i2ha;i d  D .0/ fO./e i2ha;i d : (7.7.7) H) lim k!1 Rn

Rn

O O Again, for f 2 S.Rn /, the sequence f .a C k /. / ! f .a/. / as k ! 1 and O O jf .a C k /. /j  kf k1 j. /j with kf k1 D sup 2Rn ;k2N jf .a C k /j R R O O H) Rn kf k1 j. /jd D kf k1 Rn j. /jd < C1, since O 2 S.Rn / H) R O O 2 L1 .Rn / H) Rn j. /jd < C1. Hence, again by Lebesgue’s Dominated Convergence Theorem B.3.2.2 (Appendix B), we have, from (7.7.4):       Z Z Z O O O lim . /d D . / d D f aC lim f a C f .a/. /d : k k k!1 Rn Rn k!1 Rn (7.7.8)

Section 7.7 Fourier inversion theorem in S.Rn /

From (7.7.7) and (7.7.8), Z Z i2ha;i O f ./e d  D f .a/ .0/ Rn

421

O . /d 8 2 S.Rn /:

(7.7.9)

Rn

R 2 2 O O D e kk and .0/ D 1, Rn . /d D In particular, for .x/ D e kxk , ./ R 2 k k d D 1 (the Gauss integral) – see Example 7.1.5. Rn e R From (7.7.9), we get Rn fO./e Ci2ha;i d  D f .a/ H) .F F f /.a/ D f .a/ 8a 2 Rn (by (7.7.3)) H) F F f D f in S.Rn /. Remark 7.7.1. 



The proof shows that if f 2 L1 .Rn / is continuous and bounded in Rn and fO 2 L1 .Rn /, then F ŒfO D f everywhere. Subsequently, when Fourier transforms of tempered distributions are introduced, it can be shown that the condition of the boundedness of f is superfluous.

Corollary 7.7.1. Let F W S.Rn / ! S.Rn /. If F  D 0 for  2 S.Rn /, then  D 0 in S.Rn /. Isomorphism of Fourier transform on S.Rn / Theorem 7.7.2. Fourier transform F W S.Rn / ! S.Rn / is a topological isomorphism from S.Rn / onto S.Rn /. Proof. From Theorems 7.6.1 and 7.7.1, F is a linear bijective mapping from S.Rn / onto S.Rn /, and hence an algebraic isomorphism from S.Rn / onto S.Rn /. By Theorem 7.6.1, F and its inverse F 1 D F W S.Rn / ! S.Rn / are continuous from S.Rn / onto S.Rn /. Hence, F is a topological isomorphism from S.Rn / onto S.Rn /. Application 2

We give here an alternative proof of Example 7.1.2 of the function f .x/ D e x , whose Fourier transform is the function itself, i.e. f D fO D F f , if we neglect the role of variables x and . 2 Let f .x/ D e x be the function on R which satisfies the differential equation f 0 .x/ C 2xf .x/ D 0

H)

f 0 .x/ C i.i 2x/f .x/ D 0 8x 2 R: (7.7.10)

2 But f .x/ D e x 2 S.R/ H) xf .x/ 2 S.R/ H) F Œf .x/ D fO./ and c ./ exist and belong to S.R/. But F Œf 0 .x/ C i.i 2x/f .x/ D F Œxf .x/ D xf O F Œ0 D 0, F Œf 0 .x/ D i 2 fO./ and F Œ.i 2x/f .x/ D d f ./ H) i 2 fO./C

d

i.fO/0 ./ D 0 H) .fO/0 ./ C 2 fO./ D 0.

Chapter 7 Fourier transforms of functions of L1 .Rn / and S.Rn /

422

Thus, f and fO satisfy the same first-order differential equation (7.7.10) in variables x and  respectively. Hence, a particular solution of the first-order equation in variable 2 2  is e  , and its general solution is given by fO./ D C e  , C being a constant. Now, the constant C from the condition that C D fO.0/. In fact, fO./ D R 1 we determine i2x dx 1 f .x/e R1 R1 2 H) fO.0/ D 1 f .x/dx D 1 e x d x D 1 (Gauss integral) 2 H) C D 1 H) fO./ D e  H) f D fO. Plancherel–Parseval theorem on isometry in S.Rn / Theorem 7.7.3. Let S.Rn /  L2 .Rn / be equipped with inner product h  ;  iL2 .Rn / R induced by L2 .Rn /, i.e. 8; 2 S.Rn /, h; iL2 .Rn / D Rn .x/ .x/d x, where .x/ is the complex conjugate of .x/. Then, 8; 2 S.Rn / with O D F ; O D F 2 S.Rn /, O O iL2 .Rn / , i.e.  h; iL2 .Rn / D h; Z

Z

O O ./d I ./

.x/ .x/d x D Rn 

(7.7.11)

Rn

O L2 .Rn / , i.e. kkL2 .Rn / D kk Z

 12 Z j.x/j d x D

Rn

2

2

O j./j d

 12 :

(7.7.12)

Rn

Proof. Let ; 2 S.Rn /. Then 2 S.Rn / and 9 2 S.Rn / such that D F D O since F is an isomorphism , on S.Rn /. Hence, D F F DR F N D F . From R R O Property 8 in (7.1.26), Rn ./ ./d  D Rn .x/ .x/d O x D Rn .x/ .x/d x. R R O O O But ./ D F ./ D ./ H) Rn ./ ./d  D Rn .x/ .x/d x O O iL2 .Rn / . H) h; iL2 .Rn / D h; O kk2 2 n D h; iL2 .Rn / D h; O i O L2 .Rn / D In particular, for D  with O D , L .R /

O 22 n kk L .R / O L2 .Rn / . H) kkL2 .Rn / D kk

Chapter 8

Fourier transforms of distributions and Sobolev spaces of arbitrary order H S .Rn/

8.1

Motivation for a possible definition of the Fourier transform of a distribution

0 n Let f 2 L1 .Rn /. Then f 2 L1loc .Rn / H) R f 2 423gD .R / defines a regular n n distribution on R : 8 2 D.R /, hf; i D Rn f .x/.x/d x. Since f 2 L1 .Rn /, its Fourier transform fO D F f is bounded and continuous in n R and fO./ ! 0 as kk ! 1 (see (7.1.6), (7.1.24), (7.1.36)), and consequently fO 2 L1loc .Rn / H) fO 2 D 0 .Rn / defines a distribution on Rn , i.e. 8 2 D.Rn /,

Z

Z

Z

hF f; i D

.F f /././d  D Z

Rn

f .x/e Rn

i2hx;i

 d x ./d 

Rn

f .x/./e i2hx;i d xd ;

D R2n

since the multiple integral over the product space R2n exists. In fact, je i2hx;i f .x/./j D jf .x/./j is integrable on R2n as the product of an integrable (summable) function of x on Rn with another integrable (summable) function  of  with compact support in Rn . Consequently, we can apply Fubini’s Theorem 7.1.2C to change the order of integration and write Z hF f; i D

f .x/./e Z

i2hx;i

Z d xd  D

R2n

Rn

f .x/.F /.x/d x D hf; F i

D

Z f .x/

./e

i2hx;i

 d dx

Rn

8 2 D.Rn /:

(8.1.1)

Rn

(8.1.1) holds, even if  … D.Rn /, but  must belong to L1 .Rn /, since  2 L1 .Rn / O O H) .x/ D F Œ.x/ is bounded and continuous in Rn and .x/ ! 0 as kxk ! 1, and ˇZ ˇ ˇ ˇ

Rn

ˇ Z ˇ ˇ f .x/.F /.x/d xˇ  sup j.F /.x/j x2Rn

jf .x/jd x < C1: Rn

(8.1.2)

424

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Hence, the two formulae: Z hF f; i D Z

fO././d ;

(8.1.3)

O f .x/.x/d x

(8.1.4)

Rn

hf; F i D Rn

are well defined and equal for  2 L1 .Rn /. Thus, (8.1.1) suggests that a possible definition of the Fourier transform F T for a distribution T 2 D 0 .Rn / should be hF T; i D hT; F i

8 2 D.Rn / with  D ./;

(8.1.5)

if the expressions on both sides of this equality (8.1.5) are meaningful. Unfortunately, the right-hand side of (8.1.5) has no meaning for arbitrary  2 D.Rn /, since, for  2 D.Rn /, F  … D.Rn / in general. In fact,we have the following result: If  2 D.Rn /;  ¤ 0;

then F  … D.Rn / [7]:

(8.1.6)

Thus, the Fourier transform F T of arbitrary distribution T 2 D 0 .Rn / does not exist. (8.1.7) n But the right-hand side of (8.1.5) becomes meaningful for  2 S.R /, since, by Theorem 7.6.1, 8 2 S.Rn /; F  2 S.Rn / with S.Rn / ,! L1 .Rn /. Hence, we are going to identify a particular subclass of distributions containing L1 .Rn / as a subspace. This particular subclass of distributions, described by Laurent Schwartz, are called tempered distributions, or distributions tempérés in French.

8.2

Space S 0 .Rn / of tempered distributions

8.2.1 Tempered distributions Definition 8.2.1. A distribution T 2 D 0 .Rn / is called a tempered distribution on Rn if and only if T can be extended to a continuous linear functional on S.Rn /, and the extended continuous, linear functional will still be denoted by T . Every tempered distribution T is a distribution in D 0 .Rn /. Precisely speaking, if T is a tempered distribution, then its restriction to D.Rn / denoted by T #D.Rn / 2 D 0 .Rn /, since D.Rn / ,! S.Rn / and D.Rn / is dense in S.Rn /. Characterization of tempered distributions We state the results, which follow from Definition 8.2.1 and the notion of convergence in S.Rn / (see Definition 7.2.5), in the form of a proposition.

Section 8.2 Space S 0 .Rn / of tempered distributions

425

Proposition 8.2.1. A distribution T 2 D 0 .Rn / is a tempered distribution if and only if I. T is a linear functional on S.Rn /: T ./ 2 C 8 2 S.Rn /, T .˛1 1 C ˛2 2 / D ˛1 T .1 / C ˛2 T .2 /

8i 2 S.Rn /; 8˛i 2 CI (8.2.1)

II. T is continuous on S.Rn /: 9 a constant C > 0, independent of , and 9 multiindex ˛; ˇ such that jT ./j  C q˛;ˇ ./ D C sup jx˛ @ˇ .x/j

8 2 S.Rn /;

(8.2.2)

x2Rn

or, equivalently, 9C > 0, independent of , and 9l; m 2 N0 such that  ./ D C sup sup j.1 C kxk2 /l @ˇ .x/j jT ./j  C ql;m

8 2 S.Rn /:

jˇjm x2Rn

(8.2.3) Equivalent notations are: T ./ D hT; i D ŒT; 

8 2 S.Rn /:

(8.2.4)

Algebraic properties of tempered distributions Sum of tempered distributions The sum T1 C T2 of tempered distributions T1 and T2 is a tempered distribution defined by: .T1 C T2 /./ D T1 ./ C T2 ./

8 2 S.Rn /;

(8.2.5)

equivalently written as hT1 C T2 ; i D hT1 ; i C hT2 ; i8 2 S.Rn /. Multiplication by a number The product ˛T of the multiplication of a tempered distribution T by a complex number ˛ 2 C is a tempered distribution defined by: .˛T /./ D ˛T ./

8 2 S.Rn /;

(8.2.6)

equivalently written as h˛T; i D ˛hT; i 8 2 S.Rn /. Null tempered distribution A tempered distribution T is called a null tempered distribution, denoted by 0 2 S 0 .Rn /, if and only if T ./ D 0 8 2 S.Rn /, i.e. T D 0 in S 0 .Rn /

H)

T ./ D 0 8 2 S.Rn /:

(8.2.7)

Equality of tempered distributions Two tempered distributions T1 and T2 are called equal in S 0 .Rn / if and only if T1 ./ D T2 ./ 8 2 S.Rn /, i.e. T1 D T2 in S 0 .Rn /

”

T1 ./ D T2 ./ 8 2 S.Rn /:

(8.2.8)

426

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8.2.2 Space S 0 .Rn / Definition 8.2.2. Let the sum of tempered distributions, multiplication of a tempered distribution by a number and the null tempered distribution be defined by (8.2.5), (8.2.6) and (8.2.7) respectively. Then the set of all tempered distributions T on Rn form a linear space called the space of tempered distributions on Rn , which is the dual space of S.Rn / and hence denoted by S 0 .Rn /. The celebrated French mathematician Laurent Schwartz introduced the space S 0 .Rn / of tempered distributions [7], [8]. Every tempered distribution T 2 S 0 .Rn / is a distribution in D 0 .Rn /, i.e. T 2 0 S .Rn / H) T 2 D 0 .Rn / H) the algebraic inclusion S 0 .Rn /  D 0 .Rn / (8.2.9) (see imbedding results later). But every distribution T 2 D 0 .Rn / is not a tempered distribution in S 0 .Rn /, i.e. D 0 .Rn / 6 S 0 .Rn /. 2 2 For example, for n D 1, e x 2 D 0 .1; 1Œ/, but e x … S 0 .1; 1Œ/. (8.2.10) 2 2 In fact, e x 2 L1loc .1; 1Œ/ H) e x defines a distribution Tex2 2 D 0 .1; 1Œ/ R 1 x2 by hTex2 ; i D 1 e .x/dx 8 2 D.1; 1Œ/. But hTex2 ; i is not defined 8 2 S.1; 1Œ/. R1 2 2 2 2 For instance, for .x/ D e x , hTex2 ; e x i D 1 e x :e x dx D 1 H) Tex2 2

is not a tempered distribution on 1; 1Œ, i.e. e x … S 0 .1; 1Œ/.

(8.2.11)

8.2.3 Examples of tempered distributions of S 0 .Rn / R 1. Function f 2 L1 .Rn /, integrable on Rn , i.e. Rn jf .x/jd x < C1, defines a tempered distribution Tf 2 S 0 .Rn / by Z Tf ./ D hf; i D f .x/.x/d x 8 2 S.Rn /: (8.2.12) Rn

In fact, the integral in (8.2.12) exists 8 2 S.Rn /, since this integral exists for bounded  and functions  2 S.Rn / are indeed bounded in Rn : 8x 2 Rn , j.x/j  sup j.x/j D q0;0 ./ < C1 x2Rn

(see (7.2.12)) and ˇZ ˇ Z ˇ ˇ ˇ ˇ f .x/.x/d xˇ  q0;0 ./ ˇ Rn

jf .x/jd x < C1

8f 2 L1 .Rn /:

Rn

Moreover, we have shown later (see p. 433) that L1 .Rn / ,! S 0 .Rn /, the imbedding operator ,! being a continuous one. Hence, we identify f 2 L1 .Rn / with Tf 2 S 0 .Rn / defined by (8.2.12) and can write f D Tf 2 S 0 .Rn /;

i.e. L1 .Rn /  S 0 .Rn /:

(8.2.13)

Section 8.2 Space S 0 .Rn / of tempered distributions

427

2. Every bounded function f in Rn defines a tempered distribution Tf 2 S 0 .Rn / by the formula (8.2.12). In fact, f is bounded in Rn H) 9M > 0 such n n 1 n that jf .x/j  M R 8x 2 R , and  2 S.R / H)  2 L .R / (by Proposition 7.2.2), i.e. Rn j.x/jd x < C1. Hence, ˇZ ˇ Z ˇ ˇ ˇ ˇ jTf ./j D ˇ f .x/.x/d xˇ  sup j.x/jd x < C1 8 2 S.Rn /: Rn

x2Rn

Rn

(8.2.14) The simplest examples of bounded functions on R D 1; 1Œ are the Heaviside function H.x/ D 1 for x > 0 and H.x/ D 0 for x < 0, constant function 2 f .x/ D C 8x 2 R; e kxk , trigonometric functions sin ˛x, cos ˇx 8˛; ˇ 2 R, etc. All these functions define tempered distributions by (8.2.12). 3. Every locally integrable function f 2 L1loc .Rn / with slow or polynomial growth at infinity defines a tempered distribution Tf 2 S 0 .Rn / by the integral (8.2.12). R f is locally integrable in Rn H) V jf .x/jd x < C1 8 compact subsets K of K Rn . f has slow/polynomial growth at infinity H) 9 an integer k 2 N0 and a constant C > 0 such that jf .x/j  C.1 C kxk2 /k H)

jf .x/j .1Ckxk2 /k

for kxk ! 1;

 C for kxk ! 1.

(8.2.15)

 In fact, for  2 S.Rn /, j.1Ckxk2 /kCn .x/j  qkCn;0 ./ < C1 (by (7.2.16)) q

./

kCn;0 n and jf .x/.x/j  C .1 C kxk2 /k j.x/j H) j.x/j  .1Ckxk 2 /kCn 8x 2 R (using (8.2.14))

H)

jf .x/.x/j  C..1 C kxk2 /k / 

 ./ qkCn;0

.1 C kxk2 /kCn

DC

 ./ qkCn;0

.1 C kxk2 /n

:

Hence, ˇZ ˇ ˇ ˇ ˇ jTf ./j D jhf; ij D ˇ f .x/.x/d xˇˇ Rn Z dx   ./  C qkCn;0 ./ n < C1;  C qkCn;0 2 /n n .1 C kxk R R dx n (see (7.2.26)). since Rn .1Ckxk 2 /n   Hence, every locally integrable function f 2 L1loc .Rn / with slow growth at infinity defines a tempered distribution Tf 2 S 0 .Rn /, and we identify f with

428

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Tf by writing Tf D f 2 S 0 .Rn /. The term tempered corresponds to this slow increase at infinity, and the corresponding distributions are thus called tempered. Every locally summable function f 2 L1loc .Rn / defines a regular distribution R 0 n Tf 2 D .R / by Tf ./ D Rn f .x/.x/d x 8 2 D.Rn /, but f 2 L1loc .Rn / does not define a tempered distribution of S 0 .Rn / in general. 2

For example, for n D 1; f .x/ D e x 2 L1loc .1; 1Œ/ defines a regular distri2 2 bution such that e x 2 D 0 .1; 1Œ/, but e x does not define a tempered distri2 bution, i.e. e x … S 0 .1; 1Œ/ (see (8.2.10)). Similarly, e x 2 D 0 .1; 1Œ/, but e x … S 0 .1; 1Œ/. Function f .x/ D e x cos.e x / 8x 2 R does not have slow/polynomial growth at infinity, since supx2R .je x cos.e x /j=.1 C jxj2 /k / D 1 8k 2 N, although e x cos.e x / 2 L1loc .1; 1Œ/, whereas g.x/ D sin.e x / 8x 2 R has slow/ polynomial growth at infinity, since j sin.e x /j  1 8k 2 N 2 k x2R .1 C jxj / sup

and

sin.e x / 2 L1loc .1; 1Œ/:

(8.2.16)

Hence, sin.e x / defines a tempered distribution, i.e. sin.e x / 2 S 0 .1; 1Œ/. Now we show that e x cos.e x / also defines a tempered distribution. In fact, integrating by parts, we have, 8 2 S.1; 1Œ/, ˇZ 1 ˇ ˇ Z 1 ˇ ˇ ˇ ˇ ˇ jTf ./j D ˇˇ e x cos.e x /.x/dx ˇˇ D ˇˇ sin.e x / 0 .x/dx ˇˇ Z

1 1

j sin.e x /j j 0 .x/jdx 

 1

 C sup ..1 C x 2 /j 0 .x/j/  x2R

with C D

Z

1

1

j 0 .x/jdx

1  C q1;1 ./

R1

dx 1 1Cx 2 , S 0 .1; 1Œ/

H) Tf 2 by Proposition 8.2.1 H) e x cos.e x / is a tempered 0 distribution in S .1; 1Œ/. Moreover, we will show later that sin.e x / 2 S 0 .1; 1Œ/ will imply that d .sin.e x // D e x cos.e x / 2 S 0 .1; 1Œ/ (see Section 8.2.5 later). Thus, dx 0 S .Rn / contains functions which may not have slow/polynomial growth at infinity. 4. Every distribution T 2 E 0 .Rn /  D 0 .Rn / with compact support in Rn is a tempered distribution of S 0 .Rn /. In fact, T 2 D 0 .Rn / has compact support H) T ./ is defined 8 2 C 1 .Rn /. But S.Rn /  C 1 .Rn / H) T ./ is well defined 8 2 S.Rn / (see (5.6.3)). For example, Dirac distribution ıa D ı.x  a/ 2 D 0 .Rn / with mass/charge/

Section 8.2 Space S 0 .Rn / of tempered distributions

429

force etc. concentrated at a 2 Rn has compact support D ¹aº. Hence, ıa D ı.x  a/ 2 S 0 .Rn / is defined by: 8 2 S.Rn /:

hıa ; i D hı.x  a/; i D .a/ 5. Functions H)

(8.2.17)

2 S.Rn / define tempered distributions, i.e. 2 S.Rn / Z T 2 S 0 .Rn / with T ./ D .x/.x/d x 8 2 S.Rn /: Rn

(8.2.18)  In fact, from (7.2.26), S.Rn / ,! L1 .Rn / with kkL1 .Rn /  C qn;0 ./ 8 2 n S.R / ˇZ ˇ Z ˇ ˇ ˇ ˇ H) jT ./j D ˇ .x/.x/d xˇ  sup j .x/j j.x/jd x Rn

Rn

x2Rn

  C q0;0 . /qn;0 ./ < C1

8 2 S.Rn /

H) T is a tempered distribution. The result also follows from (1) by virtue of the imbedding S.Rn / ,! L1 .Rn /, and also from (2) by virtue of the boundedness of functions 2 S.Rn /. Hence, S.Rn /  S 0 .Rn /: 2

2

(8.2.19) 2

2

For n D 1, .x/ D x m e x , sin ˛x e x , cos ˇxe x , e x , etc. belongs to S.R/, and consequently belongs to S 0 .R/. (8.2.20) 8 multi-index ˛, 2

x˛ e kxk 2 S.Rn /

H)

2

x˛ e kxk 2 S 0 .Rn /:

(8.2.21)

8.2.4 Convergence of sequences in S 0 .Rn / Definition 8.2.3. A sequence .Tk /1 of tempered distributions Tk 2 S 0 .Rn / is said kD1 to converge to the tempered distribution T 2 S 0 .Rn / if and only if hTk ; i ! hT; i in C 8 2 S.Rn / as k ! 1:

(8.2.22)

Then we write Tk ! T in S 0 .Rn / as k ! 1, i.e. Tk ! T in S 0 .Rn /

H)

hTk ; i ! hT; i in C as k ! 18 2 S.Rn /: (8.2.23)

Proposition 8.2.2. Let .Tk /1 be a sequence of tempered distributions Tk 2 S 0 .Rn / kD1 8k 2 N such that limk!1 hTk ; i exists in C 8 2 S.Rn /. Then the sequence has a limit in S 0 .Rn /, i.e. 9 a unique T 2 S 0 .Rn / such that .Tk /1 kD1 hTk ; i ! hT; i in C as k ! 1 8 2 S.Rn /:

(8.2.24)

430

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Convergence of series of tempered distributions in S 0 .Rn / P Definition 8.2.4. Let 1 Tk be a series of tempered distributions Tk 2 S 0 .Rn / PkD1 1 D N in 8k 2 N. Let SN P kD1 Tk 8N 2 N. If the sequence .SN /N D1 converges P 1 0 n 0 n S .R /, the series kD1 Tk is said to convergeP in S .R /, or equivalently 1 T kD1 k converges in S 0 .Rn /, if and only if the series 1 kD1 hTk ; i converges in C 8 2 S.Rn /. P 0 n Then the series 1 kD1 Tk is called summable in S .R /. P1 0 n 0 n P1 kD1 Tk converges in S .R / ” 9 a unique T 2 S .R / such that T D kD1 Tk with hT; i D

1 X

hTk ; i

8 2 S.Rn /:

(8.2.25)

kD1

P

i2I

Ti , I being a set of indices, converges in S 0 .Rn / X ” hTi ; i converges in C 8  2 S.Rn /:

(8.2.26)

i2I

Example 8.2.1. For real number a > 0, consider the following two series in S 0 .R/: P1 n (A) nD0 a ın ; P1 n (B) nD1 a ın , where ın is the Dirac distribution with mass/charge/force etc. concentrated at n D 0; ˙1; ˙2; : : : defined by hın ; i D .n/ 8 2 S.R/, 8n D 0; ˙1; ˙2; : : : Show that 1. the series (A) converges in S 0 .R/ for 0 < a  1 and diverges for a > 1; 2. the series (B) converges in S 0 .R/ only for a D 1. Solution. 1. From Proposition 7.2.1, 8 2 S.R/, 9M > 0 such that 8k; l 2 N0 , jx k  .l/ .x/j  M 8x 2 R. Hence, 8 2 S.R/, 9M > 0 such that 8k 2 N0 , M jx k .x/j  M 8x 2 R H) j.x/j  jxj k 8x ¤ 0 8k 2 N H) j.n/j  M nk 8n 2 N, 8k 2 N and j.0/j  M . Hence, 8n 2 N, jhan ın ; ij D jan .n/j  M an nk 8k 2 N and, for n D 0, jhı0 ; ij D j.0/j  M . Then ˇ X ˇX N N X ˇ ˇ N n n ˇ ˇ ha ı ; i a jhı ; ij D jhı ; ij C an jhın ; ij  n n 0 ˇ ˇ nD0

nD0

nD1

  N X n k a n M 1C 8k 2 N; 8N 2 N; 8 2 S.R/: nD1

Section 8.2 Space S 0 .Rn / of tempered distributions

431

In particular, this inequality holds 8k  2 and 8N 2 N. But 8k  2, the PN n k 1 sequence .M.1 C nD1 P1a n n //N D1 converges in R as N ! 1 8 (strictly positive) a  1. Then nD0 ha ın ; i converges (absolutely) in R 8a  1 and P n ı converges in 8 2 S.R/. Hence, by Definition 8.2.4, the series 1 a n nD0 S 0 .R/ for 0 < a  1. Case a > 1: Let b 2 R be any strictly positive real number such that 1 < b < a, with a=b > 1. Then it is possible to construct a  2 S.R/ such that for x > 0, .x/ D b x (which is a C 1 -function with rapid decay at infinity, since x k b x ! 0 as x ! 1 8k 2 N). Then han ın ; i D an .n/ D an b n D . ab /n 8n 2 N with a=b > 1, and N X

an .n/ D 1 C

nD0

N  n X a nD1

b

D

N  n X a nD0

b

!1

as N ! 1:

P1

n nD0 a ın

Hence,P the series diverges in S 0 .R/ for a > 1. Combining the two 1 n cases, nD0 a ın converges in S 0 .R/ for 0 < a  1 and diverges in S 0 .R/ for a > 1. P1 P n n 2. in S 0 .R/Pif and only if both 1 nD1 nD1 a ın P1 a ınn will be convergent 1 0 n 0 and nD0 a ın converge in S .R/. But nD0 a ın is convergent in S .R/ for a  1 and divergent in S 0 .R/ for a > 1. Similarly, we can show that P 1 an ı converges in S 0 .R/ for a  1 and diverges in S 0 .R/ for a < 1. nD1 P P1 n n ı n ı ; ij D jan .n/j  In fact, nD1 an ın D 1 n and jha n nD1 a M k n k M a n D an n 8 2 S.R/, 8n 2 N, 8k 2 N. Following the steps of P n ı ; i the proof of the convergence of series (A), we can show that 1 ha n nD1 P P 1 n ı n converges for a1  1, i.e. for a  1. Hence, 1 n D nD1 a nD1 a ın P1 0 n converges in S .R/ for a  1. Therefore, both the nD1 a ın and Pseries P1 1 n 0 n nD0 a ın converge in S .R/ only for a D 1, i.e. nD1 a ın converges in S 0 .R/ only for a D 1. P Example 8.2.2. Consider the series 1 nD0 ak ık , where ak 2 R 8k 2 N0 , ık is the Dirac distribution with concentration at the point k 2 N0 . Show the conditions under which the series converges in S 0 .R/. P1 P1 0 .R/ if and only if Proof. a ı converges in S D k k kD0 kD0 ak hık ; i P1 kD0 ak .k/ converges 8 2 S.R/. But 8 fixed  2 S.R/, 9C > 0 such that j.x/j  C jxjl 8l 2 N, 8x P ¤ 0 H) j.k/j  C k l 8l 2 N, 8k 2 N. Hence, 8 fixed  2 S.R/, 1 C1 k p 8k 2 N kD0 ak .k/ will converge if jak j P .lp/ / for some fixed C1 > 0 and p 2 N, since the majoring series M.1 C 1 kD1 k will converge with l  p  2, M D max¹ja0 k.0/j; C C1 º. 2

Remark 8.2.1. For ak D e k , the series 2 can not write e k  C1 k p 8k 2 N.

P1

kD0 ak ık

2

will diverge, since for e k , we

432

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8.2.5 Derivatives of tempered distributions Every tempered distribution T 2 S 0 .Rn / is infinitely differentiable In fact, T 2 S 0 .Rn / H) 8 multi-index ˛, @˛ T 2 S 0 .Rn / and the derivative @˛ T 2 S 0 .Rn / is defined by h@˛ T; i D .1/j˛j hT; @˛ i

8 2 S.Rn /:

(8.2.27)

Justification  2 S.Rn / H) @˛  2 S.Rn / 8j˛j 2 N H) hT; @˛ i is well defined 8T 2 S 0 .Rn / H) the left-hand side h@˛ T; i is also well defined 8 2 S.Rn /, and @˛ T is continuous on S.Rn / H) @˛ T 2 S 0 .Rn / is a tempered distribution 8j˛j 2 N. For n D 1, f .x/ D sin.e x / 8x 2 R H) sin.e x / is a locally summable function of slow/polynomial growth at infinity (see (3)) H) sin.e x / 2 S 0 .1; 1Œ/ is a tempered distribution. Then, 8 2 S.1; 1Œ/,   Z 1 d x d sin.e /; D sin.e x / .x/dx dx dx 1 Z 1 D e x cos.e x /.x/dx 8 2 S.1; 1Œ/ .by integration by parts/ 1   d x x x D he cos.e /; i D  .sin .e //;  dx d H) dx .sin.e x // D e x cos.e x / 2 S 0 .1; 1Œ/, since sin.e x / 2 S 0 .1; 1Œ/ implies that all its derivatives also belong to S 0 .1; 1Œ/. 8˛, @˛ W S 0 .Rn / ! S 0 .Rn / is linear and continuous. (8.2.28)

Proof. Linearity: h@˛ .1 T1 C 2 T2 /; i D .1/j˛j h1 T1 C 2 T2 ; @˛ i D .1/j˛j Œ1 hT1 ; @˛ i C 2 hT2 ; @˛ i D .1/j˛j 1 hT1 ; @˛ i C .1/j˛j 2 hT2 ; @˛ i D 1 .1/j˛j hT1 ; @˛ i C 2 .1/j˛j hT2 ; @˛ i D 1 h@˛ T1 ; i C 2 h@˛ T2 ; i D h1 @˛ T1 C 2 @˛ T2 ; i

8 2 S.Rn /

H) @˛ .1 T1 C 2 T2 / D 1 @˛ T1 C 2 @˛ T2 8T1 ; T2 2 S 0 .Rn /, 81 ; 2 2 C. Continuity of operation of differentiation on S 0 .Rn /: 8j˛j 2 N, @˛ W  2 S.Rn / 7! @˛  2 S.Rn / is linear and continuous by (7.3.5). We are to show that Tk ! T in S 0 .Rn / H) @˛ Tk ! @˛ T in S 0 .Rn / as k ! 1.

Section 8.2 Space S 0 .Rn / of tempered distributions

433

In fact, h@˛ T  @˛ Tk ; i D h@˛ .T  Tk /; i D .1/j˛j h.T  Tk /; @˛ i H) jh@˛ T  @˛ Tk ; ij D jhT  Tk ; @˛ ij ! 0 as k ! 1, since Tk ! T in 0 S .Rn / H) @˛ Tk ! @˛ T in S 0 .Rn / as k ! 1 8j˛j 2 N, H) @˛ W S 0 .Rn / ! S 0 .Rn / is continuous on S 0 .Rn /. Multiplication of a tempered distribution by a polynomial T 2 S 0 .Rn /, p a polynomial in n variables x1 ; x2 ; : : : ; xn H) pT 2 S 0 .Rn / defined by hpT; i D hT; pi 8 2 S.Rn /. (8.2.29) Indeed,  2 S.Rn / H) p 2 S.Rn / 8 polynomials p (by Proposition 7.2.1) H) hT; pi is well defined 8T 2 S 0 .Rn /, 8 2 S.Rn / H) hpT; i is well defined 8 2 S.Rn / H) pT 2 S 0 .Rn /. Structure of the elements of S 0 .Rn / We agree to accept the following result without proof (see [8, p. 239]). Theorem 8.2.1. A distribution T 2 D 0 .Rn / is a tempered distribution of S 0 .Rn / if and only if there exists an integer m 2 N0 , a multi-index ˛ and a bounded continuous function f in Rn such that T D @˛ Œ.1 C kxk2 /m f  with hT; i D h@˛ Œ.1 C kxk2 /m f ; i D .1/j˛j h.1 C kxk2 /m f; @˛ i D .1/j˛j hf; .1 C kxk2 /m @˛ i

8 2 S.Rn /:

(8.2.30)

Imbedding results 1. Lp .Rn / ,! S 0 .Rn /, 1  p  1, the imbedding operator ,! being a continuous one, i.e. fk ! f in Lp .Rn /

H)

Tfk ! Tf in S 0 .Rn / as k ! 1:

(8.2.31)

Proof. Algebraic inclusion: Lp .Rn /  S 0 .Rn /8p 2 Œ1; 1. For p D 1, it has been shown in (8.2.12); now we prove it for general p. Let f 2 Lp .Rn /; we are to show that ,!f D Tf 2 S 0 .Rn /. In fact, if, for f 2 R jf .x/j Lp .Rn /, Rn .1Ckxk 2 /n d x < C1, then f will define a tempered distribution R 0 n Tf 2 S .R / by Tf ./ D Rn f .x/.x/d x 8 2 S.Rn /. Applying Hölder’s inequality and the proof of (7.4.1), we get   Z   jf .x/j 1   p n d x  kf k L .R /  2 n .1 C kxk2 /n Lq .Rn / Rn .1 C kxk /  kf kLp .Rn /  n=q < C1 (with

1 q

C

1 p

D 1, q D 1 for p D 1).

434

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Hence, f 2 Lp .Rn / H) ,!f D Tf 2 S 0 .Rn / H) Lp .Rn /  S 0 .Rn / 8p 2 Œ1; 1. ,! W Lp .Rn / ! S 0 .Rn / is linear: ,!.˛1 f1 C ˛2 f2 / D T˛1 f1 C˛2 f2 D ˛1 Tf1 C ˛2 Tf2 D ˛1 ,!f1 C ˛2 ,!f2

8˛1 ; ˛2 2 C; 8f1 ; f2 2 Lp .Rn /:

Continuity of ,!W Lp .Rn / ! S 0 .Rn /: For this we are to show that fk ! f in Lp .Rn / H) Tfk ! Tf in S 0 .Rn /, where Tfk D ,!fk ; Tf D ,!f . Set Tk D Tfk 8k 2 N, T D Tf . Then, 8 2 S.Rn /, ˇZ ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ jhT; i  hTk ; ij D ˇ f d x  fk d xˇ D ˇ .f  fk /d xˇˇ Rn Rn Rn Z jf .x/  fk .x/j j.1 C kxk2 /n .x/jd x  2 /n n .1 C kxk R Z jf .x/  fk .x/j 2 n dx  sup j.1 C kxk / .x/j 2 n Rn .1 C kxk / x2Rn   qn;0 ./ n=q kf  fk kLp .Rn / ! 0

as k ! 1;

since fk ! f in Lp .Rn /. Hence, Tk ! T in S 0 .Rn / H) Tfk ! Tf in S 0 .Rn /. Therefore, ,! W Lp .Rn / ! S 0 .Rn / is continuous. 2. S 0 .Rn / ,! D 0 .Rn / with continuous imbedding operator ,!.

(8.2.32)

Proof. Since D.Rn / is a dense subspace of S.Rn / (see (7.5.1)) and D.Rn / ,! S.Rn /, the imbedding being a continuous one (see (7.4.2)), we have S 0 .Rn / ,! D 0 .Rn /, the imbedding operator ,! being a continuous one (see also (4.2.7)).

3. D.Rn / ,! S.Rn / ,! Lp .Rn / ,! S 0 .Rn / ,! D 0 .Rn /,

(8.2.33)

each imbedding operator ,! being continuous from the space on its left-hand side into the space on its right-hand side. Proof. Combining the imbedding results (7.4.1), (7.4.2), (8.2.31) and (8.2.32), we get (8.2.33).

435

Section 8.3 Fourier transform of tempered distributions

8.3

Fourier transform of tempered distributions

Definition 8.3.1. Let T 2 S 0 .Rn / be a tempered distribution. Then its Fourier transform F T 2 S 0 .Rn / and its co-transform FN T 2 S 0 .Rn / are tempered distributions defined by: hF T; i D hT; F i

8 2 S.Rn /;

(8.3.1)

hFN T; i D hT; FN i

8 2 S.Rn /

(8.3.2)

((8.3.1)–(8.3.2) were, in fact, suggested in (8.1.5)), with Z Z ./e i2h;xi d ; .FN /.x/ D ./e i2h;xi d : .F /.x/ D Rn

(8.3.3)

Rn

Justification By Theorem 7.7.2, F W S.Rn / ! S.Rn / is an isomorphism from S.Rn / onto S.Rn /. Hence, 8 2 S.Rn /, F  2 S.Rn / and consequently, 8T 2 S 0 .Rn /, hT; F i is well defined 8 2 S.Rn /. Moreover, by Theorem 7.6.1, F  2 S.Rn / and hT; F i represents a continuous, linear functional on S.Rn /. Hence, the left-hand side hF T; i of (8.3.1) is well defined as a continuous, linear functional on S.Rn /, and F T 2 S 0 .Rn / is also a tempered distribution. Replacing F by FN and repeating the same arguments, we can justify that FN T 2 S 0 .Rn / and the formula (8.3.2) is well defined. The Fourier transform defined on S 0 .Rn / by (8.3.1) extends that defined on L1 .Rn /. Proof. 8f 2 L1 .Rn /  S 0 .Rn / and 8 2 S.Rn /  L1 .Rn /, with F  2 S.Rn / by Theorem 7.6.1, Z Z hf; F i D f .x/.F /.x/d x D .F f /././d  D hF f; i Rn

Rn

(by (7.1.28)). Fourier transforms of functions of L2 .Rn / Since L2 .Rn / ,! S 0 .Rn / (see (8.2.31)), every function f 2 L2 .Rn /1 defines a tempered distribution Tf D f 2 S 0 .Rn /, and hence its Fourier transform F f 2 S 0 .Rn / exists in the sense of (tempered) distribution. Long before the introduction of tempered distributions by Laurent Schwartz, Fourier transforms of functions f 2 L2 .Rn / were defined by Plancherel–Riesz by a method of classical functional analysis. But their (Fourier transforms of L2 -functions) definition in the framework of tempered distributions is quite simple and we state it here. 1 L2 .Rn /

6 L1 .Rn /. For example, f .x/ D

p 1 1Cx 2

2 L2 .R/, but f … L1 .R/ (see (7.1.10)).

436

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Theorem 8.3.1 (Plancherel–Riesz). Fourier transform F W S.Rn / ! S.Rn / (resp. co-transform F W S.Rn / ! S.Rn /) can be extended to a unitary operator (see Definition A.15.1.1 and (A.15.1.3)–(A.15.1.4) in Appendix A) from L2 .Rn / onto L2 .Rn / (this extended operator will still be denoted by the same notation F (resp. F )) such that F W f 2 L2 .Rn / 7! F f 2 L2 .Rn / (resp. F W f 2 L2 .Rn / 7! F f 2 L2 .Rn /) satisfies the following properties: I. hF f; F giL2 .Rn / D hf; giL2 .Rn / 8f; g 2 L2 .Rn / (h  ;  iL2 .Rn / being the inner product in L2 .Rn /), (8.3.4) 2 n 2 n i.e. F is a unitary operator from L .R / onto L .R /; II. kF f kL2 .Rn / D kf kL2 .RZn / 8f 2 L2 .Rn /;

III. hF f; iS 0 .Rn /S.Rn / D

(8.3.5)

Z

.F f /././d  D Rn

f .x/.F /.x/d x Rn

D hf; F iS 0 .Rn /S.Rn /

8f 2 L2 .Rn /; 8 2 S.Rn /; (8.3.6)

h  ;  iS 0 .Rn /S.Rn / being the duality between S 0 .Rn / and S.Rn /; Replacing F by F in the results above, the corresponding formulae for F are obtained. IV. F F f D F F f D f 8f 2 L2 .Rn /.

(8.3.7)

Proof. By the Plancherel–Riesz Theorem 7.7.3 on S.Rn /, F W S.Rn / ! S.Rn / is a unitary operator from S.Rn / onto S.Rn / equipped with the norm of L2 .Rn /, i.e. hF ; F iL2 .Rn / D h; iL2 .Rn / 8; 2 S.Rn /, and kF kL2 .Rn / D n n kkL2 .Rn / 8 2 S.R /. Since S.R / is dense in L2 .Rn / in the norm of L2 .Rn / (see Proposition 7.5.1), by the principle of extension by density (see Theorem A.8.2.1 in Appendix A), F W S.Rn / ! S.Rn / is extended to a unique continuous linear operator from L2 .Rn / onto L2 .Rn / such that this extended operator is still denoted by the same notation F , i.e. F W f 2 L2 .Rn / 7! F f 2 L2 .Rn /. I. Again, by virtue of the density of S.Rn / in L2 .Rn /, 8f; g 2 L2 .Rn /, 9 sequences .k / and . k / in S.Rn / such that k ! f , k ! g in L2 .Rn / as k ! 1 H) limk!1 hk ; k iL2 .Rn / D hf; giL2 .Rn / . In fact, using the Schwarz inequality, jhf; giL2 .Rn /  hk ;  .kf kL2 .Rn / kg

k iL2 .Rn / j

 jhf; g 

k kL2 .Rn / Ckf

k iL2 .Rn / j

k kL2 .Rn / k

C jhf  k ;

k kL2 .Rn / /

k iL2 .Rn / j

! 0 as k ! 1;

since k ! g H) kg  k kL2 .Rn / ! 0, k ! f H) kf  k kL2 .Rn / ! 0, k k k  k k  gk C kgk ! kgkL2 .Rn / as k ! 1. Hence, limk!1 hk ; k iL2 .Rn / D hf; giL2 .Rn / . Since F is a continuous linear operator from L2 .Rn / onto L2 .Rn /, k ! f , k ! g in L2 .Rn / H)

437

Section 8.3 Fourier transform of tempered distributions

F k ! F f; F

k

! F g in L2 .Rn / as k ! 1. Hence,

lim hF k ; F

k!1

k iL2 .Rn /

D hF f; F giL2 .Rn / :

(8.3.8)

But again by Plancherel–Riesz Theorem 8.3.1 on S.Rn /, hF k ; F hk ; k iL2 .Rn / 8k 2 N H)

lim hF k ; F

k!1

k iL2 .Rn /

D lim hk ; k!1

k iL2 .Rn /

k iL2 .Rn /

D

D hf; giL2 .Rn / :

Thus, using (8.3.8), hF f; F giL2 .Rn / D hf; giL2 .Rn / 8f; g 2 L2 .Rn / H) F W L2 .Rn / ! L2 .Rn / is a unitary operator from L2 .Rn / onto L2 .Rn /. II. For f D g, hF f; F f iL2 .Rn / D hf; f iL2 .Rn / H) kF f kL2 .Rn / D kf kL2 .Rn / 8f 2 L2 .Rn /. III. For f 2 L2 .Rn /, let .k / be a sequence in S.Rn / such that k ! f in L2 .Rn / as k ! 1. Then, 8 2 S.Rn /, hf; iL2 .Rn / D lim hk ; iL2 .Rn / D lim hF k ; F k!1

D hF f; F

k!1

iL2 .Rn /

iL2 .Rn / 8f 2 L2 .Rn /:

(8.3.9)

R R Hence, from (8.3.9), Rn f .x/ .x/d x D Rn .F f /./.F /./d  8 2 S.Rn /. By Theorem 7.7.2 on isomorphism, 9 2 S.Rn / such that .x/ D .F / H) F D F .F .// D  (using (7.1.17) and the Inversion Theorem 7.7.1 on R R S.Rn /) H) F D  H) Rn f .x/.F /.x/d x D Rn .F f /././d  Z Z H) hF f; iS 0 .Rn /S.Rn / D .F f /././d / D f .x/.F /.x/d x Rn

D hf; F iS 0 .Rn /S.Rn /

2

n

Rn n

8f 2 L .R /; 8 2 S.R /:

IV. Since S.Rn / is dense in L2 .Rn /, for .k / in S.Rn / with k ! f in L2 .Rn /, by the Inversion Theorem 7.7.1 on S.Rn /, we have F F k D F F k D k 8k 2 N. Hence, k ! f in L2 .Rn / H) F k ! F f in L2 .Rn / H) F .F k / ! F .F f / in L2 .Rn / by virtue of the continuity of F on L2 .Rn /. But F .F k / D k ! f in L2 .Rn /. Hence, F .F f / D f 8f 2 L2 .Rn /, since the limit is unique. Similarly, F F f D f 8f 2 L2 .Rn /.

Remark 8.3.1. By an abuse of definitions, one often writes Z f .x/e i2hx;i d x 8f 2 L2 .Rn /; .F f /./ D Rn

(8.3.10)

438

Chapter 8 Fourier transforms of distributions and Sobolev spaces

which, in fact, must be understood in the following sense: for f R2 L2 .Rn /, F f 2 L2 .Rn / is the limit of the sequence .fk /k2N defined by fk ./ D Bk f .x/e i2hx;i d  in L2 .Rn /; .Bk /k2N being a sequence of relatively compact subsets which tend to Rn as k ! 1. Obviously, this formula (8.3.10) is meaningless in the usual sense in general, since the integral in (8.3.10) may not be defined for f 2 L2 .Rn /. For example, for f .x/ D p 1 2 2 L2 .R/, the integral in (8.3.10) does not exist, 1Cx 1 .R/ (see (7.1.10)). But for p 1 p 1 … L 2 L2 .R/, its Fourier transform 1Cx 2 1Cx 2 R R F Œ p 1 2  can be defined by (8.3.6): R F . p 1 2 /./d  D R p 1 2 .F /.x/dx 1Cx 1Cx 1Cx 8 2 S.R/, where p 1 2 F  2 L1 .R/ H) the right-hand side integral is well 1Cx

since

defined 8 2 S.R/. Isometric isomorphism of Fourier transforms on L2 .Rn / Corollary 8.3.1. F W L2 .Rn / ! L2 .Rn / defined by (8.3.6) is an isometric isomorphism (see Definition A.8.3.2 in Appendix A) from L2 .Rn / onto L2 .Rn /. Proof. The result follows from the linearity and isometry (8.3.5) of F W L2 .Rn / ! L2 .Rn /.

8.3.1 Fourier transforms of Dirac distributions and their derivatives Using the definitions of Fourier transform F and co-transform FN of tempered distributions of S 0 .Rn / in (8.3.1) and (8.3.2) respectively, we will find the Fourier transforms and co-transforms of Dirac distributions and their derivatives. For alternative methods, see Section 8.4 later. 1. T D ı D ı.x/ 2 E 0 .Rn /  D 0 .Rn / is the Dirac distribution with mass/charge/ force etc. concentrated at x D 0 2 Rn , i.e. with compact support ¹0º. Hence, ı D ı.x/ 2 S 0 .Rn /  D 0 .Rn /, with hı; i D hı.x/; .x/i D .0/8 2 S.Rn /  E 0 .Rn /, is a tempered distribution and its Fourier transform .F ı/./ and co-transform .FN ı/./ are defined by (8.3.1) and (8.3.2) respectively. hF ı; i D hı; F i D hı.x/; .F /.x/i D .F /.0/ Z Z D ./e i2h;0i d  D 1  ./d  D h1; i 8 2 S.Rn / Rn

H)

Rn 0

n

F ı D F Œı.x/ D 1 in S .R /:

hFN ı; i D hı; FN i D hı.x/; .FN /.x/i D FN .0/ Z ./e i2h;0i d  D Rn Z D ./1d  D h1; i 8 2 S.Rn / Rn

(8.3.11)

439

Section 8.3 Fourier transform of tempered distributions

H)

FN ı D F Œı.x/ D 1 in S 0 .Rn /:

(8.3.12)

˛ j˛j ˛ hF Œ@˛ x .ı/; i D h@x .ı/; F i D .1/ hı.x/; @x .F /.x/i

(using definition of derivative in (8.2.27)) D .1/j˛j Œ@˛ x .F /.0/

(applying the definition of ı D ı.x/)

D .1/j˛j .F Œ.i 2/˛ .//.0/ Z j˛j D .1/ .i 2/˛ ./e i2h;0i d  n R Z D .i 2/˛ ./d  Rn

D hT.i2/˛ ; i D h.i 2/˛ ; i

8 2 S.Rn /;

j˛j

@ ˛ ˛1 ˛n where @˛ x D @x1 ˛1 @x2 ˛2 @xn ˛n ; .i 2/ D .i 21 / : : : .i 2n / ; ˛ ˛ @x .F / D F Œ.i 2 .// is given by Theorem 7.1.3,

H)

˛ ˛ F Œ@˛ x .ı/ D F Œ@x .ı.x// D .i 2/

in S 0 .Rn /8˛:

(8.3.13)

˛ j˛j ˛ N N hF Œ@˛ x .ı/; i D h@x .ı/; F i D .1/ hı.x/; @x .F /.x/i

N D .1/j˛j Œ@˛ x .F /.0/ D .1/j˛j .FN Œ.i 2/˛ .//.0/ Z j˛j D .1/ .i 2/˛ ./e i2h;0i d  Rn Z .i 2/˛ ./d  D Rn

D hT.i2/˛ ; i D h.i 2/˛ ; i H)

˛ N ˛ FN Œ@˛ x .ı/ D F Œ@x .ı.x// D .i 2/

8 2 S.Rn /

in S 0 .Rn / 8˛:

(8.3.14)

2. T D ıa D ı.x  a/ 2 D 0 .Rn / is the Dirac distribution with mass/charge/force etc. concentrated at x D a 2 Rn , i.e. with compact support ¹aº. Hence, ıa D ı.x  a/ 2 S 0 .Rn /  E 0 .Rn /  D 0 .Rn / is a tempered distribution, and its Fourier transform .F ıa /./ D .F Œı.x  a/.// and co-transform .FN ıa /./ D .F Œı.x  a//./ are defined by (8.3.1) and (8.3.2) respectively: hF ıa ; i D hıa ; F i D hı.x  a/; .F /.x/i D .F /.a/ Z D ./e i2h;ai d  D he i2h;ai ; i 8 2 S.Rn / Rn

H)

F ıa D F Œı.x  a/ D e i2h;ai in S 0 .Rn /;

from which (8.3.11) is obtained for a D 0.

(8.3.15)

440

Chapter 8 Fourier transforms of distributions and Sobolev spaces

hFN ıa ; i D hıa ; FN i D .FN /.a/ D he i2h;ai ; i 8 2 S.Rn / H) FN ıa D FN Œı.x  a/ D e i2h;ai in S 0 .Rn /, from which we get (8.3.12) for a D 0. ˛ j˛j ˛ hF Œ@˛ x .ıa /; i D h@x .ıa /; F i D .1/ hıa ; @x ŒF i

D .1/j˛j hıa ; @˛ x ŒF .x/i D .1/j˛j hı.x  a/; .F Œ.i 2/˛ .//.x/i D .1/j˛j .F Œ.i 2/˛ .//.a/ Z D .i 2/˛ e i2h;ai ./d  Rn

D hT.i2/˛ ei2h;ai ; i D h.i 2/˛ e i2h;ai ; i

8 2 S.Rn /

(see the proof of (8.3.13) for the intermediate steps) H)

˛ ˛ i2h;ai F Œ@˛ in S 0 .Rn /; x .ıa / D F Œ@x .ı.x  a// D .i 2/ e (8.3.16)

from which (8.3.13) is obtained for a D 0. ˛ j˛j ˛ N N hFN Œ@˛ a .ıa /; i D h@x .ıa /; F i D .1/ hıa ; @x ŒF i

D h.i 2/˛ e i2h;ai ; i

8 2 S.Rn /

(see the proofs of (8.3.13) and (8.3.16)) H)

˛ i2h;ai FN Œ@˛ in S 0 .Rn /; x .ıa / D .i 2/ e

(8.3.17)

from which (8.3.14) is obtained for a D 0.

8.3.2 Inversion theorem for Fourier transforms on S 0 .Rn / Theorem 8.3.2 (Fourier Inversion Theorem). Let F W S 0 .Rn / ! S 0 .Rn / and F W S 0 .Rn / ! S 0 .Rn / be defined by (8.3.1) and (8.3.2), respectively: 8 2 S.Rn /, 8T 2 S 0 .Rn /, hF T; i D hT; F i, hFN T; i D hT; FN i. Then FN D F 1 (resp. F D FN 1 ) is the inverse to transform F (resp. FN ) and F is an isomorphism from S 0 .Rn / onto S 0 .Rn /, 8T 2 S 0 .Rn /;

FN F T D F FN T D T:

(8.3.18)

Proof. From the Fourier Inversion Theorem 7.7.1 on S.Rn /; FN F  D F FN  8 2 S.Rn /. But 8T 2 S 0 .Rn /, F T; FN T 2 S 0 .Rn / and are defined by (8.3.1), (8.3.2), respectively. Hence, FN .F T /; F .FN T / 2 S 0 .Rn / and are given by:

Section 8.3 Fourier transform of tempered distributions

441

hFN F T; i D hF T; FN i D hT; F FN i D hT; i (since F FN  D ) 8 2 S.Rn / H) FN F T D T 8T 2 S 0 .Rn /; hF FN T; i D hFN T; F i D hT; FN F i D hT; i (since FN F  D ) 8 2 S.Rn / H) F FN T D T 8T 2 S 0 .Rn /. Continuity of F (resp. F 1 ): Tk ! T in S 0 .Rn / H) F Tk ! F T in S 0 .Rn / 1 (resp. Tk ! T in S 0 .Rn / H) F 1 Tk D F Tk ! F 1 T D F T in S 0 .Rn /) as k ! 1. Hence, F is an isomorphism from S 0 .Rn / onto itself. Corollary 8.3.2. I. F T D S H) FN S D T ; FN S D T H) F T D S . II. F T D 0 H) T D 0; FN S D 0 H) S D 0. III. .F T /_ D F T ; F TL D F T ; F F T D TL ,

(8.3.19) (8.3.20) (8.3.21)

L where TL and .F T /_ are defined by: 8 2 S.Rn / with .x/ D .x/, _ L h.F T / ; i D hF T; i L D hT; F i. L hTL ; i D hT; i, Proof. I. F T D S H) FN S D FN F T D T , FN S D T H) F T D F FN S D S . II. F T D S D 0 H) T D FN S D FN 0 D 0; FN S D T D 0 H) S D F T D 0. L D hT; F i L D hT; FN i D hFN T; i H) .F T /_ D III. h.F T /_ ; i D hF T; i FN T ; hFN TL ; i D hTL ; FN i D hTL ; .F /_ i D hT; F iDhF T; i H) FN TL DF T ; F F T D F FN TL D TL . Application

From (8.3.11) and (8.3.12), F ı D FN ı D 1. Hence, F 1 D F .FN ı/ D ıI

FN 1 D FN .F ı/ D ı:

(8.3.22)

8.3.3 Fourier transform of even and odd tempered distributions Even and odd tempered distributions (see also Definition 1.7.1) A tempered distribution T 2 S 0 .R/ is called 

L D T ./ 8 2 S.R/; even if and only if T ./

(8.3.23)



L D T ./ 8 2 S.R/, odd if and only if T ./

(8.3.24)

L where .x/ D .x/ 8 2 S.R/.

442

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Examples of even tempered distributions on R are even functions f 2 S 0 .R/, evenorder derivatives ı .2m/ of the Dirac distribution ı, etc. (see (1.7.3)–(1.7.4)). Examples of odd tempered distributions on R are odd functions f 2 S 0 .R/, oddorder derivatives ı .2mC1/ of the Dirac distribution ı, etc. (see (1.7.6)–(1.7.7)). L 8T 2 S 0 .R/, T D TE C T0 such that 8 2 S.R/, TE ./ D 12 ŒT ./ C T ./, 1 L T0 ./ D 2 ŒT ./T ./ are unique even and odd tempered distributions respectively. The proof is exactly similar to that for (1.7.9) with  2 D.R/ replaced by  2 S.R/. Example 8.3.1. Show that c:p:v: x1 is an odd tempered distribution. Proof. 

 Z Z L 1 .x/ .x/ dx D lim dx c:p:v: ; L D lim x x "!0C jxj" x "!0C jxj"  Z "  Z 1 .x/ .x/ dx C dx D lim x x "!0C 1 "  Z " Z 1 .y/ .y/dy dy C D lim y "!0C 1 y "  Z "  Z 1 .y/ .y/ dy  dy D lim  y "!0C 1 y "   Z .x/ 1 dx D  c:p:v: ;  8 2 S.R/ D  lim x "!0C jxj" x

H) c:p:v: x1 is an odd tempered distribution. Fourier transforms of even and odd distributions T 2 S 0 .R/ is even (resp. odd) H) FN T D F T (resp. FN T D F T ) in S 0 .R/. (8.3.25) Proof. hFN T; i D hT; FN i 8 2 S.R/ (by definition). Z Z i2x N .F /./ D .x/e dx D .x/e i2x. / dx R

R _

D .F /./ D .F / ./ 8 2 Rn H) FN  D .F /_ 8 2 S 0 .R/. Hence, hFN T; i D hT; .F /_ i D hT; F i (since T is even) D hF T; i 8 2 S.R/ H) FN T D F T in S 0 .R/. Similarly, for odd distributions T 2 S 0 .R/, hFN T; i D hT; .F /_ i D hT; F i (since T odd) D hF T; i H) FN T D F T in S 0 .R/. T 2 S 0 .R/ is even H) F T is even (see also Property 4(a) in (7.1.19)); T 2 S 0 .R/ is odd H) F T is odd.

(8.3.26)

443

Section 8.3 Fourier transform of tempered distributions

L D hT; F i L D Proof. From (8.3.25), 8 even T 2 S 0 .R/, FN T D F T . But hF T; i _ hT; .F / i, since Z 1 Z i2x L L .x/e dx D .x/e i2.x/. / dx .F /./ D R 1

Z D

1

.y/e i2y. / dy D .F /./ D .F /_ ./

1

L D hT; .F /_ i D hT; F i (T is H) F L D .F /_ 8 2 S.R/. Hence, hF T; i even) D hF T; i 8 2 S.R/ H) F T is even. L D hT; F i L D hT; .F /_ i D hT; F i (since T is For odd T 2 S 0 .R/; hF T; i L D hF T; i 8 2 S.R/ H) F T is odd) D hF T; i 8 2 S.R/ H) hF T; i odd.

Fourier transform of homogeneous tempered distributions A tempered distribution T 2 S 0 .Rn / is called homogeneous of degree d 2 R if and only if, 8 > 0, hT;  i D .nCd / hT; i

8 2 S.Rn /;

(8.3.27)

where  .x/ D .x/ 8x 2 Rn (see (1.10.41) and (1.10.39) for the definition of homogeneous distributions of D 0 .Rn /). Proposition 8.3.1. Let T 2 S 0 .Rn / be a homogeneous tempered distribution of degree d 2 R. Then its Fourier transform F T 2 S 0 .Rn / is a homogeneous tempered distribution of degree n  d . Proof. T 2 S 0 .Rn / is homogeneous of degree d 2 R H) hT;  i D .nCd / hT; i 8 > 0, 8 2 S.Rn /, hF T;  i D hT; F  i 8 2 S.Rn /, where Z Z i2hx;i  .x/e dx D .x/e i2hx;i d x F  D Rn Rn Z  1 D .y/e i2hy;  i n d y (by change of variables: y D x with jJ j D n )  Rn    1 1 D n .F / D n .F /1= ./:    Hence, hF T;  i D hT; 1n .F /1= i D n hT; .F /1= i. But hT; .F /1= i D . 1 /.nCd / hT; F i (since T is homogeneous) H) hF T;  i D n  .nCd / hT; F i D d hT; F i 8 2 S.Rn / H) F T is homogeneous of degree p with .n C p/ D d H) p D n  d .

444

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Example 8.3.2. Show that z1 with z D x C iy defines a tempered distribution T 2 S 0 .R2 / such that T and its Fourier transform F T 2 S 0 .R2 / are homogeneous of degree 1. Proof.

1 jzj

1 .x 2 Cy 2 /1=2 .x 2 C y 2 /1=2

D

is integrable in the neighbourhood of the origin, since 1=r

with r D is integrable in the neighbourhood of the origin. Hence, 1 1 1 2 /. Moreover, 1 is bounded for jzj  1. Hence, T D 1 2 S 0 .R2 /. D 2 L .R loc r jzj jzj jzj Z 1  1 .x; y/dxdy D .; / 2 d d hT;  i D 2 2 .x C iy/  C i  R R Z 1 1 D .; /d d D 1 hT; i D .21/ hT; i;  R2  C i Z

which is obtained by the change of variables  D x,  D y with jJ j D 1=2 , H) T D z1 2 S 0 .R2 / with z D x C iy is homogeneous of degree d D 1 by definition in (8.3.27). F T D F . z1 / 2 S 0 .R2 / is also a homogeneous tempered distribution of order p D n  d D 2  .1/ D 1 by Proposition 8.3.1. Term-by-term differentiation of series of tempered distributions For series of tempered distributions, we have a theorem similar to Theorem 2.11.1: P 0 n /8k 2 Theorem 8.3.3. LetP 1 kD1 Tk be a series of tempered distributions Tk 2 S P.R 1 1 0 n ˛ N with its sum T D kD1 Tk in S .R /. Then, 8 multi-index ˛, @ T D kD1 @˛ Tk in S 0 .Rn /. Proof. The proof is similar to that of Theorem 2.11.1, with ‘ 2 D./’ replaced by ‘ 2 S.Rn /’ and ‘D 0 ./’ by ‘S 0 .Rn /’. Fourier transform of a convergent series of tempered distributions P1 distributions Tk 2 S 0 .Rn / Theorem 8.3.4. Let P1 P1 kD1 Tk be0 a nseries of tempered 8k 2 N with T D kD1 Tk in S .R /. Then F T D kD1 F Tk in S 0 .Rn /. PN 0 n Proof. Set SN D kD1 Tk such that SN ! T in S .R / as N ! 1. But 0 n 0 n F P W S .R / !PS .R / is continuous, i.e. SN ! T in S 0 .Rn / H) F SN D N 0 n FŒ N kD1 Tk  D kD1 F Tk ! F T in S .R / as N ! 1, from which the result follows.

445

Section 8.4 Fourier transform of distributions with compact support

8.4

Fourier transform of distributions with compact support

Let T 2 E 0 .Rn /  S 0 .Rn /  D 0 .Rn / be a distribution with compact support. Then, for T D Tx D T .x/ 2 E 0 .Rn / and 8 fixed  2 Rn , e i2hx;i 2 C 1 .Rn / D E.Rn / is a function of x and hTx ; e i2hx;i iE 0 .Rn /E.Rn / is well defined as a function of  2 Rn . Hence, we set b./ D hTx ; e i2hx;i iE 0 .Rn /E.Rn / : T

(8.4.1)

But for fixed x 2 Rn , e i2hx;i 2 C 1 .Rn / as a function of  H)

b./ 2 C 1 .Rn / D E.Rn / as a function of  T

H) 8 multi-index ˛ D .˛1 ; ˛2 ; : : : ; ˛n / with @˛  D

@j˛j ˛1 ˛ ˛ @ 1 @ 2 2 :::@ n n

(8.4.2)

,

˛ i2hx;i i2hx;i b i D hTx ; @˛ /i D hTx ; .i 2x/˛ e i2hx;i i @˛  T ./ D @ hTx ; e  .e

b./; D .i 2x/˛ hTx ; e i2hx;i i D .i 2x/˛ T

(8.4.3)

where .i 2x/˛ D .i 2x1 /˛1 .i 2x2 /˛2 : : : .i 2xn /˛n , since

@j˛j ˛ ˛ @ 1 1 :::@ n n

Œe i2.x1 1 Cx2 2 CCxn n /  D .i 2x1 /˛1 : : : .i 2xn /˛n : b D F T , i.e. the Fourier transform of T 2 E 0 .Rn /: T Since T 2 E 0 .Rn / H) T 2 S 0 .Rn /, F T 2 S 0 .Rn / is defined, 8 2 S.Rn /, by:  Z hF T; i D hT; F i D hTx ; .F /.x/i D Tx ;  Z D Tx ;

 d

Rn

 .x; /d 

./e

i2h;xi

with

.x; / D ./e i2h;xi :

(8.4.4)

Rn

Since T D Tx is independent of , T can be written in the tensor product form: T D Tx ˝ 1 with 1 ./ D 18 2 Rn , and hTx ˝ 1 ; .x; /i D hTx ; h1 ; .x; /ii D h1 ; hTx ; .x; /ii;

(8.4.5)

which follows from the definition of the tensor product Tx ˝ 1 of distributions in (6.1.19). But  Z hTx ; h1 ; .x; /ii D Tx ;

Rn



 Z 1 ./ .x; /d  D Tx ;

Rn

 .x; /d  D hF T; i;

446

Chapter 8 Fourier transforms of distributions and Sobolev spaces

which is obtained from (8.4.4). Again, Z h1 ; hTx ; .x; /ii D 1 ./hTx ; ./e i2h;xi id  n R Z Z i2h;xi b././d  T hTx ; e i./d  D D Rn

Rn

b; i (using (8.4.1)): D hT

(8.4.6)

b. b; i 8 2 S.Rn / H) F T D T Then, from (8.4.4)–(8.4.6), we have hF T; i D hT Now, we summarize the results: Proposition 8.4.1. For distributions T 2 E 0 .Rn / with compact support, the Fourier transform F T 2 C 1 .Rn / and co-transform F T 2 C 1 .Rn / are defined by b./ D hTx ; e i2hx;i iI .F T /./ D T

(8.4.7)

.F T /./ D hTx ; e i2hx;i i:

(8.4.8)

Fourier transforms of derivatives of distributions T 2 E 0 .Rn / with compact support and derivatives of their Fourier transforms 9 ˛ F Œ@˛ x T  D .i 2/ .F T /./I > > > > > ˛ = F Œ@˛ T  D .i 2/ .F T /./I x ˛ @˛  .F T /./ D F Œ.i 2x/ T I ˛ @˛  .F T /./ D F Œ.i 2x/ T :

> > > > > ;

(8.4.9)

Proof. ˛ i2hx;i i2hx;i i D .1/j˛j hT; @˛ /i F Œ@˛ x T ./ D h@x T; e x .e

D .1/j˛j hT; .i 2/˛ e i2hx;i i D .1/j˛j .i 2/˛ hT; e i2hx;i i D .i 2/˛ .F T /./; ˛

since @x@ ˛i i e i2.x1 1 Cx2 2 CCxn n / D .i 2i /˛i e i2.x1 1 CCxn n / and i .i 2/˛ D .i 21 /˛1 .i 22 /˛2 : : : .i 2n /˛n : ˛ i2hx;i i2hx;i F Œ@˛ i D .1/j˛j hT; @˛ /i x T ./ D h@x T; e x .e

D .1/j˛j hT; .i 2/˛ e i2hx;i i D .i 2/˛ F T ./: ˛ i2hx;i i2hx;i @˛ iDhTx ; @˛ /iDhTx ; .i 2x/˛ e i2hx;i i  .F T /./ D @ hTx ; e  .e

D h.i 2x/˛ Tx ; e i2hx;i i D F Œ.i 2x/˛ T ;

447

Section 8.4 Fourier transform of distributions with compact support

since Tx D T 2 E 0 .Rn /, the product .i 2x/˛ T 2 E 0 .Rn / is a distribution with compact support 8 multi-index ˛ and .i 2x/˛ 2 C 1 .Rn /. ˛ i2hx;i i2hx;i @˛ i D hTx ; @˛ /i  .F T /./ D @ hTx ; e  .e

D h.i 2x/˛ Tx ; e i2hx;i i D F ..i 2x/˛ T /: Theorem 8.4.1 (Paley–Wiener–Schwartz). Let T 2 E 0 .Rn / be a distribution with compact support in Rn . Then its Fourier transform F T can be extended to an entire function F in C n given by F .z/ D hTx ; e i2hx;zi i, such that F #Rn ./ D hTx ; e i2hx;i i D .F T /./ 8 2 Rn . Moreover, 9 constants C; M > 0 and an integer n0 2 N such that, 8z 2 C n , jF .z/j  C.1 C kzk/n0 e M k Im.z/k . Conversely, 8 entire functions F satisfying this inequality in C n , 9T 2 E 0 .Rn / such that .F T /.z/ D F .z/ 8z 2 C n . Example 8.4.1. Let T 2 E 0 .Rn / be a distribution with compact support in Rn such that hT; x˛ i D 0 8 multi-index ˛ D .˛1 ; ˛2 ;    ; ˛n / with x˛ D x1 ˛1 x2 ˛2    xn ˛n , j˛j D ˛1 C ˛2 C    C ˛n . Then, using the Paley–Wiener–Schwartz Theorem 8.4.1, prove that T D 0 in E 0 .Rn /. Proof. Since T 2 E 0 .Rn /, by the Paley–Wiener–Schwartz Theorem 8.4.1, TO D F T can be extended to an entire function F on C n defined by: F .z/ D hT; e i2hx;zi i By defining

@ @zk

D

1 @ 2 . @ k

8z D z1 ; z2 ; : : : ; zn 2 C n :

 i @ @ / with zk D k C ik , 1  k  n, the differk

entiation is reduced to that with respect to real variables k , k such that hT;

@ i2hx;zi e i. @zk

@F .z/ @zk

D

Hence, 8 multi-index ˛,

˛ i2hx;zi @˛ /i D .i 2/j˛j hT; x˛ e i2hx;zi i z F .z/ D hT; @z .e j˛j ˛ 0 j˛j ˛ H) 8˛, @˛ z F .0/ D .i 2/ hT; x  e i D .i 2/ hT; x i D 0 . P @˛z F .0/ ˛ But F is an entire function H) F .z/ D ˛ ˛Š  z 8z 2 C n , where ˛Š D ˛1 Š˛2 Š  ˛n Š, z˛ D z1˛1 z2˛2    zn˛n . H) F .z/ D 0 8z 2 C n H) F #Rn D TO D 0 in S 0 .Rn / H) T D F TO D 0 in S 0 .Rn /.

Examples of Fourier transforms of Dirac distributions and their derivatives Since the Dirac distribution (measure) ı D ıx D ı.x/ 2 E 0 .Rn / with mass/charge/ force etc. concentrated at x D 0 has compact support ¹0º, we can apply Proposition 8.4.1 to find its Fourier transform, and also the Fourier transform of its derivatives (see (8.3.11)–(8.3.17) for finding these directly from Definition 8.3.1).

448

Chapter 8 Fourier transforms of distributions and Sobolev spaces

1. .F ı/./ D b ı./ D hıx ; e i2hx;i i D e i2h0;i D e 0 D 1 8 2 Rn H) F ı D F Œıx  D F Œı.x/ D 1 (see (8.3.11)), where ıx (resp. ı.x)) denotes that the Dirac distribution ı with concentration at x D 0 is associated with the variable x. .F ı/./ D hıx ; e i2hx;i i D e i2h0;i D 1 8 2 Rn H) F ı D F Œıx  D F Œı.x/ D 1 (see (8.3.12)). ˛ i2hx;i i2hx;i F Œ@˛ i D .1/j˛j hıx ; @˛ /i x .ı/./ D h@x .ı/; e x .e

D .1/j˛j hıx ; .i 2/˛ e i2hx;i i D .1/j˛j .i 2/˛ hıx ; e i2hx;i i D .i 2/˛  1 D .i 2/˛

(see (8.3.13))

˛ ˛ ˛ H) F Œ@˛ x .ı/ D F Œ@x .ıx / D F Œ@x .ı.x// D .i 2/ . ˛ i2hx;i F Œ@˛ i D .1/j˛j hıx ; .i 2/˛ e i2hx;i i D .i 2/˛ x .ı/ D h@x .ı/; e ˛ ˛ H) F Œ@˛ x .ı/ D F Œ@x .ı.x// D .i 2/ (see (8.3.14)). ˛ i2hx;i i2hx;i @˛ i D hıx ; @˛ /i  .F ı/./ D @ hıx ; e  .e

D hıx ; .i 2x/˛  e i2hx;i i D i 20  e i20 D 0 8˛ 6D 0; which also follows from F ı./ D 1 H) @˛  1 D 0. Here, ıx denotes that Dirac distribution ı is associated with x, whereas F ı is associated with . ˛ @˛  .F ı/./ D @  1 D 0;

since F ı D 1 (see (8.3.22)):

2. For Dirac distribution (measure) ıa D ı.x  a/ with mass/charge/force concentrated at a, ıa 2 E 0 .Rn / with compact support ¹aº, we have .F ıa /./ D b ı a ./ D hıa ; e i2hx;i i D hı.x  a/; e i2hx;i i D e i2ha;i

(see (8.3.15));

from which .F ı/./ D 1 is obtained for a D 0. Here, ı.x  a/ denotes Dirac distribution ıa with concentration at a, which is associated with x, whereas F ıa is associated with . .F ıa /./ D hıa ; e i2hx;i i D e i2ha;i , from which .F ı/./ D 1 is obtained for a D 0 (see (8.3.12)). ˛ i2hx;i i2hx;i Œ@˛ i D .1/j˛j hıa ; @˛ /i x .ıa /./ D h@x .ıa /; e x .e

D .1/j˛j hıa ; .i 2/˛ e i2hx;i i D .i 2/˛ hıa ; e i2hx;i i D .i 2/˛ e i2ha;i

(see (8.3.14));

449

Section 8.4 Fourier transform of distributions with compact support ˛ from which F Œ@˛ x ı D .i 2/ is obtained for a D 0. ˛ i2hx;i F Œ@˛ i D .1/j˛j .i 2/˛ hıa ; e i2hx;i i x .ıa /./ D h@x .ıa /; e

D .i 2/˛ e i2ha;i ; ˛ from which F Œ@˛ x ı D .i 2/ is obtained for a D 0. ˛ @˛  ŒF ı./ D @ 1 D 0I

˛ @˛  ŒF ı./ D @ 1 D 0:

F Œe i2hx;ai ./ D F Œ.F ıa .//.x/ D F F ıa ./ D ıa ./:

3.

(8.4.10)

˛ ˛ F Œ.i 2x/˛ ./ D F Œ.FN .@˛  ı.///.x/ D F F .@ ı.// D @ ı./: (8.4.11)

Fourier transform of ıSR 2 S 0 .R3 /, the Dirac distribution of charges concentrated on a sphere SR  R3 Example 8.4.2. Let ıSR denote a simple layer of charges on a sphere SR of radius R with centre at the origin. Find the Fourier transform F ŒıSR . Solution. ıSR is a Dirac distribution with compact support SR  R3 . Hence, ıRSR 2 S 0 .R3 / and its Fourier transform is given by F ŒıSR  D hıSR ; e i2hx;i i D i2hx;i dS . Choosing spherical coordinates R; ;  such that the  coincides SR e with the axis of the cone D constant in order that 8x 2 SR , hx; i D kxkkk cos D Rkk cos , kk D .12 C 22 C 32 /1=2 , we have Z F ŒıSR  D

2 0

Z

Z



e i2Rkk cos  R2 sin d d

0 2

Z



e i2Rkk cos  d.i 2Rkk cos / i 2Rkk 0 0 1 R D 2R2  e i2Rkk cos jD .e CiRkk  e iRkk / D0 D i 2Rkk i kk R 2R D 2i sin.2Rkk/ D sin.2Rkk/ i kk kk D R2

H) F ŒıSR  D

d

2R kk

sin.2Rkk/.

Moreover, we have the important results used extensively in electrical engineering. Example 8.4.3. Show that R1 1. ı./ D 2 0 cos 2xdx; R1 2. @ı D 2 0 2x sin 2xdx. @

(8.4.12)

450

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Solution.

R i2x dx D 2 1 cos 2xdx. 1 e 0 R1 R1 i2x D 1 .i 2x/e dx D 2 0 2x sin 2xdx.

1. ı./ D F Œ1.x/ D 2.

@ı @

D F .i 2x/

R1

The integral in (1) (resp. (2)) is the limit in S 0 .R/ [8] of the corresponding integral R1 RA over the finite interval ŒA; A with A > 0, i.e. 1 .: : : /dx D limA!1 A .: : : /dx in S 0 .R/ (see also Examples 1.8.1–1.8.4 and Example 2.10.8). (2) can be Robtained directly from (1) by differentiating with respect to  under the integral sign .

8.5

Fourier transform of convolution of distributions

First of all, we summarize the convolution results (see Chapter 6 for details). Convolution  ;

2 S.Rn / of ;

2 S.Rn /

2 S.Rn / Z

H)

.

/.x/ D

.t/ .x  t/d t:

(8.5.1)

Rn

In fact, for  2 S.Rn /, the mapping from S.Rn / into S.Rn /.

2 S.Rn / 7! 

2 S.Rn / is continuous

Convolution  T 2 S.Rn / of 2 S.Rn / and distribution T 2 E 0 .Rn / with compact support 8 2 C 1 .Rn / D E.Rn / and 8 distributions T 2 E 0 .Rn / with compact support,  T D T  2 C 1 .Rn / is defined by (6.4.1): . T /.x/ D .T /.x/ D hT .t/; .x  t/i D hTL .t/; .x C t/i

(8.5.2)

L (hTL .t/; .t/i D hT .t/; .t/i D hT .t/; .t/i, and for fixed x 2 Rn , x .t/ D L .x C t/ H) x .t/ D x .t/ D .x  t/ and hTL .t/; .x C t/i D hTL .t/; x .t/i D hT .t/; L x .t/i D hT .t/; .x  t/i).  2 S.Rn /  C 1 .Rn /, T 2 E 0 .Rn / H)  T 2 S.Rn / defined by: . T /.x/ D hTL .t/; .x C t/i: Convolution  T 2 S 0 .Rn / of 2 S.Rn / and T 2 S 0 .Rn / T 2 S 0 .Rn / H)  T 2 S 0 .Rn / is defined by: h T; i D hT; L

i

8

2 S.Rn /:

(8.5.3)  2 S.Rn /, (8.5.4)

In fact,  2 S.Rn / H) L 2 S.Rn / H) L 2 S.Rn / by (8.5.1) H) the righthand side of (8.5.4) is well defined 8T 2 S 0 .Rn / H) the left-hand side of (8.5.4) is well defined as a continuous linear functional on S.Rn / H)  T 2 S 0 .Rn /.

451

Section 8.5 Fourier transform of convolution of distributions

Convolution S  T 2 E 0 .Rn / of distribution S; T 2 E 0 .Rn / with compact supports S; T 2 E 0 .Rn / are distributions with compact supports H) S T 2 D 0 .Rn / with compact support H) S T 2 E 0 .Rn / H) S T 2 S 0 .Rn / defined by: hS T; i D hS. / ˝ T ./; . C /i

8 2 S.Rn /:

(8.5.5)

Convolution S  T 2 S 0 .Rn / of tempered distribution S 2 S 0 .Rn / and distribution T 2 E 0 .Rn / with compact support S 2 S 0 .Rn /, T 2 E 0 .Rn / H) S T 2 S 0 .Rn / defined by 8 2 S.Rn /:

hT S; i D hS; TL i

(8.5.6)

In fact, T 2 E 0 .Rn / H) TL 2 E 0 .Rn / H) TL  2 S.Rn / by (8.5.3) and  2 S.Rn / 7! TL  is continuous from S.Rn / into S.Rn / H) the right-hand side of (8.5.6) is well defined 8S 2 S 0 .Rn / H) the left-hand side of (8.5.6) is also well defined as a continuous linear functional on S.Rn / and T S 2 S 0 .Rn /.

8.5.1 Fourier transforms of convolutions Fourier transform of convolution 

2 S.Rn / of ;

2 S.Rn /

Theorem 8.5.1. 8; 2 S.Rn /, the following relations hold in S.Rn /: I. F . / D .FN / .FN /; F . / D F  F ;

(8.5.7)

/; FN .

(8.5.8)

II. F .

/ D .F /  .F

/ D .FN /  .FN /.

(the Fourier transform of the convolution  forms).

is the product of their Fourier trans-

Proof. I. From Riesz’s formula in Corollary 7.1.1 we have, 8 fixed a 2 Rn , 8f; g 2 S.Rn /  L1 .Rn / (changing variables: y D x C a with jJ j D 1, J being the Jacobian), Z F Œ.F f /./g./.a/ D .F f /./g./e i2ha;i d  n R Z f .a C x/.F g/.x/d x D Rn Z Z D f .y/.F g/.y  a/d y D f .y/.F g/_ .a  y/d y Rn _

D .f .F g/ /.a//

Rn _

(since .F g/ .a  y/ D .F g/..a  y//

H)

FN Œ.F f /./g./.a/ D .f .F g/_ /.a/

H)

FN ŒF fg D f .F g/_ D f FN g;

8fixed a 2 Rn (8.5.9)

452

Chapter 8 Fourier transforms of distributions and Sobolev spaces

since .F g/_ .x/ D .F g/.x/ D

Z

g./e i2h;xi d 

Rn

Z

g./e i2h;xi d  D .FN g/.x/

D

8x 2 Rn :

Rn

By Theorem 7.7.2, F is an isomorphism from S.Rn / onto S.Rn /, and hence, for 8f 2 S.Rn /, 9 a unique  2 S.Rn / such that F f D  H) FN  D f . Set D g. Then, from (8.5.9), FN Œ.F f /g D FN Œ  D FN  FN 2 S.Rn /. F Œ  D F  F : For f; g 2 S.Rn /, (8.3.2) has the form: Z Z .F f /./g./d  D f .x/F g.x/d x: Rn

Rn

For a f , Z Z F .a f /./g./d  D .a f /.x/.F g/.x/d x Rn Rn Z D f .x C a/.F g/.x/d x Rn Z D f .y/F g..a  y//d y D f .F g/_ Rn

R

H) Rn F .a f /./g./d  D f F g, since F g D .F g/_ H) .F g/_ D .F g/_ D F g. But Z Z F .a f /./ D f .x C a/e i2hx;i d x D f .y/  e i2hya;i d y Rn Rn Z D e i2h;ai f .y/e i2hy;i d y D e i2h;ai .F f /./: Rn

R Hence, Rn F .a f /./g./d  D Rn .F f /./g./e i2h;ai d  D f F g H) F Œ.F f /./g.a/ D .f F g/.a/ 8a H) F Œ.F f /./g D f F g. Set D g and F f D . Then F  D F F f D f H) F Œ  D F  F . II. ; 2 S.Rn / H)  2 S.Rn / and FN . / D FN  FN 2 S.Rn /. Hence, taking the Fourier transform of both sides, we get F FN Œ  D F ŒFN  FN  8; 2 S.Rn / H)  D F ŒFN  FN . Set 1 D FN  and 1 D FN . Then  D F 1 , D F 1 H) .F 1 /.F 1 / D F .1 1 /, (1 ; 1 2 S.Rn / being arbitrary elements of S.Rn /). F .1 1 / D F 1 F 1 H) FN F .1 1 / D F .F 1 F 1 /. Set F  D 1 , F D 1 H) F 1 D , F 1 D . Then .F /  .F / D F . /, since F F 1 1 D 1 1 . R

453

Section 8.5 Fourier transform of convolution of distributions

Example 8.5.1. Let f and f be the normal probability distributions of Gauss defined by: 2 2 1 1  x x p e 2 2 ; f .x/ D p e 2 2 ;  2  2 ;  being the standard deviations of the distributions f and f , respectively. Then f f D fp 2 C 2 , i.e.

f .x/ D

2 2 x2 1 1 1   x x p e 2 2 p e 2 2 D p p e 2. 2 C 2 / :  2  2  2 C  2 2 2

2

Solution. From Example 7.1.2, F Œe x ./ D e  . Then, using (7.1.19), F Œf .x/./ D fO./ H) F Œf .kx/ D k1 fO. k / for real k ¤ 0, we have: r q  2 k   2 2 1 . p k / .  x/2 kx 2  F Œe e k ./ D F Œe ./ D q e D k k  

x2 2 2

s

H)

F Œe

./ D

H)

F Œf .x/./ D

Similarly, F Œf .x/ D e 2 Hence .e 2

2 2 2

1 2 2

e

p 2 2 2 D  2e 2 

(k D

1 ) 2 2

2 1 2 2 2  x p F Œe 2 2 ./ D e 2  :  2

2 2 2

/.e 2



2 2   1 2 2

.

2 2 2

2

2

2

2

/ D e 2 . C / D F Œfp 2 C 2 .x/   x2 1  2 C 2 / 2. DF p : p e  2 C  2 2 Then, since f 2 S.R/, f 2 S.R/ and f f 2 S.R/ by the Convolution Theorem 8.5.1 on S.R/: F Œf   F Œf  D F Œf f  D F Œfp 2 C 2  H)

F F Œf f  D F F Œfp 2 C 2 

H)

f f D fp 2 C 2

(by Theorem 8.3.2).

Fourier Transform of Convolution  T 2 S 0 .Rn / with 2 S.Rn /, T 2 S 0 .Rn / Theorem 8.5.2. 8 2 S.Rn /, 8T 2 S 0 .Rn /, Fourier transform F and co-transform FN satisfy the following reciprocal relations in S 0 .Rn /: I. F . T / D .F /.F T /; FN . T / D .FN /.FN T /; (8.5.10) II. F .T / D .F / .F T /; FN .T / D FN  FN T . (8.5.11)

454

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof. I.  2 S.Rn /, T 2 S 0 .Rn / H)  T 2 S 0 .Rn / and is defined by (8.5.4): h T; i D hT; L i 8 2 S.Rn /. Hence, by the definition of Fourier transforms in S 0 .Rn /, 8 2 S.Rn /, hF . T /; i D h T; F

i D hT; L F

D hFN F T; L F D hF T; FN .L F

L F But ;

2 S.Rn / H) L F H)

FN .L F

i /i

i

(since FN F T D T by Theorem 8.3.2) (by definition of FN in (8.3.2)): (8.5.11a)

2 S.Rn / by (8.5.1)

L FN F / D .FN /. L D .FN /

/

(by Theorem 8.5.1)

D .F / ;

since L .FN /.x/ D

Z Z

i2h;xi L ./e d D

Z

Rn

./e i2h;xi d  Rn

./e i2h;xi d  D F 

D Rn

(by change of variables,  D ). Then, from (8.5.11a), hF . T /; i D hF T; FN .L F D h.F /.F T /; i

/i D hF T; .F / i 8

2 S.Rn /

H) F . T / D .F /.F T / in S 0 .Rn /. Replacing F by FN and vice versa, and ‘i ’ by ‘i ’ in this proof, we get the proof of FN . T / D .FN /  .FN T /. II. Set T1 D F T , 1 D F . Then, applying the Fourier inversion Theorems 7.7.1 and 8.3.2 on S.Rn / and S 0 .Rn /, respectively, we have FN T1 D T; FN 1 D . Hence, F .T / D F Œ.FN 1 /  .FN T1 / D F ŒFN .1 T1 / D F FN .1 T1 / D 1 T1 D F  F T: Again, replacing F by FN and vice versa in this proof, we get FN .T / D FN  FN T .

455

Section 8.5 Fourier transform of convolution of distributions

Fourier transform of the convolution S  T 2 E 0 .Rn / of S; T 2 E 0 .Rn / S; T 2 E 0 .Rn / are distributions with compact support H) S T 2 E 0 .Rn / (i.e. a distribution with compact support) H) F .S T / is defined by (8.4.8): F .S T /./ D hS T; e i2hx;i i D hS.y/ ˝ T .z/; e i2hyCz;i i D hS.y/; hT .z/; e i2hz;i e i2hy;i ii D hS.y/; e i2hy;i i  hT .z/; e i2hz;i i D hS.y/; e i2hy;i i  hT .z/; e i2hz;i i D .F S /.F T / (since S; T 2 E 0 .Rn / and we can take out hT .z/; e i2hz;i i, which is independent of y) H)

F .S T / D .F S/  .F T /

8S; T 2 E 0 .Rn /:

(8.5.12)

Fourier transform of the convolution S  T 2 S 0 .Rn / of S 2 S 0 .Rn / and T 2 E 0 .Rn / with compact support Theorem 8.5.3. 8 tempered distributions S 2 S 0 .Rn / and 8 distributions T 2 E 0 .Rn / with compact support, we have: F .T S/ D .F T /.F S /

(8.5.13)

FN .T S/ D .FN T /  .FN S /:

(8.5.14)

Remark 8.5.1. (8.5.12) follows from (8.5.13), since S 2 E 0 .Rn / H) S 2 S 0 .Rn /. Proof. For T 2 E 0 .Rn / and S 2 S 0 .Rn /, T S 2 S 0 .Rn / by (8.5.6). Then, by the definition of F in S 0 .Rn /, hF .T S/; i D hT S; F i D hS; TL F i D hFN F S; TL F i D hF S; FN .TL F /i

(by (8.4.6))

(by Theorem 8.3.2) (by definition of FN )

D hF S; .FN TL /.FN F /i (by Theorem 8.5.2) D hF S; .F T /i D h.F T /.F S /; i

8 2 S.Rn /;

since FN TL D F T (hFN TL ; i D hTL ; FN i D hT; .FN /_ i D hT; F i D hF T; i 8 2 S.Rn / and for .FN /_ D F  (see (7.1.17) ) H) F .T S / D .F T /.F S /. Similarly, (8.5.14) can be proved. Example 8.5.2. Prove that for a > 0, ˛ > 0, ˇ > 0 and the Heaviside function H , H.x/e ax

x ˛1 x ˇ 1 x ˛Cˇ 1 H.x/e ax D H.x/e ax : .˛/ .ˇ/ .˛ C ˇ/

(8.5.14a)

456

Chapter 8 Fourier transforms of distributions and Sobolev spaces ˛1

ˇ1

Proof. For a > 0, ˛ > 0, ˇ > 0, H.x/e ax x.˛/ ; H.x/e ax x.ˇ / 2 L1 .R/ with ˛1

ˇ1

unbounded support H) (from (8.2.12)) H.x/e jajx x.˛/ ; H.x/e jajx x.ˇ / 2 S 0 .R/ ˇ1

ˇ1

(see Remark 6.3.7). .H.x/e jajx x.ˇ / H.x/e jajx x.ˇ / / 2 L1 .R/  S 0 .R/. Hence, ˛1

Theorem 8.5.3 can be applied. In fact, from Example 7.1.6, F ŒH.x/e ax x.˛/  D 1 /˛ for a > 0, ˛ > 0 H) for a > 0, ˛ > 0, ˇ > 0, . aCi2

 ˛Cˇ ˛  ˇ   ˛Cˇ 1  1 1 1 ax x D D F H.x/e .˛ C ˇ/ a C i 2 a C i 2 a C i 2    ˛1 ˇ 1  ax x ax x F H.x/e D F H.x/e .˛/ .ˇ/  ˛1 ˇ 1  ax x ax x D F H.x/e H.x/e .˛/ .ˇ/ by the Convolution Theorem 8.5.3, from which the result follows:   ˛1 ˇ 1  ˛Cˇ 1  ax x ax x ax x H.x/e D F F H.x/e F F H.x/e .˛/ .ˇ/ .˛ C ˇ/ H)

H.x/e ax

x ˛1 x ˇ 1 x ˛Cˇ 1 H.x/e ax D H.x/e ax : .˛/ .ˇ/ .˛ C ˇ/

Remark 8.5.2. This proof with the help of the Fourier transform holds only for a > 0, ˛1 i.e. for a < 0 we cannot apply the Fourier transform since H.x/e jajx x.˛/ is not a tempered distribution and, consequently, has no Fourier transform as a tempered distribution. But the result (8.5.14a) has been proved for any complex a in (6.3.6). Fourier transforms of non-tempered distributions have been studied in the framework of generalized functions by Gelfand and Schilov [1] (see also Schwartz [8, p. 233]). Multiplier Set M .Rn / for S.Rn / Definition 8.5.1. The set M .Rn / or simply M defined by

M .Rn / D ¹f W f 2 C 1 .Rn /; 8 2 S.Rn /;  7! f  is a continuous, linear mapping from S.Rn / into S.Rn /º (8.5.15) is called the multiplier set for S.Rn /. For example, f .x/ D .i 2x/˛ 2 M .Rn /. Proposition 8.5.1. Let f 2 M .Rn /. Then the mapping T 7! f T is a continuous, linear mapping from S 0 .Rn / into S 0 .Rn /.

457

Section 8.5 Fourier transform of convolution of distributions 





f 2 M .Rn / H) fL; @˛ f; x˛ f 2 M .Rn / 8˛, and 8a 2 Rn , a f 2 M .Rn /. (8.5.16) f 2 M .Rn / ” @˛ f 2 C 1 .Rn / with slow growth at infinity (see also @˛ f n (8.2.15)), i.e. 9k 2 N0 such that .1Ckxk 2 /k is bounded in R 8j˛j 2 N0 . f 2 M .Rn / H) f 2 S 0 .Rn /.

0 .Rn / of distributions with rapid decay Space C

Definition 8.5.2. The set C0 .Rn / of tempered distributions,

C0 .Rn / D ¹T W T 2 S 0 .Rn /; T  2 S.Rn /;  2 S.Rn / 7!  T 2 S.Rn / is continuous from S.Rn / into S.Rn /º (8.5.17) is called the space of distributions with rapid decay. For T 2 C0 .Rn /, TL 2 C0 .Rn / and TL  2 S.Rn / 8 2 S.Rn / such that 8S 2 S 0 .Rn /, hT S; i D hS; TL i 8 2 S.Rn /. Hence, 8T 2 C0 .Rn /, T S 2 S 0 .Rn / 8S 2 S 0 .Rn /:

(8.5.18)

0 .Rn / Properties of C

1. C0 .Rn /  S 0 .Rn / is a subspace of S 0 .Rn /.

(8.5.19)

2. E 0 .Rn /  C0 .Rn / is a subspace of C0 .Rn /.

(8.5.20)

3. 4.

C0 .Rn / contains locally integrable functions with T 2 C0 .Rn / H) TL ; a T; @˛ T; x˛ T 2 C0 .Rn /

rapid decay. 8a 2 Rn , 8 multi-index ˛. (8.5.21)

Theorem 8.5.4. Let f 2 M .Rn /, T 2 C0 .Rn / and S 2 S 0 .Rn /. Then: I. F f 2 C0 .Rn /; II. F T 2 M .Rn /; III. F Œf S  D .F f / .F S/ and F ŒS T  D F ŒS   F ŒT .

(8.5.22) (8.5.23)

Proof. I. f 2 M .Rn / H) f  2 S.Rn / 8 2 S.Rn / by definition. Set T1 D F f . Then, for 2 S.Rn /, T1 D F ŒFN  F f D F Œ.FN /f  2 S.Rn / (by (8.5.11a)), and the mapping 7! T1 is continuous from S.Rn / into S.Rn / (since it is a composition of three continuous mappings: 7! F 7! F f 7! F ŒF f ) for T1 2 S 0 .Rn /. Hence T1 D F f 2 C0 .Rn / by definition (8.5.17).

458

Chapter 8 Fourier transforms of distributions and Sobolev spaces

II. T 2 C0 .Rn / H)  T 2 S.Rn / and  7!  T is continuous from S.Rn / into S.Rn /. Set f1 D F T . For 2 S.Rn /, f1 D F FN f1 D n N N F .F /  .F T / D F .F T / 2 S.R / and the mapping 2 S.Rn / 7! f1 2 S.Rn / is continuous from S.Rn / into S.Rn / (since it is a composition of three continuous mappings: 7! F 7! F T 7! F .F T /). Hence, f1 D F T 2 M .Rn / by definition (8.5.15), since f1 2 S.Rn /. III. 8T 2 C0 .Rn /, 8S 2 S 0 .Rn /, T S 2 S 0 .Rn / by (8.5.18). Following the steps of the proof of Theorem 8.5.2, we have, 8T 2 C0 .Rn /, 8S 2 S 0 .Rn /, hF .T S/; i D hT S; F i D hS; TL F i D hFN F S; TL F i D hF S; FN .TL F /i D hF S; .FN TL /  .FN F /i D hF S; .F T /i 8 2 S.Rn /

D h.F T /.F S/; i

H) F .T S/ D .F T /  .F S/ (the second equality in (8.5.23)), since hF TL ; i D hTL ; F i D hT; .F /_ i D hT; F i

(by (7.1.17))

D hF T; i 8 2 S.Rn / H) F TL D F T in S 0 .Rn /. Set R D F S 2 S 0 .Rn /. Then T D F f H) FN T D f and FN R D S H) f S D .FN T /.FN R/ D FN .T R/ H) F .f S / D F FN .T R/ D T R D F f F S (the first equality in (8.5.23)).

8.6

Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions

Theorem 8.6.1. Let T 2 S 0 .Rn / be a tempered distribution on Rn . Then the following relations hold: I. 8 multi-index ˛, ˛ F Œ@˛ x T  D .i 2/ F T I

F Œx˛ T  D .1/j˛j

˛ @˛  .F T /./ D F Œ.i 2x/ T I

1 @˛ .F T /I .i 2/j˛j 

(8.6.1)

II. 8a 2 Rn , F Œa T  D e i2h;ai F T I where @˛ x D

@j˛j ˛ ˛ ˛ , @x1 1 @x2 2 :::@xn n

@˛  D

a .F T / D F Œe i2hx;ai T ; @j˛j ˛ ˛ ˛ . @ 1 1 @ 2 2 :::@ n n

(8.6.2)

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

459

Proof. ˛ ˛ I. 8T 2 S 0 .Rn /  D 0 .Rn /, ı T D T H) @˛ x T D @x .ı T / D @x ı T (see (6.3.24), (6.7.4))

H)

˛ ˛ F Œ@˛ x T  D F Œ@x ı T  D F .@x ı/  F T ˛

D .i 2/ F T ˛

(by (8.3.13)):

˛

F Œ.i 2x/ T  D F Œ.i 2x/  F T D @˛ ı F T D

@˛  .ı

(by (8.5.13))

(by (8.5.23))

(by (8.4.11))

F T / D @˛  .F T /./

(by (6.3.22), (6.7.4)):

F Œ.i 2x/˛ T  D .1/j˛j .i 2/j˛j F Œx˛ T  D @˛  .F T / 1 ˛ H) F Œx˛ T  D .1/j˛j .i2/ j˛j @ .F T /.

F Œa T  D F .ıa T /

II.

(by (6.3.29))

D F .ıa /  .F T / D e i2h;ai .F T / (by (8.3.15)): F .e i2hx;ai T / D F .e i2hx;ai / F T D ıa F T D a .F T /

(by (6.3.29));

since F .e i2hx;ai / D ıa by (8.4.10). Example 8.6.1. Let u 2 S 0 .R/ such that show that

dku dx k

d 4u dx 4

C u 2 L2 .R/ 8 constant  > 0. Then

2 L2 .R/ for 0  k  4.

Proof. From Corollary 8.3.1, F W L2 .R/ ! L2 .R/ is an isometric isomorphism. 4 4 Hence, ddxu4 C u 2 L2 .R/ H) F Πddxu4 C u 2 L2 .R/  S 0 .R/. u 2 S 0 .R/ H) d 4u dx 4

4

2 S 0 .R/ H) F Œ ddxu4 C u D .i 2/4 uO C uO D Œ.2/4  4 C uO by Theo4

 4 C 1uO 2 L2 .R/ ” . 4 C 1/uO 2 L2 .R/, rem 8.6.1. Hence, 8 > 0, Œ .2/  since, in particular, it must hold for  D .2/4 > 0 and 8 2 R, jjk  1 C  4 for 0  k  4. In fact,  2 R H) jj  1 or jj > 1. Then jj  1 H) jjk  1 for 0  k  4 H) jjk  1 C  4 for 0  k  4, and jj jjk   4 R > 1 H) k 4 k 2d   for 0  k  4 H) jj  1 C  for 0  k  4. Hence, R j.i 2/ u./j O R R 2k 2k O 2 d   .2/2k 4 2 O 2 d  < C1, since . 4 C 1/u O2 R .2/ jj ju./j R .1 C  / ju./j k

L2 .R/ H) .i 2/k uO 2 L2 .R/ for 0  k  4. But F Πddx ku  D .i 2/k uO 2 L2 .R/ for 0  k  4 H)

dku dx 4

2 L2 .R/ for 0  k  4 by Corollary 8.3.1.

460

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Fourier transform of polynomial P.x/ Example 8.6.2. 1. Show that every polynomial P .x/ D a0 Ca1 xC  Can x n with real or complex coefficients ak , 0  k  n, defines a tempered distribution. 2. Find the Fourier transform of P .x/. Proof. 1. 8 2R S.R/, P .x/.x/ 2 S.R/ 8 polynomials P .x/ H) P .x/ 2 L1 .Rn / H) R P .x/.x/dx is well defined 8 2 S.R/ as a continuous linear functional on S.R/. Hence, P .x/ defines a tempered distribution TP by: Z hTP ; i D P .x/.x/dx 8 2 S.R/: R

F Œx k  D F Œx k  1 D .1/k

2.

D .1/k

1 @k ı .i 2/k

D .1/k

ı .k/ : .i 2/k

1 @k .F Œ1/ .i 2/k (by (8.3.22))

By virtue of the linearity of F , F ŒP .x/ D a0 F .1/ C a1 F Œx C    C an F Œx n  D a0 ı C a1 D

n X

.1/ .1/ .1/2 .2/ .1/n n ı C a2 ı C    C a ı n i 2 .i 2/2 .i 2/n

.1/k

kD0

ak ı .k/ .i 2/k

with ı .0/ D ı, hı .k/ ; i D .1/k hı;  .k/ ./i D .1/k  .k/ .0/

8 2 S.R/:

Fourier transform of zn with z D x C iy in S 0 .R2 / Example 8.6.3. 1. Find the Fourier transform of the tempered distribution T D z n 2 S 0 .R2 / with z D x C iy 8n 2 N0 . 2. Using the identity z: z1 D 1, an elementary solution of the Cauchy–Riemann @ @ C i @y / with zN D x  iy and the homogeneity of z1 , operator @ D @@zN D 12 . @x prove that F . z1 /./ D  i with  D  C i.

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

461

Proof. n

jxCiyj n D .x C iy/n 1. Since 9l 2 N such that sup.x;y/2R2 Œ1C.x 2 Cy 2 /l < C1 and z is a polynomial in two variables x; y with complex coefficients and with slow growth at infinity (8.2.15), z n defines a tempered distribution in R2 , i.e. T D z n 2 S 0 .R2 / and F .z n / 2 S 0 .R2 / is well defined.

But F .z/ D F Œx C iy D F Œx C i F Œy D F Œx:1 C i F Œy  1   1 @ 1 @ 1 @ @ D .F 1/ C i .F 1/ D  Ci ı i 2 @ i 2 @ i 2 @ @         @ @ 1 @ @ 1 n Ci ı  Ci ı H) F .z / D  i 2 @ @ i 2 @ @     1 @ @   Ci ı i 2 @ @     @ n 1 n @ Ci D  Œı ı    ı i 2 @ @

(by repeated applications of (8.5.23)). Since ı ı D ı H) ı ı ı D ı H) : : : H) ı ı    ı D ı, we have     @ n 1 n @ Ci ı: F .z n / D  i 2 @ @

2. T D z1 2 S 0 .R2 /, since z1 2 L1loc .R2 / and is bounded for jzj  1. Then z  T D 1 H) F .z  T / D F 1 D ı (by (8.3.22))

H)

  1 @ @ F Œ.x C iy/T  D F ŒxT  C i F ŒyT  D  Ci F T D ı: i 2 @ @

1 @ @ / D ı, where @ D 12 . @x C i @y / is the Now we will show that @. z1 / D @. xCiy Cauchy–Riemann operator [15]. In fact, for x D r cos , y D r sin , J D r,

462

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Q /, @ D 1 . @ C i @ /, .x; y/ D .r cos , r sin / D .r; 2 @x @y @ @Q @r @Q @

@Q @Q sin

D : C  D cos  ; @x @r @x @ @x @r @ r @ @Q @r @Q @

@Q @Q cos

D C  D sin C ; @y @r @y @ @y @r @ r       1 1 @ ;  D  ; @ z z  Q   Z 1 Z 2 1 @ @Q sin

1 D cos C  2 0 r.cos C i sin / @r @

r 0  Q  @ @Q cos

Ci sin C rdrd 8 2 D.R2 / @r @ r  Q Z Z 1 1 2 1 @ D .cos C i sin / 2 0 cos C i sin @r 0  i @Q C .cos C i sin / drd

r @

  Z Z 1 1 2 @Q i @Q D C drd

2 0 @r r @

0 Z 1 Q   Z 2 Q  Z Z @ @ i 11 1 2 dr d  d dr: D 2 0 @r 2 0 r @

0 0 Q / D .0; 0/, .r; Q / is 2-periodic and .r; Q / ! 0 But, 8 2 D.R2 /, .0; R 2 @Q R 1 @Q as r ! 1, H) 0 @r dr D .0; 0/, 0 @ d D 0. Hence,     Z 1 1 1 2 @ Œ.0; 0/d D .0; 0/  2 ; D  z 2 0 2 D .0; 0/ D hı; i

8 2 D.R2 /:

1 is an elementary solution of the Cauchy– H) @. z1 / D ı in D 0 .R2 / H) z 1 @ @ 1 / D ı in D 0 .R2 /. But ı 2 Riemann operator @ D 2 . @ C i @ /, since @. z 1 @ @ 1 @ Œ @x Ci @y . z1 / D ı H)  i2 Œ @x C S 0 .R2 / H) @. z1 / D ı in S 0 .R2 /. Then 2 @ i 0 2 0 2 . i i @y z / D ı in S .R / H)  z 2 S .R / is an elementary solution of 1 @ @ . @x C i @y / in S 0 .R2 /. Hence, for  D  C i, E0 D  i is an ele i2 1 @ @ . @ C i @ /.F T / D ı (8.6.2a). The general solution mentary solution of  i2 1 @ @ . @ C i @ /.F T / D 0 is an of the corresponding homogeneous equation  i2 analytic function F D F ./ with  D  C i. Hence, F T D F .1=z/ D

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

463

E0 C F D  i C F ./. But T D z1 is homogeneous of degree 1 H) by Proposition 8.3.1, F T is also homogeneous of degree 1 (see Example 8.3.2). Since  i with  D  C i is homogeneous of degree 1, analytic F is homogeneous of degree 1 H) F  0, i.e. F ./ D 0 8 2 C with  D  C i. Hence,   1 i F D z 

with  D  C i:

In fact, if we suppose that F 6D 0, then F ./ D 1 F ./ 8, 8 > 0. In particular, for  D 1, F ./ D 1 F .1/ 8 > 0. But analytic F is a C 1 -function with lim!0C ;2RC F ./ D 1, which is impossible. Hence, F  0 (See also [1] (page 168, pages 370–379) for more details). Fourier transforms of c:p:v: x1 , Heaviside function H.x/ Example 8.6.4. Find the Fourier transform of T D c:p:v: x1 . Then, using this result, find F H and F H , H being the Heaviside function with H.x/ D 1 for x > 0 and D 0 for x < 0. Solution. From Example 8.3.1, c:p:v: x1 is an odd tempered distribution, and Z 1 1 x dx hx  c:p:v: ; i D hc:p:v: ; xi D lim x x "!0C jxj" x Z Z D lim .x/dx D .x/dx D h1; i "!0 jxj"

8 2 S.R/

R

H) x  c:p:v: x1 D 1 in S 0 .R/ H) F .xT / D F 1 with T D c:p:v: x1 H) 1 d b D i 2ı H) .F T / D ı (by Theorem 8.6.1 and (8.3.22)) H) dd T .1/ i2 d b./ D i 2H./ C C (see Example 2.7.4), where H./ is the Heaviside funcT b D F (c:p:v: 1 ) is a unique tion in  and C is a constant to be determined, since T x tempered distribution. From Example 8.3.1, c:p:v: x1 is an odd tempered distribution b is also odd by (8.3.26) H) T b is an odd function of  H) H) F .c:p:v: x1 / D T b./ D T b./ H) i 2H./ C C D i 2H./  C H) i 2  1 C C D C T for  > 0, and C D i 2  C for  < 0 H) C D i . Hence, bDF T



1 c:p:v: x



´ i  D i 2H./ C i  D i

b D i 2.F H / C F .i /: FT

for  > 0 for  < 0;

464

Chapter 8 Fourier transforms of distributions and Sobolev spaces

b D TL and F .i / D i F Œ1 D i ı (by (8.3.22)). But from (8.3.21), F F T D F T Hence, .F H /.x/ D 

1 b  F .i / D  1 ŒTL  i ı D  1 TL C 1 ı: ŒF T i 2 i 2 i 2 2

b D F T is odd H) F T b D TL is odd H) TL D T . In T D c:p:v: x1 is odd H) T L D hTL ; i D hT; i L 8 2 S.R/ H) TL D T H) .F H /.x/ D fact, hTL ; i 1 1 1 1  i2 .T / C 2 ı D 2 ı C i2 c:p:v: x1 . F F T D F .i 2H C i / D i 2F H C i F 1 H) T D i 2F H C i ı 1 i (by (8.3.22)) H) FN H D i2 T C i2 ı with T D c:p:v: x1 H) .F H /.x/ D 1 1 1 2 ı  i2 c:p:v: x . 1 1 c:p:v: x1 , .F H /.x/ D 12 ı  i2 c:p:v: x1 . Thus, .F H /.x/ D 12 ı C i2 Example 8.6.5. Consider the function H.x/e ax , which is integrable on R for a > 0, H.x/ being the Heaviside function. 1. Find the Fourier transform of H.x/e ax . 2. Show the values of a for which H.x/e ax 2 S 0 .R/. 3. Find the Fourier transform F H and co-transform FN H of the Heaviside function H.x/ using the Fourier transform of H.x/e ax and the identity H.x/ C H.x/ D 1. Solution. 1. For a > 0, H.x/e ax 2 L1 .R/ H) F ŒH.x/e ax  exists and is given by: Z 1 Z 1 ax ax i2x D H.x/e e dx D e .aCi2 /x dx F ŒH.x/e 1

D

ˇ e .aCi2 /x ˇxD1

ˇ .a C i 2/ ˇ

xD0

0

D

1 ; .a C i 2/

since e ax ! 0 as x ! 1, H) F ŒH.x/e ax  D 2. For a  0,

H.x/e ax

2

S 0 .R/,

since H 2

1 aCi2

8a > 0.

S 0 .R/.

1 1 3. lima!0 H.x/e ax D H.x/ … L1 .R/ and lima!0 aCi2 D i2 … L1 .R/. 0 0 Since Fourier transform F : S .R/ ! S .R/ is continuous from S 0 .R/ onto 1 S 0 .R/, we can find F ŒH.x/ by taking the limit of aCi2 in S 0 .R/ as a ! 0C , 1 i.e. F ŒH.x/ D lima!0C aCi2 in S 0 .R/. In fact, H.x/e ax 2 S 0 .R/ 8a  0 H) H.x/e ax ! H.x/ in S 0 .R/ as a ! 0C H) F ŒH.x/e ax  ! F ŒH.x/ in S 0 .R/ as a ! 0C . Moreover, dd : S 0 .R/ ! S 0 .R/ is also continuous on S 0 .R/.

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

But dd ln.aCi 2/ D i 2/ 8a > 0.

i2 aCi2

2 S 0 .R/ 8a > 0 H) d d

Hence, by virtue of the continuity of

1 aCi2

D

465

1 d i2 d Œln.aC

on S 0 .R/ (see (8.2.28)),

  1 2 1 d 2 2 2 1=2 lim D lim ln.a C 4  / C i arctan i 2 a!0C d  a a!0C a C i 2 ²  ³ 1 d 2 lim ln.a2 C 4 2  2 /1=2 C i arctan D i 2 d  a!0C a   1 d  D ln 2jj C i ŒH./  H./ ; i 2 d  2 2 since for  > 0, a ! 0C H) 2 a ! C1 H) arctan. a / ! =2, for 2  < 0, a ! 0C H) 2 a ! 1 H) arctan. a / ! =2



H)

H)

 lim arctan 2 a a!0C

D

=2 =2

for  > 0 for  < 0

D .=2/ŒH./  H./ 8 2 R    1 d  1 D ln.2jj/ C i H./  H./ lim i 2 d  2 a!0C a C i 2    1 1 D c:p:v: C i .ı./ C ı.// i 2  2 1 1 1 D c:p:v: C ı; i.2/  2

since dd ln.2jj/ D ple 2.3.6), 

´



d Œln 2 d

C ln jj D 0 C

d lnjj d

D c:p:v: 1 (see Exam-

   Z 0 dH.x/ d d ;  D  H.x/; dx D .0/ D hı; i 1 D dx dx dx 1

H)

dH.x/ dx

D ı H) F ŒH.x/ D

1 i2

8

c:p:v: 1 C 12 ı.

H.x/ C H.x/ D 1 and F ŒH.x/ C F ŒH.x/ D F Œ1 D ı. But H.x/ D HL .x/ H) F ŒHL .x/ D F ŒH.x/. Hence, F H C FN H D ı in S 0 .R/.

466

Chapter 8 Fourier transforms of distributions and Sobolev spaces

But

Z

Z

1

1

hF H  F H; i D

e 1

i2x

e

i2x

 dx ./d 

0 1Z 1

Z

D 2i

sin 2x./d dx 0

1 Z M

Z

D 2i lim

M !1 0 "!0C

8 2 S.R/

sin 2x./d dx; j j"

where the limit is to be understood in the sense of distribution. Applying Fubini’s Theorem 7.1.2C, we can interchange the order of integration and get, by integration with respect to x: Z cos M 2  1 hF H  F H; i D i 2 lim ./d  M !1 j j" 2 C "!0   Z 1 i i 1 lim ./d  D D c:p:v: ;  ;  "!0C j j"    since limM !1 cos M 2 D 0 in the distribution sense (see (1.8.12)). H) F H  F H D  i c:p:v: 1 D  i c:p:v: 1 in S 0 .R/ and F H C F H D ı i i c:p:v: 1 , .FN H /./ D 12 ı C 2 c:p:v: 1 in S 0 .R/ H) .F H /./ D 12 ı  2 (see also Example 8.6.4). Fourier Transforms F Œjxj and F ŒPf 12 Example 8.6.6. Find F Œjxj and F ŒPf 12 , where Pf 12 is defined by (1.4.19). Solution. Consider the identity jxj D xH.x/  xH.x/, where H is the Heaviside function with H.x/ D 1 for x > 0 and D 0 for x < 0, since the right-hand side equals x for x > 0 and x for x < 0. Hence, F Œjxj D F ŒxH.x/  F ŒxH.x/ D

1 d 1 d .F H /./  ŒF H.x/: i 2 d  i 2 d 

1 But, from Example 8.6.5, .F H /./ D 12 ı C i2 c:p:v: 1 . F ŒH.x/ D F HL D F H . In fact, hF HL ; i D hHL ; F i D /_ i D hH; F i R hH; .Fi2x. / R D hF H; i 8 2 _ dx D R .x/e i2x dx D S.R/, since .F / D .F /./ D R .x/e F . 1 c:p:v: 1 . Hence, From Example 8.6.5, F H./ D 12 ı  i2     1 d 1 1 1 1 d 1 1 1 F Œjxj D ıC c:p:v: C ı c:p:v: i 2 d  2 i 2  i 2 d  2 i 2      2 1 1 1 d 1 d 1 .c:p:v: / .c:p:v: / D D D  Pf 2 (see (2.3.32)) 2 2 2 d  4 2 d   2 

467

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

H) F .jxj/ D  21 2 Pf. 12 / H) F F .jxj/ D jxj L D jxj D  21 2 F ŒPf 12  (see (8.3.21)) H) F ŒPf 12  D 2 2 jxj. 2

Fourier transform of e x , e ix

2

2

2

Instead of finding the Fourier transforms of e x and e i2x directly from the definition, we will find them with the help of differential equations as shown in the following example. Example 8.6.7. 1. Find the tempered distribution solutions of  6D 0.

du Cxu dx

D 0 in S 0 .R/ for complex

2. Let u 2 S 0 .R/ be a tempered distribution solution of the equation in .1/ and uO D F u be its Fourier transform. Find the differential equation satisfied by uO 2 S 0 .R/. Then, show the tempered distribution solutions of this equation for u. O 2

3. Applying the results of .1/ and .2/, find the Fourier transforms F Œe x  and 2 Finally, using these results, find the Fourier transform of F Œe ix . 2 2 Œe .x Cy /Ci2y  in S 0 .R2 /. Solution. 1. From Theorem 2.7.2, the distribution solutions of the usual solutions: u D C e   2 C, Re./  0, e

2  x 2

2

R

xdx

L1 .R/

D Ce

2  x 2

du dx

C xu D 0 in D 0 .R/ are

, where C is a constant. For

is a tempered distribution in S 0 .R/. Hence, x 2

8 2 C with Re./  0, u D C e  2 2 S 0 .R/ are tempered distribution solutions of the equation u0 C xu D 0. For  2 C with Re./ < 0, classix 2

cal solutions u D C e  2 are not tempered distributions on R and hence not tempered distribution solutions. R1 2. For Re./  0, F u is well defined and given by F u D 1 u.x/e i2x dx 1 d and F Œ du  D i 2F u, F Œxu D  i2 .F u/. Set uO D F u. Then,  dd uO D dx d  H) i 2.i 2/uO D i 2F Œxu D i 2F Œxu and i 2.F u/ D F Œ du dx du d uO du 2 i 2F Œ dx  H)  d C 4  uO D i 2F Œ dx C xu D i 2F 0 D 0. Thus, uO satisfies the differential equation  dd u O C 4 2  uO D 0. For  6D 0,

d uO d

C

4 2  uO 

D 0 H) uO D C1 e 

4 2 

R

d

D C1 e

2 2  2 

2 2  2 

is the

general solution, C1 being a constant. For Re./  0, e 2 S 0 .R/. For  D 0, the differential equation reduces to  uO D 0 H) uO D C2 ı 2 S 0 .R/; C2 being a constant.

468

Chapter 8 Fourier transforms of distributions and Sobolev spaces 2 2  2

2

3. Choose  D 2, C D 1, i.e. u D e x . Then uO D F u D C1 e  2 D 2 2 C1 e  , where C1 is an unknown constant, which will be determined now. Z 1 Z 1 p 2 O D .F u/.0/ D u.x/dx D e x dx D : C1 D u.0/ 1

1

Thus, 2

F Œe x  D

p

e 

2 2

: 2

Now choose  D i 2 with Re./ D 0, C D 1, i.e. u.x/ D e ix . 2 2

2

2

Then uO D F u D C1 e  i2 D C1 e i D C1 u./, u./ D complex conjugate of u./, C1 2 R. F uO D F F u D u and F uO D F ŒC1 u D C1 F Œu D C1 F u D C1 C1 u D C12 u H) C12 u D u H) C12 D 1 and C1 D 1, since C1 > 0. 2

2

F Œe ix  D e i : e .x

2 Cy 2 /Ci2y

H)

is integrable on R2 , i.e. belongs to L1 .R2 / “ 2 2 2 2 e .x Cy /Ci2y e i2.x Cy / dxdy F Œe .x Cy /Ci2y .; / D R2

Z

 Z 2 x 2 i2x D e e d  e y Ci2y e i2y dy R R Z 2 2 e y e i2. 1=/y dy D F Œe x ./  R

p 2 2 2 D . e  /  F Œe y .  1=/ p 2 2 p 2 2 D e   e  . 1=/ D e 

2 2

D e 

2 2

F Œe

e

 2 . 2 C

1 2 =/ 2

 e 1  e .

.x 2 Cy 2 /Ci2y

2 2 2 /

.; / D e

:

 2 2

:e 1  e .

2 2 2 /

:

Fourier transform of sin x, cos x, H.x/ sin x Example 8.6.8. 1. Find the Fourier transform of sin x and cos x. Then, find F ŒH.x/ sin x, where H is the Heaviside function.

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

469

2. With the help of Fourier transforms, find the tempered distribution solution of 2 the differential equation ddxu2 C u D H.x/ sin x for the admissible real values of . Solution. ix ix ix ix 1. sin x D e e , cos x D e Ce are tempered distributions in S 0 .R/. In 2i 2 fact, for example, ˇZ 1 ˇ Z 1 ˇ ˇ ˇ jT ./j D jhsin x; ij D ˇ sin x.x/dx ˇˇ  j.x/jdx 0. Then a2 C 4 2  2 is a C 1 -function and a2 C 4 2  2 6D 0 1 1 -function on R and we can write 8 2 R. Hence, a2 C4 2 2 is also a C uO D

1 F a2 C4 2 2

ŒH.x/ sin x. Then u D F F u D F uO   1 F ŒH.x/ sin x H) u D F  2 a C 4 2  2   1 DF  2 F F ŒH.x/ sin x; a C 4 2  2

where it is assumed that the convolution is well defined, even though neither distribution has compact support. Under this assumption, we have, 8a > 0,   1 uDF  2 H.x/ sin x: a C 4 2  2 We can further simplify using the result of Example 8.6.5. 1 1 1 1 In fact, a2 C4 2 2 D 2a Œ aCi2 C ai2         1 1 1 1 F D H) F  2 CF a C 4 2  2 2a a C i 2 a  i 2 1 D  ŒH.x/e ax C H.x/e ax ; 2a 1 1 since, from Example 8.6.5, F ŒH.x/e ax  D aCi2 , F ŒH.x/e ax  D ai2 .

Hence, for  D a2 < 0 with a > 0, uD

1 ŒH.x/e ax C H.x/e ax  H.x/ sin x: 2a

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

Fourier transform of

471

1 rk

Let 1

r D r.x/ D kxk D .x12 C x22 C    C xn2 / 2 f .x/ D

8x D .x1 ; x2 ; : : : ; xn / 2 Rn

1 : rk

and (8.6.3)

(8.6.4) For 0 < k < n, f 2 S 0 .Rn / and F Œf  2 S 0 .Rn /. R For 0 < k < n, 8 compact subsets K  Rn , K r1k d x < C1 H) f 2 L1loc .Rn / H) f D r1k 2 D 0 .Rn / is a distribution on Rn . Let 1B be the characteristic function of the unit ball B D B.0I 1/ D ¹x W kxk < 1º in Rn , B { D ¹x W kxk  1º being the complement of B in Rn , i.e. ´ 1 for x 2 B .i.e. for kxk < 1/ 1B .x/ D 0 for x 2 B { .i.e. for kxk  1/: Then f D 1B f C .1  1B /f D fB C fB { , where ´ f .x/ for kxk < 1 fB .x/ D 1B .x/f .x/ D 0 for kxk  1; and fB { .x/ D f .x/  fB .x/ 8x 2 Rn . R R For 0 < k < n, Rn fB .x/d x D kxk n, i.e. n2 < k < n, F Œf  D fO is a function: fB { is a square integrable function on Rn , i.e. fB { 2 L2 .Rn /, since Z Rn

2

jfB { .x/j d x D

Z kxk1

1 r

d x D Sn 2k

Z

1 1

1 r 2k

r n1 dr < C1

for 2k C .n  1/ C 1 < 0, i.e. for 2k > n, Sn being the surface area of the n-dimensional unit sphere, which is obtained as the result of transformation into spherical coordinates H) F ŒfB {  D fOB { 2 L2 .Rn / (by Theorem 8.3.1). fB 2 L1 .Rn / H) F ŒfB  D fOB is a continuous, bounded function in Rn with kfOk1 ! 0

472

Chapter 8 Fourier transforms of distributions and Sobolev spaces

as kk ! 0 (see (7.1.24), (7.1.33), (7.1.36)). Hence, F Œf  D fOB C fOB { D fO is a function, which is locally square integrable on Rn . For n2 < k < n, F Œf  D fO is a radial (spherically symmetric) function in S 0 .Rn /. f .r/ D r1k H) f is a homogeneous radial function of degree k 2 r H) f .r/ D 1 k D k : 1k D k f .r/ in r H) F Œf  D fO is a homoge.r/

r

neous radial function of degree .n C .k// D k  n in (see Proposition 8.3.1) Ck;n H) fO D nk , where Ck;n is a constant to be determined now. 2 2 Calculation of Ck;n : For  D e  , O D F ΠD e  r (Example 7.1.2), hF Πr1k ; R Ck;n 2 R  r 2 2 2 Ck;n e  i D h r1k ; e  r i with F Πr1k  D nk H) Rn nk e d  D Rn e r k d x H)

Ck;n D

R1

2

e  r r n1 dr 0 rk R 1  2 Sn 0 enk n1 d

Sn

where Z

1

D

I1 ; I2



1

nk I1 D e r dr D  nk 2 0 2 2   Z 1 k 1 2 I2 D e  k1 d D  k 2 0 2 2  r 2 nk1

 and

are obtained first by transformation into spherical coordinates and then again by inq t dt 2 , we get troducing new variables. In fact, setting t D  r , r D  , dr D p 2 t

p Z 1 Z 1 . nk . t/nk1 1 t t nk 2 / 2 1 dt D I1 D e p e t ; p dt D nk nk . /nk1 2  t 0 2 2 0 2 2 q ds , we get and setting s D  2 , D s , ds D 2p s I2 D

Z

1 k

2 2

1

k

e s s 2 1 ds D

. k2 / k

:

2 2

0

Here, k

Ck;n D Finally, for

n 2

2 2 . nk 2 / 2

nk 2

. k2 /

n

D  k 2

. nk 2 / . k2 /

:

< k < n,  F

1 rk



n

D  k 2

. nk 2 / . k2 /

:

1

nk

:

(8.6.5)

473

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

Case 0 < 2k < n, i.e. 0 < k < n=2: Set p D n  k. Then, k < n2 H) p D n  k > n  n2 D n2 H) n2 < p < n, i.e. we are again in the earlier case for r1p H) F Πr1p  D Cp;n 1 1 1 1 1 _ np H) F Πr nk  D Cp;n k H) F F Πr nk  D Cp;n F Πk  D Cp;n F Πk  D Cp;n F Π1k  H)

1 r nk

1 D Cp;n Ck;n r nk H) Cp;n Ck;n D 1 H) n 2,

Π1k 

1

1 Cp;n D Ck;n D r nk .

1

Ck;n .

F D Cp;n r nk InterchangHence, for 0 < 2k < n, i.e. 0 < k < ing the notation with r and vice versa (Schwartz [8] uses the same notation r instead of ), we get, for 0 < 2k < n, i.e. 0 < k < n2 , 



and

n 2

1 F k r Thus, for 0 < k <  F

1 n

r2



n 2

n

 k 2 . nk Ck;n 2 / 1 : D nk D k

nk . 2 / < k < n, the formula (8.6.5) holds. Finally, for k D n2 ,



1 D limn F k r k! 2



 D limn

n

 k 2 . nk 2 /

k! 2

. k2 /

  nk :

1

(8.6.6)

Combining (8.6.5) and (8.6.6), we get F Πr1k  for 0 < k < n: 

1 F k r



n

D

 k 2 . nk 2 / . k2 /



1

nk

:

(8.6.7)

Singular values of k in r1k : Formula (8.6.5) with 0 < k < n does not hold for nk nk k 2 D p and k D 2p with integer p  0, since . 2 / (resp. . 2 /) has poles nk at 2 D p (resp. k D 2p). Hence, for these singular values of k D n C 2p (resp. k D 2p) with p  0, the formula for F Œ r1k  is to be modified by taking limits (starting with non-singular values of k as in (8.6.7)) and given by [8, p. 258]: for k D 2p with p 2 N0 ,    p 2p F Œr  D  2 ı; (8.6.8) 4 where the notation ‘Pf’ is useless, since 2p  0;  D

@2 @ 12

C  C

@2 ; @ n2

hı; ./i D .0/ 8 2 S.Rn /. For k D n C 2p with p 2 N0 , n C 2p > n, ‘Pf’ is to be used and we get, from [8, p. 258],    n 1 .1/p 2p  2 C2p F Pf nC2p  2

D r . n2 C p/ pŠ     1 1 1 1  0 . n2 C p/ 1 C 1 C C  C  C C ;  ln  2 2 p 2 . n2 C p/ (8.6.9)

474

Chapter 8 Fourier transforms of distributions and Sobolev spaces

1 where C is Euler’s constant, Pf. r nC2p / is defined by (3.3.42), the sum .1C 12 C  C p1 / must be replaced by 0 for p D 0. For k D n (i.e. p D 0) we get, from (8.6.9), p      1 2. /n 1 1 C 1  0 . n2 / F Pf n D  ln C A with A D ln  C r . n2 /

 2 2 . n2 / (8.6.10) p n        1 1 2. / H) F F Pf n  F ln D C AF Œ1 n r . 2 /

p       2. /n 1 1 H) Pf n D  F ln C Aı ; r . n2 /

since ./ L D ./ D ./, F Œln. 1 / D F Œln. 1 /. Replacing by r and vice versa, we can write       . n2 / 1 1 F ln  Aı (8.6.11) D p n Pf n r

2. / with A defined in (8.6.10). For n D 1,

  1 1 F Œln jxj D  Pf  .C C ln 2/ı; 2 jj

Œ8

(8.6.12)

C being Euler’s constant. Fourier transform of the tensor product T.x/ ˝ S.y/ of tempered distributions T.x/ and S.y/ Let T .x/ and S.y/ (Tx , Sy being alternative notations – see Chapter 6) be tempered distributions with Fourier transforms TO ./ D F T ./, SO . / D F S. /, and T .x/ ˝ S.y/ be their tensor product (see Chapter 6). Then the Fourier transform F ŒT .x/ ˝ O S.y/ of the tensor product T .x/ ˝ S.y/ is equal to the tensor product of TO ./ ˝ S. / of their Fourier transforms, i.e. F ŒT .x/ ˝ S.y/ D TO ./ ˝ SO . /:

(8.6.13)

For this,P it is sufficient [1] to show that the equation (8.6.13) holds for functions .; / D i i ./ i . / with i ./; i . / 2 D.Rn /. In fact,    X  X F ŒT .x/ ˝ S.y/; i ./ i . / D T .x/ ˝ S.y/; F i ./ i . / i

i

  X O O D T .x/ ˝ S.y/; i .x/ i .y/ i

475

Section 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives

with O i .x/ D .F i /.x/, O i .y/ D .F i /.y/ (see (7.1.15))    X X O O D T .x/; S.y/; i .x/ i .y/ D hT .x/; O i .x/i  hS.y/; O i .y/i i

i

X D hF T ./; i ./ihF S. /;

 X . /i D F T ./ ˝ F S. /; i ./ i

i

 . / i

i

O H) F ŒT .x/ ˝ S.y/ D TO ./ ˝ S. /. Example 8.6.9. 1. For ´ f .x; y/ D H.x/ ˝ H.y/ D

1 for x > 0 and y > 0 0 otherwise,

F Œf .x; y/ D F ŒH.x/ ˝ H.y/ D F H./ ˝ F H./     1 1 1 1 1 1 ı./ C c:p:v: ˝ ı./ C c:p:v: ; D 2 i 2  2 i 2  which is obtained from Example 8.6.4, ı./ (resp. ı./) being the Dirac distribution with force/charge/mass concentrated at  D 0 (resp.  D 0). 2. For T D ı.x/, F Œı.x/ ˝ S.y/ D F ı./  F S. / D 1./ ˝ SO . /. 3. For T D ı.x/, S.y/ D 1.y/, F Œı.x/ ˝ 1.y/ D 1./ ˝ ı./. 4. For T D T .x1 ; x2 ; : : : ; xn / D 1.x1 ; x2 ; : : : ; xk / ˝ S.xkC1 ; : : : ; xn /, i.e. T is independent of the variables x1 ; x2 ; : : : ; xk , (see (6.1.18)), .F T /.1 ; : : : ; n / D F Œ1.x1 ; x2 ; : : : ; xk / ˝ S.xkC1 ; : : : ; xn / D .F 1/.1 ; : : : ; k / ˝ F S.kC1 ; : : : ; n / D ı.1 ; : : : ; k / ˝ F S.kC1 ; : : : ; n /; i.e. Fourier transform F T D TO of a distribution T independent of the variables x1 ; x2 ; : : : ; xk (see (6.1.18)) is the extension of the distribution SO .kC1 ; kC2 ; : : : ; n / defined on the subspace of the variables kC1 ; kC2 ; : : : ; n to ı.1 ; : : : ; k / ˝ SO .kC1 ; : : : ; n / defined on the whole space of variables 1 ; 2 ;    ; n .

476

8.7

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Fourier transform methods for differential equations and elementary solutions in S 0 .Rn /

Differential operators and Fourier transforms For the sake of convenience and good notation, we will differentiate between the partial differential operators @˛ and D ˛ . @˛ D @.˛1 ;˛2 ;:::;˛n / D @˛1 1 @˛2 2    @˛nn D with @˛kk D

@˛k , @xk

@j˛j : : : @xn˛n

@x1˛1 @x2˛2

1  k  n;

D ˛ D D .˛1 ;˛2 ;:::;˛n / D D1˛1 D2˛2 : : : Dn˛n      1 1 1 ˛1 ˛2 ˛n D @ @ @   .i 2/˛1 1 .i 2/˛2 2 .i 2/˛n n 1 1 D @˛1 @˛2    @˛nn D @˛ with .i 2/j˛j 1 2 .i 2/j˛j 1 1 @˛ k ˛k D ˛k D @ D ; 1  k  n: .i 2/˛k k .i 2/˛k @xk˛k

(8.7.1)

P ˛ n With each polynomial P ./ D j˛jm a˛  of degree  m in R with constant coefficients a˛ 2 C 8j˛j  m, weP associate the partial differential operator P .D/ with constant coefficients, P .D/ D j˛jm a˛ D ˛ , where D ˛ is defined by (8.7.1). Similarly, we can associate another partial differential operator P .@/ with the polynomial P ./, defined by: X

P .@/ D

a ˛ @˛ D

j˛jm

Then P .@/ D

P

j˛jm a˛ @

˛

P .D/ D

X

a˛

j˛jm

@j˛j : : : : @xn˛n

@x1˛1 @x2˛2

is related to P .D/ by: X

X

a˛ D ˛ D

j˛jm

j˛jm

a˛

1 @˛ : .i 2/j˛j

(8.7.2)

Definition 8.7.1A. A linear partial differential operator A with constant coefficients a˛ 2 C 8j˛j  m defined by Au D P .@/u D

X j˛jm

a ˛ @˛ u D

X j˛jm

a˛

@j˛j    @xn˛n

@x1˛1 @x2˛2

8u 2 D 0 .Rn /

477

Section 8.7 Fourier transform methods for differential equations

is called elliptic in Rn if and only if Pm ./ D

X

a˛  ˛ ¤ 0 8 ¤ 0 in Rn ;

j˛jDm

where Pm ./ is a homogeneous polynomial of degree equal to m in the n variables 1 ; 2 ; : : : ; n . Fourier transform of P.D/u Let u 2 S 0 .Rn /. Then the Fourier transform of P .D/u (resp. P .@/u) is defined by: X

F ŒP .D/u D

a˛ F ŒD ˛ u D

j˛jm

X

X j˛jm

a˛

1 F Œ@˛ u .i 2/j˛j

X F Œu D a˛  ˛ u./ O D P ./u./; O (8.7.3) .i 2/j˛j j˛jm j˛jm X X X ˛ a˛ F Œ@ uD a˛ .i 2/˛ F ŒuD a˛ .i 2/j˛j  ˛ u./; O F ŒP .@/u D D

a˛

.i 2/˛

j˛jm

j˛jm

j˛jm

(8.7.4) where j˛j D ˛1 C ˛2 C    C ˛n ,  ˛ D 1˛1 2˛2    n˛n , uO D F Œu. Example 8.7.1. For u 2 S 0 .R2 /, find the Fourier transform of 1. u; P2 P2 2. iD1 j D1

@ @xj

2

u @u .aij @x / C a0 u D aij @x@ @x C a0 u with constant coefficients i

i

j

a0 , aij 2 R, 1  i; j  2;2 3. u. Solution. 1.

 @2 u @2 u C 2 D F Œ@.2;0/ u C @.0;2/ u F Œ u D F @x12 @x2 

D .i 2/2 12 20 F Œu C .i 2/2 10 22 F Œu D 4 2 Œ12 C 22 uO H)

F Πu D 4 2 2 uO with 2 D 12 C 22 :

(8.7.5)

2 Here we have followed on the right-hand side of the equality Einstein’s summation convention with respect to twice-repeated indices to avoid the summation sign.

478 2.

Chapter 8 Fourier transforms of distributions and Sobolev spaces

  @2 u @2 u @2 u F a11 2 C .a12 C a21 / C a22 2 C a0 u @x1 @x2 @x1 @x2  2   2   2  @ u @ u @ u C .a12 C a21 /F C a22 F C a0 F Œu: D a11 F 2 @x1 @x2 @x1 @x22  2  @ u @2 u .2;0/ D .i 2/2 12 uO D 4 2 12 u; D@ u H) F O 2 @x1 @x12  2  @2 u @ u D .i 2/2 1 2 uO D 4 2 1 2 u; D @.1;1/ u H) F O @x1 @x2 @x1 @x2  2  @ u @2 u .0;2/ D @ u H) F O D .i 2/2 22 uO D 4 2 22 u: 2 @x2 @x22 Hence,

F

X 2 X 2 iD1 j D1

 @2 u aij C a0 u D 4 2 Œa11 12 C a12 1 2 @xi @xj C a21 2 1 C a22 22 uO C a0 uO   2 2 X X 2 O (8.7.6) aij i j C a0 u: D 4 iD1 j D1

3.



 @4 u @4 u @4 u F C 2 2 2 C 4 D F Œ@.4;0/ u C [email protected];2/ u C @.0;4/ u 4 @x1 @x1 @x2 @x2 D .i 2/4C0 14 20 uO C 2.i 2/2C2 12 22 uO C .i 2/0C4 10 24 uO D 16 4 Œ14 C 212 22 C 24 uO D 16 4 4 uO

with 2 D 12 C 22 :

(8.7.7)

P Theorem 8.7.1. Let P .D/ D j˛jm a˛ D ˛ with D ˛ defined by (8.7.1) be a partial differential operator in Rn such that P ./ ¤ 0 8 ¤ 0 in Rn . Then show that the kernel of P .D/ in S 0 .Rn / consists of polynomials, i.e. every solution of the equation P .D/u D 0 in S 0 .Rn / is a polynomial. Proof. Let u 2 S 0 .Rn /. Then the kernel of P .D/ in S 0 .Rn / D ¹u W u 2 S 0 .Rn /, P .D/u D 0º. But P .D/u D 0 in S 0 .Rn / H) F ŒP .D/u D 0 H) P ./u./ O D0 0 .Rn / with in S 0 .Rn / by (8.7.3). P ./ ¤ 0 for  2 Rn n ¹0º and uO 2 S 0 .Rn /  D P u# O Rn n¹0º D 0 H) supp.u/ O D ¹0º. Hence, by Theorem 5.7.1, uO D j˛jm0 C˛ @˛  ı./, m0 being the order of u, O C˛ 2 C 8˛, ı./ being the Dirac distribution with

Section 8.7 Fourier transform methods for differential equations

concentration at  D 0. But uO 2 S 0 .Rn / X H) F uO D C˛ F Œ@˛  ı

(by (8.3.14))

j˛jm0

D

X

479

C˛ .i 2x/˛ D

j˛jm0

X

C˛ .i 2/j˛j x˛ :

j˛jm0

P ˛ j˛j 8j˛j  m is a Hence, u D F uO D 0 j˛jm0 d˛ x , where d˛ D C˛ .i 2/ polynomial of degree  m0 . P Example 8.7.2. Let P .D/ D j˛jm a˛ D ˛ with D ˛ defined by (8.7.1) be a partial differential operator with constant coefficients a˛ 2 C 8j˛j  m and D ˛ defined by (8.7.1) such that the corresponding polynomial P ./ is not identically zero in Rn . If u 2 E 0 .Rn / (i.e. a distribution with compact support) such that P .D/u D 0, then u D 0. Solution. Let u 2 E 0 .Rn /. Then u 2 S 0 .Rn / H) F u D uO 2 S 0 .Rn / and uO 2 1 C .Rn / by Proposition 8.4.1. Again, P .D/u D 0 H) F ŒP .D/u D P ./u./ O D 0 (by (8.7.3)). Let Z denote the set of zeros of P ./, i.e. Z D ¹ W  2 Rn , P ./ D 0º. Then P ./ ¤ 0 8 2 Rn n Z H) u./ O D 0 8 2 Rn n Z. But the set Z of zeros of a polynomial is a closed set with an empty interior. Hence, Rn n Z is O D0 dense in Rn , in which the continuous function uO 2 C 1 .Rn / vanishes H) u./ 8 2 Rn H) F Œu O D 0 H) u D 0 in S 0 .Rn /. Hypoelliptic operator A Following Lions and Magenes [15], we define: Definition 8.7.1B. A linear differential A with coefficients aij 2 C 1 ./ is called hypoelliptic if and only if u 2 D 0 ./ and Au 2 C 1 ./ implies that u 2 C 1 ./. A is elliptic H) A is hypoelliptic. For more details, we refer to Hörmander [5]. Example 8.7.3. Let P .D/ be the operator defined by (8.7.2). If P .D/ is elliptic in Rn , i.e. X Pm ./ D a˛  ˛ ¤ 0 8 ¤ 0 in Rn ; (8.7.8) j˛jDm

then show that n 1. 9˛0 > 0 such that jPm ./j  ˛0 kkm Rn 8  2 R ; n 2. 9˛1 > 0 and R > 0 such that jP ./j  ˛1 kkm Rn 8kkR > R.

(8.7.9) (8.7.10)

480

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof.

P ˛ 1. The homogeneous polynomial Pm ./ D j˛jDm a˛  is continuous on the O D . 2 C  2 C    C n2 / 12 D 1º, unit sphere S D S.0I 1/ D ¹O W O 2 Rn , kk 1 2 O  C which is a compact subset of Rn . Hence, 9C > 0 such that jPm ./j   n O D Pm . /  C 8 ¤ 0 8O 2 S  R . 8 ¤ 0, kk D O 2 S H) Pm ./ kk n in R . But for  ¤ 0,     X X   ˛ 1 Pm D a˛ D a˛ ˛ kk kk kkj˛j j˛jDm

D

1 kkm

j˛jDm

X

a˛  ˛ D

j˛jDm

1 Pm ./ kkm

O D 1 m jPm ./j  C > 0 H) Pm ./ kk H) 8 2 Rn , jPm ./j  ˛0 kkm , with ˛0 D C > 0, since for  D 0 the inequality becomes an equality with both sides equal to zero. P 2. P a˛  ˛ D Pm ./ C Pm1 ./ C    C P0 ./ with Pk ./ D P./ D j˛jm ˛ j˛jDk a˛  8k D 0; 1; 2; : : : ; m. But  X  X jPk ./j  ja˛ jkkj˛j D ja˛ j kkk D Ck kkk j˛jDk

with Ck D

P

j˛jDk

j˛jDk

ja˛ j.

jPm ./j D jP ./ 

m1 X kD0

Pk ./j  jP ./j C

m1 X

jPk ./j

kD0

P Pm1  m k H) jP ./j  jPm ./j  m1 kD0 jPk ./j  C kk  kD0 C kk with  2 C D max0km1 ¹Ck º. Hence, for 1  R < kk  kk      m kkm kkm1  kkm1  kk H) jP ./j  C kkm  mC  kk D R D R R mC  C mC  C m m .C  R /kk  2 kk , if R  2 for sufficiently large R. Then the inequality (8.7.10) holds with ˛1 D C2 > 0 for sufficiently large R > 0 such  that mC R  ˛1 . Remark 8.7.1. Example 8.7.3 suggests to redefine the ellipticity:Let A  P .D/ be the linear partial differential operator with constant coefficients defined P by (8.7.2). Then A is called elliptic in Rn if and only if (8.7.8) holds, i.e. Pm ./D j˛jDm a˛  ˛¤ 0 8 ¤ 0 in Rn , or equivalently (8.7.9) holds, i.e. 9˛0 > 0 such that jPm ./j  n ˛0 kkm Rn 8  2 R .

481

Section 8.7 Fourier transform methods for differential equations

Laplacian .F / and iterated Laplacian k .F / of Fourier transform F

with 2 S.Rn / Let x D

@2 @x12

C

@2 @x22

CC

Z

@2 2 @xn

and kx D .

@2 @x12

C

@2 @x22

CC

@2 k 2/ . @xn

Then we have

Z

i2h;xi

x .F /.x/ D x ./e d D ./ x e i2h;xi d  Rn Rn Z D ./Œ.i 21 /2 C.i 22 /2 C  C.i 2n /2 e i2h;xi d  Rn Z .4 2 /.12 C 22 C    C n2 /./e i2h;xi d  D n R Z D .4 2 2 /./e i2h;xi d  DF Œ4 2 2 ./.x/ 8 2 S.Rn /: Rn

Z x .F /.0/ D kx .F /.x/ D

Rn

Z

Rn

Z D

Rn

Z D

(8.7.11) 4 2 2 ./d 

with 2 D 12 C 22 C    C n2 :

(8.7.12)

./ kx .e i2h;xi /d  Œ.4 2 /.12 C 22 C    C n2 /k ./ e i2h;xi d  .4 2 2 /k ./e i2h;xi d  DF Œ.4 2 2 /k  8 2 S.Rn /:

Rn

kx .F /.0/ D

Z

(8.7.13) .4 2 2 /k ./d 

8 2 S.Rn /:

(8.7.14)

Rn

 k Fourier transforms F Œ. 4 2 / ı , F Œ.1  m F Œ. C / ı with  > 0

 m / ı , 4 2

F Œ.  /m ı ,

        1 1 hı; .F /i D x .F /.0/ ı ; D ı; F  D F 2 2 2 4 4 4 4 2 Z 1 4 2 2 ./d  D h 2 ; i 8 2 S.Rn / (by (8.7.12)) D 4 2 Rn    H) F (8.7.15) ı D  2 in S 0 .Rn / 4 2

482

Chapter 8 Fourier transforms of distributions and Sobolev spaces

with 2 D 12 C 22 C    C n2 .         k k F ı ; D ı; F  4 2 4 2 1 D hı; k .F /i (since .1/2k D 1) .4 2 /k Z 1 1 k D .F /.0/ D .4 2 2 /k ./d (by (8.7.14)) .4 2 /k x .4 2 /k Rn Z

2k ./d  D .1/k H) H)

Rn

  k hF ı ; i D h.1/k 2k ; i8 2 S.Rn / 4 2    k F ı D .1/k 2k in S 0 .Rn / 4 2 

(8.7.16)

with 2 D 12 C 22 C    C n2 .    k ı D 2k in S 0 .Rn /: (8.7.17) F  4 2         F 1 ı DF Œı  F ı DF ı ı D1 C 2 in S 0 .Rn /: 4 2 4 2 4 2 (8.7.18)   m      m X k k m.m  1/    .m  k C 1/ ı D F 1 C .1/ ı F 1 4 2 kŠ 4 2 kD1

m X

D F Œı C

k m.m

.1/

kD1

D1C

m X

.1/k

kD1

D1C

 1/    .m  k C 1/ F kŠ



4 2

m.m  1/    .m  k C 1/ .1/k 2k kŠ

k  ı

(by (8.7.16))

m X m.m  1/    .m  k C 1/ 2 k . / kŠ

kD1

D .1 C 2 /m

in S 0 .Rn /:

(8.7.19)

Similarly, we have F Œ.  /m ı D .4 2 2  /m m

2 2

m

F Œ. C / ı D .4 C /

in S 0 .Rn /I

(8.7.20)

0

(8.7.21)

n

in S .R /:

483

Section 8.7 Fourier transform methods for differential equations

1 Fourier transforms F Œ . r n2 / for n ¤ 2, F Œ .ln 1r / for n D 2, F Œc:p:v: x1 for n D 1

For n ¤ 2, n  2 < 0, we are not in a singular situation, and hence the notation Pf in 1 front of r n2 is useless. Consequently, using (8.6.6) and (8.7.5) with uO D F Œı D 1, we have          1 1 1 1 2 2 F n2 D F ı n2 D F Œ ı  F n2 D .4 /  F n2 r r r r n

D 4 2 2

 n2 2 . nnC2 / 2 . n2 2 /



1

n

D

4 2  2  2 n

2. 2 /

.n.n2// n2  n n  2 2 2

D .n  2/2  n D .n  2/Sn ; Sn D . 2 /  n2

(8.7.22)

from which we can retrieve the formula (3.3.15), since 



1 r n2

   1 D F F n2 D .n  2/Sn F Œ1 D .n  2/Sn ı; r

F Œ1 D ı:

For n D 2, using (8.7.5) and (8.6.11),

H)

         1 1 1 2 2 2 2 .1/ F ln Pf 2  Aı D .4 /F ln D .4 / r r 2

  2 1 4 2

 Pf 2 C A4 2 2 ı D 2

   1 F ln D 2; (8.7.23) r

from which we retrieve (3.3.14), since F Œ1 D ı; 2  Pf. 12 / D 1 and h 2 ı; ./i D hı; 2 ./i D 0 8 2 S.R2 / H) 2 ı D 0 H) A4 2 2 ı D 0 in S 0 .R2 /. For n D 1, (6.3.24), (8.3.13) and (8.6.12) give 

   d d F ln jxj D F ı  F Œln jxj D i 2F Œln jxj dx dx       1 1 D i  Pf  ŒC C ln 2i 2ı D i  Pf ; jj jj since ı D 0 (hı; ./i D hı; ./i D 0 8 2 S.R/ H) ı D 0).

484

Chapter 8 Fourier transforms of distributions and Sobolev spaces

ln jxj D c:p:v: x1 . Hence, ´     for  > 0 Pf. 1 / 1 d F c:p:v: DF ln jxj D i  1 x dx Pf.  / for  < 0  ´  1 i  for  > 0 D (see Example 8.6.4), H) F c:p:v: x Ci  for  < 0

From Example 2.3.6,

d dx

(8.7.24)

1 since  Pf. ˙ / D ˙1. m

Fourier transform F Œ.1 C r 2 / 2 m

The Fourier transform F Œ.1Cr 2 / 2  for m > n2 is given by [8]: 8m ¤ 0; 2; 4; : : : , p   2. /m .mn/ 1 r 2 K .nm/ .2 r/ D F m 2 . m .1 C r 2 / 2 2/ p m .mn/ 2. / D PfŒr 2 K .nm/ .2 r/ D Lm ; (8.7.25) m 2 . 2 / where Lm is already defined in (3.3.57), since m

1 0 n m 2 S .R / 8m 2 N. .1Cr 2 / 2 .1 C r 2 /k . Then, from (8.6.9),

For m D 2k with k 2 N0 , .1 C r 2 / 2 D   k F Œ.1 C r 2 /k  D 1  ı D L2k (see (3.3.58)): 4 2 Pk k.k1/.kpC1/ 2 p In fact, .1 C r 2 /k D 1 C pD1 .r / pŠ H)

(8.7.26)

  k X k.k  1/    .k  p C 1/ 2p r F Œ.1 C r 2 /k  D F 1 C pŠ pD1

k X k.k  1/    .k  p C 1/ F Œr 2p  D F Œ1 C pŠ pD1

  k X p k.k  1/    .k  p C 1/ DıC  2 ı pŠ 4

(by (8.6.9))

pD1

    k X k.k  1/    .k  p C 1/ p D 1C .1/p ı pŠ 4 2 pD1



D 1 4 2

k ı:

Section 8.7 Fourier transform methods for differential equations

485

In particular, for k D 0, F Œ1 D ıI



 for k D 1, F Œ.1 C r / D 1  ı; 4 2 2

(8.7.27)

ı D ı./, which we already know. Problems of division of tempered distribution by polynomials Let P ./ be a polynomial in  D .1 ; 2 ; : : : ; n / and S 2 S 0 .Rn / be a given tempered distribution. Then the question is: does there exist a tempered distribution T 2 S 0 .Rn / such that P ./T D S in S 0 .Rn /? In other words, can we write T D PS./ 2 S 0 .Rn /? In fact, we are asking whether S can be divided by a polynomial P ./. If the answer to these equivalent questions is affirmative, then every partial differential equation with constant coefficients, P .@/T D S

in S 0 .Rn /;

(8.7.28)

where X

P .@/ D

a ˛ @˛

(8.7.29)

j˛jm

is a linear partial operator with constant coefficients a˛ 2 R 8 multi-index ˛ with j˛j  m, will have a tempered distribution solution T 2 S 0 .Rn /. In fact, taking the Fourier transform of both sides of equation (8.7.28), we have, from P (8.7.4), F ŒP .@/T  D F ŒS in S 0 .Rn / H) P  ./F ŒT  D SO with P  ./ D j˛jm a˛ O .i 2/j˛j  ˛ H) F ŒT  D TO D S 2 S 0 .Rn / is well defined by the assumption P ./

that the division by polynomial P  ./ is well defined. Then T D F ŒTO  D F



 SO 2 S 0 .Rn / P  ./

(8.7.30)

is a tempered distribution solution of equation (8.7.28). This affirmative result for division of a tempered distribution by a polynomial was proved by Hörmander [31]. Thus, we have: Theorem 8.7.2. 8S 2 S 0 .Rn /, the partial differential equation with constant coefficients in (8.7.28) has a tempered distribution solution T 2 S 0 .Rn / defined by (8.7.30). Example 8.7.4. For given f 2 E 0 .R3 / (i.e. with compact support in R3 ), find u 2 S 0 .R3 / such that  u D f in S 0 .R3 /.

486

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Solution. Taking the Fourier transform of both sides of the equation, F Œ u D O O F Œf  H) 4 2 2 uO D fO H) uO D f2 2 H) u D F Œu O D F Œ f2 2 . 4 

4 

In particular, for f D ı 2 E 0 .R3 /, i.e.  u D ı, fO D F Œı D 1 and  uDF

3    1 1 1 1 1  2 2  . 12 / 1  D 2 S 0 .R3 / F  D D 2 2 2 2 2 4 4

4 .1/ r 4 r

(using (8.6.5)). Thus, we retrieve the elementary solution u D  41 r of given in Theorem 3.3.2 for n D 3. Example 8.7.5. Show that the homogeneous elliptic partial differential equation .1 

k / T D 0 in S 0 .Rn / has no solution in S 0 .Rn / other than the trivial solution T D 0 4 2 in S 0 .Rn /.

k 0 n Proof. F Œ.1  4 2 / T  D F Œ0 D 0 in S .R /.

k

k

k But .1  4 2 / T D .1  4 2 / .ı T / D .1  4 2 / ı T . Hence,       k k T DF 1 ı  F ŒT  D .1 C 2 /k F ŒT  D 0 F 1 4 2 4 2

using the Convolution Theorem 8.5.3 and (8.7.19). Then F ŒT  D 0, since .1 C 2 / does not vanish in Rn H) T D 0 in S 0 .Rn /.

k In fact, .1  4 2 / T D 0 has only the null tempered solution. But this equation has O

an infinite number of non-tempered solutions T D e 2hh;xi 2 D 0 .Rn / [8, p. 282] with O Rn D .h2 C h2 C    C h2 / 12 D 1, since, for T 2 D 0 .Rn /, hO D .h1 ; h2 ; : : : ; hn /, khk n 1 2  2  2 @ @ O O T D C  C 2 e 2hh;xi D Œ.2h1 /2 C .2h2 /2 C  C .2hn /2 e 2hh;xi 2 @xn @x1 O

O

D 4 2 Œ.h1 /2 C .h2 /2 C    C .hn /2 e 2hh;xi D 4 2 e 2hh;xi 2 D 0 .Rn / O D 1), and .1  (khk O e 2hh;xi

/T 4 2

D .1 

4 2 2hhO ;xi /e 4 2

O D1 D 0 in D 0 .Rn / 8hO with khk

and … S 0 .Rn /.

k 0 n 2hhO ;xi … S 0 .Rn / 8h O 2 Rn , Hence, .1  4 2 / T D 0 in D .R / with T D e O D 1. khk Convolution equations in S 0 .Rn / Consider the following convolution equation: for given A; B 2 S 0 .Rn /, find T 2 S 0 .Rn / such that A T DB

in S 0 .Rn /;

(8.7.31)

487

Section 8.7 Fourier transform methods for differential equations

where it is assumed that A T is well defined and A T 2 S 0 .Rn /. Then, taking the Fourier transform of both sides of (8.7.31), we have F ŒA T  D F ŒB

H)

O F ŒA  F ŒT  D F ŒB D B:

(8.7.32)

If F ŒA is a polynomial, then 9 a tempered distribution T 2 S 0 .Rn / defined by 

O  B T D F ŒTO  D F 2 S 0 .Rn /; F ŒA

(8.7.33)

which follows from Hörmander’s results [31]. Remark 8.7.2. Łojasiewicz (see [8, p. 126]) solved the problem of division of distributions by analytic functions. Elementary (or fundamental) tempered distribution solution of a linear operator with constant coefficients Definition 8.7.1C. A tempered distribution E 2 S 0 .Rn / is called an elementary or fundamental solution of the linear operator P .@/ in (8.7.28)–(8.7.29) if and only if P .@/E D ı

in S 0 .Rn /:

(8.7.34)

Theorem 8.7.3. Every partial differential operator P .@/ with constant coefficients as in (8.7.29) has at least one elementary tempered solution E 2 S 0 .Rn /. Proof. By Theorem 8.7.2, for S D ı 2 S 0 .Rn / in equation (8.7.28), elementary O D F Œ 1  2 solution T D E 2 S 0 .Rn / of P .@/ exists and is given by E D F ŒE P ./ P S 0 .Rn /, with P  ./ D j˛jm a˛ .i 2/j˛j  ˛ . In fact, taking Fourier transforms of both sides of (8.7.34), we have P  ./F ŒE D F Œı D 1 H) F ŒE D EO D P 1./ 2 S 0 .Rn /, which is well defined by virtue of Hörmander’s result [31] on division by the polynomial P  ./. Then an elementary solution is O DF E D F ŒE



  _ 1 1 2 S 0 .Rn /: DF P  ./ P  ./

Elementary solution E 2 S 0 .Rn / is not unique in general. If 9E0 2 S 0 .Rn / such that P .@/E0 D 0 in S 0 .Rn /, then E C E0 2 S 0 .Rn / is also an elementary solution. In fact, P .@/ŒE C E0  D P .@/E C P .@/E0 D ı C 0 D ı in S 0 .Rn /.

488

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Example 8.7.6. Find an elementary solution E 2 S 0 .Rn / of the following operators with the help of Fourier transforms: 1. the iterated Laplace operator k for k  1;

k 2. the elliptic operator .1  4 2 / with k  1. Solution. 1. k Ek D ı in S 0 .Rn /. But ı Ek D Ek H) k Ek D k .ı Ek / D k ı Ek D ı H) F Œ k ı Ek  D F Œı H) F Œ k ı  F ŒEk  D 1 (by Theorem 8.5.3). Then, using (8.7.16), F Œ k ı D .4 2 2 /k with 2 D 12 C 22 C    C n2 H) .4 2 2 /k EO k D 1 with EO k D F ŒEk  H) EO k D .1/k 22k1 2k Pf. 12k / H) elementary solution E D Ek D F ŒEO k , since

./ L D ./ D ./ H) F ŒPf. 12k / D F ŒPf. 12k /. Hence,    1 1 k E D Ek D .1/ 2k 2k F Pf 2k ; 2 

(8.7.35)

where F ŒPf. 12k / is given by two different formulae (8.6.7) and (8.6.9) for two different cases: Case I: If n is odd or if n is even, but 2k < n, we are not in a singular case and the symbol Pf is useless, so can be dropped for 2k < n. Then we get, from (8.6.7) and (8.7.35), n    2k 2 . n2k 1 1 1 1 2 / E D Ek D .1/k 2k 2k F 2k D .1/k 2k 2k  n2k 2k 2 

2  r . 2 / D .1/k

. n2  k/ n

22k  2 .k  1/Š

 r 2kn

for k  1:

(8.7.36)

For n D 3, k D 1, p . 32  1/ 1 1   1 ; (8.7.37) E D E1 D .1/  D D 3 3 4 r 22  2 0Š r 4 2 r which has been obtained in (3.3.22). Case II: For even n and 2k  n, we are in a singular case and get, from (8.6.9) and (8.7.35) with 2p D 2k  n  0:    1 1 k E D Ek D .1/ 2k 2k F Pf nC.2kn/ 2 

  n n 1  2 C2kn  2  .1/k 2 2kn 1 k D .1/ 2k 2k r ln C B r 2  . n2 C k  n2 /.k  n2 /Š n

D

.1/ 2 n 2

22k1  .k  1/Š.k 

n 2 /Š

r 2kn ln

1 C Br 2kn ; r

(8.7.38)

489

Section 8.7 Fourier transform methods for differential equations

where  BD



n

.1/ 2 n

22k1  2 .k  1/Š.k  n2 /Š   1 1 1 1 1 C C  C  ln C  2 2 k

 n 2

C

1  0 .k/ C 2 .k/

 (8.7.39)

is a constant, i.e., independent of x. But this constant B is of no importance and can be replaced by 0, since k .Br 2kn / D 0, which leads to the addition of polyharmonic polynomials to the elementary solution. These polyharmonic polynomials can be neglected to consider the simplest part of (8.7.38): n

E D Ek D

.1/ 2 n

22k1  2 .k

 1/Š.k 

n 2 /Š

1 r 2kn ln : r

(8.7.40)

For n D 2; k D 1 with 2k D n, we get from (8.7.40) that E D E1 D

1 1 1  ln D ln r 2 r 2

(8.7.41)

is an elementary solution of the Laplace operator (see (3.3.23)), since ln 1r D  ln r. For n D 2; k D 2 with 2k > n, we get from (8.7.40) that E D E3 D

1 23 1Š1Š

 r 42 ln

1 1 2 1 1 D  r 2 ln D r ln r r 8 r 8

(8.7.42)

is an elementary solution of the biharmonic operator in two variables (see also (3.3.65)). For n D 2; k D 3 with 2k > n, E D E3 D

1 4 1 1 4 1 1  r 4 ln D  r ln D r ln r 25 .3  1/Š.3  1/Š r 128 r 128 (8.7.43)

is an elementary solution of in two variables (see also (3.3.67)). For n D 2; k D 4 with 2k > n, we get from (8.7.40) that 1 1 1 1  r 6 ln D  r 6 ln 27 .4  1/Š.4  1/Š r 128  6  6 r 1 1 1 1 D r 6 ln D  r 6 ln (8.7.44) 4608 r 4608 r

E D E4 D

is an elementary solution of in two variables (see also (3.3.68)).

490

Chapter 8 Fourier transforms of distributions and Sobolev spaces

2. .1 

k / Ek 4 2

H)

D ı in S 0 .Rn /

      k k F 1 .ı Ek / D F 1  ı  F ŒEk  D F Œı D 1: 4 2 4 2

k 2 k 2 k O O From (8.7.19), F Œ.1  4 2 / ı D .1 C / H) .1 C / Ek D 1 H) Ek D 1 1 H) F ŒEO k  D F Œ .1C2 /k , since ./ L D ./. Using (8.7.25) [8] .1C2 /k with m replaced by 2k,

 E D Ek D F D

1 .1 C 2 /k



2

D

2k 2

. 2k 2 /

r

2kn 2

K n2k .2 r/ 2

2 k k n r 2 K n2 k .2 r/; .k  1/Š

(8.7.45)

where K n2 k from the theory of Bessel functions is defined by (3.3.60). This is the only tempered solution. Elementary solution of .  /k with  > 0 .  /k Ek D ı in S 0 .Rn /. Using (8.7.20), we have F ŒEO k  D Ek D F

H)



1 2 .4 2  /k





.1/k

DF k

 1C

4 2 2 

  .1/k 1 2 F with N D p : E D Ek D k 2 k  .1 C N / 

 k

(8.7.46)

Elementary solution of elasticity operator and Stokes operator Without proof, we accept the elementary solutions of the elasticity operator and the Stokes operator given in [41]. The elasticity operator A is defined by 3 X @ .Au/i D ij .u/; @xj

1  i  3;

(8.7.47)

j D1

where u D .u1 ; u2 ; u3 / with ui .x/ the displacement vector field; "ij .u/ D 12 Œui;j C P uj;i , 1  i  3, is the strain tensor field; ij .u/ D  3lD1 "l l .u/ C 2"ij .u/ is the stress tensor field,  and  being Lamé’s parameters, satisfying the equilibrium P equations j3D1 @x@ .ij .u//.x/ D fi .x/ for x 2 R3 , 1  i  3, f D .f1 ; f2 ; f3 / j being a force vector field.

491

Section 8.7 Fourier transform methods for differential equations

An elementary solution of the elasticity operator A is a 3  3 symmetric matrix E D .Eij / with Eij .x/ D Ej i .x/ such that AE D ıI; 1  i; j  3;

(8.7.48)

where ı is the Dirac distribution with force concentrated at 0 and I is the identity matrix of order 3. Then an elementary solution E D .Eij / of A is given by [41, pp. 68–69]:   1 @.kxk/ @.kxk/ ; . C 3/ıij C . C / Eij .x/ D Ej i .x/ D  8. C 2/kxk @xi @xj (8.7.49) ´ 1 for i D j ıij D 1  i; j  3; kxk2 D x12 C x22 C x32 : 0 for i ¤ j; For further details, we refer to [41]. An elementary solution of the Stokes operator A defining the flow of incompressible viscous fluid in R3 with weak velocity (unknowns being the velocity field u D .u1 ; u2 ; u3 / and the pressure p at a point x 2 R3 ) is a 4  4 matrix E D Ei;j ; 1  i; j  4 such that AE D ıI;

(8.7.50)

ı being the Dirac distribution with concentration at 0 2 R3 , I being the identity matrix of order 4. Then an elementary solution E D .Ei;j /; 1  i; j  4, of the Stokes operator A is given by [41, pp. 72–77]:     1 @2 @2 1  .kxk/ C 2ıij .kxk/ ; Eij .x/ D 8 @xi @xj kxk @xi @xj

1  i; j  3; (8.7.51)

ıij D 1 for i D j and ıij D 0 for i ¤ j , kxk2 D x12 C x22 C x32 ; 1 @ E4i .x/ D Ei4 .x/ D Pi .x/ D 4 @xi E44 .x/ D ı;



 1 ; kxk

1  i  3I

ı being the Dirac distribution.

(8.7.52) (8.7.53)

For further details, we refer to [41]. Remark 8.7.3. The author has not checked the correctness of the result in (8.7.49) (resp. (8.7.51)–(8.7.53)), which is left to the reader as an exercise.

492

8.8

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Laplace transform of distributions on R

8.8.1 Space D 0C Definition 8.8.1. D 0C  D 0C .R/ is the space of distributions T 2 D 0 .R/ with supp.T /  RC 0 D ¹x W x 2 R, x  0º, i.e. D 0C D D 0C .R/ D ¹T W T 2 D 0 .R/ with supp.T /  RC 0 º:

(8.8.1)

Let T 2 D 0C and 0 2 R such that e  0 x T 2 S 0 .R/. Since e  0 x is a C 1 function on R, the product e  0 x T is a well defined distribution on R, and is, in fact, a tempered distribution by assumption. Then, T 2 D 0C ; 0 2 R; e  0 x T 2 S 0 .R/

H)

e  x T 2 S 0 .R/ 8 > 0 : (8.8.2)

In fact, e  x T D e .  0 /x e  x T with e  0 x T 2 S 0 .R/,  > 0 . Although for  > 0 , e .  0 /x … M (the multiplier set, see Definition 8.5.1), but ˛.x/e .  0 /x 2 M such that ˛.x/e .  0 /x  2 S.R/, where ˛.x/ is defined by ˛ 2 C 1 .R/;

˛.x/ D 1

8x 2 U I

(8.8.3)

with supp.T /  U , supp.˛/ is bounded on the left. Since e  0 x T 2 S 0 .R/ and ˛.x/e .  0 /x  2 S.R/ with  > 0 , he  0 x T; ˛.x/ .  0 /x i is well defined (8 2 S.R/ with  >  ) and independent of the choice e 0 of the function ˛.x/. For example, let ˇ be another function such that ˇ 2 C 1 .R/, ˇ.x/ D 1 8x 2 U with supp.T /  U and supp.ˇ/ is bounded on the left. Then ˛.x/  ˇ.x/ D 0 8x 2 supp.T / H) he  0 x T; .˛.x/  ˇ.x//e .  0 /x i D 0 8 2 S.R/ with  > 0 H) he  0 x T; ˛.x/e .  0 /x i D he  0 x T; ˇ.x/e .  0 /x i 8 2 S.R/. Then, for  > 0 , he  x T; i D he  0 x T; e .  0 /x i D he  0 x T; ˛.x/e .  0 /x i 8 2 S.R/ with ˛.x/e .  0 /x  2 S.R/. Hence, e  x T 2 S 0 .R/ is well defined 8 > 0 .

Laplace transform L Definition 8.8.2. Let T 2 D 0C  D 0C .R/ be a distribution such that e  0 x T 2 S 0 .R/. Then the Laplace transform of T , denoted by LT or L.T /, is defined, 8 > 0 , by: .LT /.p/ D hT; e px i

with p D  C i 2 C;  > 0 :

L.T /.p/  .LT /.p/ 8p with Re.p/ > 0 .

(8.8.4)

493

Section 8.8 Laplace transform of distributions on R

Justification Although e px … S.R/ for Re.p/ D  > 0 , we can define a function ˛.x/ by (8.8.3) such that ˛.x/e .p 0 /x 2 S.R/. Then, (8.8.4) can be rewritten as hT; e px i D he  0 x T; ˛.x/e .p 0 /x i;

(8.8.5)

which is well defined by virtue of the assumptions. Example 8.8.1. 1. LŒı D 1; 2. LŒıa  D e ap .a  0/; 3. LŒı .m/  D p m ; 4. LŒH.x/ D 1=p for  D Re.p/ > 0; R1  0 x f 2 5. LŒf  D 0 f .x/e px dx for f 2 L1 .R/ with supp.f /  RC 0 and e 0 S .R/; R1 6. LŒH.x/ ln x D  ln ppC , C D Euler’s constant D  0 e y ln ydy. Solution. 1. LŒı.p/ D hı; e px i D e p.x/ jxD0 D 1; 2. LŒıa .p/ D hıa ; e px i D e px jxDa D e ap for a  0; m

d px /i D .1/m Œ.1/m p m  3. LŒı .m/ .p/ D hı .m/ ; e px i D .1/m hı; dx m .e px m e xD0 D p 8m 2 N; R1 px 1 4. LŒH.x/.p/ D hH; e px i D 0 1:e px dx D  e p j1 0 D p for Re.p/ > 0I R1 5. LŒf .p/ D hTf ; e px i D 0 f .x/e px dx with Re.p/ > 0 , since f .x/ D 0 for almost all x < 0;

6. T D H.x/ ln x H) T 2 S 0 .R/ and e  0 x T 2 S 0 .R/ for 0 D 0. Then, for Re.p/ > 0, LŒH.x/ ln x.p/ D hH.x/ ln x; e px i D dx. For real p > 0, set y D px. Then x D py , dx D dy p Z H) 0

1

R1 0

ln xe px

  Z y y dy 1 1 D .ln y  ln p/e y dy e p p p 0 0   Z 1 Z 1 1  ln p  C y y D  ln p ; e dy C ln y  e dy D p p 0 0

ln xe px dx D

where C D  constant.

R1 0

Z

1

ln

ln y  e y dy D limn!1 Œ1 C

1 2

C  C

1 n

 ln n D Euler’s

494

Chapter 8 Fourier transforms of distributions and Sobolev spaces

L is linear L.T1 /.p/; L.T2 /.p/ exist for Re.p/ > 0 H) LŒ˛1 T1 C ˛2 T2 .p/ exists for Re.p/ > 0 and 8˛1 ; ˛2 2 R; LŒ˛1 T1 C ˛2 T2 .p/ D h˛1 T1 C ˛2 T2 ; e px i D ˛1 hT1 ; e px i C ˛2 hT2 ; e px i D ˛1 L.T1 /.p/ C ˛2 L.T2 /.p/ 8T1 ; T2 2 D 0C ; 8˛1 ; ˛2 2 R: (8.8.6) Properties of the Laplace transform L Property 1: Relation withR Fourier transform Instead of the definition of Fourier 1 transform F f in (7.1.2) ( 1 f .x/e i2x dx) followed up to now, we will define the Fourier transform F f here by (7.1.12): Z

1

g3 ./ D .F f /./ D

f .x/e ix dx;

(8.8.7)

g3 ./e ix d ;

(8.8.8)

1

with the co-transform FN defined by: 1 f .x/ D FN g3 D 2

Z

1 1

so that a simple relation between Fourier transform and Laplace transform can be obtained. Under the definition in (8.8.7), the Fourier transform F T of a tempered distribution T 2 S 0 .R/ can be defined by the same formula (8.3.1): hF T; i D hT; F i where F  is defined by .F /.x/ D

8 2 S.R/;

R1

1 ./e

ix d .

(8.8.9)

Then we have:

Proposition 8.8.1. Let T 2 D 0C  D 0C .R/ be a distribution with support in RC 0 such that e  0 x T 2 S 0 .R/ for some 0 2 R. Then the Laplace transform L is related to the Fourier transform F by: 8 > 0 , .LT /.p/ D F Œe  x T ./

with p D  C i 2 C:

(8.8.10)

Proof. Set F .p/ D L.T /.p/ with p D  C i, Re.p/ D  > 0 . Let ˛ 2 C 1 .R/ be such that ˛.x/ D 1 8x 2 U , supp.T /  U , supp.˛/ is bounded on the left. Then F .p/ D hT; e px i D he  0 x T .x/; e .  0 /x e i x i D he

 0 x

T .x/; ˛.x/e

.  0 /x i x

e

i:

(since p D  C i)

495

Section 8.8 Laplace transform of distributions on R

8 2 S.R/, Z hF; i D he  0 x T .x/; ˛.x/e .  0 /x e i x i./d R

D he  0 x T .x/ ˝ ./; ˛.x/e .  0 /x e i x i D he  0 x T .x/; h./; ˛.x/e .  0 /x e i x ii D he  0 x e .  0 /x T .x/˛.x/h./; e i x ii D he  x T .x/h./; e i x ii D he  x T .x/; F .x/i D hF Œe  x T ./; ./i

8 2 S.R/

H) F D F Œe  x T  H) L.T /. C i/ D F Œe  x T ./, since F .p/ D L.T /.p/.

Property 2 Let T 2 D 0C  D 0C .R/ with e  0 x T 2 S 0 .R/ for some 0 2 R. Then, without proof we agree to accept that F .p/ D .LT /.p/ is an analytic function of p D  C i 2 C with Re.p/ D  > 0 . For example, T D ı 2 D 0C .R/ with e  0 x ı 2 S 0 .R/ 80 2 R, and hence F .p/ D LŒı.p/ D 1 is an analytic function of p D  C i 2 C. Property 3  L

 d mT .p/ D p m L.T /.p/ 8m 2 N; dx m

(8.8.11)

where the derivative is in the sense of distribution. In fact,   m   m   d T d T px d m px m ; e .e / L .p/ D D .1/ T; dx m dx m dx m D .1/m hT; .p/m :e px i D p m hT; e px i D p m LŒT .p/: Property 4 

 dm L.T / .p/ D LŒ.x/m T .p/: dp m

(8.8.12)

Indeed,  m    d dm d m px px L.T / .p/ D hT; e i D T; .e / dp m dp m dp m D hT; .x/m e px i D h.x/m T; e px i D LŒ.x/m T .p/:

496

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Property 5: Convolution S  T of distributions S; T 2 D 0C Proposition 8.8.2. Let S; T 2 D 0C  D 0C .R/ such that e  1 x S 2S 0 .R/, e  2 x T 2 S 0 .R/ for some 1 ; 2 2 R and L.S /.p/ and L.T /.p/ exist for Re.p/ > 1 and Re.p/ > 2 respectively. Then their convolution S T 2 D 0C is defined and given, 8 2 D.R/, by: hS T; i D hSx ˝ Ty ; .x C y/i D hSx ; hTy ; .x C y/ii:

(8.8.13)

Proof. 8 fixed x 2 supp.S /; hTy ; .x C y/i is well defined, since x .y/ D .x C y/ has compact support in y. Again, the function x 7! hTy ; .x C y/i is a C 1 function with compact support. Indeed, x 2 supp.S / H) x  0, y 2 supp.T / H) y  0 and .x C y/ 2 supp./ H) 9A > 0 such that jx C yj  A. Hence, we have 0  x  x C y  A. So, the formula (8.8.13) is meaningful. Thus, S T 2 D 0 .R/ is well defined, with supp.S T /  supp.S / C supp.T /  Œ0; 1Œ

H)

S T 2 D 0C :

(8.8.14)

Since ı 2 D 0C , ı T D T 2 D 0C

8T 2 D 0C :

(8.8.15)

8.8.2 Distribution T 1 2 D 0C (see also convolution algebra A D D 0C (6.9.15b)) Definition 8.8.3. Let T 2 D 0C . Then the unique distribution S 2 D 0C which satisfies the convolution equation T S D ı, ı being the Dirac distribution, is denoted by T 1 , i.e. T T 1 D ı:

(8.8.16)

Example 8.8.2. Find 1. H 1 ; 2. .ı 0 /1 ; 3. .ı 0  ı/1 ; where ı 0 D Solution.

dı . dx

d dı 1. H S D ı H) dx .H S/ D dx H) S dH D ı 0 H) S ı D ı 0 H) S D ı 0 dx 1 0 by virtue of (8.8.15). Hence, H D ı .

2. S ı 0 D ı. But S ı 0 D S 0 ı D ı H) S 0 D ı H) S D H H) .ı 0 /1 D H .

497

Section 8.8 Laplace transform of distributions on R

3. S .ı 0  ı/ D ı H) S ı 0  S ı D ı H)  S/ ı D ı. . dS dx

dS dx

ı  S D ı H)

Set S D e x T . Then e x ŒT C ddxT  T  D ı H) e x ddxT D ı H) ddxT D e x ı D ı, since he x ı; i D hı; e x i D .e x .x//jxD0 D .0/ D hı; i 8 2 D.R/ H) e x ı D ı. But

dT dx

D ı H) T D H , since

dH dx

D ı. Hence, .ı 0 ı/1 D S D e x H.x/.

Theorem 8.8.1 (Convolution Theorem). Let S; T 2 D 0C  D 0C .R/ such that e  1 x S 2 S 0 .R/, e  2 x T 2 S 0 .R/ for some 1 ; 2 2 R and L.S /.p/ and L.T /.p/ are defined for Re.p/ > 1 and Re.p/ > 2 respectively. Then, for Re.p/ > max¹1 ; 2 º, L.S T / D L.S /  L.T /:

(8.8.17)

Proof. For Re.p/ > max¹1 ; 2 º, L.S T /.p/ D hS T; e px i D hSx ; hTy ; e p.xCy/ ii D hSx ; e px hTy ; e py ii D hSx ; e px i  hTy ; e py i D L.S /  L.T /; since Re.p/ > both 1 and 2 . Property 6 L.T /.p/ D 0 for Re.p/ > 

H)

T D 0:

(8.8.18)

8.8.3 Inverse L1 of Laplace transform L Property 7 If F .p/ D L.T /.p/ for p 2 C with Re.p/ > 0 , L1 .F .p// D T .x/. Example 8.8.3. 1. Find L.T /.p/, when (a) T D Pf. H.x/ x /; (b) T D x k e ˛x H.x/ for k 2 N0 , ˛ 2 C with Re.p/ > Re.˛/; (c) T D H.x/ sin x; (d) T D H.x/e ˛x cos ˇx with Re.p/ > ˛; (e) T D H.x/e ˛x sin ˇx with Re.p/ > Re.˛/.

(8.8.19)

498

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Then, using these results, find p / with Re.p/ > 1; (f) L1 . pC1 2

Ci (g) L1 . p2p3pC2 /.

2. Find T 2 D 0C which satisfies .xe x H.x// T D H.x/ sin x. Solution. 1(a) From (1.4.26),    Z 1   .x/ H.x/ ;  D lim dx C .0/ ln " : hT; i D Pf x x "!0C "

(8.8.20)

Set S D H.x/ ln x. Then, 8 2 D.R/,     Z 1 dS d ;  D  S; ln x 0 .x/dx D dx dx 0  Z 1 0 D  lim ln x .x/dx (by Lebesgue’s Theorem) "!0C



D  lim

"!0C

"

Z

1

D lim

"!0C

"

Z

1

.x/ dx x "  .x/ dx C ln "  ."/ : x

.ln x  .x//j1 " 



But from (1.2.41) we get ."/ D .0/ C " ."/ H) ."/ ln."/ D .0/ ln " C " ln "  ."/ H) lim"!0C ln "."/ D lim"!0C .0/ ln " C lim"!0C " ln "  ."/ D lim"!0C .0/ ln ", where " ln " ."/ ! 0 as " ! 0C , since " ln " ! 0 and ."/ ! .0/ as " ! 0C . Hence,    Z 1 d .x/ ŒH.x/ ln x;  D lim dx C .0/ ln " : (8.8.21) dx x "!0C " From (8.8.20) and (8.8.21),       d H.x/ ŒH.x/ ln x;  D Pf ;  8 2 D.R/ dx x   H.x/ d ŒH.x/ ln xDPf H) in D 0 .R/ with supp.Pf. H.x/ x //  Œ0; 1Œ dx x      H.x/ d H) L Pf DL .H.x/ ln x/ x dx D pL.H.x/ ln x/ (by (8.8.11)) Dp

 ln p  C D .ln p C C / (see (6) in Example 8.8.1): p

499

Section 8.8 Laplace transform of distributions on R

1(b) k ˛x

LŒx e

k ˛x

H.x/ D hx e

H.x/; e

px

Z

1

iD 0

„ Ik D x k

x k e .˛p/x dx ; ƒ‚ …

Re.p/ > Re.˛/:

Ik

ˇ Z 1 k ˇ  x k1 e .˛p/x dx ˛  p ˇxD0 ˛p 0 „ ƒ‚ …

e .˛p/x ˇxD1

Ik1

k Ik1 (since e .˛p/x ! 0 as x ! 1 for Re.p/ > Re.˛/) p˛ Z 1 k k1 kŠ kŠ D   Ik2 D    D I D e .˛p/x dx 0 p˛ p˛ .p  ˛/k .p  ˛/k 0 ˇ kŠ kŠ e .˛p/x ˇˇxD1 kŠ 1 D D  D : .p  ˛/k ˛  p ˇxD0 .p  ˛/k .p  ˛/ .p  ˛/kC1

D

Hence, for Re.p/ > Re.˛/, LŒx k e ˛x H.x/.p/ D

kŠ : .p  ˛/kC1

(8.8.22)

1(c) 1 1 ix Œ.e  e ix /H.x/ D Œe ix H.x/  e ix H.x/ 2i 2i 1 L.H.x/ sin x/ D ŒL.e ix H.x//  L.e ix H.x/: 2i

T D H.x/ sin x D

From (b) with k D 0, ˛ D i , L.e ix H.x// D 1 Re.p/ > 0, L.e ix H.x// D pCi .

1 pi ,

and with k D 0, ˛ D i ,

Hence,   1 1 1 1 L.H.x/ sin x/ D  D 2 : 2i p  i p C i p C1 1(d) For ˛ > 0, T D H.x/e ˛x cos ˇx D H.x/e ˛x

e iˇx C e iˇx 2

1 D ŒH.x/e .˛iˇ /x C H.x/  e .˛Ciˇ /x  2

(8.8.23)

500

Chapter 8 Fourier transforms of distributions and Sobolev spaces

H)

1 LŒH.x/e ˛x cos ˇx D ŒL.H.x/e .˛iˇ /x / C L.H.x/e .˛Ciˇ / /.p/ 2   1 1 1 C D 2 p C .˛  iˇ/ p C .˛ C iˇ/ 2.p C ˛/ 1 D  2 .p C ˛/2 C ˇ 2 pC˛ D for Re.p/ > ˛: .p C ˛/2 C ˇ 2

1(e) Similarly, 1 L.H.x/e ˛x sin ˇx/.p/D ŒL.H.x/  e .˛iˇ /x /  L.H.x/e .˛Ciˇ /x /.p/ 2i   1 1 1  D 2i p C .˛  iˇ/ p C .˛ C iˇ/ 2iˇ ˇ 1  D for Re.p/ > ˛: D 2i .p C ˛/2 C ˇ 2 .p C ˛/2 C ˇ 2 1 1(f) From (b), we get, for k D 0, ˛ D 1, L.H.x/e x /.p/ D pC1 with Re.p/ > 1. From (8.8.11),   d p x L .H.x/e / .p/ D pL.H.x/e x /.p/ D with Re.p/ > 1 dx pC1      p d d D L1 L .H.x/e x / D .H.x/e x / H) L1 pC1 dx dx dH  H.x/e x D e x  ı  H.x/e x ; D e x  dx

ı D ı.x/ being the Dirac distribution with concentration at 0. But he x ı; i D hı; e x i D .e x .x//jxD0 D .0/ D hı; i H) e x ı D ı. Hence, p L1 . pC1 / D ı  H.x/e x for Re.p/ > 1. 1(g)

p 2 Ci p 2 3pC2

H)

D L

p 2 Ci .p2/.p1/

1



D

p2 p2

p2 C i p 2  3p C 2



C

i p2



p2 p1



i p1

   p2 1 1 DL C iL p2 p2   2   p i 1 1 L : (8.8.24) L p1 p1 1



Then, from (8.8.22) with k D 0, ˛ D 2, L.e 2x H.x// D 1 ˛ D 1, L.e x H.x// D p1 .

1 p2 ,

and k D 0,

501

Section 8.8 Laplace transform of distributions on R

Then 

 d 2 2x p2 2 2x L Œe H.x/ D p L.e H.x// D dx 2 p2

(by (8.8.11))

and 

 d2 x p2 2 x L Œe H.x/ D p L.e H.x// D dx 2 p1

(by (8.8.11)):

Hence, from (8.8.24),   p2 C i d 2 2x L1 2 Œe H.x/ C i e 2x H.x/ D p  3p C 2 dx 2 

d2 x Œe H.x/  i Œe x H.x/: dx 2

(8.8.25)

But d 2x Œe H.x/ D e 2x ı C 2e 2x H.x/ D ı C 2H.x/e 2x ; dx d x Œe H.x/ D e x ı C e x H.x/ D ı C H.x/e x ; dx d 2 2x Œe H.x/ D ı 0 C 2ıe 2x C 4H.x/e 2x D ı 0 C 2ı C 4H.x/e 2x ; dx 2 d2 x Œe H.x/ D ı 0 C ıe x C H.x/e x D ı 0 C ı C H.x/e x ; dx 2 where ı D ı.x/ is the Dirac distribution with concentration at 0, and ı 0 D he kx ı; i D hı; e kx i D .e kx .x//jxD0 D .0/ D hı; i

dı . dx

8 2 D.R/

H) e kx ı D ı. So, from (8.8.25),   p2 C i L1 2 D .ı 0 C 2ı C 4e 2x H.x/ C i e 2x H.x// p  3p C 2 C .ı 0  ı  e x H.x//  i e x H.x/ D ı C .4 C i /e 2x H.x/  .1 C i /e x H.x/: 2. LŒxe x H.x/ T  D LŒH.x/ sin x. From (8.8.17) and (8.8.22), LŒxe x H.x/ T .p/ D LŒxe x H.x/.p/  LŒT .p/ D

1Š  L.T /.p/: .p  1/2

502

Chapter 8 Fourier transforms of distributions and Sobolev spaces

From (8.8.23), L.H.x/ sin x/ D H)

L.T /.p/ D

1 . p 2 C1

Then

1 L.T /.p/ .p1/2

D

1 p 2 C1

p 2  2p C 1 p .p  1/2 D D12 2 : 2 2 p C1 p C1 p C1

But 

 d p L .H.x/ sin x/ .p/ D 2 dx p C1     p p 1 1 1 H) T D L D L .1/  2L 12 2 p C1 p2 C 1 d H) T D ı  2 ŒH.x/ sin x D ı  2 sin x  ı  2H.x/ cos x dx D ı  0  2H.x/ cos x D ı  2H.x/ cos x d (since dx H.x/ D ı, and hsin xı; i D hı; sin x.x/i D Œsin x.x/xD0 D 0 8 2 D.R/ H) ı sin x D 0 in D 0 .R/).

For more details, we refer to [6], [7], [8].

8.9

Applications

8.9.1 Sobolev spaces H s .Rn / In Section 2.15 of Chapter 2, we defined Sobolev spaces H m ./ of integral order m 2 N for   Rn . Now we are in a position to define Sobolev spaces H s .Rn / for arbitrary order s 2 R with the help of Fourier transforms. From this general definition we can retrieve the space H m .Rn / for  D Rn in Definition 2.15.1; this is proved in Proposition 8.9.2 of this section. Now we begin with the definition of Sobolev spaces H s .Rn / of arbitrary order s 2 R. Definition 8.9.1. 8s 2 R, the set H s .Rn / defined by H s .Rn / D ¹u W u 2 S 0 .Rn /; .1 C kk2 /s=2 uO 2 L2 .Rn /; u./ O D .F u/./; kk2 D 12 C 12 C    C n2 º

(8.9.1)

is called the Sobolev space of arbitrary order s 2 R on Rn , which is equipped with inner product h  ;  is;Rn and norm k  ks;Rn given by: 8u; v 2 H s .Rn / with uO D F u, vO D F v, Z NO .1 C kk2 /s u./ O v./d ; (8.9.2) hu; viH s .Rn / D hu; vis;Rn D Rn

503

Section 8.9 Applications

NO where v./ is the complex conjugate of v./; O 1=2

Z

kukH s .Rn / D kuks;Rn D hu; uis;Rn D

2 .1 C kk2 /s ju./j O d

 12 :

(8.9.3)

Rn

Theorem 8.9.1. 8s 2 R, H s .Rn / equipped with the inner product h  ;  is;Rn defined by (8.9.2) is a Hilbert space. Proof. For the proof we are to show that H s .Rn / is complete, i.e. every Cauchy sequence in H s .Rn / is convergent in H s .Rn /. Let .uk / be any Cauchy sequence in H s .Rn //, i.e. Z 2 kuk  um ks;Rn D .1 C kk2 /s juO k  uO m j2 d  ! 0 as k; m ! 1 Rn

H)

s

2 k.1 C kk2 / 2 .uO k  uO m /kL 2 .Rn / Z D .1 C kk2 /s juO k  uO m j2 d  ! 0 as k; m ! 1 Rn

H) ..1 C kk2 /s=2 uO k / is a Cauchy sequence in L2 .Rn / which is a complete space s H) 9w 2 L2 .Rn / such that .1 C kk2 / 2 uO k ! w in L2 .Rn / as k ! 1. But s L2 .Rn / ,! S 0 .Rn / by (8.2.31) H) w 2 S 0 .Rn / H) .1 C kk2 / 2 w 2 S 0 .Rn /. By Theorem 8.3.2, F W S 0 .Rn / ! S 0 .Rn / is an isomorphism H) 9u 2 S 0 .Rn / s such that F u D uO D .1 C kk2 / 2 w 2 S 0 .Rn /. u 2 S 0 .Rn / H) .1 C kk2 /s=2 uO 2 s S 0 .Rn /. But .1Ckk2 /s=2 uO D w 2 L2 .Rn / H) u 2 H s .Rn / and .1Ckk2 / 2 uO k ! s w D .1 C kk2 / 2 uO in L2 .Rn / as k ! 1 Z 2 H) ku  uk ks;Rn D .1 C kk2 /s juO  uO k j2 d  ! 0 as k ! 1 Rn

H) the Cauchy sequence .uk / converges to u 2 H s .Rn /. Hence, H s .Rn / is complete, i.e. a Hilbert space.

8.9.2 Imbedding result Proposition 8.9.1. For s1  s2 , H s1 .Rn / ,! H s2 .Rn / with kuks2 ;Rn  kuks1 ;Rn

8u 2 H s1 .Rn /:

(8.9.4)

C kk2 /s2  .1 C kk2 /s1 . Let u 2 H s1 .Rn /. Then, RProof. For s12 s s2 , .1 2 d  < C1. But .1 C kk2 /s2 ju./j 2  .1 C kk2 /s1 ju./j 2 1 O O O Rn .1 C kk / ju./j n a.e. in R Z Z 2 2 H) .1 C kk2 /s2 ju./j O d  .1 C kk2 /s1 ju./j O d  < C1: Rn

Rn

504

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Hence, u 2 H s1 .Rn /  S 0 .Rn / H) .1 C kk2 /s1 =2 uO 2 L2 .Rn / H) .1 C kk2 /s2 =2 uO 2 L2 .Rn / H) u 2 H s2 .Rn / with kuks2 ;Rn  kuks1 ;Rn H) H s1 .Rn /  H s2 .Rn / with kuks2 ;Rn  kuks1 ;Rn H) H s1 .Rn / ,! H s2 .Rn / with kuks2 ;Rn  kuks1 ;Rn 8u 2 H s1 .Rn /. Remark 8.9.1. For s  0, H s .Rn / ,! L2 .Rn / D H 0 .Rn /. Hence, for s  0, H s .Rn / can equivalently be defined by: Definition 8.9.2. For s  0, H s .Rn / D ¹u W u 2 L2 .Rn /; .1 C kk2 /s=2 uO 2 L2 .Rn /º

(8.9.5)

with hu; vis;Rn and k  ks;Rn defined by (8.9.2) and (8.9.3) respectively. Example 8.9.1. Show that the partial differential operator  Ck 2 W H sC2 .Rn / ! H s .Rn / is an isomorphism from H sC2 .Rn / onto H s .Rn / 8 real k ¤ 0, 8s 2 R, 2 2 D @ 2 C    C @ 2 being the n-dimensional Laplace operator. @x1

@xn

Solution. Set A D  C k 2 defined by 8u 2 H sC2 .Rn /; Au   u C k 2 u 2 s H .Rn / 8 real k ¤ 0, 8s 2 R. Continuity of A W H sC2 .Rn / ! H s .Rn /: Let u 2 H sC2 .Rn /  S 0 .Rn /. Hence, F Œ u C k 2 u D .4 2 kk2 C k 2 /uO 2 S 0 .Rn / with uO D F u H)

O .1 C kk2 /s=2 jF Πu C k 2 uj D .1 C kk2 /s=2 j.4 2 kk2 C k 2 /j  juj:

But .1 C kk2 /s=2 .4 2 kk2 C k 2 /  max¹4 2 ; k 2 º.kk2 C 1/.1 C kk2 /s=2 D C.1 C kk2 /s=2C1 ; with C D max¹4 2 ; k 2 º > 0. Hence, .1 C kk2 /s=2 jF Œ u C k 2 uj  C.1 C kk2 /s=2C1 juj: O u 2 H sC2 .Rn / H) from definition of H sC2 .Rn /, .1Ckk2 /s=2C1 uO 2 L2 .Rn / H) .1 C kk2 /s=2 F Œ u C k 2 u 2 L2 .Rn / H) . u C k 2 u/ 2 H s .Rn /. Thus, u 2 H sC2 .Rn / H)  u C k 2 u 2 H s .Rn /. Moreover, 2 k  u C k 2 uk2s;Rn D k.1 C kk2 /s=2 jF Œ u C k 2 ujkL 2 .Rn / 2  C 2 k.1 C kk/s=2C1 jujk O L 2 .Rn /

H) k  u C k 2 uks;Rn  C kuksC2;Rn with C > 0, independent of u,

505

Section 8.9 Applications

H) A D  u C k 2 W H sC2 .Rn / ! H s .Rn / is continuous from H sC2 .Rn / into H s .Rn /. A W H sC2 .Rn / ! H s .Rn / is injective: For u 2 H sC2 .Rn /, Au D  u C k 2 u D 0 in H s .Rn / H) .4 2 kk2 C k 2 /uO D 0 in S 0 .Rn / H) uO D 0 in S 0 .Rn /, since .4 2 kk2 C k 2 / ¤ 0 8 2 Rn H) FN uO D 0 H) u D 0 in S 0 .Rn / H) u D 0 in H sC2 .Rn /. A W H sC2 .Rn / ! H s .Rn / is surjective: Let f 2 H s .Rn /. Then f 2 S 0 .Rn / H) fO 2 S 0 .Rn / H) .4 2 kk2 C k 2 /1 fO 2 S 0 .Rn /. Set v D .4 2 kk2 C k 2 /1 fO. Then v 2 S 0 .Rn / H) fO D .4 2 kk2 C k 2 /v 2 S 0 .Rn / H) f D FN fO D FN Œ.4 2 kk2 C k 2 /v D . C k 2 /FN v. Define u D FN v 2 S 0 .Rn /. Then . C k 2 /u D f 2 H s .Rn /. But F Œ. C k 2 /u D .4 2 kk2 C k 2 /uO D fO H)

O D .1 C kk2 /s=2C1 juj 

.1 C kk2 /s=2C1 O jf j .4 2 kk2 C k 2 /

.1 C kk2 /  .1 C kk2 /s=2 jfOj  C1 .1 C kk2 /s=2 jfOj; .4 2 kk2 C k 2 /

since .4 2 kk2 C k 2 /  min¹4 2 ; k 2 º.1 C kk2 / 1 1 1 H) 4 2 kk 2 Ck 2  C1 .1Ckk2 / with C1 D min¹4 2 ;k 2 º > 0. Hence, Z Z 2 sC2 2 2 .1 C kk / juj O d   C1 .1 C kk2 /s jfOj2 d  < C1 Rn

Rn

H) u 2 H sC2 .Rn /. Thus, 8f 2 H s .Rn /, 9u 2 H sC2 .Rn / such that Au D  u C k 2 u D f in H s .Rn / H)  C k 2 is surjective from H sC2 .Rn / onto H s .Rn /. Hence, A D  C k 2 is bijective from H sC2 .Rn / onto H s .Rn /. But H sC2 .Rn / and H s .Rn / are Banach spaces, and A D  Ck 2 is a linear, bijective mapping from Banach space H sC2 .Rn / onto Banach space H s .Rn /. Then, by Corollary A.8.1.1 of the Open Mapping Theorem A.8.1.3 in Appendix A, the continuity of the inverse follows. Hence,  C k 2 is an isomorphism from H sC2 .Rn / onto H s .Rn /. Now we state two important lemmas which will be needed later. Lemma 8.9.1 ([40]). For 1  j˛j  m, the following inequality holds: 8 2 Rn , 9C D C.m/ > 0 such that n Y iD1

  m n X i2˛i  1 C i2 C 1C iD1

X 1j˛jm

Y n iD1

i2˛i

 :

506

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof. 0  ˛1 C ˛2 C    C ˛n  m. Set 1 C 12 C    C n2 D P  1. Hence, P m  P ˛1 P ˛2    P ˛n . But P  i2 8i D 1; 2; : : : ; n H) P ˛i  i2˛i 8i D 1; 2; : : : ; n  m n X H) 12˛1 22˛2    n2˛n  P ˛1 P ˛2    P ˛n  P m D 1 C i2 iD1

H)

n Y



i2˛i  1 C

iD1

n X

i2

m :

iD1

For the proof of the second inequality, we apply binomial expansion. In fact, .1 C 12 C    C n2 /m D

ˇ ˇX ˇ1 m X X



ˇ ˇ1 ˇ X n2

ˇ D0 ˇ1 D0 ˇ2 D0

2ˇ1

Cˇ;ˇ1 ;:::ˇn1 1

2ˇ

n1 2.ˇ ˇ1 ˇn1 /    n1 n ;

ˇn1 D0

where Cˇ;ˇ1 ;:::;ˇn1 are constants with C0;0;:::;0 D 1, ˇ1 C ˇ2 C    C ˇn1 C .ˇ  ˇ1  ˇ2      ˇn1 / D ˇ  m   m X 12˛1 22˛2    n2˛n H) .1 C 12 C    C n2 /m  C 1 C j˛jD1

with C  max¹Cˇ;ˇ1 ;:::;ˇn1 º > 0. Lemma 8.9.2. For 1  j˛j  m, the following inequality holds: 8 2 Rn , 9C D C.m/ > 0 such that   m  n n n X Y Y X 2˛i 2˛i 2   1C i C 1C i : (8.9.6) iD1

iD1

j˛jDm

iD1

Proof. For the proof of the first inequality, see the proof of Lemma 8.9.1. For the proof of the second inequality, set  m   n n X Y X ı 2˛i 2 f ./ D 1 C > 0 8 2 Rn : 1C i i iD1

j˛jDm

iD1

Moreover, f is continuous in Rn . Hence, in every compact subset of Rn ; f is Pn 2 2 bounded. Set r D iD1 i . Then, .1 C r 2 /m ; Q 1 C j˛jDm . niD1 i2˛i / P in which the denominator behaves as 1 C r 2m D 1 C . niD1 ji j2 /m . Hence, 9r0 > 0 such that 8r  r0 , f ./  C1 and for 0  r  r0 , 9C2 > 0 such that f ./  C2 . Then f ./  C D max¹C1 ; C2 º H) the result (8.9.6). f ./ D

P

507

Section 8.9 Applications

8.9.3 Sobolev spaces H m .Rn / of integral order m on Rn H m .Rn / with m 2 N can be defined in two different ways: 1. One based on Definition 2.15.1 in (2.15.1)–(2.15.3), which we agree to denote temporarily by a different notation H m .Rn / (instead of the natural one H m .Rn / with  D Rn ): H m .Rn / D ¹u W u 2 L2 .Rn /; @˛ u 2 L2 .Rn / 8j˛j  mº:

(8.9.7)

2. The other one based on Fourier transforms given by Definition 8.9.1 with (8.9.1)–(8.9.3) or by (8.9.5), which we will denote by H m .Rn / with s D m. Then we will show that H m .Rn /  H m .Rn /. In fact, we have: Theorem 8.9.2. For s D m 2 N, H m .Rn / defined by (8.9.1)/ (8.9.5): H m .Rn / D ¹u W u 2 L2 .Rn /; .1 C kk2 /m=2 uO 2 L2 .Rn / with uO D F uº (8.9.8) coincides with the space H m .Rn / defined by (8.9.7), i.e. H m .Rn /  H m .Rn /; where derivatives @˛ u are in the distributional sense: 8 2 D.Rn /, Z Z ˛ j˛j @ u.x/.x/d x D .1/ u.x/@˛ .x/d x: Rn

Rn

Then 

X

kukH .Rn / D

0j˛jm

1=2 j@ u.x/j d x

Z

˛

2

(see (2.15.10))

(8.9.9)

(see (8.9.3))

(8.9.10)

Rn

and Z kukm;Rn D

2 m

2

1=2

.1 C kk / ju./j O d Rn

are equivalent norms in H m .Rn /, i.e. 9C1 ; C2 > 0 such that C1 kukH m .Rn /  kukm;Rn  C2 kukH m .Rn /

8u 2 H m .Rn /:

(8.9.11)

Proof. H m .Rn /  H m .Rn /: Let u 2 H m .Rn /; we will show that u 2 H m .Rn /. m 2  .1 C kk2 /m  Then u 2 H m .Rn / H) .1 C kk2 / 2 uO 2 L2 .Rn /. Hence, ju./j O 2 n ju./j O a.e. in R Z Z 2 H) ju.j O 2d   .1 C kk2 /m ju./j O d  < C1 Rn

H) uO 2 L2 .Rn /.

Rn

508

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Using Lemma 8.9.1, we have, 8 multi-index ˛, Y 2 n 2 2 j ˛ u./j O D i2˛i ju./j O  .1 C kk2 /m ju./j O X

H)

a.e. in Rn

iD1 2 2 Œ12˛1 22˛2    n2˛n ju./j O  CN .1 C kk2 /m ju./j O

a.e. in Rn

0j˛jm

with CN D CN .m/ > 0 Z X Z 2 H) 12˛1 22˛2    n2˛n ju./j O d   CN 0j˛jm

Rn

2 .1 C kk2 /m ju./j O d

Rn

H)  ˛ uO 2 L2 .Rn / 8 multi-index ˛ with j˛j  m H)  ˛ uO 2 S 0 .Rn / 8j˛j < m. ˛ O D .i 2/j˛j  ˛ u O H) F .@˛ But from Theorem 8.6.1, F .@˛ x u/ D .i 2/ u x u/ D ˛ j˛j 2 n .i 2/  uO 2 L .R / 8 multi-index ˛ with j˛j  m H) by the Plancherel–Riesz 2 n Theorem 8.3.1, @˛ x u 2 L .R / with 2 ˛ 2 j˛j ˛ 2 O L k@˛ 2 .Rn / x ukL2 .Rn / D kF .@x u/kL2 .Rn / D k.i 2/  uk 2 D .2/2j˛j k ˛ uk O L 2 .Rn /

H)

kuk2H m .Rn / D

8j˛j  m X 2 k@˛ x ukL2 .Rn /

0j˛jm

D

X

2j˛j

.2/

0j˛jm

Z

 C0 D

Z Rn

2 12˛1 22˛2    n2˛n ju./j O d

2 .1 C kk2 /m ju./j O d

Rn C0 kuk2m;Rn

< C1 with C0 D .2/2m CN > 0

H) u 2 H m .Rn / with kuk2H m .Rn /  C0 kuk2m;Rn H) u 2 H m .Rn / with p kukH .Rn /  C0 kukm;Rn with C0 D C 0 > 0. Hence, u 2 H m .Rn / H) u 2 H m .Rn / H)

H m .Rn /  H m .Rn /

with C1 kukH m .Rn /  kukm;Rn with C1 D

1 > 0: C0 (8.9.12)

H m .Rn /  H m .Rn /: Let u 2 H m .Rn /; we will show that u 2 H m .Rn /. Then u 2 H m .Rn / H) u 2 L2 .Rn /, @˛ u 2 L2 .Rn / 8 multi-index ˛ with j˛j  m. Using Lemma 8.9.1 again, we get   X 2 2 .1 C kk2 /m ju.j O 2  C ju./j O C j ˛ u./j O ; (8.9.13) 1j˛jm

509

Section 8.9 Applications

Q where . ˛ /2 D 12˛1 22˛2    n2˛n D niD1 i2˛i . But from Theorem 8.6.1, F .@˛ x u/ D ˛ 2 n .i 2/ uO 2 L .R / 8j˛j  m and, by the Plancherel–Riesz Theorem 8.3.1, ˛ j˛j ˛ k@˛ O L2 .Rn / x ukL2 .Rn / D kF .@x u/kL2 .Rn / D .2/ k uk

H) H)

8j˛j  m

1 k@˛ ukL2 .Rn / 8j˛j  m .2/j˛j x Z Z 1 2 2  2˛ ju./j O d D j@˛ x u.x/j d x 8j˛j  m: 2j˛j n n .2/ R R

k ˛ uk O L2 .Rn / D

Hence, from (8.9.13) and (8.9.14), Z Z 2 m 2 Q .1 C kk / ju./j O d  C Rn

ju.x/j d x C Rn

1j˛jm

D CQ kuk2H m .Rn / < C1;

 j@ u.x/j d x

Z

X

2

(8.9.14)

˛

2

Rn

CQ > 0

H) u 2 H m .Rn / with kuk2m;Rn  CQ kuk2H m .Rn / . Then, u 2 H m .Rn / p H) u 2 H m .Rn / with kukm;Rn  C2 kukH m .Rn / with C2 D CQ > 0 H)

H m .Rn /  H m .Rn / with kukm;Rn  C2 kukH m .Rn / :

(8.9.15)

Combining (8.9.12) and (8.9.15), we have H m .Rn /  H m .Rn / and H m .Rn /  H m .Rn / H) H m .Rn /  H m .Rn / with C1 kukH m .Rn /  kukm;Rn  C2 kukH m .Rn / . Theorem 8.9.3. For non-integer s > 0 with s D Œs C , Œs 2 N0 being the integral part of s,  2 0; 1Œ being the fractional part, H s .Rn / defined by (8.9.1) coincides with the space defined by ¹u W u 2 H Œs .Rn /; @˛ u 2 H  .Rn / 8j˛j D Œsº (see also (8.10.71)):

(8.9.16)

Then the mapping s

n

u 2 H .R / 7!



kuk2Œs;Rn

C

X

k@

˛

uk2;Rn

 12 (8.9.17)

j˛jDŒs

defines a norm in H s .Rn / equivalent to the original norm k  ks;Rn , i.e. 9C1 ; C2 > 0 such that, 8u 2 H s .Rn /,   12 X 2 ˛ 2 n C1 kuks;R  kukŒs;Rn C k@ uk;Rn  C2 kuks;Rn ; (8.9.18) j˛jDŒs

where k  ks;Rn , k  kŒs;Rn and k  k;Rn denote norms in H s .Rn /, H Œs .Rn / and H  .Rn /, respectively. k  kŒs;.Rn / and k  k;.Rn / are also defined by (8.9.3) with ‘s’ replaced by ‘Œs’ and ‘’, respectively.

510

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof. (H)): Let u 2 H s .Rn /. Since 0  Œs < s, H s .Rn / ,! H Œs .Rn / and kukŒs;Rn  kuks;Rn by Proposition 8.9.1. It remains to show that @˛ u 2 H  .Rn / 8j˛j D Œs. From (8.6.1), F .@˛ u/ D @˛ u D .i 2/˛ uO 8j˛j D Œs. H s .Rn / ,! S 0 .Rn / H) u 2 S 0 .Rn / H) @˛ u 2 S 0 .Rn / and

b

b

2 .1 C kk2 / j@˛ uj2 D .1 C kk2 / j.i 2/˛ j2 ju./j O 2  .2/2j˛j .1 C kk2 /ŒsC ju./j O ;

Q since  2˛ D niD1 i2˛i  .1 C kk2 /j˛j with j˛j D Œs by Lemma 8.9.1. Hence, 8j˛j D Œs, s D Œs C , Z Z 2  ˛ 2 2Œs 2 .1 C kk / j@ uj d   .2/ .1 C kk2 /s ju./j O d ;

b

Rn

Rn

H)

2 2Œs 2 k@˛ ukH kukH  .Rn /  .2/ s .Rn /

H)

k@˛ uk;Rn  .2/Œs kuks;Rn

8j˛j D Œs:

(8.9.19)

Hence, 8j˛j D Œs, @˛ u 2 H  .Rn /. Thus, we have proved that u 2 H s .Rn / H) u 2 H Œs .Rn /, @˛ u 2 H  .Rn / 8j˛j D Œs with     X X 2 ˛ 2 2 2j˛j 2 k@ uk;Rn  kuks;Rn C .2/ kuks;Rn kukŒs;Rn C j˛jDŒs

j˛jDŒs

  X 2 2j˛j  kuks;Rn 1 C .2/ D C22 kuk2s;Rn j˛jDŒs

H)

  12 X kuk2Œs;Rn C k@˛ uk2;Rn  C2 kuks;Rn j˛jDŒs

P 1 with C2 D .1 C j˛jDŒs .2/2j˛j / 2 > 0, i.e. the second inequality in (8.9.18) holds. ((H): Let u 2 H Œs .Rn / with @˛ u 2 H  .Rn / 8j˛j D Œs. We are to show that u 2 H s .Rn /, satisfying the first inequality in (8.9.18). u 2 H Œs .Rn / H) u 2 S 0 .Rn / and .1 C kk2 /Œs=2 uO 2 L2 .Rn /. By Proposition 8.9.1, H Œs .Rn / ,! L2 .Rn / 8Œs  0. Then u 2 H Œs .Rn / H) u 2 L2 .Rn / H) uO 2 L2 .Rn / by the Plancherel– O 2 D .1 C kk2 / ..1 C kk2 /Œs /juj O 2. Riesz Theorem 8.3.1. Besides, .1 C kk2 /s juj Using the second inequality in (8.9.6):   n X Y .1 C kk2 /Œs  C 1 C ji j2˛i : j˛jDŒs

iD1

511

Section 8.9 Applications

b

But from (8.6.1), .i 2/˛ uO D @˛ u and, by assumption, @˛ u 2 H  .Rn / 8j˛j D Œs. Hence,   n X Y .1 C kk2 /Œs juj O2 O 2 C 1C ji j2˛i juj j˛jDŒs

iD1

  X 2 ˛ 2 D C juj O C j uj O : j˛jDŒs

b

b

˛ 2 1 ˛ j.2/j˛j  ˛ uj O D j@˛ uj H) j ˛ uj O D .2/ O D j˛j j@ uj H) j uj P P ˛ 2 1 ˛ 2 O D j˛jDŒs .2/2j˛j j j@ uj j˛jDŒs j uj

H)

b

b

 X .1 C kk / juj O  .1 C kk / C juj O 2C 2 s

2

2 

j˛jDŒs

j@˛ uj2 .2/2j˛j



˛ uj2 j@c .2/2j˛j

H)

a.e. in Rn

b

  X 2 2  2 2  ˛ 2  C0 .1 C kk / juj O C .1 C kk / j@ uj j˛jDŒs

Z H)

.1 C kk2 /s juj O 2d  Z X Z  C02 .1 C kk2 / juj O 2d  C Rn

Rn

H)

s

j˛jDŒs

n

u 2 H .R /

with

kuk2s;Rn



C02



b

.1 C kk2 / j@˛ uj2 d 

Rn

kuk2;Rn

C

X

k@

˛

uk2;Rn





j˛jDŒs

H)

kuk

s;Rn

 1=2 X 2 ˛ 2  C0 kuk;Rn C k@ uk;Rn :

(8.9.20)

j˛jDŒs

For Œs D 0, we get the trivial case, since Œs D j˛j D 0 and s D  2 0; 1Œ, and consequently H s .Rn /  H  .Rn /, and hence u 2 H  .Rn / ,! H Œs .Rn /  L2 .Rn /:

(8.9.21)

For Œs  1,  2 0; 1Œ, H Œs .Rn / ,! H  .Rn /;

with kuk;Rn  kukŒs;Rn

by (8.9.4):

(8.9.22)

Hence, from (8.9.20) and (8.9.22), we get the first inequality in (8.9.18): for Œs  1,   12 X 2 ˛ 2 n kuks;R  C0 kukŒs;Rn C j k@ uk;Rn : (8.9.23) j˛jDŒs

Finally, from (8.9.19)–(8.9.23), the result (8.9.18) follows.

512

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8.9.4 Sobolev’s Imbedding Theorem (see also imbedding results in Section 8.12) Now we will identify conditions under which functions of H s .Rn / will be continuous in Rn , or will have continuous derivatives in the usual pointwise sense in Rn . For this we will need: Pn

2 1=2 Lemma 8.9.3. Let r D r./ D , ˆ D ˆ.r/  0 8r 2 0; 1Œ and iD1 i f D f ./ D ˆ.r.//  08 2 Rn be integrable on Rn . Then Z 1 Z f ./d  D nVn ˆ.r/r n1 dr; (8.9.24) Rn

0

where Vn is the volume of an n-dimensional unit sphere in Rn [7, page 121]. Theorem 8.9.4 (Sobolev’s Imbedding Theorem). If s 

n 2

> k with k 2 N0 , then

H s .Rn / ,! C k .Rn /;

(8.9.25)

i.e. u 2 H s .Rn / H) u 2 C k .Rn / with kukC k .Rn /  C kukH s .Rn / for some C > 0 and limkxk!1 j@˛ u.x/j D 0 8j˛j  k, where C k .Rn / is equipped with the notion of uniform convergence on every compact subset of Rn for all derivatives of order j˛j  k and kukC k .Rn / D max0j˛jk supx2Rn j@˛ u.x/j. Proof. Let u 2 H s .Rn /. Then, by the definition of H s .Rn /, u 2 S 0 .Rn / H) @˛ u 2 S 0 .Rn /. We are to show that for s  n2 > k, @˛ u 2 C 0 .Rn / 8j˛j  k. For this, if F Œ@˛ u 2 L1 .Rn / 8j˛j  k, we can apply (7.1.7) and Property 6 in (7.1.24) of Fourier transforms, which also hold for the Fourier co-transform FN , of functions of L1 .Rn /, i.e. w˛ D @˛ u D F Œ@˛ u 2 L1 .Rn /  S 0 .Rn /,

b

H)

FN w˛ D FN F Œ@˛ u D @˛ u 2 C 0 .Rn /

b

(8.9.26)

with k@˛ ukC 0 .Rn / D supx2Rn j@˛ u.x/j D k@˛ uk1  k@˛ ukL1 .Rn / 8j˛j  k (since FN F Œ@˛ u D @˛ u 2 S 0 .Rn // by the Fourier Inversion Theorem 8.3.2 on S 0 .Rn // along with the Riemann–Lebesgue Property 11 in (7.1.36): lim FN w˛ D

kxk!1

lim @˛ u.x/ D 0

kxk!1

b

8j˛j  k:

(8.9.27)

Hence, it is sufficient to show that @˛ u 2 L1 .Rn / 8j˛j  k. Then, (8.9.26) and (8.9.27) will follow. In fact, by Theorem 8.6.1, @˛ u D F Œ@˛ u D .i 2/˛ uO 2 S 0 .Rn /, where

b

.i 2/˛ D .i 21 /˛1 .i 22 /˛2    .i 2n /˛n D .i /j˛j .2/j˛j 1˛1 2˛2    n˛n D .i /j˛j .2/j˛j  ˛ :

513

Section 8.9 Applications

b

Hence, to prove that @˛ u 2 L1 .Rn / 8j˛j  k, we are to show that Z Z Z j@˛ u./jd  D j.i 2/˛ u./jd O  D .2/j˛j j ˛ u./jd O  n n n R R R Z j˛j D .2/ j ˛ j  ju./jd O  < C1: (8.9.28)

2

Rn

But j ˛ j D j1 j˛1  j2 j˛2    jn j˛n  kk˛1 kk˛2    kk˛n D kkj˛j and s

s

j ˛ ku.j O  kkj˛j .1 C kk2 / 2 .1 C kk2 / 2 ju./j O : „ ƒ‚ …„ ƒ‚ … f ./

g./

For u 2 H s .Rn /, g./ D .1 C kk2 /s=2 uO 2 L2 .Rn /. Now, if f ./ 2 L2 .Rn /, then, by Hölder’s inequality, their product f ./g./ 2 L1 .Rn /, and consequently  ˛ uO 2 L1 .Rn / and we have, 8j˛j  k, Z Z jj˛ ju./jd O  kkj˛j .1 C kk2 /s=2 .1 C kk2 /s=2 ju./jd O  Rn

Rn

Z 

kk

2j˛j

 12  Z

2 s

2 s

.1 C kk / d 

2

 12

.1 C kk / ju./j O d

Rn

< C1;

Rn

(8.9.29)

b

and finally, from (8.9.28) and (8.9.29), @˛ u 2 L1 .Rn / with k@buk ˛

Z

L1 .Rn /



kk

2j˛j

2 s

.1 C kk / d 

Rn

 12 kuks;Rn :

Now the whole proof reduces to the proof that f ./ D kkj˛j .1 C kk2 /s=2 2 For this we apply Lemma 8.9.3 with r D kk to show that Z Z 1 2j˛j 2 s kk .1 C kk / d  D nVn .1 C r 2 /s r 2j˛j r n1 dr

L2 .Rn /.

Rn

Z

D nVn „0

0 1

2 s 2j˛jCn1

.1 C r / r ƒ‚ I1

Z

1

2 s 2j˛jCn1

dr C .1 C r / r … „1 ƒ‚ I2

 dr < C1; …

i.e. kkj˛j .1 C kk2 /s=2 2 L2 .Rn / ifRboth I1 < C1; I2 < C1. 1 For 0 < r < 1, .1Cr 2 /s  1, I1  0 r 2j˛jCn1 dr < C1 if .2j˛jCn1/C1 > 0 H) 2j˛j C n > 0, which holds 8j˛j.

514

Chapter 8 Fourier transforms of distributions and Sobolev spaces

For R1 < r < 1, .1 C r 2 /  r 2 H) .1 C r 2 /s  r 2s H) 1 I2  1 r 2sC2j˛jCn1 dr < C1, if .2s C 2j˛j C n  1/ C 1 < 0 H) 2s C 2j˛j C n < 0 H) 2s  n > 2j˛j H) s  n2 > j˛j 8j˛j  k. Hence, kkj˛j .1 C kk2 /s=2 2 L2 .Rn / if s  n2 > j˛j with j˛j  k H) @˛ u 2 L1 .Rn / if s  n2 > j˛j with j˛j  k H) from (8.9.26), @˛ u 2 C 0 .Rn / with k@˛ ukC 0 .Rn /  k@˛ ukL1 .Rn / 8j˛j  k. Hence u 2 C k .Rn / and, using (8.9.28) and (8.9.29),

b

b

b

kukC k .Rn / D max k@˛ uk1  max k@˛ ukL1 .Rn /  C kukH s .Rn / ; 0j˛jk

0j˛jk

and from (8.9.27) lim @˛ u.x/ D 0

8j˛j  k:

kxk!1

Remark 8.9.2. At this point some explanations on the results of the imbedding theorem are in order. Elements of H s .Rn / are equivalence classes of functions defined almost everywhere in Rn , i.e. functions of an equivalence class may differ on a set of points with n-dimensional Lebesgue measure equal to zero. Then, by virtue of the imbedding of type (8.9.25), we mean that every equivalence class Œu 2 H s .Rn / must contain a function u (i) belonging to C k .Rn /, C k .Rn / being the target space of imbedding, and (ii) bounded in C k .Rn / by C kuks;Rn . Hence, the imbedding H s .Rn / ,! C k .Rn / for s  n2 > k H) each u 2 H s .Rn / can be considered as a function, which can be redefined on a set of points with ndimensional Lebesgue measure equal to zero in Rn such that the modified function uQ thus obtained has the properties: uQ D u in H s .Rn /, uQ 2 C k .Rn /, kuk Q C k .Rn /  C kuk Q H s .Rn / . Example 8.9.2. 1. For n D 1, s D 1, H 1 .R/, s  H) H 1 .R/ ,! C 0 .R/;

n 2

D1

1 2

>0

i.e. functions of H 1 .R/ are continuous on R: (8.9.30)

2. For n D 2, s D 1, H 1 .R2 /, s

2 n D 1  D 0: 2 2

(8.9.31)

Hence, from Theorem 8.9.4, we cannot say anything. In fact, function u in Example 2.15.2 belongs to H 1 .R2 /, but there does not exist uQ 2 C 0 .R2 / such that uQ D u in H 1 .R2 / with kuk Q C 0 .R2 /  C kuk Q H 1 .R2 / (see Remark 8.9.2). Hence, functions of H 1 .R2 / are not continuous in general.

515

Section 8.9 Applications

3. For n D 2, s > 1, s  n2 D s  1 > 0 H) H s .R2 / ,! C 0 .R2 /, i.e. for s > 1, functions of H s .R2 / are continuous in R. In particular, for s D 2, H 2 .R2 / ,! C 0 .R2 /;

i.e. functions of H 2 .R2 / are continuous in R2 : (8.9.32)

Translation property of imbedding results in Sobolev spaces s

n 2

>k2N

H)

H s .Rn / ,! C k .Rn /

H)

H sCm .Rn / ,! C kCm .Rn / 8m 2 N: (8.9.33)

In fact, s  n2 > k H) .s C m/  n2 > k C m 8m 2 N. Hence, H s .Rn / ,! C k .Rn / H) H sCm .Rn / ,! C kCm .Rn /. Theorem 8.9.5. If s 

n 2

D  2 0; 1Œ, H s .Rn / ,! C 0; .Rn /, i.e.

u 2 H s .Rn /

H)

u 2 C 0; .Rn /;

(8.9.34)

u being a -Hölder continuous function in Rn , 8x; y 2 Rn , 9C > 0 such that ju.x C y/  u.x/j  C kyk ;

(8.9.35)

and 9CQ > 0 such that kukC 0; .Rn / D kukC 0 .Rn / C sup

x2Rn y¤0

ju.x C y/  u.x/j  CQ kukH s .Rn / : kyk

(8.9.36)

Proof. We give the scheme of the proof. u.x C y/ D .y u/.x/ H)

F Œy u D e i2hy;i F Œu D e i2hy;i uO .by.7:1:23//

H)

u.x C y/ D FN Œe i2hy;  i u. O  /.x/

H)

u.x C y/  u.x/ D FN Œ.e i2hy;  i  1/u. O  /.x/ ˇ ˇZ ˇ ˇ i2hy;i i2h;xi ˇ u./Œe O  1e d  ˇˇ ju.x C y/  u.x/j D ˇ

H)

Z

Rn

 Rn

je i2hy;i  1j kks ju./jd O : kks

(8.9.37)

516

Chapter 8 Fourier transforms of distributions and Sobolev spaces

kks  .1 C kk2 /s=2 H) kks ju./j O 2 L2 .Rn /, since u 2 H s .Rn /. Z

je i2hy;i  1j2 d   C12 kyk2sn kk2s

I D Rn

with C1 D C1 .s; n/ > 0.

Avoiding the long proof of the estimate, we show again the scheme of the proof (see also [42]). The integral I is invariant under rotation, since a rotation preserves the inner product h  ;  i and norm k  k in Rn , the Jacobian J of such a transformation being 1. Moreover, for y ¤ 0, 9 a rotation which transforms y into a vector collinear with eO 1 D .1; 0; : : : ; 0/. Hence, we choose y D .jhj; 0; : : : ; 0/ with kyk D jhj collinear 1 with the first basis vector eO 1 , and set  D jhj 2 Rn with D .1 ; 2 ; : : : ; n /, jJ j D

1 jhjn

and h ¤ 0. Then hy; i D jhj  Z

je i2 1  1j2

I D Rn

with C12 

R

1 k k2s jhj2s



1 jhj

C 0 C    C 0 D 1 , and

1 d  C12 jhj2sn D C12 kyk2sn ; jhjn

je i21 1j2 d (see Remark 8.9.3). k k2s 2 n 2 L .R /, which was shown earlier.

Rn

Hence,

je i2hy;i 1j kks

2 L2 .Rn /

and kks ju./j O H) by Hölder’s inequality, from (8.9.37), we have Z

je i2hy;i  1j kks ju./jd O  s n kk R  12  Z  12 Z je i2hy;i  1j2 2s 2 d kk ju./j O d  kk2s Rn Rn

ju.x C y/  u.x/j 

1

 C1 .kyk2sn / 2 kuks;Rn D C1 kyk  kuks;Rn ; R R 2d   2 s O 2 d  D kuk2 since Rn kk2s ju./j O Rn .1 C kk / ju./j s;Rn . Then, from n (8.9.37), for s  2 D  2 0; 1Œ and y ¤ 0, 9C1 > 0, independent of u, such that ju.xCy/u.x/j  C1 kuks;Rn 8x 2 Rn . Hence, u 2 C 0; .Rn / and for the semi-norm kyk j  j in C 0; .Rn /, jujC 0; .Rn / D sup

x2Rn y¤0

ju.x C y/  u.x/j  C1 kukH s .Rn / : kyk

But u 2 H s .Rn / with s  n2 D  > 0 H) u 2 C 0 .Rn / with kukC 0 .Rn /  C2 kukH s .Rn / . Then kukC 0; .Rn / D kukC 0 .Rn / C jujC 0;.Rn /  C kukH s .Rn / ; with C D C1 C C2 > 0.

517

Section 8.9 Applications

Remark 8.9.3. Although we have shown in the proof of Theorem 8.9.5 that Z je i2hy;i  1j2 d   C12 kyk2sn ; I D kk2s Rn in fact, we have: Lemma 8.9.4. For .s  n2 / 2 0; 1Œ, the following result holds: 9C D C.s; n/ > 0 such that Z je i2hy;i  1j2 I.y; 2s/ D d  D C kyk2sn : kk2s Rn Then, for s 2 0; 1Œ, “ Z ju.x C y/  u.x/j2 2 d xd y D C kk2s ju./j O d : nC2s n kyk R Rn Rn Proof. Assuming the first result, I.y; 2s/ D C kyk2sn , we prove the second equality. O  /.x/ From the proof of Theorem 8.9.5, u.x C y/  u.x/ D F Œ.e i2hy;  i  1/u. H) F Œu.x C y/  u.x/./ D .e i2hy;i  1/u./ O a.e. in Rn . By the Plancherel–Riesz Theorem 8.3.1, 8y 2 Rn , Z Z 2 ju.x C y/  u.x/j d x D jF Œu.x C y/  u.x/./j2 d  Rn Rn Z 2 D je i2hy;i  1j2 ju./j O d : Rn

The functions .x; y/ 2 Rn  Rn 7! i2hy;i

2

ju.xCy/u.x/j2 kyknC2s

 0 (a.e.), .; / 2 Rn 

2

1j ju./j O Rn 7! je  0 (a.e.) are integrable on Rn  Rn , and applying kyknC2s Fubini’s Theorem 7.1.2C, “ Z Z ju.x C y/  u.x/j2 dy d xd y D ju.x C y/  u.x/j2 d x nC2s kyknC2s Rn kyk Rn Rn Rn Z Z dy 2 je i2hy;i  1j2 ju./j O d D nC2s Rn kyk Rn Z Z je i2hy;i  1j2 2 ju./j O d d y: D kyknC2s Rn Rn

Set 2s1 D n C 2s > n. Then 0 < 2s1  n D 2s < 2, since s 2 0; 1Œ. Hence, for .s1  n2 / 2 0; 1Œ, from the first result of this lemma, Z Rn

je i2hy;i  1j2 d y D I.; 2s1 / D C kk2s1 n D C kk2s kyk2s1

518

Chapter 8 Fourier transforms of distributions and Sobolev spaces

and “ Rn Rn

ju.x C y/  u.x/j2 d xd y D C kyknC2s

Z

2 kk2s ju./j O d : Rn

Now we prove the first result, I.y; 2s/ D C kyk2sn . In fact, from the proof of Theorem 8.9.5, we have xi D jhj 2 Rn ; kyk D jhj; hy; i D 1 ; jJ j D jhj1n and Z I.y; 2s/ D

je i2hy;i  1j2 d  D jhj2sn kk2s Rn

Z

je i2 1  1j2 d D jhj2sn I.2s/: k k2s Rn „ ƒ‚ … I.2s/

Now we will show that for s  n2 D  2 0; 1Œ, I.2s/ < C1. Again, we change the variables: 1 D t1 ;

2 D t1 t2 ;

:::;

n D t1 tn ;

such that D .1 ; 2 ; : : : ; n /, ˇ ˇ1 ˇ ˇ ˇ t2 ˇ ˇ J. ; t/ D ˇt3 ˇ ˇ :: ˇ: ˇ ˇt n

ˇ 0 ˇˇ :: ˇˇ :ˇ :: ˇ D t n1 : ˇˇ 1 ˇ 0 ˇˇ 0    0 t1 ˇ

0   : t1 : : : 0 t1 : : :: : : : : : : :

and Z

je i2t1  1j2 Pn jt jn1 d t with d t D dt1    dtn 2s .1 C 2 /s 1 n jt j jt j 1 R iD2 i Z Z i2t 2 1 je  1j 1 Pn D dt1  dt2 : : : dtn D I1  I2 : 2snC1 n1 jt j .1 C iD2 jti j2 /s 1 „R ƒ‚ … „R ƒ‚ …

I.2s/ D

I1

I2

Hence, I.2s/ < C1 if I1 < C1 and I2 < C1. Convergence of I1 : 

at infinity: if .2s  n C 1/ C 1 < 0 H) 2s  n > 0 H) s  holds, since s  n2 D  2 0; 1Œ; i2t



n 2

> 0 which

at the origin: je i2t1  1j2 t12 (since e t11 1 ! finite limit) H) .2s  n C 12/C1 > 0 H) 2sCnC2 > 0 H) 2 > 2sn H) 1 > s n2 D  2 0; 1Œ.

Hence, for s 

n 2

D  2 0; 1Œ; I1 < C1.

519

Section 8.9 Applications

Convergence of I2 with n  2: I2 D

R

1 Rn1 .1C2 /s dt2 : : : dtn ,

for which we can R1 1 apply Lemma 8.9.3. Then I2 converges, if the corresponding integral 0 .1C 2 /s

n2 d converges at the origin and at infinity for n  2: 



1 n2  n2 H) at the origin: For 0 < < 1, 1 C 2  1 H) .1C 2 /s convergence for n  2 C 1 > 0, i.e. for n > 1, which always holds;

at infinity: For 1  < 1, .1 C 2 /s  2s H) the integral converges if .2s Cn2/C1 < 0 H) 2s Cn1 < 0 H) 2s Cn < 1 H) s  n2 >  12 , which always holds, since s  n2 D  2 0; 1Œ.

Hence, for s  n2 D  2 0; 1Œ, I2 < C1. Thus, I.y; 2s/ D C kyk2sn with C D C.s; n/ D I.2s/, since kyk D jhj. Alternative definition of H s .Rn / for s 2 0; 1Œ Theorem 8.9.5 allows us to give an alternative definition of H s .Rn / for s 2 0; 1Œ (see also Definition 8.10.6) instead of the general definition of H s .Rn / in (8.9.1) or (8.9.5). Definition 8.9.3. For 0 < s < 1, H s .Rn / is defined by the space ² ³ “ ju.x C y/  u.x/j2 s n 2 n d xd y < C1 H .R / D u W u 2 L .R /; kyknC2s Rn Rn (8.9.38) equipped with the norm given by  Z 2 kukL2 .Rn / C

Z Rn Rn

 12 ju.x C y/  u.x/j2 d xd y ; kyknC2s

(8.9.39)

which is equivalent to the original norm k  ks;Rn in (8.9.3), i.e. 9C1 ; C2 > 0 such that  “ 2 C C1 kuks;Rn  kukL 2 .Rn /

Justification of Definition 8.9.3

Rn Rn

 12 ju.x C y/  u.x/j2 d xd y  C2 kuks;Rn : kyknC2s (8.9.40)

For this we will need the following result:

Lemma 8.9.5. 8  0, 8s  0, 9˛1 ; ˛2 > 0 such that ˛1 

.1 C /s  ˛2 H) ˛1 .1 C s /  .1 C /s  ˛2 .1 C s /: 1 C s

Proof. The proof is similar to that of Lemma 8.9.2, with necessary modifications.

520

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Now we begin with the justification of Definition 8.9.3. Set D kk2 with 1 C s D 1 C kk2s , .1 C /s D .1 C kk2 /s . Then, using Lemma 8.9.5, ˛1 .1 C kk2s /  .1 C kk2 /s  ˛2 .1 C kk2s /. But for s  0, u 2 H s .Rn / H) u 2 L2 .Rn / ” uO 2 L2 .Rn / by the Plancherel–Riesz Theorem 8.3.1. Hence, we have 2 2 2 ˛1 .1 C kk2s /ju./j O  .1 C kk2 /s ju./j O  ˛2 .1 C kk2s /ju./j O a.e. in Rn  Z Z Z 2 2 2 H) ˛1 ju./j O d C kk2s ju./j O d  .1 C kk2 /s ju./j O d Rn

Z

 ˛2

Rn

2 ju./j O d C

Rn

Z

 2s 2 kk ju./j O d :

Rn

Rn

But from Lemma 8.9.4, 9C > 0 such that “ Z ju.x C y/  u.x/j2 1 2 kk2s ju./j O d D d xd y: C Rn Rn kyknC2s Rn Then,  Z Z ju.x C y/  u.x/j2 1 2 ˛1 ju./j O d C d xd y  kukH s .Rn / C Rn Rn kyknC2s Rn  Z Z Z ju.x C y/  u.x/j2 1 2  ˛2 ju./j O d C d xd y C Rn Rn kyknC2s Rn 1=2  Z Z ju.x C y/  u.x/j2 2 ˇ1 kukL C d xd y  kukH s .Rn / 2 .Rn / kyknC2s Rn Rn 1=2  Z Z ju.x C y/  u.x/j2 2  ˇ2 kukL2 .Rn / C d xd y ; kyknC2s Rn Rn Z

H)

2

where kukL2 .Rn / D kuk O L2 .Rn / by the Plancherel–Riesz Theorem 8.3.1, ˇ12 D min¹˛1 ; ˛C1 º > 0, ˇ22 D max¹˛2 ; ˛C2 º > 0. Hence,  Z 2 C1 kukH s .Rn /  kukL C 2 .Rn /

Z Rn Rn

 12 ju.x C y/  u.x/j2 d xd y kyknC2s

 C2 kukH s .Rn / ; with C1 D

1 , ˇ2

C2 D

1 ˇ1

> 0.

Alternative definition of H s .Rn / for s > 0 Instead of the definition of H s .Rn / for non-integral s D Œs C  > 0 given by Theorem 8.9.3, the following definition is a convenient one:

521

Section 8.9 Applications

Definition 8.9.4. For s > 0 with s D Œs C , Œs D k 2 N0 being the integral part of s,  2 0; 1Œ the fractional part, H s .Rn /  H Œs .Rn / defined by (8.9.16)–(8.9.18) can be defined alternatively by (see also Definition 8.10.7): H s .Rn / D ¹u W u 2 H Œs .Rn /; @˛ u 2 H  .Rn / 8j˛j  k; H  .Rn / is defined by (8.9.38)–(8.9.39)º: Then the mapping  X “ 2 u 2 H .R / 7! kukH Œs .Rn / C s

n

0j˛jk

 j@˛ u.x C y/@˛ u.x/j2 d xd y 1=2 kyknC2s Rn Rn

defines a norm equivalent to the original norm k  ks;Rn given by (8.9.3): 9C1 ; C2 > 0 such that C1 kuks;Rn

 2  kukH Œs .Rn / C

X “ 0j˛jk

Rn Rn

1=2 j@˛ u.x C y/  @˛ u.x/j2 d xd y kyknC2s

 C2 kuks;Rn :

(8.9.41)

8.9.5 Imbedding result: S.Rn / ,! H S .Rn / Proposition 8.9.2. 8s 2 R, S.Rn / ,! H s .Rn /;

(8.9.42)

the imbedding ,! being a continuous one from S.Rn / into H s .Rn /. Proof. S.Rn /  H s .Rn /: Let  2 S.Rn /  S 0 .Rn /. Then, from Theorem 7.6.1, O D s F  2 S.Rn /. But O 2 S.Rn / H) .1 C kk2 / 2 O 2 S.Rn / ,! L2 .Rn / by s Proposition 7.4.1 H) .1 C kk2 / 2 O 2 L2 .Rn / H)  2 H s .Rn / H) S.Rn /  H s .Rn /. Continuity of ,!: Let .m / be any sequence in S.Rn / such that m ! 0 in S.Rn / as m ! 1. Then, by Theorem 7.6.1, O m D F m ! 0 in S.Rn / as m ! 1 H) .1 C s s kk2 / 2 O m ! 0 in S.Rn / ,! L2 .Rn / by Proposition 7.4.1 H) .1 C kk2 / 2 O m ! s 0 in L2 .Rn / as m ! 1 H) km ks;Rn D k.1 C kk2 / 2 O m kL2 .Rn / ! 0 as m ! 1 H) ,!m D m 2 H s .Rn / and m ! 0 in H s .Rn / as m ! 1 H) ,! W S.Rn / ! H s .Rn / is continuous.

522

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8.9.6 Density results H S .Rn / Theorem 8.9.6. I. 8s 2 R, S.Rn / is dense in H s .Rn /;

(8.9.43)

II. 8s 2 R, D.Rn / is dense in H s .Rn /.

(8.9.44)

III.

D.Rn /

,!

S.Rn /

,!

H s .Rn /

with dense, continuous imbeddings.

(8.9.45)

Proof. I. Let u 2 H s .Rn /. Then, from (8.9.1), u 2 S 0 .Rn / and .1 C kk2 /s=2 uO 2 L2 .Rn /. Since D.Rn / is dense in L2 .Rn / by (1.2.26), 9 a sequence .k / in D.Rn / such that k ! .1 C kk2 /s=2 uO in L2 .Rn / as k ! 1. Define k by O k ./ D k ./ n n s 8k 2 N, 8 2 R , which is well defined, since k 2 D.R / H) 2 .1Ckk / 2 k s

.1Ckk2 / 2

H)

2 D.Rn /  S.Rn / as

1 s .1Ckk2 / 2

O k 2 S.Rn /

FN ΠO k  D

H)

2 C 1 .Rn /, k

2 S.Rn / 8k 2 N

(8.9.46)

by Theorem 7.7.1. s But O k 2 S.Rn / H) .1 C kk2 /s=2 Ok 2 S.Rn / H) .1 C kk2 / 2 O k 2 L2 .Rn /, since S.Rn / ,! L2 .Rn / by (7.4.1). Hence, k 2 H s .Rn / 8k 2 N. s s Again, by construction, .1 C kk2 / 2 O k D k ! .1 C kk2 / 2 uO in L2 .Rn / as k ! 1 H) k ! u in H s .Rn / as k ! 1 Hence, 8u 2 H s .Rn /, 9. k / in S.Rn / such that k ! u in H s .Rn /, i.e. S.Rn / is dense in H s .Rn /. In other words, by virtue of (8.9.42), S.Rn / ,! H s .Rn /

with a dense, continuous imbedding.

(8.9.47)

II. From Propositions 7.4.1 and 7.5.1, D.Rn / ,! S.Rn / with dense, continuous imbeddings, which together with (8.9.42) gives the continuous imbeddings in D.Rn / ,! S.Rn / ,! H s .Rn / in (8.9.47). Since S.Rn / is dense in H s .Rn / for s 2 R, for u 2 H s .Rn /, 8" > 0, 9 2 S.Rn / such that ku 

" ks;Rn < : 2

(8.9.48)

By Proposition 7.5.1, D.Rn / is a dense subspace of S.Rn /. Hence, for 2 S.Rn / satisfying (8.9.48), 9 a sequence .m / in D.Rn / such that m ! in S.Rn / as m ! 1, which in turn implies that m ! in H s .Rn / by virtue of the continuous imbedding S.Rn / ,! H s .Rn / (by Proposition 8.9.2), i.e. 8" > 0, 9m0 ."/ 2 N such that k

 m ks;Rn
0, 9m0 ."/ 2 N such that ku  m ks;Rn  ku  i.e.

ks;Rn C k

 m ks;Rn
0, %" .x/ D "1n %. x" / with %" 2 D.Rn /, %" .x/  0 8x 2 Rn , Rn %" .x/d x D 1, volume measure .supp.%" // ! 0 as " ! 0C . Then uQ 2 L2 .Rn / H) %" uQ 2 D.Rn /, since %" 2 D.Rn / 8" > 0 (by Corollary 6.2.1) H) .%" u/# Q Rn 2 D.RnC / C n m Q Rn º">0 in 8" > 0. Thus, 8u 2 H .RC / we have constructed a family ¹.%" u/# C

D.RnC /. (8.10.33) n Hence, the proof will be complete if we can show that .%" u/# Q R ! u in H m .RnC / as " ! 0C .

C

Section 8.10 Sobolev spaces on  ¤ Rn revisited

555

From (8.10.30), uQ 2 L2 .Rn /  D 0 .Rn / H) @˛ uQ 2 D 0 .Rn / 8j˛j 2 N, and u 2 H m .RnC / H) @˛ u 2 L2 .RnC / 8j˛j  m H) its null extension @˛ u 2 L2 .Rn / 

e

e

D 0 .Rn / 8j˛j  m, i.e. @˛ u D @˛ u in RnC and D 0 in Rn n RnC . Hence, their

e

difference @˛ uQ  @˛ u will be a distribution T˛ 8j˛j  m, i.e.

e

T˛ D @˛ uQ  @˛ u 2 D 0 .Rn /:

(8.10.34)

supp.T˛ /  Rn0 : For this, it suffices to show that T #.Rn /{ D 0, (.Rn0 /{ D the 0

complement of Rn0 in Rn ). In fact, 8 2 D..Rn0 /{ /,

e

hT˛ ; i D h@˛ uQ  @˛ u; iD 0 ..Rn /{ /D..Rn /{ /

e

0

0

˛

D h@ u; Q iD 0 ..Rn /{ /D..Rn /{ /  h@˛ u; i 0 0 Z j˛j ˛ D .1/ hu; Q @ iD 0 ..Rn /{ /D..Rn /{ /  0

j˛j

D .1/

Z

Z

˛

Rn C

0

u@ d x 

Rn C

(8.10.35) Rn C

@˛ ud x D 0

@˛ ud x

(8.10.36)

8 2 D..Rn0 /{ /:

(8.10.37) R Since u 2 H m .RnC / H) 8j˛j  m, Rn @˛ ud x D .1/j˛j Rn u@˛ d x 8 2 C C R R D.RnC /, 8j˛j  m, .1/j˛j Rn u@˛ d x  Rn @˛ ud x D 0 8 2 D..Rn0 /{ / with R C R C ˛ Q d x  Rn @˛ ud x D 0 8 2 D..Rn0 /{ / with supp./  RnC and .1/j˛j Rn u@   supp./  Rn . (8.10.38) n { Hence, from (8.10.37) and (8.10.38), hT˛ ; i D 0 8 2 D..R0 / / with support in n RC or Rn 8j˛j  m, i.e. 8 2 D.RnC [ Rn / H) T˛ D 0 in D 0 ..Rn0 /{ / by (5.1.1) (see Figure 8.3) R

e

H)

T˛ #.Rn /{ D 0 0

H)

supp.T˛ /  ..Rn0 /{ /{  Rn0 8j˛j  m: (8.10.39)

8" > 0, %" 2 E 0 .Rn / is a distribution with compact support in Rn H) 8" > 0, T˛ %" is well defined and T˛ %" 2 C 1 .Rn /  D 0 .Rn / 8j˛j  m by Theorem 6.4.1. Moreover, 8 sufficiently small " > 0, .T˛ %" /#Rn D 0 C

in D 0 .RnC /:

(8.10.40)

Since supp.%/  Rn by (8.10.31), % can be chosen such that supp.%" /  Rn for all sufficiently small " > 0. Then, by Theorem 6.2.2, 8j˛j  m, supp.T˛ %" /  supp.T˛ / C supp.%" /  Rn0 C Rn  Rn

556

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Figure 8.3 T˛ D 0 in D 0 ..Rn0 /{ /

for all sufficiently small " > 0, since supp.T˛ /  Rn0 and supp.%" /  Rn for all sufficiently small " > 0. Hence, for all sufficiently small " > 0, T˛ %" D 0 in .Rn /{  RnC H) .T˛ %" /#Rn D 0 in D 0 .RnC /. C An alternative proof of (8.10.40) is as follows: In fact, 8 2 D.Rn / with supp./  RnC , by the definition of convolution of distributions in (6.3.14), we have, 8" > 0, 8j˛j  m, hT˛ %" ; iD 0 .Rn /D.Rn / D hT˛ .x/; h%" .y/; .x C y/ii D hT˛ .x/;

" .x/iD 0 .Rn /D.Rn / ;

where Z 0 n n .x/ D h% .y/; .x C y/i D %" .y/.x C y/d y " " D .R /D.R / Rn Z %" .y/.x C y/d y: D supp.%" /\supp..xCy//

Since supp.%/  Rn by the definition of % in (8.10.32) and supp./  RnC , 8y 2 supp.%" /, .x C y/ D 0 for x C y D .x1 C y1 ; : : : ; xn C yn / … supp./, i.e. for x C y with xn C yn  0. Then 9ı > 0, dependent on the choice of the support of % satisfying (8.10.32) and also on the choice of , such that 8" > 0, Z .x/ D %" .y/.x C y/d y D 0 8x 2 Rn with jxn j  ı: " supp.%" /\supp..xCy//

(8.10.41)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

557

xn

y(x) = 0

d O

y(x) = 0

"/

Figure 8.4 supp.

n-1

^x 0 ú

d

 the complement of ¹x W x 2 Rn with jxn j  ıº

Hence, 8" > 0, " 2 D.Rn / with supp. " /  complement of ¹x W x 2 Rn with jxn j  ıº (see Figure 8.4), hT˛ ; " iD 0 .Rn /D.Rn / D 0, since supp.T˛ /  Rn0 (by (8.10.39)) and " .x/ D 0 8x 2 Rn0 (by (8.10.41)) H) 8" > 0, hT˛ %" ; iD 0 .Rn /D.Rn / D hT˛ ; " iD 0 .Rn /D.Rn / D 0 8 2 D.Rn / with supp./  RnC H) 8" > 0, 8j˛j  m, h.T˛ %" /#Rn ; iD 0 .Rn /D.Rn / D 0 C 8 2 D.RnC / (see (5.1.1)), H) 8 sufficiently small " > 0, 8j˛j  m, .T˛ %" /#Rn D 0 in D 0 .RnC /. C

e

˛ .@˛ Q Rn D .%" @˛ x .%" u//# y u/#Rn with @x D C

C

In fact, we have @˛ x .%"

Z

@j˛j : @y1 ˛1 :::@yn ˛n

Z

%" .x  y/u.y/d Q yD @˛ Q y x Œ%" .x  y/u.y/d Rn Z @˛ Q y D .1/j˛j y Œ%" .x  y/u.y/d Rn Z D .1/2j˛j %" .x  y/@˛ u.y/d Q y D .%" @˛ Q y u/.x/;

u/.x/ Q D

@˛ x

@j˛j , @˛ D @x1 ˛1 : : : @xn ˛n y

Rn

Rn

since %" 2 D.Rn / 8" > 0. Hence,

e

@˛ Q Rn D .%" @˛ Q Rn D .%" .T˛ C @˛ x .%" u/# y u/# y u//#Rn C

C

e

D .%" T˛ C %" @˛ y u/#Rn

C

C

(by (8.10.34))

558

Chapter 8 Fourier transforms of distributions and Sobolev spaces

e

e

H) @˛ Q Rn D .T˛ %" /#Rn C .%" @˛ D .%" @˛ , since x .%" u/# y u/#Rn y u/#Rn C C C C .T˛ %" /#Rn D 0 8 sufficiently small " > 0 by (8.10.40). Then, for u 2 H m .RnC /,

e

C

e

e

2 L2 .RnC / 8j˛j  m H) @˛ u 2 L2 .Rn / H) .%" @˛ u/#Rn ! @˛ u#Rn D C C @˛ u 2 L2 .RnC / as " ! 0C H) 8j˛j  m, @˛ .%" u/# Q Rn D .%" @˛ u/#Rn ! @˛ u 2 L2 .RnC / as " ! 0C C C (by Theorem 6.8.2, Property II) H) .%" u/# Q Rn ! u in H m .RnC / as " ! 0C . C Hence, u 2 H m .RnC / and, from (8.10.33), 9.%" u/# Q Rn 2 D.RnC / 8" > 0 such C that .%" u/# Q Rn ! u in H m .RnC / as " ! 0C . Consequently, D.RnC / is dense in C H m .RnC /. For an alternative proof of this theorem using cut-off functions (but not @˛ u

e

convolutions), see Lions [14].

8.10.3 m-extension property of RnC Theorem 8.10.4. RnC has the m-extension property for m  0. Proof. n /, let u Case m D 0: For u 2 H 0 .RnC /  L2 .RC Q be the null extension to Rn of u. Then Z

2 ju.x/j Q dx D Rn

Z Rn C

ju.x/j2 d x C

Z Rn nRn C

Z 0d x D

Rn C

ju.x/j2 d x < C1

Q L2 .Rn / D kukL2 .RnC / . Hence, define the 0-extension H) uQ 2 L2 .Rn / with kuk operator P W L2 .RnC / ! L2 .Rn / by P u D uQ and .P u/#Rn D u with C

kP ukL2 .Rn / D kukL2 .RnC /

8u 2 L2 .RnC /:

(8.10.42)

Case m  1: First of all, we will show that 9P W D.RnC / ! H m .Rn / such that 8u 2 D.RnC /, P u 2 H m .Rn / and P is a continuous, linear operator from D.RnC / equipped with the norm k  kH m .RnC / defined in (8.10.29) into H m .Rn /. Then the result will follow from the density of D.RnC / in H m .RnC / by Theorem 8.10.3. Let u 2 D.RnC /. We set x D .x1 ; : : : ; xn1 ; xn / D .y; z/ 2 Rn with y D .x1 ; : : : ; xn1 / 2 Rn1 and z D xn 2 R. Define P u by ´ u.y; z/ .P u/.y; z/ D v.y; z/

for z  0; for z < 0;

(8.10.43)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

559

where v.y; z/ is defined in terms of the values ¹u.y; kz/ºm with z < 0, which are kD1 n well defined for u 2 D.RC / as follows: 8y D .x1 ; : : : ; xn1 / 2 Rn1 , z D xn < 0, v.y; z/ D 1 u.y; z/ C 2 u.y; 2z/ C    C m u.y; mz/ D

m X

k u.y; kz/;

kD1

(8.10.44) k being constants, which are to be determined from the condition that P u2H m .Rn /. Remark 8.10.3. Other choices of v.y; z/ are also possible. For example, v.y; z/ D

m X kD1

  1 k u y;  z : k

(8.10.45) j

.P u/ .y; z/ of order In order that P u 2 H m .Rn /, the usual partial derivatives Œ @ @z j j  m  1 with respect to z in the pointwise sense must be continuous across the m boundary   Rn0 , i.e. for z D 0, since otherwise the distributional derivative @ @z.Pmu/ of mth order with respect to z will involve the Dirac distribution ı concentrated on m  (i.e. on Rn0 ), and then @ @z.Pmu/ will not belong to L2 .Rn / and, consequently, P u will not belong to H m .Rn / (see Chapter 3 for more details). Thus, for u 2 D.RnC /, we must have, for 0  j  m  1,  j   j   j  @ .P u/ @ u @ u .y; z/ D .y; z/ D .y; z/ @z j @z j @z j zD0 zD0C zD0  j   j  @ v @ u H) .y; z/ D .y; 0/ for 0  j  m  1: (8.10.46) @z j @z j  zD0

Then, using the definition of v.y; z/ in (8.10.44), we have: j D0

H)

v.y; z/#zD0 D lim v.y; z/ D z!0

 j D1

H)

m X

k u.y; 0/I

kD1

   m X @v @u .y; z/ .y; 0/ I D .k/k @z @z zD0 kD1

:: :  j Dm1

H)

 m1   m X @m1 v @ u m1 .y; z/ D .k/  .y; 0/ : k @z m1 @z m1 zD0 kD1

(8.10.47)

560

Chapter 8 Fourier transforms of distributions and Sobolev spaces

From (8.10.46) and (8.10.47), we have m X

k u.y; 0/ D u.y; 0/

m X

H)

kD1 m X

k D 1I

kD1



 @u @u .y; 0/ .k/k Π.y; 0/ D @z @z

m X

H)

kD1

.k/k D 1I

kD1

:: : m X kD1

.k/m1 k



  m1  @ @m1 u u .y; 0/ D .y; 0/ @z m1 @z m1

which we write in matrix form: 2 1 1  6 1 2   6 2 6 .1/2 .2/   6 6 :: :: 4 : : .1/m1 .2/m1

H)

m X

kD1

3 2 3 1 1 7 6 2 7 617 76 7 6 7 7 6 3 7 617 76 7 D 6 7: 7 6 :: 7 6 :: 7 5 4 : 5 4:5 m1 m    .m/ 1 1 m .m/2 :: :

.k/m1 k D 1;

32

(8.10.48)

This is the well-known Vandermonde system of equations with x0 D 1, x1 D 2, : : : , xm1 D m, having a unique solution which completely determines .k /m . kD1 n Since u 2 D.RC /, u 2 C 1 .RnC /

and

v 2 C 1 .Rn /

(8.10.49)

by definition (8.10.44). From the condition (8.10.46), the partial derivative j .P u/ Œ @ @z .y; z/ of order j  m  1 in the usual pointwise sense is continuous in j n RC [ Rn  Rn , i.e. P u 2 C0m1 .Rn / with compact support in Rn , since u 2 D.RnC / has compact support in Rn . Hence, the distributional derivative @˛ .P u/ 2 L2 .Rn /

8j˛j  m  1:

(8.10.50)

Moreover, the usual partial derivatives Œ@˛ P u.y; z/ 2 L2 .Rn / 8j˛j  m (although defined in Rn n Rn0 , the n-dimensional Lebesgue volume measure of Rn0 is zero). m It remains to show that the mth-order distributional derivative @ @z.Pmu/ coincides m with the mth-order usual partial derivative Œ @ @z.Pmu/ .y; z/ in the pointwise sense, m Œ @ @z.Pmu/ .y; z/ being an element of L2 .Rn /. Then, combining with (8.10.50), we shall have P u 2 H m .Rn /. Hence, we show that  m  ´ @m u @ .P u/ @m .P u/ Œ m .y; z/ for z > 0 in D 0 .Rn /: (8.10.51) D .y; z/ D @@zm v m m @z @z Œ @z m .y; z/ for z < 0

Section 8.10 Sobolev spaces on  ¤ Rn revisited

561

In fact, 8 2 D.Rn /,     m1 @m  .P u/ @ m1 @ P u; m D .1/ ; @z D 0 .Rn /D.Rn / @z m1 @z D 0 .Rn /D.Rn /     Z  m1 Z  m1 @ @ @ @ u v m1 d ydz C d ydz D .1/ .y; z/ .y; z/ @z m1 @z @z m1 @z Rn Rn  C Z  Z 1  @ .m1/ D .1/m1 .u dy .y; z/.y; z// dz Rn1 0C @z   Z  Z 0   Z 1 m @ u @ .m1/ .v  .y; z/ .y; z/dz C d y .y; z/.y; z// dz m Rn1 1 @z 0C @z   Z 0  m @ v  .y; z/ .y; z/dz m 1 @z     Z Z 1  m Z 0  m @ u @ v D.1/m .y; z/ .y; z/dz .y; z/  .y; z/dz dy m m Rn1 1 @z 0C @z Z  m1 Œ.u.m1/ .y; z/.y; z//jzD1 C.v .m1/ .y; z/.y; z//jzD0 C .1/ zD1 d y zD0C n1 R  Z  m @ .P u/ D .1/m .y; z/ .y; z/d ydz @z m Rn (using (8.10.51)), since .y; z/#zD˙1 D 0, .y; 0C / D .y; 0 / D .y; 0/, u.m1/ .y; 0/ D v .m1/ .y; 0 / by (8.10.46), m m H) hP u; @@z m i D .1/m hŒ @ @z.Pmu/ .y; z/; i 8 2 D.Rn / m m ” @ @z.Pmu/ D Œ @ @z.Pmu/ .y; z/ 2 D 0 .Rn / (by the definition of distributional derivatives) with @m .P u/ 2 L2 .Rn /: @z m

(8.10.52)

Hence, P u 2 H m .Rn / 8u 2 D.RnC /. From the definition (8.10.43)–(8.10.44) of P u, it is obvious that P W D.RnC / ! H m .Rn / is linear. To complete the proof, we are to show that kP ukm;Rn  C0 kukm;RnC 8u 2 D.RnC / for some C0 > 0. In fact, 8j˛j  m, Z

˛

2

j@ .P u/j d x D Rn

Z

˛

Rn C

2

Z

j@ uj d ydz C

j@˛ vj2 d ydz;

(8.10.53)

Rn 

where the distributional derivatives and the usual derivatives in the pointwise sense are identical in L2 .Rn /8j˛j  m.

562

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Set y D y, z D k with 1  k  m,  < 0, Jacobian J D k given by: ˇ ˇ ˇ1 0    0 ˇ ˇ ˇ ˇ : : : : :: ˇ ˇ0 : : : ˇ ˇ D k: J D ˇˇ : : ˇ ˇ :: : : 1 0 ˇ ˇ ˇ ˇ0    0 k ˇ Define u.y; z/  u.y; k/  vk .y; / with  D  k1 z < 0, y D .x1 ; x2 ; : : : ; xn1 / 2 Rn1 . @ k k Hence, @u .y; z/ D @v .y; .z// @z D  k1 @v .y; / @z @ @ ˛n

˛n

H) @@z ˛nu .y; z/ D . k1 /˛n @@ ˛vnk .y; / H) 8˛ D .˛1 ; ˛2 ; : : : ; ˛n1 ; ˛n / with j˛j  m,  ˛n    ˛n   @˛1 CC˛n1 @ u @ vk 1 ˛n @˛1 CC˛n1 .y; z/ D  .y; / ˛1 ˛n1 ˛1 ˛n1 ˛ n k @ ˛n @x1 : : : @xn1 @z @x1 : : : @xn1 1 j@˛ vk .y; /j2 with  < 0, y D .x1 ; : : : ; xn1 /, j˛j  m. H) j@˛ u.y; z/j2 D k 2˛ n R R 1 j@˛ vk .y; /j2 jJ jd yd . H) Rn j@˛ u.y; z/j2 d ydz D Rn k 2˛ n  C Replacing  by z and putting jJ j D k, we get Z Z j@˛ vk .y; z/j2 d ydz D k 2˛n 1 j@˛ u.y; z/j2 d ydz: (8.10.54) Rn C

Rn 

P Then, from (8.10.44), (8.10.53) and (8.10.54), 8j˛j  m, jv.y; z/j  m kD1 jk j  jvk .y; z/j for z D xn < 0, and Z Z ˛ 2 j@ .P u/j d ydz  j@˛ uj2 d ydz C 2m1 . max ¹jk jº/2 m X kD1

k 2˛n 1

1km

Rn C

Rn

Z Rn C

j@˛ uj2 d ydz  C

Z Rn C

j@˛ uj2 d ydz

with C D C.1 ; 2 ; : : : ; m ; ˛n ; m/ > 0 X Z X Z ˛ 2 j@ .P u/j d ydz  C1 H) j˛jm

Rn

j˛jm

Rn C

j@˛ uj2 d ydz

H) kP uk2m;Rn  C1 kuk2m;Rn 8u 2 D.RnC / H) 9C0 > 0 with C0 D C

p

C1 such

that kP ukH m .Rn /  C0 kukH m .RnC / . Thus, the linear operator P W D.RnC / ! H m .Rn / is continuous from D.RnC / equipped with the norm k  kH m .RnC / into H m .Rn /. But D.RnC / is dense in H m .RnC / by Theorem 8.10.3. Hence, P has a unique, continuous, linear extension PQ (which will be ultimately denoted by the earlier original notation P itself) from H m .RnC / into H m .Rn /. Thus, RnC has the m-extension property.

Section 8.10 Sobolev spaces on  ¤ Rn revisited

563

Babitch–Nikolski extension The extension of u 2 D.RnC / to P u 2 H m .Rn / defined by (8.10.43)–(8.10.44) will be called the Babitch–Nikolski extension ([16], [43]) here, although some authors, for example Grisvard [18], call this the Nikolski technique of extension by reflection. m-extension property of RnC Define RnC D ¹x W x 2 Rn ; xi > 0; 1  i  nºI RnC D ¹x W x 2 Rn ; xi  0; 1  i  nºI Rn D ¹x W x 2 Rn ; xi < 0; 1  i  nºI Rn D ¹x W x 2 Rn ; xi  0; 1  i  nºI Rn0 D ¹x W x 2 Rn ; xi D 0; 1  i  nºI RnC D RnC [ Rn0 I

Rn D Rn [ Rn0 :

(8.10.55)

For n D 2, R2C is the first quadrant, R2 the third quadrant in R2 , R2C being the upper half-plane in R2 . Then R2C; D ¹x W x 2 R2C , x1 < 0º is the second quadrant in R2 (this last notation is not standard). Proposition 8.10.4. D.RnC / is a dense subspace of H m .RnC / with  kukH m .RnC / D

X

k@

˛

2 ukL 2 .RnC /

1=2 :

0j˛jm

Proof. The proof is exactly similar to that of Theorem 8.10.3 with minor modifications: 



Choose % 2 D.Rn / with supp.%/  Rn and define %" D supp.%" /  Rn for all sufficiently small " > 0;

1 x "n %. " /

such that

8j˛j  m, T˛ %" 2 C 1 .Rn / with supp.T˛ %" /  supp.T˛ / C supp.%" /  Rn H) T˛ %" #RnC D 0 8j˛j  m and for all sufficiently small " > 0, the other steps being almost the same, with RnC replaced by RnC .

Proposition 8.10.5. RnC has the m-extension property. Proof. For the sake of simplicity, consider the case n D 2, i.e. R2C is the first quadrant in R2 . Let u 2 D.R2C /. Following the steps of the Babitch–Nikolski extension

564

Chapter 8 Fourier transforms of distributions and Sobolev spaces

procedure in (8.10.44)–(8.10.52), we define 8 ˆ 0; (8.10.56)

P j are uniquely determined by solving m where ¹k ºm kD1 .k/ k D 1 for j D kD1 2 ˛ 2 0; 1; : : : ; m  1 (see (8.10.48)) and @ .P1 u/ 2 L .RC / 8j˛j  m with Z Z ˛ 2 j@ P1 uj dx1 dx2 D j@˛ uj2 dx1 dx2 R2C Wx2 >0

R2C Wx1 >0;x2 >0

Z

C Z

R2C; Wx1 0

C R2C

j@˛ vj2 dx1 dx2

j@˛ uj2 dx1 dx2 ;

since from (8.10.56) and (8.10.54), Z j@˛ u.kx1 ; x2 /j2 dx1 dx2 R2C; Wx1 0

D k 2˛1 1

Z

R2C Wx

j@˛ u.x1 ; x2 /j2 dx1 dx2 1 >0;x2 >0

.k .x1 ; x2 / D u.kx1 ; x2 //

8˛ D .˛1 ; ˛2 / with j˛j D ˛1 C ˛2  m (see (8.10.54)). Hence, 8u 2 D.R2C /, kP1 ukH m .R2 /  C0 kukH m .R2C / . Then, by virtue of the C

density of D.R2C / in H m .R2C /, 9 a unique, continuous, linear extension of P1 to H m .R2C /, which will still be denoted by P1 2 L.H m .R2C /I H m .R2C // such that 8u 2 H m .R2C /, P1 u 2 H m .R2C / with .P1 u/#R2C D u and kP1 ukH m .R2 /  C C0 kukH m .R2C / . Again, by Theorem 8.10.4, 9 an m-extension operator P2 W P1 u 2 H m .R2C / 7! P2 .P1 u/ 2 H m .R2 / with .P2 .P1 ; u//#R2 D P1 u and C

kP2 .P1 u/kH m .R2 /  C1 kP1 ukH m .R2 /  C1 C0 kukH m R2C 8u 2 H m .R2C /. DeC

fine P D P2 ı P1 . Then P W H m .R2C / ! H m .R2 / with P u D P2 ı P1 u D P2 .P1 u/ 2 H m .R2 /, P u#R2C D .P2 .P1 u//#R2C D .P1 u/#R2C D u, and kP ukH m .R2 /  C2 kukH m .R2C / with C2 D C1 C0 > 0. Hence P 2 L.H m .R2C /; H m .R2 // is the required m-extension operator. See Figure 8.5 for an illustration of this proof.

Remark 8.10.4. In many situations, repeated applications of the Babitch–Nikolski extension technique, possibly including a change of variables, will give the required

Section 8.10 Sobolev spaces on  ¤ Rn revisited m

2+

H (ú )

565

m

2

H (ú+ ) (by Babitch-Nikolsky

P1

extension by reflection )

P=

P

2

P2

BP

1 m

2

H (ú ) (by Babitch-Nikolsky extension by reflection ) Figure 8.5 The m-extension operator P of R2C constructed by Babitch–Nikolsky extension

m-extension operator. For example, for  D ¹x W x 2 Rn , xn1 > 0, xn > 0º, we can define the m-extension operators P1 W H m ./ ! H m .RnC / with RnC D ¹x W x 2 Rn ; xn > 0º and P2 W H m .RnC / ! H m .Rn / such that P D P2 ı P1 is the required m-extension operator from H m ./ into H m .Rn /. Similarly, we have: Proposition 8.10.6. Let  be an infinite angular sector in R2 with angular measure 0 < ! < =2. Then  has the m-extension property. Proof. Without loss of generality, we consider the angular sector  with vertex at the origin and bounded by the x1 -axis, x2 D 0, x1 > 0, and the half-line x2 D mx1 , x1 > 0, m > 0, i.e.  D ¹.x1 ; x2 / W x1 > 0, 0 < x2 < mx1 , m > 0º. x2

x2

x2

x2

=

m

x1 x = (x1, x2) F

W

x2

F(x) = (x1, x2)

w O

x1

x1

O

x1

x1

· W = { (x1, x2) | x1 > 0, 0 < x2 < mx1, m>0} · 0 < w < p/2 Figure 8.6 Defining an m-extension operator through a change of variables

Let F W R2 ! R2 be an invertible linear mapping (see Figure 8.6) defined, 8x D .x1 ; x2 / 2 R2 , by: ! !   x1 1 1 1=m (8.10.57) F.x/ D D 2 R2 0 1 x2 2

566

Chapter 8 Fourier transforms of distributions and Sobolev spaces

such that 8x 2 , F.x/ D  2 R2C ; 8 D .1 ; 2 / 2 R2C , F1 ./ D x 2 , where, 8 2 R2C , ! !   x1 1 1 1=m 1 F ./ D D 2 : (8.10.58) 0 1 2 x2 Then the linear mappings F# D F W  ! R2C , F1 W R2C !  are continuously, infinitely differentiable, i.e. F and F1 are C 1 -diffeomorphisms. Let u 2 D./. 8.x1 ; x2 / 2 , u.x1 ; x2 / D u.F1 .1 ; 2 // D .u ı F1 /.1 ; 2 /. Define v D uıF1 and u D v ıF with u.x1 ; x2 / D v.1 ; 2 /. Then, for u 2 H m ./, v 2 H m .R2C / with kukH m ./ D kv ı FkH m .R2C /  kvkH m .R2C / ; kvkH m .R2C / D ku ı F1 kH m ./  kukH m ./ ; where  D .m; F; F1 / > 0. By Proposition 8.10.5, 9 an extension operator P1 2 L.H m .R2C /; H m .R2 // such that .P1 v/#R2C .1 ; 2 / D v.1 ; 2 / D u.x1 ; x2 / with .x1 ; x2 / 2  and kP1 vkH m .R2 /  C kvkH m .R2C / . Define P u D P1 .u ı F1 / 2 H m .R2 / such that .P u/# .x1 ; x2 / D .P1 .u ı F1 //#R2C .1 ; 2 / D .P1 v/#R2C .1 ; 2 / D v.1 ; 2 / D u.x1 ; x2 / for almost all x 2 , i.e. P u# D u. Moreover, kP ukH m .R2 / D kP1 vkH m .R2 /  C kvkH m .R2C /  C kukH m ./ H) kP ukH m .R2 /  C0 kukH m ./ . Hence, P W H m ./ ! H m .R2 / is the required m-extension operator. m-extension property of a non-convex planar sector with ! D 3=2 Now we will consider the example of a non-convex planar sector with angular measure ! D 3=2. Proposition 8.10.7. Let  be a planar sector with angular measure ! D  has the m-extension property.

3 2 .

Then

Proof. We orient the coordinate axes in such a way that  D ¹.x1 ; x2 / W x1 > 0 or x2 > 0º 1 D ¹x W x1 > 0º;

(see Figure 8.7),

2 D ¹x W x2 > 0º D R2C ;

(8.10.59)

3 D ¹x W x1 > 0; x2 < 0º:

Let u 2 H m ./ such that v D u#2 2 H m .2 /. Since 2 D R2C , by Theorem 8.10.4 9 an m-extension operator P1 2 L.H m .2 /I H m .R2 // such that P1 v D v 2 H m .R2 / with v #2 D v. Then .u  v # /#2 D u#2  v #2 D v  v D 0. e e e e

Section 8.10 Sobolev spaces on  ¤ Rn revisited

x2

· w = 3p/2 · W : x1>0 or x2>0 w

O

· w=p · W1 : x1>0

x2

x2

w x1

· Case: w = 3p/2

O

567 · w=p · W2 : x2>0

x2

· w = p/2 · W3 : x1>0, x2 0. Then, e again by Theorem 8.10.5, 9 an m-extension operator P2 2 L.H m .1 /I H m .R2 // such that P2 w D w 2 H m .R2 / with w #2 D w D 0, i.e. for x2 > 0. Finally, e 2 /. Then P W H m ./ ! H m .R2 / is a define P u D u byeu D v C w 2 H m .R e e e e continuous, linear operator. Moreover, for 3 W x1 > 0, x2 < 0 and 2 W x2 > 0, u#3 D v #3 C w #3 D v #3 C w#3 1 D v #3 C .u#3  v #3 / D u#3 e e on  (since and u#2e D v #e C w #e D v C 0 D v D u#e2 , i.e. P u D u a.e. 2 2 e e e m m 2 area meas. n .2 [ 3 // D 0) and P u 2 L.H ./I H .R // is the required m-extension operator of  W x1 > 0 or x2 > 0. m-extension property of polygonal domains in R2 Since the extension results for planar sectors, quadrants, half-planes, etc. can be locally ‘patched’ or ‘glued’ together, the results given in the earlier Propositions and Theorems can be used to construct the required extension for any polygonal domain (see also Grisvard [18]). Proposition 8.10.8. Every bounded polygonal domain   R2 has the m-extension property. Proof. We will give the scheme of the proof using a partition of unity S (see ApN 2 pendix C). Since  is compact in R , 9 a finite open cover ¹i ºiD0 of   N iD0 i S such that   N  ,  \  D  \ S , 0  i  N , where     R2 , i i i 0 iD1 i S0 D R2 and Si , 1  i  N , is an infinite angular sector, possibly also a quadrant or a half-plane, etc., for which we have already shown the extension results – see Figure 8.8 for an illustration. With this finite open cover ¹i ºN associate a partition ¹i ºN iD0 of  we iD0 of unity PN PN with the properties: iD0 i D 1 in  (i.e. iD0 i .x1 ; x2 / D 1 8.x1 ; x2 / 2 /; i 2 D.Rn / with supp.i /  i (i.e. i 2 D.i /, 0  i  N ) (see Appendix C). Let u 2 H m ./ and uQ be its null extension to R2 . Then, for 1  i  N , denoting the restriction i u# Q Si by i u, Q we have i u# Q Si D i uQ 2 H m .Si / with N supp.i u/ Q  Si , ¹Si ºiD1 being infinite angular sectors. For i D 0, S0 D R2 and let 0 uQ D w0 . Then w0 2 H m .R2 /, since supp.0 u/ Q D supp.0 / \ supp.u/ Q 

568

Chapter 8 Fourier transforms of distributions and Sobolev spaces

N Figure 8.8 Finite open cover ¹i ºN iD0 with i \  D i \ S i , 1  i  N , ¹Si ºiD1 being infinite angular sectors

supp.0 /  0    R2 , the result follows from Theorem 2.15.5. For i D 0, the extension operator P0 2 L.H m ./I H m .R2 // is defined by P0 u D w0 2 H m .R2 / with w0 D 0 uQ for u 2 H m ./. Hence, for 1  i  N , 9 an extension operator Pi 2 L.H m .Si /I H m .R2 // and 9wi 2 H m .R2 / such that Pi .i u/ Q D wi 2 H m .R2 / with Pi .i u/# Q Si D wi #Si D i u, Q 8u 2 H m ./. Then the required m-extension operaP PN tor P 2 L.H m ./I H m .R2 // is defined by P u D w0 C N iD1 wi D iD0 wi D m 2 w 2 H .R /. In fact, .x1 ; x2 / 2  H) 9i with 0  i  N such that .x1 ; x2 / 2 Si and .Pi .i u//.x Q Q 1 ; x2 /. Then 1 ; x2 / D wi .x1 ; x2 / D i u.x

.P u/.x1 ; x2 /Dw.x1 ; x2 / D

N X

wi .x1 ; x2 / D

iD0

D

X N

N X

.i u/.x Q 1 ; x2 /

iD0

 i .x1 ; x2 / u.x Q 1 ; x2 /D1  u.x Q 1 ; x2 / D u.x1 ; x2 /

iD0

for almost all .x1 ; x2 / 2  H) .P u/# D u a.e. on .

Section 8.10 Sobolev spaces on  ¤ Rn revisited

569

8.10.4 m-extension property of C m -regular domains  C m -regular domains  Definition 8.10.4. An open subset   Rn with boundary  is called C m -regular with m 2 N if and only if 1.  is bounded; 2.  is locally on one side of its boundary  (i.e.  is a one-sided domain); 3.  is an .n  1/-dimensional manifold belonging to C m -class (see Appendix D). Some explanations on Properties 2 and 3 are in order. Property 2 excludes the possibility of  having a slit/cut/crack. For example,  D ¹.x1 ; x2 / W 0 < jx1 j < 1, 0 < x2 < 1º  R2 is a bounded domain in R2 with a slit/cut along the boundary segment 0 W x1 D 0, 0 < x2 < 1 (see Remark 8.10.2). Hence, it is locally on two sides of the boundary segment 0 , i.e. it is a two-sided domain locally along 0 and Property 2 does not hold. Similarly,  D ¹.x1 ; x2 / W x12 C x22 < 1, x2 ¤ 0 for 0 < x1 < 1º  R2 has a cut/crack along 0 W x2 D 0, 0 < x1 < 1 and is locally on both sides of 0 . Hence, Property 2 does not hold in this case, either. For more examples, see Dauge [44]. Property 3: we define an .n  1/-dimensional boundary  of C m -class in a manner suitable for Theorem 8.10.5 as follows (for an alternative definition and other details, see Appendix D). Boundary  of C m -class Definition 8.10.5. Let ¹i ºN iD0 be a finite open cover of the compact set  D  [ n   R such that 0  , and ¹i ºN iD1 be an open cover of the boundary  of  (see Appendix C). Then  belongs to C m -class with m 2 N if and only if. 8i D 1; 2; : : : ; N , 9 an m-times continuously differentiable and bounded (i.e. the derivatives of all orders  m are continuous and bounded), invertible mapping ˆ i : b  Rn ; ˆ i W x 2 i 7! ˆ i .x/ D  2 Q

(8.10.60)

b is a hypercube in Rn W Q b D ¹ W  D .1 ; : : : ; n / 2 Rn , jj j < 1 where Q b can be chosen (and will be chosen) 8j D 1; 2; : : : ; nº. 8i , the same hypercube Q such that its inverse 1 b ˆ 1 i W  2 Q 7! ˆ i ./ D x 2 i

(8.10.61)

b onto is also an m-times continuously differentiable and bounded mapping from Q b \ Rn  Q bC i and the following additional conditions hold: ˆ i .i \ / D Q C n n n1 with RC D ¹ W  D .b ; n / 2 R with b  D .1 ; : : : ; n1 / 2 R , n > 0º, n1 b b b b , n D 0º  Q0 . (8.10.62) ˆ i .i \ / D Q \ ¹ W  D .; n / with  2 R

570

Chapter 8 Fourier transforms of distributions and Sobolev spaces

m-extension property of C m -regular bounded domain  Theorem 8.10.5. Let   Rn be a bounded open subset of Rn with a C m -regular boundary  for m  1. Then  has the m-extension property. Proof. Since  D  [   Rn is a compact subset of Rn , 9 a finite open SNcover ¹i ºN of  with open subsets  , 0  i  N , such that   ,   i 0 iD0 i iD0 S N and   iD1 i ; see Figure 8.9. Then, by Theorem C.1.1.1 (Appendix C), 9 a C01 -partition ¹i ºN iD0 of unity 1 PN (i.e. iD0 i .x/ D 1 8x 2 ) subordinate to this finite open cover ¹i ºN iD0 with i 2 D.i / and supp.i / D Ki  i , 0  i  N . Let u 2 H m ./. Considering the null extension Qi to Rn of i and then taking the restriction Qi # 2 D./, 0  i  N , we have ui D Qi #  u 2 H m ./, which we agree to denote by ui D i u 2 H m ./ with i 2 D.i /; u 2 H m ./, supp.ui / D supp.i / \ supp.u/  supp.i / D Ki  i ; 0  i  N: (8.10.63) Then N X iD0

ui D

X N

 i u D 1  u D u in :

(8.10.64)

iD0

Figure 8.9 Finite open cover ¹i ºN iD0 with 0  ,   SN iD1 i

SN

iD0

i ,  

SN

iD1

i ,  

For i D 0, u0 2 H m ./ with supp.u0 / D K0  0   H) its null extension uQ 0 D 0 u 2 H m .Rn / with uQ 0 .x/ D u0 .x/ for x 2 0 and uQ 0 .x/ D 0 for x 2 Rn n 0 .

e

Section 8.10 Sobolev spaces on  ¤ Rn revisited

571

Define P0 W H m ./ ! H m .Rn / by P0 u D uQ 0

8u 2 H m ./;

(8.10.65)

since u 2 H m ./ 7! 0 u D u0 2 H m ./ with supp.u0 / D K0  0  , and consequently uQ 0 D 0 u 2 H m .Rn / (see Theorem 2.15.5). Then P0 is a continuous, linear operator from H m ./ into H m .Rn / with P0 u# D uQ 0 # D u0 a.e. on . b (such a choice of Q b 8 fixed i can be made) 8i D 1; 2; : : : ; N , let ˆ i W i ! Q b Ki and Li being such that ˆ i .Ki / D Li with Ki D supp.i /  i , Li  Q, b respectively, where ˆ i is a C m -diffeomorphism5 from compact subsets of i and Q, b such that the derivatives of all orders  m of the components  i and i i onto Q k k i n b (1  k  n) of ˆ i D .ki /nkD1 and ‰ i D ˆ 1 i D . k /kD1 are bounded in i and Q, b with supp.ui ı ˆ 1 /  respectively. Then, 8i D 1; 2; : : : ; N , ui ı ˆ 1 2 H m .Q/ i i b define vi D .ui ı ˆ 1 /# bC D Q b \ Rn . Li  Q; 8i D 1; 2; : : : ; N with Q i C b QC C m m b b b Then vi 2 H .QC / with supp.vi /  Li D Li \ QC  Q, since ui 2 H ./ b and the derivatives of all orders and ˆ i is a C m -diffeomorphism from i onto Q 1 b respectively,  m of the components of ˆ i and of ˆ i are bounded in i and Q 1 1 b Then, supp.ui /  Ki D ˆ i .Li / and, consequently, supp.ui ı ˆ i /  Li  Q.

e

C C

8i D 1; 2; : : : ; N , the null extension ui ı ˆ1 to Rn of ui ı ˆ1 is defined by i i b and ui ı ˆ1 ./ D 0 for  2 Rn n Q. b Since ui ı ˆ1 ./ D ui ı ˆ1 ./ for  2 Q

C

C

i i i 1 1 m b ui ıˆi 2 H .Q/ and supp.ui ı ˆi / D supp.ui ıˆ1 i /  Li m n n H .R / (by Theorem 2.15.5), whose restriction ui ı ˆ1 i #RC

C

C

C

b ui ı ˆ1 2  Q, i 2 H m .RnC /. Then,

n m n for vi D .ui ı ˆ1 , define vQi D ui ı ˆ1 i /#b i #R such that vQi 2 H .RC / with Q C

C Li ,

C

b C ; see Figure 8.10. b C , LC D Li \ Q D Li \ Q supp.vQi / D supp.vi /  i Hence, by Theorem 8.10.4 on the m-extension property of RnC , 9 a unique Babitch– Nikolski extension P vQ i 2 H m .Rn / of vQ i 2 H m .RnC / with LC i

b supp.P vQ i / D Li  Q

8 fixed i;

(8.10.66)

b with supp.P vQ i / D L  Q. b 2 H m .Q/ H) .P vQ i /#b i Q b is a C m -diffeomorphism with the derivatives of the compoBut ˆ i W i ! Q i nents k ; 1  k  n, of ˆ i bounded in i . Hence, 8i D 1; 2; : : : ; N , ..P vQ i /ıˆ i / 2 5 C m -diffeomorphism:

i onto b Q if and only if

8 fixed i , 1  i  N , ˆ i W i ! b Q is called a C m -diffeomorphism from

1. ˆ i D .1i ; 2i ; : : : ; ni / is a bijection and of C m -class (i.e. the derivatives of all orders  m of the components ki , 1  k  n, of ˆ i are continuous from i onto b Q); i i b 2 ; : : : ; n / W Q ! i is i components k , 1  k  n, of ‰ i

D ‰i D . 2. its inverse ˆ 1 i of all orders  m of the

i 1;

also of C m -class (i.e. the derivatives are continuous from b Q onto i ).

572

Chapter 8 Fourier transforms of distributions and Sobolev spaces

D

H m .i / with supp.P vQ i ı ˆ i /  i . Let .P vQ i ı ˆ i / be the null extension to Rn of P vQ i ı ˆ i , i.e. .P vQ i ı ˆ i /.x/ D .P vQ i ı ˆ i /.x/ for x 2 i and .P vQ i ı ˆ i /.x/ D 0 for x 2 Rn n i . Then .P vQ i ı ˆ i / 2 H m .Rn / (see the proof of Theorem 2.15.5). 8u 2 H m ./, define

D D

C

C D .uC ı ˆ /#

D

Pi u D P vQ i ı ˆ i 2 H m .Rn /;

, vQ i with vi D .ui ı ˆ 1 i /#b Q C

1 i

i

Rn C

(8.10.67)

8i D 1; 2; : : : ; N .

Figure 8.10 C m -regular bounded domain  with .n  1/-dimensional boundary  of C m class m  1

8 fixed i D 1; 2; : : : ; N , Pi W H m ./ ! H m .Rn / is linear and continuous with the property

C

.Pi u/# D .P vQ ı ˆ i /# D ui

a.e. on ;

(8.10.68)

ui 2 H m ./ with supp.ui /  i (by (8.10.63)). Now we define Pm W H m ./ ! H m .Rn / by Pm u D P0 u C

N X

Pi u 8u 2 H m ./;

(8.10.69)

iD1

where P0 and Pi , 1  i  N , are defined by (8.10.65) and (8.10.68) respectively.

Section 8.10 Sobolev spaces on  ¤ Rn revisited

573

P PN Then, 8u 2 H m ./, .Pm u/# D .P0 u/# C N iD1 .Pi u/# D u0 C iD1 ui D u a.e. on  by (8.10.64), and Pm W H m ./ ! H m .Rn / is a linear, continuous operator from H m ./ into H m .Rn / by virtue of the linearity and continuity of P0 and ¹Pi ºN iD1 with .Pm u/# D u a.e. on . Hence, Pm is an m-extension operator of . Thus, any C m -regular domain with m  1 has the m-extension property. b For fixed i , 1  i  N , Justification of (8.10.66), supp.P vQ i / D Li  Q C b LC D Li \ Q b C , LC D vQ i 2 H m .RnC / with supp.vQ i / D supp.v/  Li  Q, i i b C , and let P vQ i 2 H m .Rn / be its Babitch–Nikolski extension to Rn . Then Li \ Q b In fact, by virtue of the density of D.Rn / in H m .Rn /, supp.P vQ i / D L  Q. C

i

C

in D.RnC / such that k ! vQ i in H m .RnC / as k ! 1. But 9 a sequence . k /1 kD1 n 8 k 2 D.RC /, its natural extension (denoted by k itself) k 2 D.Rn / may not b Hence, we are to construct a new sequence .wk /1 in D.Rn / have the support in Q. kD1 (wk #Rn 2 D.RnC / also denoted by wk itself) with the desired additional properties C

C

as follows. Let  2 D.Rn / with  D 1 in Li , 0  ./  1 8 2 Rn , supp./  b (see Theorem 6.2.4 for such a ). Then  vQ i D vQ i 2 H m .Rn / 8 E0  E 0  Q C b 8k 2 N fixed i . Set wk D  k 2 D.RnC / with supp.wk /  supp./  E0  Q and wk ! vQ i in H m .RnC / as k ! 1, since the mapping k 7!  k D wk is continuous on D.RnC / in the norm of H m .RnC /, k ! vQ i H)  k !  vQ i D vQ i 2 H m .RnC / as k ! 1. But for fixed i , 1  i  N , by the construction of Babitch–Nikolski extension (8.10.43), for wk D  k 2 D.RnC /:  

b 8k 2 N; P wk 2 H m .Rn / with supp.P wk /  E0  Q P wk ! P vQ i in H m .Rn / (since P 2 L.H m .RnC /; H m .Rn // and wk ! vQ i in H m .RnC /) H) P wk ! P vQ i in L2 .Rn / as k ! 1 H) supp.P vQ i / D Li  b 8 fixed i . E0  Q

In fact, for fixed i , 2 kP vQ i P wk kL 2 .Rn / D lim

k!1

Z E0

jP vQ i P wk j2 d C

Z Rn nE0

 jP vQ i 0j2 d  D0:

R

jP vQ i j2 d  D 0 H) P vQ i D 0 a.e. on Rn n E0 H) Li D b supp.P vQ i /  E0  Q. Hence,

Rn nE0

8.10.5 Space H s ./ with s 2 RC ,   Rn For an arbitrary open set   Rn and s D Œs C  with Œs D the integral part of s D m 2 N0 and  D the fractional part of s, 0 <  < 1, an alternative definition of H s .Rn / was given in Definition 8.9.4 as the set ¹u W u 2 H Œs .Rn /, @˛ u 2 H  .Rn /

574

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8j˛j D Œs D mº, where H  .Rn / is defined by (8.9.38)–(8.9.39) for 0 <  < 1: ² “ H  .Rn / D u W u 2 L2 .Rn /;

Rn Rn x¤y

³ ju.x/  u.y/j2 d xd y < C1 : kx  yknC2

We will now extend this definition to arbitrary open sets   Rn by replacing Rn by  (see Grisvard [17], [18], [19], Neˇcas [16], etc.).

Case 0 < s < 1 with s D  Definition 8.10.6. For an arbitrary open set   Rn and 0 < s D  < 1, H  ./ is defined by ² “ H  ./ D u W u 2 L2 ./;

 x¤y

³ ju.x/  u.y/j2 d xd y < C1 ; kx  yknC2

which is equipped with the inner product h  ;  i; , norm k  k; and semi-norm j  j; , for u; v 2 H  ./, given by: “ hu; vi; D hu; viL2 ./ C

 x¤y

.u.x/  u.y//  .v.x/  v.y// d xd yI kx  yknC2 (8.10.70a)

2 kuk2; D hu; ui; D kukL 2 ./ C

juj2;

“ D

 x¤y

“  x¤y

ju.x/  u.y/j2 d xd yI kx  yknC2

ju.x/  u.y/j2 d xd y: kx  yknC2

(8.10.70b)

(8.10.70c)

Case s > 0 Definition 8.10.7. For an arbitrary open set   Rn and for s D Œs C  with Œs D m 2 N0 , 0 <  < 1, H s ./ is defined by (Grisvard [17], [18], [19], Neˇcas [16]): H s ./ D ¹u W u 2 H m ./; @˛ u 2 H  ./ 8j˛j D m D Œsº;

(8.10.71)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

575

which is equipped with the inner product h  ;  is; , norm k  ks; and semi-norm j  js; : hu; vis; D hu; viH m ./ X “ C j˛jDm

D

X

 x¤y

.@˛ u.x/  @˛ u.y//  .@˛ v.x/  @˛ v.y// d xd y kx  yknC2

h@˛ u; @˛ viL2 .

0j˛jm

C

X “

j˛jDm

 x¤y

.@˛ u.x/  @˛ u.y//  .@˛ v.x/  @˛ v.y// d xd yI kx  yknC2 (8.10.72)

kuk2s; D

X

2 k@˛ ukL 2 ./ C

0j˛jm

X “ j˛jDm

 x¤y

j@˛ u.x/  @˛ u.y/j2 d xd yI kx  yknC2 (8.10.73)

juj2s; D

X “ j˛jDm

 x¤y

j@˛ u.x/  @˛ u.y/j2 d xd y: kx  yknC2

(8.10.74)

Theorem 8.10.6. For s > 0, H s ./ equipped with the inner product h  ;  is; is a Hilbert space. Proof. Let s D Œs C  with the integral part Œs D m 2 N0 and fractional part , s 0 <  < 1. Let .un /1 nD1 be a Cauchy sequence in H ./. We are to show that s .un / converges to u 2 H ./ as n ! 1. Since .un / is a Cauchy sequence in H s ./, from the definition of H s ./, .un / is a Cauchy sequence in H m ./ with m D Œs 2 N0 and, 8j˛j D m, I Œ@˛ ul  @˛ un  ! 0 as l; n ! 1, i.e. “ j@˛ .ul  un /.x/  @˛ .ul  un /.y/j2 ˛ ˛ d xd y ! 0 I Œ@ ul  @ un  D  kx  yknC2 x¤y

(8.10.75) as l; n ! 1, 8j˛j D m, 0 <  < 1. Since H m ./ is a Hilbert space by Theorem 2.15.1, 9u 2 H m ./ such that un ! u 2 L2 ./ and @˛ un ! @˛ u in L2 ./ 8j˛j  m. Then, we can apply the following result. Lemma 8.10.1 ([20, Theorem 2, p. 305], [27, Theorem 3.12, p. 68]). If .un /1 nD1 converges to u 2 Lp ./ such that ku  un kLp ./ ! 0 as n ! 1, 1  p  1, then 9 a subsequence .unk /1 of the sequence .un / which converges pointwise almost kD1 everywhere to u, i.e. limk!1 unk .x/ D u.x/ a.e. on .

576

Chapter 8 Fourier transforms of distributions and Sobolev spaces

2 Hence, by Lemma 8.10.1, 9 a subsequence .@˛ unk / of .@˛ un /1 nD1 in L ./ such that

lim @˛ unk .x/ D @˛ u.x/

k!1

a.e. on  8j˛j D m:

Then, for x ¤ y, 8 fixed ˛ with j˛j D m, the sequence  ˛   ˛  j@ unk .x/  @˛ unk .y/j2 j@ u.x/  @˛ u.y/j2 ! kx  yknC2 kx  yknC2

(8.10.76)

(8.10.77)

a.e. on    as k ! 1. In fact, j.@˛ u.x/  @˛ u.y//  .@˛ unk .x/  @˛ unk .y//j  j@˛ u.x/  @˛ unk .x/j C j@˛ u.y/  @˛ unk .y/j ! 0 a.e. in    as k ! 1 8j˛j D m (by virtue of (8.10.76)) H) 8j˛j D m, lim j@˛ unk .x/  @˛ unk .y/j D j@˛ u.x/  @˛ u.y/j

k!1

a.e. in    H) 8j˛j D m, j@˛ unk .x/  @˛ unk .y/j2 j@˛ u.x/  @˛ u.y/j2 D kx  yknC2 kx  yknC2 k!1 lim

a.e. in    for x ¤ y H) 8j˛j D m, lim inf k!1

j@˛ u.x/  @˛ u.y/j2 j@˛ unk .x/  @˛ unk .y/j2 D kx  yknC2 kx  yknC2

(8.10.78)

a.e. in    for x ¤ y. Since .un / is a Cauchy sequence in H s ./ satisfying (8.10.75), i.e. 8" > 0, 9n0 D n0 ."/ 2 N such that 8j˛j D m, I Œ@˛ ul  @˛ un  < " 8 l; n > n0 ."/ 2 N (8.10.78a) ˛ ˛ H) I Œ@ ul  @ unk  < " 8l > n0 , 8k > k0 with nk > nk0 > n0 , 8j˛j D m H) I Œ@˛ unk  is bounded 8k 2 N, i.e. 8k 2 N, 9M > 0 such that, 8j˛j D m, “ j@˛ unk .x/  @˛ unk .y/j2 ˛ d xd y  M < C1 I Œ@ unk  D kx  yknC2  (8.10.79) “ ˛ ˛ 2 j@ unk .x/  @ unk .y/j H) lim inf I Œ@˛ unk  D lim inf d xd y kx  yknC2 k!1 k!1   M < C1

8j˛j D m:

(8.10.80)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

577

j@˛ unk .x/@˛ unk .y/j2 1 Hence, for the sequence of non-negative functions intekD1 kxyknC2 grable on  satisfying (8.10.78) and (8.10.79), we can apply Fatou’s Lemma: Lemma 8.10.2. R Let .fn / be a sequence of non-negative, integrable functions on  (i.e. fn  0,  fn .x/d x  M < 1 8n 2 N) such that Z lim inf fn .x/d x < 1; lim inf fn .x/ D f .x/ a.e. on : n!1



n!1

Then R I. f is integrable on , i.e.  f .x/d x < 1; R R II.  f .x/d x  lim infn!1  fn .x/d x. R R In other words,  lim infn!1 fn .x/d x  lim infn!1  fn .x/d x. Remark 8.10.5. If limn!1 fn .x/ D f .x/ a.e. on , then lim infn!1 fn .x/ D limn!1 fn .x/ D f .x/ a.e. on  by the property of the limit inferior (see Definitions A.0.8.1 and A.0.8.2 in Appendix A). Consequently, by virtue of (8.10.78) and (8.10.79), I and II of Fatou’s Lemma 8.10.2 ˛ u.x/@˛ u.y/j2 follow, i.e. j@ kxyk is integrable on    8j˛j D m and nC2 “ j@˛ u.x/  @˛ u.y/j2 d xd ylim inf I Œ@˛ unk  < 1; 8j˛j D m: I Œ@˛ uD  kx  yknC2 k!1 x¤y

Hence, u 2 H m ./ with m D Œs and I Œ@˛ u < 1 8j˛j D m and for 0 <  < 1 H) u 2 H s ./. Now we are to show that for u 2 H s ./, “ j@˛ .u  un /.x/  @˛ .u  un /.y/j2 ˛ ˛ d xd y ! 0 I Œ@ u  @ un  D kx  yknC2 x¤y as n ! 1, 8j˛j D m. In fact, from (8.10.78a), we also have I Œ@˛ ulk  @˛ un  < " 8n > n0 , 8k > k0 , with lk > lk0 > n0 8j˛j D m. We fix n > n0 . Then, 8 fixed n > n0 , I Œ@˛ ulk  @˛ un  < " 8k > k0 with lk > lk0 > n0 8j˛j D m H)

lim inf I Œ@˛ ulk  @˛ un  < " 8 fixed n > n0 ; 8j˛j D m: k!1

(8.10.81)

Applying (8.10.76) with lk D nk 8k 2 N, 8 fixed n > n0 , lim infk!1 @˛ .ulk  un /.x/ D @˛ .u  un /.x/ a.e. on  8j˛j D m. Then, lim inf k!1

j@˛ .ulk  un /.x/  @˛ .ulk  un /.y/j2 j@˛ .u  un /.x/  @˛ .u  un /.y/j2 D kx  yknC2 kx  yknC2 (8.10.82)

a.e. on    8 fixed n > n0 , 8j˛j D m (see (8.10.78)).

578

Chapter 8 Fourier transforms of distributions and Sobolev spaces

j@˛ .ulk un /.x/@˛ .ulk un /.y/j2 1 Again, for the sequence 8 fixed n > n0 satkD1 kxyknC2 isfying (8.10.81) and (8.10.82), we can apply Fatou’s Lemma 8.10.2 once more and get, 8" > 0, 9n0 D n0 ."/ 2 N such that, 8j˛j D m, “ j@˛ .u  un /.x/  @˛ .u  un /.y/j2 ˛ ˛ I Œ@ u  @ un  D d xd y  kx  yknC2 x¤y

 lim inf I Œ@˛ ulk  @˛ un  < " 8n > n0 k!1

H) limn!1 I Œ@˛ u  @˛ un  D 0 8j˛j D m. Thus we have proved that for u 2 H s ./, ku  un kH m ./ ! 0 with m D Œs and I Œ@˛ u  @˛ un  ! 0 as n ! 1 8j˛j D m H) un ! u in H s ./ as n ! 1. Hence, H s ./ is a Hilbert space. Equivalent norm in Hilbert space H s ./ with s > 0 In some situations it may be convenient to use the norm in H s ./ with s D Œs C , Œs D m 2 N0 , 0   < 1, defined by (see also Definition 8.9.4 and (8.9.41)): X 2 2 2 kukH k@˛ ukH s ./ D kukH m ./ C  ./ 1j˛jm

D

X

k@

˛

2 ukL 2 ./

X

C

0j˛jm

“

1j˛jm

 x¤y

j@˛ u.x/  @˛ u.y/j2 d xd y kx  yknC2 (8.10.83)

which is equivalent to the original norm: 2 kukH s ./

D

X

k@

˛

2 ukL 2 ./

C

0j˛jm

X “ j˛jDm

 x¤y

j@˛ u.x/  @˛ u.y/j2 d xd y: kx  yknC2 (8.10.84)

Hence, in the following we will use either of these two equivalent norms interchangeably, according to our convenience.

8.10.6 Density results in H s ./ Although D.Rn / is dense in H s .Rn / 8s > 0 (see Theorem 8.9.6) (in fact, this density result holds 8s 2 R), D./ is not dense in H s ./ for  ¤ Rn if s > 1=2 (see also (2.15.26d)). But we have: Theorem 8.10.7. Let   Rn be an open set of Rn with Lipschitz continuous boundN is dense in H s ./ 8s > 0, ary  (see Definition D.2.3.1, Appendix D). Then D./ N D ¹ W 9 2 D.Rn / such that  D # º (see Definition 8.10.2). where D./

Section 8.10 Sobolev spaces on  ¤ Rn revisited

579

Proof. For Lipschitz continuous boundary  of , the result follows from Theorem 8.10.1 and Theorem 8.10.9 given later, since H s ./  H s ./ for s > 0. For arbitrary , D./ \ H s ./ is dense in H s ./

(8.10.85)

for s > 0 [12]. Remark 8.10.6. Theorem 8.10.7 also holds for a continuous boundary [18]. Domains with slits do not have a continuous boundary (see Appendix D). For example,  D ¹.x1 ; x2 / W 0 < jx1 j < 1; 0 < jx2 j < 1º in Remark 8.10.2 has a slit along 0 W x1 D 0; 0 < x2 < 1 and consequently does not possess a continuous boundary , and D./ is not dense in H s ./ for s > 0 [12]. Proposition 8.10.9 ([17, p. 5], [18]). Let   Rn be a bounded domain with Lipschitz continuous boundary . Then D./ is dense in H s ./ for 0  s  1=2. In particular, D./ is dense in H 1=2 ./:

(8.10.86)

8.10.7 Dual space H s ./ For  ¤ Rn and s > 1=2, the dual space .H s .//0 of H s ./ cannot be identified with a space of distributions (see Remark 4.2.1, Chapter 4), although .H s .Rn //0  H s .Rn / is a space of distributions 8s > 0 (see Theorem 8.9.7 and Theorem 8.9.6). Consequently, we need:

8.10.8 Space H0s ./ with s > 0 Definition 8.10.8. For s > 0, H0s ./  D./ in the norm of H s ./, i.e. H0s ./ is the closure of D./ in H s ./. Theorem 8.10.8. For s > 0, H0s ./ is a Hilbert space equipped with the inner product h  ;  is; induced by H s ./. Proof. The result follows from Definition 8.10.8, since H0s ./ is a closed subspace of Hilbert space H s ./. Corollary 8.10.1. Let   Rn be a bounded open subset with Lipschitz continuous boundary . Then, for 0 < s  1=2, H0s ./ D H s ./. Proof. The result follows from Proposition 8.10.9 and Definition 8.10.8. Now we are in a position to define H s ./ for s > 0.

580

Chapter 8 Fourier transforms of distributions and Sobolev spaces

8.10.9 Space H s ./ with s > 0 Definition 8.10.9. For s > 0, H s ./ is defined as the (topological) dual space of H0s ./, i.e. .H0s .//0  H s ./ ,! D 0 ./, i.e. for s > 0, H s ./ is a space of distributions (D./ ,! H0s ./, the imbedding ,! being a dense, continuous one). For  with Lipschitz continuous boundary  and for 0 < s  1=2, D./ is dense in H s ./  H0s ./ and H s ./ D .H0s .//0 D .H s .//0 ,! D 0 ./:

(8.10.87)

N is dense in H s ./ 8s > Proposition 8.10.10 ([18]). For arbitrary   Rn , D./ 0. Imbedding result

For 0 < s1 < s2 ,

H s2 ./ ,! H s1 ./ with dense, continuous imbedding ,!:

(8.10.88)

Relation between H s ./ and H s ./ for s > 0 Theorem 8.10.9 ([18]). Let   Rn be a bounded domain with Lipschitz continuous boundary . Then, 8s > 0, 9 an s-extension operator P 2 L.H s ./; H s .Rn // such that .P u/# D u, kP ukH s .Rn /  C kukH s ./ 8u 2 H s ./ and for some C > 0. Consequently, N with norm equivalence; H s ./  H s ./

(8.10.89)

N is defined by (8.10.1). where H s ./ For bounded polygonal domains   R2 , the results of Theorem 8.10.9 hold, since bounded polygons are Lipschitz continuous. For s D m, see the independent proof of Proposition 8.10.8. Remark 8.10.7. For integral m 2 N0 and 1  p  1, W m;p ./ has been defined in (2.15.29)–(2.15.34) with W m;2 ./  H m ./, i.e. by giving a natural extension to the definition of H m ./. Similarly, Definition 8.10.6 of H s ./ for s > 0 can be given a natural extension to define the space W s;p ./ for real s > 0 and 1  p < 1 such that W s;2 ./  H s ./.

8.10.10 Space W s;p ./ for real s > 0 and 1  p < 1 Definition 8.10.10. For real s D Œs C  with Œs D m 2 N0 , 0 <  < 1, and 1  p < 1, W s;p ./ is defined as a subspace of W m;p ./ (see (2.15.28)) by: ² “ j@˛ u.x/  @˛ u.y/jp s;p m;p ./; d xd y 0 Here we collect the important properties of W s;p ./. Property 1 Since, for p D 1, by modifying the values of the functions on a set of points with measure zero, we can identify W s;1 ./ with C m; ./ and write W s;1 ./  C m; ./ with s D m C , m D Œs (see Remark 8.10.8), for s > 0, the case 1  p < 1 is of interest. Hence, for s > 0, we will discuss W s;p ./ for 1  p < 1 (in fact, for 1 < p < 1 later). Theorem 8.10.10. 8 real s > 0 and 1  p < 1, W s;p ./ equipped with the Slobodetskii norm k  ks;p; is a Banach space and separable.

582

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof. See the proof of Theorem 8.10.6 for H s ./  W s;2 ./, i.e. p D 2, since the proof for 1  p < 1 is similar to the one given for p D 2. For 1 < p < 1, W s;p ./ is reflexive for s > 0, but for p D 1; s > 0; W s;p ./ is not reflexive:

(8.10.92)

Hence, in the following we will consider only 1 < p < 1, unless otherwise stated. For p D 2, W s;2 ./ D H s ./ is a Hilbert space. Equivalent norm in W s;p ./ with s > 0, 1  p < 1 p

p

kuks;p; D kukm;p; C

X 1j˛jm

“  x¤y

8u 2 W s;p ./,

j@˛ u.x/  @˛ u.y/jp d xd y kx  yknCp

(8.10.93)

defines a norm in W s;p ./ with s D Œs C , Œs D m 2 N0 , 0   < 1, equivalent to the original Slobodetskii norm in (8.10.90a). Hence, (8.10.90a) and (8.10.93) will be used interchangeably according to our convenience. Property 2 For s D m 2 N0 , k  km;p; is given by (8.10.90b). s;p

Property 3: Space W0 ./ with s > 0, 1 < p < 1 Since D./ is not dense in W s;p ./ for  ¤ Rn or for s > 1=p, the dual space .W s;p .//0 of W s;p ./ will not be a space of distributions (see (4.2.8)). Hence, we need: s;p

Definition 8.10.11. For s > 0 and 1 < p < 1, W0 ./  D./ in the norm of s;p W s;p ./, i.e. D./ is dense in W0 ./ 8s > 0, 1 < p < 1. s;p

Theorem 8.10.11. 8s > 0 and for 1 < p < 1, W0 ./ is a Banach space equipped with the Slobodetskii norm (8.10.90a) or the equivalent norm (8.10.93) induced by W s;p ./. s;p

In general, W0 ./ ¨ W s;p ./.  D Rn

H)

s;p

W0 .Rn /  W s;p .Rn / 8s > 0; 1 < p < 1;

  Rn ; 0  s  1=2; p D 2; H0s ./  H s ./

(8.10.94)

(see Proposition 8.10.9). 1

1

For bounded  with Lipschitz continuous boundary , H02 ./  H 2 ./ (see (8.10.86) and also Property 6 below).

Section 8.10 Sobolev spaces on  ¤ Rn revisited s;p

Equivalent norm in W0 kukW s;p ./ D

 X

0

583

./ with s > 0, 1 < p < 1 k@

˛

p ukLp ./

j˛jDm

1=p X j@˛ u.x/  @˛ u.y/jp C d xd y kx  yknCp j˛jDm

(8.10.95) s;p

defines a norm in W0 ./ with s D Œs C , Œs D m 2 N0 , 0 <  < 1, equivalent to the original norm induced by W s;p ./. Hence, (8.10.90a), (8.10.93) and (8.10.95) will be used interchangeably according to our convenience, i.e. the notations k  ks;p; s;p and k  kW s;p ./ will mean any one of the equivalent norms in W0 ./. 0

Property 4: Dual space W s;q ./ with s > 0, 1

1 p

C

1 q

D 1, 1 < p < 1, 1 < q
0, 1 < p < 1, 1 < q < 1 with p1 C q1 D 1, s;p W s;q ./  .W0 .//0 , i.e. for s > 0, W s;q ./ is identified with the dual s;p s;p space .W0 .//0 of W0 ./ and is a space of distributions. Thus, for s > 0, s;p u 2 W s;q ./ H) hu; viW s;q ./W s;p ./ is well defined 8v 2 W0 ./, h  ;  i 0 s;p being the duality pairing between W s;q ./ and W0 ./. For p D q D 2 and  D Rn , see Theorem 8.9.3. For s > 0, p1 C q1 D 1, 1 < p < 1, s;p

D./ ,! W0 ./ ,! W s;q ./ ,! D 0 ./:

(8.10.96)

Property 5: Imbedding results For fixed p 2 Œ1; 1Œ and 0 < s1 < s2 , W s2 ;p ./ s ;p s ;p ,! W s1 ;p ./, W0 2 ./ ,! W0 1 ./ with dense, continuous imbedding ,!. (8.10.97) Property 6: Density results 







s;p

D./ is dense in W0 ./ for s > 0, 1 < p < 1, by Definition 8.10.11. (8.10.98a) s;p

D.Rn / is dense in W s;p .Rn / and consequently W0 .Rn /  W s;p .Rn / for s > 0, 1 < p < 1. (8.10.98b) For   Rn with Lipschitz continuous boundary , D./ is dense in W s;p ./, s;p and consequently W0 ./  W s;p ./ for 0 < s  1=p, 1 < p < 1. (8.10.98c) For   Rn with a continuous boundary , D./ D ¹ W 9 2 D.Rn / such that D # º is dense in W s;p ./ for s > 0, 1 < p < 1. (8.10.98d)

584

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Remark 8.10.10. Domains  with slits do not have a continuous boundary. For example,  D ¹.x1 ; x2 / W 0 < jx1 j < 1, 0 < jx2 j < 1º in Remark 8.10.2 does not have a continuous boundary owing to the slit along 0 W x1 D 0, 0 < x2 < 1, and (8.10.98d) does not hold. For arbitrary , N \ W s;p ./ is dense in W s;p ./ D./

(8.10.99)

8s > 0 [17], [18]. Property 7: Extension results Theorem 8.10.12. Let   Rn be a bounded open set with Lipschitz continuous boundary . Then, 8s > 0 and for 1  p < 1, 9 a continuous, linear, extension operator Ps W W s;p ./ ! W s;p .Rn / such that .Ps u/# D u and kPs uks;p;Rn  C kuks;p; 8u 2 W s;p ./ and for some C > 0. N s > 0, 1  p < 1 Property 8: Space W s;p ./, Definition 8.10.13. 8s > 0, 1  p < 1, let    Rn be an open subset and N be defined by (see also (8.10.1)): W s;p ./ N D ¹u W u 2 D 0 ./; 9w 2 W s;p .Rn / such that w# D uº: W s;p ./

(8.10.100)

N equipped with the norm k  k Then W s;p ./ s;p; defined by (see also (8.10.1)): kuks;p; D

inf

w2W s;p .Rn / w# Du

kwks;p;Rn

N 8u 2 W s;p ./

(8.10.101)

is a Banach space. Theorem 8.10.13. Let   Rn be a bounded open set with Lipschitz continuous N with boundary . Then, 8s > 0 and for 1  p < 1, W s;p ./ D W s;p ./ s;p kuks;p;  C kuks;p; for some C > 0, 8u 2 W ./. Proof. W s;p ./  W s;p ./: Let u 2 W s;p ./ for s > 0, 1  p < 1. Then, from (8.10.100), 9w 2 W s;p .Rn / such that w# D u. Hence, kuks;p; D kw# ks;p;  kwks;p;Rn < 1 H) u 2 W s;p ./, i.e. W s;p ./  W s;p ./. W s;p ./  W s;p ./: Let u 2 W s;p ./ for s > 0, 1  p < 1. Then, by Theorem 8.10.13, 9Ps u 2 W s;p .Rn / with kPs uks;p;Rn  C kuks;p; for some C > 0. Hence, from (8.10.101), kuks;p;  kPs uks;p;Rn  C kuks;p; < 1 H) u 2 W s;p ./, i.e. W s;p ./  W s;p ./. Hence, combining these results, we get W s;p ./  W s;p ./ with kuks;p;  C kuks;p; .

Section 8.10 Sobolev spaces on  ¤ Rn revisited

585

s 8.10.11 Space H00 ./ with s > 0

In some problems, for example in dealing with traces of functions in polygonal dos mains or domains with corners, we will require a subspace H00 ./ (also denoted by s s e H ./ in Grisvard [17], [18]) of H ./ for s > 0 defined in (8.10.71)–(8.10.72). s ./ of Lions–Magenes [15]. We have accepted the notation H00 Definition 8.10.14. Let    Rn be an open subset of Rn . Then, for real s > 0, the s ./ is defined by the set space H00 s H00 ./ D ¹u W u 2 H s ./ such that its null extension uQ 2 H s .Rn /º; (8.10.102)

which is equipped with the inner product h  ;  i00;s; and norm k  k00;s; given by: 8u 2 H s ./ with u.x/ Q D u.x/ 8x 2  and hu; vi00;s; D hu; Q vi Q s;Rn I

u.x/ Q D 08 x 2 Rn n ; 1=2

kuk00;s; D hu; ui00;s; D kuk Q s;Rn :

(8.10.103) (8.10.104)

s ./ be defined by (8.10.102)–(8.10.104). Then Theorem 8.10.14. For s > 0, let H00 s H00 ./ is a Hilbert space. s ./ with s > 0 Important properties of H00 s Property 1 H00 ./ ,! H s ./ ,! H s ./ 8s > 0, where H s ./ and H s ./ are defined by (8.10.1) and (8.10.71), respectively, with

kuks;  kuks;  kuk00;s;

s 8u 2 H00 ./:

(8.10.105)

m ./ (see (8.10.113) for their equality Property 2 For s D m 2 N, H0m ./  H00 for  with continuous boundary), since u 2 H0m ./ H) uQ 2 H m .Rn / by Theom m rem 2.15.5 H) u 2 H00 ./ H) H0m ./  H00 ./.

Property 3 The norm (8.10.104) is induced by H s ./ only when s D m 2 N. In fact, we have the following result: Lemma 8.10.3 (Grisvard [18]). m I. For s D m 2 N, kukH00 (8.10.106) ./ D kukm; . II. For s D m C , 0 <  < 1, the norm k  k00;s; in (8.10.104) is equivalent to mapping s s ./ 7! kukH00 u 2 H00 ./  1=2 X Z D kuk2s; C w .x/j@˛ u.x/j2 d x  kuks; ; (8.10.107) j˛jDm



586

Chapter 8 Fourier transforms of distributions and Sobolev spaces

where w .x/ is a suitable weight, i.e. s 9 c1 ; c2 > 0 such that c1 kuk00;s;  kukH00 ./  c2 kuk00;s; : (8.10.108)

In order to distinguish the formulae for equivalent norms we have used k  k00;s; s and k  kH00 ./ defined by (8.10.104) and (8.10.107), respectively. III. For bounded   Rn with Lipschitz continuous boundary , the weight function w in (8.10.107) satisfies the following equivalent condition: 9cQ1 ; cQ2 > 0 with cQ1  cQ2 such that cQ1

1 1  w .x/  cQ2 2 2

.x/

.x/

for x 2 ;

(8.10.109)

where .x/ D d.x; / D infy2 d.x; y/ D infy2 kx  ykRn 8x 2  denotes s the distance from x 2  to the boundary  of  and we have, 8u 2 H00 ./,  X Z j@˛ uj2 1=2 s dx  kukH00 cQ1 kuk2s; C ./ 2ı  j˛jDm

 X Z j@˛ uj2  2  cQ2 kuks; C dx ; 2  j˛jDm

(8.10.110) s ./ is defined by (8.10.107). where kukH00 R P ˛ uj2 s d x/1=2 , kukH00 Consequently, .kuk2s; C j˛jDm  j@2 ./ and kuk00;s; s are equivalent norms in H00 ./ for s D m C , 0 <  < 1. Finally,

c1

 X Z j@˛ uj2 1=2 2 dx  kuk00;s; kuks; C 2  j˛jDm



c2



kuk2s;

X Z j@˛ uj2 1=2 C dx 2  j˛jDm

(8.10.111) with c1 ; c2 > 0. Property 4: Density result Theorem 8.10.15 (Grisvard [18]). Let   Rn be an open subset of Rn with contins ./ D D./ in the norm uous boundary . Then, 8s > 0, D./ is dense in H00 k  k00;s; . (8.10.112)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

587

For s D m 2 N, m H00 ./ D H0m ./:

(8.10.113)

m ./ D D./ in the norm k  k m In fact, H00 00;m; , and H0 ./ D D./ in the norm m ./ D k  km; , but k  km; D k  k00;m; (by Lemma 8.10.3). Hence, D./ D H00 m H0 ./. s Property 5: Alternative characterization of H00 ./ for bounded domains with Lipschitz continuous boundary  For s > 0 with s D m C , m 2 N, 0 <  < 1, ˛ we define the space X s ./ by X s ./ D ¹u W u 2 H0s ./; @u 2 L2 ./ 8j˛j D mº equipped with the norm k  kX s ./ :  X Z j@˛ u.x/j2 1=2 2 kukX s ./ D kuks; C dx ; (8.10.114)

2  j˛jDm

where D .x/ denotes the distance of  from x 2 ;  being the Lipschitz continuous boundary of bounded   Rn . Case s  have:

1 2

¤ an integer: Using Hardy’s inequality (Grisvard [18], Tartar [42]), we

Theorem 8.10.16 (Grisvard [18]). For bounded   Rn with Lipschitz continuous boundary , and for s > 0, s  12 ¤ an integer and u 2 H0s ./, @˛ u 2 L2 ./

sj˛j In particular, for 0 < s
0 with s  12 ¤ an integer, s X s ./  H0s ./  H00 ./

(8.10.118)

with norm equivalence. In particular, for 0 < s < 12 , s H00 ./  H0s ./  H s ./:

(8.10.119)

588

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Proof. Since X s ./  H0s ./ by definition, for X s ./  H0s ./ it is sufficient to show that H0s ./  X s ./. In fact, let u 2 H0s ./. Then, for s  12 ¤ an integer, ˛ by Theorem 8.10.16, @u 2 L2 ./ 8j˛j D m,  D s  j˛j H) u 2 X s ./ H) H0s ./  X s ./. Hence, X s ./  H0s ./ with norm equivalence, i.e. k  kX s ./ and k  ks; are equivalent. For s  12 ¤ an integer, the norms k  kX s ./ and k  k00;s; are equivalent on s ./, D./  H0s ./ and k  ks; D./ by Lemma 8.10.3, since D./  H00 s and k  kX s ./ are equivalent on H0s ./. But D./ is dense in H00 ./ in the norm s k  k00;s; (by Theorem 8.10.15) and dense in H0 ./ in the norm k  kX s ./ (equivalent to k  ks; ). Hence, the completions D./ of D./ in two equivalent norms k  k00;s; and k  kX s ./ (by Lemma 8.10.3) represent the same space, i.e. D./  s H00 ./  H0s ./  X s ./ with norm equivalence. s ./  H s ./ and H s ./ D H s ./ For 0 < s < 12 , s  12 ¤ an integer H) H00 0 0 by Corollary 8.10.1, and the result follows. s Remark 8.10.11. For s  12 ¤ an integer, H00 ./ loses its importance by virtue of (8.10.118). For unbounded  D RnC D ¹x W x D .x1 ; : : : ; xn1 ; xn / 2 Rn , xn > 0º, we have: m .Rn / D H m .Rn /. Theorem 8.10.17. Let  D RnC . Then H00 C C 0

Proof. From the following facts: m 1. H0m .RnC / is a closed subspace of H00 .RnC /; m m 2. X00 .RnC / D ¹u W u 2 H00 .RnC /, supp.u/  RnC º is a dense subspace of m n H00 .RC /, which follows from Theorem 6.8.6 and Remark 6.8.1 with necessary modifications; m .RnC /  H0m .RnC /, which is proved with the help of convolutions with 3. X00 suitable regularizing functions, Theorem 6.2.2 and Remark 6.2.2; m m and from X00 .RnC /  H0m .RnC /  H00 .RnC /, the closed subspace H0m .RnC / is dense m n m n m n in H00 .RC / H) H0 .RC / D H00 .RC /.

Case s  12 D an integer: This exceptional case arises in trace theorems on bounded polygonal domains with corners and, hence, is a very important one. First of all, we have s

1 2

D an integer H) Corollary 8.10.2 does not hold

and X s ./ ¤ H0s ./. But we have the important result:

Section 8.10 Sobolev spaces on  ¤ Rn revisited

589

Theorem 8.10.18. Let   Rn be a bounded open subset of Rn with Lipschitz continuous boundary . Then, for s D m C 12 with  D 12 , s  12 D an integer s m 2 N0 , H00 ./  X s ./ with norm equivalence (see Lemma 8.10.3), where @˛ u X s ./ D ¹u W u 2 H0s ./; p 2 L2 ./ 8j˛j D mº   H0s ./

with kukX s ./ D Œkuk2s; C

P

R

j˛jDm 

(8.10.120)

1 j@˛ u.x/j2 d x 2 . 

Proof. s s H00 ./  X s ./: Let u 2 H00 ./ with s  12 D m 2 N0 . Then, by virtue of s the density of D./ in H00 ./ (Theorem 8.10.15) 9 a sequence .k /1 in D./ kD1 such that ku  k k00;s; ! 0 as k ! 1. By Lemma 8.10.3, Œku  k k2s; C R j@˛ u@˛ k j2 P ˛ d x ! 0 as k ! 1 H) k ! u in H s ./, and @pk ! j˛jDm   ˛u @p 

in L2 ./ 8j˛j D m as k ! 1. But .k / is a sequence in D./, which is dense ˛

in H0s ./ H) u 2 H0s ./ and @pu 2 L2 ./ 8j˛j D m H) u 2 X s ./ H) s ./  X s ./. H00 s ./: Let u 2 X s ./ with kuk s s X ./  H00 X ./ defined by (8.10.120). Then ˛u @p s 2 u 2 H0 ./ and  2 L ./ 8j˛j D m. But u 2 H0s ./ H) u 2 H0m ./, since

e

H0s ./ ,! H0m ./ H) @˛ uQ D @˛ u 8j˛j  m (see the proof of Theorem 2.15.5). But @˛ u 2 L2 ./ 8j˛j  m H) @˛ u D @˛ uQ 2 L2 .Rn / 8j˛j  m H) uQ 2 s H m .Rn /. To show that u 2 H00 ./, we are to prove that uQ 2 H s .Rn / with s D 1 m C 12 . Hence, uQ 2 H m .Rn / will belong to H s .Rn / if @˛ uQ 2 H 2 .Rn / 8j˛j D m,

e

e

1

which implies that @˛ uQ D @˛ u 2 H 2 .Rn / 8j˛j D m (since we have already shown

e

1

2 that @˛ uQ D @˛ u 2 L2 .Rn / 8j˛j D m), i.e. @˛ u 2 H00 ./ 8j˛j D m. For this, ˛ we are to prove that 8j˛j D m, k@ uk 1 < C1. From Lemma 8.10.3, 8 fixed 2 H00 ./

j˛j D m, k@˛ uk2

1

2 H00 ./

@˛ u C k p kL2 ./ 1

H 2 ./

D k@˛ uk2

 kuk2s; C

X j˛jDm

@˛ u 2 k p kL2 ./ D kukX (by (8.10.120)). s ./ < C1

1

2 s Hence, 8j˛j D m; @˛ u 2 H00 ./ H) u 2 H00 ./ with s 

1 2

D m 2 N0 .

s An alternative definition of H00 ./ for s > 0

As a consequence of Theorem 8.10.18, Theorem 8.10.16 and Corollary 8.10.2, we have:

590

Chapter 8 Fourier transforms of distributions and Sobolev spaces

Definition 8.10.15. For s > 0 with s D m C , m 2 N0 , 0   < 1 ( D 0 H) s ./ is defined s D m 2 N) and   Rn with Lipschitz continuous boundary , H00 by: s H00 ./

² ³ @˛ u s 2 D X ./ D u W u 2 H0 ./;  2 L ./ 8 j˛j D m

s

(8.10.121)

s with equivalent norms k  k00;s; , k  kH00 ./ and k  kX s ./ defined by (8.10.104), (8.10.107) and (8.10.120), respectively (see (8.10.111) also).

Then the following hold: 

m for s D m, with  D 0, H00 ./ D X m ./ D H0m ./ with kuk00;m; D kukm; (by (8.10.106));



for s 12 ¤ an integer, H00 ./DX s ./DH0s ./ (by (8.10.118));



for s 

1 2

(8.10.122)

s D an integer, i.e.  D 12 , H00 ./ D X s ./   H0s ./ with

1  X Z j@˛ uj2  2 2 s s D kuk D kuk C ; kukH00 X ./ ./ s;



(8.10.123)

j˛jDm

s the norms k  kH00 ./ and k  k00;s; being equivalent ones (see also Lemma 8.10.3).

Let   Rn be a bounded open subset of Rn with Lipschitz continuous boundary . Then, s D 12 ; 32 and so on H) s  12 D an integer and Theorem 8.10.18 is applicable, and 1 

1

kuk00; 1 ; 2

3 

1

2 H00 ./ D ¹u W u 2 H02 ./  H 2 ./,

3

˛u @p 

3

2

j˛jD1

and so on.

(8.10.124)

2 L2 ./ 8j˛j D 1º   H02 ./ with

1 X j@˛ u.x/j2  2 dx I D kuk 3 ; C

.x/ 2



2

1

2 L2 ./º   H02 ./ with

 12  Z ju.x/j2 2 dx I D kuk 1 ; C 2  .x/

2 H00 ./ D ¹u W u 2 H02 ./,

kuk00; 3 ;

u p 

(8.10.125)

Section 8.10 Sobolev spaces on  ¤ Rn revisited

591

s 8.10.12 Dual space .H00 .//0 for s > 0 s Since D./ is dense in H00 ./ for s > 0 by Theorem 8.10.15, the dual space s s 0 .H00 .// of H00 ./ is a space of distributions in D 0 ./. For bounded   Rn with Lipschitz continuous boundary , we have, for s > 0: 

s

1 2

¤ an integer, s ./ D H0s ./ H00



s

1 2



s .H00 .//0 D .H0s .//0 D H s ./I (8.10.126a)

D an integer, s ./   H0s ./ H00



H)

H)

s H s ./  .H00 .//0 I

(8.10.126b)

s s D./ ,! H00 ./ ,! H0s ./ ,! H s ./ ,! .H00 .//0 ,! D 0 ./ with continuous imbedding ,!. (8.10.127) s s s For .H00 .//0 with s > 0, hu; vi.H00 .//0 H00 ./ is well defined 8u 2 s .//0 and 8v 2 H s ./, h  ;  i s s 0 .H00 .H00 .// H00 ./ being the duality between 00 s s 0 (8.10.128) .H00 .// and H00 ./. s;p

Remark 8.10.12. We can define W00 ./ for s > 0, 1 < p < 1 by extending the s definition and properties of H00 ./ for p ¤ 2 as follows. s;p 8.10.13 Space W00 ./ for s > 0, 1 < p < 1 s;p

Definition 8.10.16. For s > 0, 1 < p < 1, W00 ./ is defined by: s;p

W00 ./ D ¹u W u 2 W s;p ./ such that its null extension uQ 2 W s;p .Rn /º; (8.10.129) s;2 s ./. with kuk00;s;p; D kuk Q s;p;Rn . For p D 2, W00 ./  H00 s;p

For s > 0, 1 < p < 1, W00 ./  W s;p ./ (see (8.10.100)–(8.10.101)), m;p m;p W0 ./W00 ./ for 1