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Introduction to Harmonic Analysis
S T U D E N T M AT H E M AT I C A L L I B R A R Y IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 105
Introduction to Harmonic Analysis Ricardo A. Sรกenz
American Mathematical Society Institute for Advanced Study
EDITORIAL COMMITTEE John McCleary Rosa C. Orellana (Chair)
Paul Pollack Kavita Ramanan
2020 Mathematics Subject Classi๏ฌcation. Primary 31B05, 31B10, 31B25, 42A16, 42A20, 42B10, 42B25, 28A20, 28A80.
For additional information and updates on this book, visit www.ams.org/bookpages/stml-105
Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/stml/105
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A Irene
Contents
IAS/Park City Mathematics Institute Preface Chapter 1.
xi xiii
Motivation and preliminaries
1
ยง1.1. The heat equation in equilibrium
1
ยง1.2. Holomorphic functions
3
ยง1.3. Know thy calculus
4
ยง1.4. The Dirichlet principle
9
Exercises Chapter 2.
11 Basic properties
13
ยง2.1. The mean value property
13
ยง2.2. The maximum principle
20
ยง2.3. Poisson kernel and Poisson integrals in the ball
24
ยง2.4. Isolated singularities
29
Exercises
31
Notes
34
Chapter 3.
Fourier series
35
ยง3.1. Separation of variables
35
ยง3.2. Fourier series
38 vii
viii
Contents
ยง3.3. Abel means and Poisson integrals
43
ยง3.4. Absolute convergence
48
ยง3.5. Fejรฉrโs theorem
54
ยง3.6. Mean-square convergence
59
ยง3.7. Convergence for continuous functions
62
Exercises
68
Notes
70
Chapter 4.
Poisson kernel in the half-space
71
ยง4.1. The Poisson kernel in the half-space
71
ยง4.2. Poisson integrals in the half-space
74
ยง4.3. Boundary limits
77
Exercises
80
Notes
82
Chapter 5.
Measure theory in Euclidean space
83
ยง5.1. The need for an integration theory
83
ยง5.2. Outer measure in Euclidean space
85
ยง5.3. Measurable sets and measure
89
ยง5.4. Measurable functions
96
Exercises
102
Notes
103
Chapter 6.
Lebesgue integral and Lebesgue spaces
105
ยง6.1. Integration of measurable functions
105
ยง6.2. Fubiniโs theorem
120 1
124
2
ยง6.4. The Lebesgue space ๐ฟ
130
Exercises
134
Notes
135
ยง6.3. The Lebesgue space ๐ฟ
Chapter 7.
Maximal functions
137
ยง7.1. Indefinite integrals and averages
137
ยง7.2. The HardyโLittlewood maximal function
138
Contents
ix
ยง7.3. The Lebesgue differentiation theorem
143
ยง7.4. Boundary limits of harmonic functions
145
Exercises
148
Notes
150
Chapter 8.
Fourier transform
151
ยง8.1. Integrable functions
151
ยง8.2. The Fourier inversion formula
156
ยง8.3. Mean-square convergence
159
Exercises
164
Notes
165
Chapter 9.
Hilbert transform
167
ยง9.1. The conjugate function
167
ยง9.2. Mean-square convergence
168
ยง9.3. The Hilbert transform of integrable functions
172
ยง9.4. Convergence in measure
179
Exercises
181
Notes
183
Chapter 10.
Mathematics of fractals
185
ยง10.1. Hausdorff dimension
185
ยง10.2. Self-similar sets
191
Exercises
201
Notes
202
Chapter 11.
The Laplacian on the Sierpiลski gasket
203
ยง11.1. Discrete energy on the interval
203
ยง11.2. Harmonic structure on the Sierpiลski gasket
207
ยง11.3. The Laplacian on the Sierpiลski gasket
212
Exercises
219
Notes
220
Chapter 12.
Eigenfunctions of the Laplacian
ยง12.1. Discrete eigenfunctions on the interval
223 224
x
Contents ยง12.2. Discrete eigenfunctions on the Sierpiลski gasket
227
ยง12.3. Dirichlet eigenfunctions
233
Exercises
241
Notes
242
Chapter 13.
Harmonic functions on post-critically finite sets
243
ยง13.1. Post-critically finite sets
243
ยง13.2. Harmonic structures and discrete energy
245
ยง13.3. Discrete Laplacians
250
ยง13.4. The Laplacian on a PCF set
255
Exercises
257
Notes
258
Appendix A. Some results from real analysis
259
ยงA.1. The real line
259
ยงA.2. Topology
261
ยงA.3. Riemann integration
262
ยงA.4. The Euclidean space
265
ยงA.5. Complete metric spaces
267
Acknowledgments
271
Bibliography
273
Index
277
IAS/Park City Mathematics Institute
The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the โRegional Geometry Instituteโ initiative of the National Science Foundation. In mid-1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The annual PCMI summer programs take place in Park City, Utah. Each yearโs PCMI summer program is an intensive three-week session where several different activities take place in parallel. These include individual programs for researchers, graduate students, undergraduate faculty, undergraduate students, and K-12 teachers, as well as a workshop devoted to issues surrounding equity in the mathematics classroom. Over 300 people are in attendance each year. A main goal of PCMI is to make all participants aware of the broad spectrum of mathematical activities and to promote interactions between these groups, often leading to new collaboration and new mentoring arrangements. Each summer a different research topic is chosen as the focus of the Research Program and Graduate Summer School. The Undergraduate Program typically focuses on closely related material. Lecture notes from the Graduate Summer School are published each year in the IAS/Park City Mathematics Series. Course material for the Undergraduate Program at PCMI is published intermittently under the IAS/Park City Mathematical Subseries in the Student Mathematical Library. We are very
xi
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IAS/Park City Mathematics Institute
pleased to make available to a wide audience the expanded versions of these undergraduate lectures, which we believe should be of great value to students everywhere. Rafe Mazzeo, Series Editor March 2023
Preface
This text grew out of the lecture notes for the course Introduction to Harmonic Analysis given at the Undergraduate Summer School during the 2018 Park City Mathematics Institute (IAS/PCMI). The twelve-hour course contained the basic properties of harmonic functions and the explicit solutions to the Laplace equation in special cases. It also contained a study of the behavior of harmonic functions at the boundary of domains, and introduced the Hilbert transform. The last lectures were dedicated to the construction of harmonic functions in fractals. The lectures were then extended to a full semester course at the University of Colima, aimed to junior and senior undergraduate mathematics majors. Besides a more detailed discussion on the previous topics, the course also included a discussion on Fourier series and their convergence, an introduction to Lebesgue measure and integration, the HardyโLittlewood maximal function, the Fourier transform and a more extended study of analysis on fractals, in particular the Laplacian on the Sierpiลski gasket and the construction of its eigenfunctions. The purpose of both the minicourse at IAS/PCMI and the course at Colima is to introduce the modern ideas and problems of harmonic analysis to undergraduate students, from the point of view of harmonic functions. Most books on harmonic and Fourier analysis are too advanced to be appropriate at the undergraduate level, and undergradute textbooks, as those by Thomas William Kรถrner [K8ฬ8] and Elias M. Stein and Rami Shakarchi [SS03], focus on Fourier series and their convergence, rather
xiii
xiv
Preface
than on harmonic functions and their behavior at boundary domains. So this text starts with a motivation to study harmonic functions and the Dirichlet problem. The first chapter discusses the solution to the heat equation in equilibrium, the real and imaginary parts of holomorphic functions, and the minimizing functions of energy, all of which are harmonic functions. This book is intended for junior and senior undergraduates with a basic knowledge of real analysis. It requires familiarity with the properties of complete metric and normed spaces, uniform convergence and density. The needed results are reviewed in Appendix A. It does not require knowledge of measure theory, as the text includes two chapters on measure theory, Lebesgue integration, and approximation theorems. It does requires knowledge of linear algebra, in particular familiarity with vector spaces, subspaces, linear operators and orthogonality. Complex analysis is not required but it is recommended, as a couple of calculations of the Fourier transform of some functions are easily done using line integrals and the residue theorem. The text can be roughly partitioned in three parts. The first part, Chapters 1โ4, discusses the basic properties of harmonic functions and the problem of their behavior at the boundary of their domains. Basic properties as the mean value property, the maximum principle or the classification of singularities are discussed in Chapter 2. Explicit solutions to the Dirichlet problem, in terms of Poisson integrals, are discussed for the ball (Chapter 2) and the half-space (Chapter 4). We also discuss the problem of the behavior at the boundary of these domains in those chapters, for the case of continuous boundary values. In Chapter 3, we discuss the solution of the Dirichlet problem in the disk using Fourier series, so in this chapter we also discuss the problem of their convergence. In particular, we discuss Abel means, Fejรฉrโs theorem and mean-square convergence. We end Chapter 3 with the construction of an example of a continuous function with divergent Fourier series at a point. The second part, Chapters 5โ9, discusses the problem of the behavior of harmonic functions at the boundary of their domains, in particular the half-space, for noncontinuous functions. This requires the use of measure theory, and thus is developed in Chapters 5 and 6. This is not intended to be a comprehensive course in measure theory, but a brief introduction to the main ideas of Lebesgue integration and the basic
Preface
xv
properties of the Lebesgue spaces ๐ฟ1 and ๐ฟ2 . In Chapter 7 we discuss the HardyโLittlewood maximal function and the problem of almost everywhere convergence of Poisson integrals of integrable functions at the boundary of the half-space. In Chapter 9 we introduce the Hilbert transform, which describes the limits at the boundary of the conjugate function to the Poisson integral. We discuss the ๐ฟ1 theory of the Hilbert transform and the concept of operators of weak type. In order to study the ๐ฟ2 theory of the Hilbert transform, we introduce the Fourier transform in the previous Chapter 8, where we discuss its basic properties, as the RiemannโLebesgue lemma for integrable functions and the Plancherel theory of square integrable functions. In the third part, Chapters 10โ13, we discuss the theory of harmonic functions on fractals. We start with the fundamental ideas of self-similarity and Hausdorff dimension in Chapter 10, and then proceed to the study of harmonic analysis, the Laplacian and its eigenfunctions, on the Sierpiลski gasket, in Chapters 11 and 12. We discuss in these chapters the construction of a harmonic structure and the harmonic functions by interpolation. We describe the construction of the Laplacian and an algorithm to construct its eigenfunctions. In particular, we explicitly describe the Dirichlet eigenfunctions on the Sierpiลski gasket. We discuss in Chapter 13 harmonic functions on more general self-similar sets. This text can be used in an introductory course on Harmonic Analysis in several ways. In Colima, semesters are sixteen weeks long, so one has enough time to cover almost all of the material,1 but for shorter semesters we can choose accordingly to the interests and previous knowledge of the audience. Chapters 1โ9 provide an introduction to classical harmonic analysis, and students who already took a course on measure theory may skip Chapters 5 and 6. For a course on analysis on fractals, one may choose Chapters 1โ3 and then move on to Chapters 10โ13, as the first chapters serve as motivation for the study of harmonic functions and eigenfunctions of the Laplacian. Each chapter has a list of exercises and bibliographic and historical notes. Ricardo A. Sรกenz Colima, Mexico, October 2022
1 The Fall 2020 course was transmitted online and is available at the page https://www. facebook.com/HarmonicAnalysis
Chapter 1
Motivation and preliminaries
1.1. The heat equation in equilibrium In this chapter we discuss a number of motivations for the study of harmonic functions, with examples taken from physics to complex analysis. We start in this section with a deduction of the heat equation in equilibrium, using the original argument given by Joseph Fourier in his seminal work Analytical Theory of Heat [Fou55]. Consider the propagation of heat through a solid in space. For example, you can consider a potato in the oven, receiving heat in part of its peel. If you wait sufficiently long, the temperature inside the potato will be in equilibrium; though not necessarily constant in its interior, it will not depend on time. Let ๐ be a small cube inside this solid, which we describe with edges parallel to the axes in โ3 . Suppose two of its opposite vertices are given by (๐ฅ0 , ๐ฆ0 , ๐ง0 ) and (๐ฅ0 +๐, ๐ฆ0 +๐, ๐ง0 +๐) for some small ๐ > 0, and we consider the propagation of heat in ๐, with temperature function ๐ข(๐ฅ, ๐ฆ, ๐ง, ๐ก). Since we are assuming the system is in equilibrium, the temperature does not depend on time, so it is then a function ๐ข(๐ฅ, ๐ฆ, ๐ง) in ๐. We also assume ๐ข is a smooth function in a neighborhood of ๐ (that is, an open set that contains ๐).
1
2
1. Motivation and preliminaries
Figure 1.1. The small cube ๐, with heat propagating in the ๐ฅ direction.
By Newtonโs law of heat flow, the amount of heat that enters through the side ๐ฅ = ๐ฅ0 of ๐ (the left side in Figure 1.1) is proportional to the change in temperature, from hotter to colder, in the ๐ฅ direction on this side, so it is given by ๐๐ข โ๐พ๐2 (๐ฅ0 , ๐ฆ0 , ๐ง0 ), ๐๐ฅ where ๐2 is the surface area of the left side and the proportionality constant ๐พ > 0, which depends only on the material of the solid, is called the conductivity constant. The amount of heat that exits through the side ๐ฅ = ๐ฅ0 + ๐ of ๐ is then given by ๐๐ข (๐ฅ + ๐, ๐ฆ0 , ๐ง0 ). ๐๐ฅ 0 The quantity of heat accumulated in ๐ as a consequence of propagation in the ๐ฅ direction is the difference between these two quantities, โ๐พ๐2
๐๐ข ๐๐ข (๐ฅ0 , ๐ฆ0 , ๐ง0 ) โ ( โ ๐พ๐2 (๐ฅ0 + ๐, ๐ฆ0 , ๐ง0 )) ๐๐ฅ ๐๐ฅ ๐๐ข 2 ๐๐ข (๐ฅ , ๐ฆ , ๐ง )). = ๐พ๐ ( (๐ฅ0 + ๐, ๐ฆ0 , ๐ง0 ) โ ๐๐ฅ ๐๐ฅ 0 0 0 By the mean value theorem, there exists 0 < ๐ฟ < ๐ such that โ ๐พ๐2
๐๐ข ๐๐ข ๐2 ๐ข (๐ฅ0 + ๐, ๐ฆ0 , ๐ง0 ) โ (๐ฅ0 , ๐ฆ0 , ๐ง0 ) = ๐ 2 (๐ฅ0 + ๐ฟ, ๐ฆ0 , ๐ง0 ), ๐๐ฅ ๐๐ฅ ๐๐ฅ so the propagation of heat through ๐ in the ๐ฅ direction is then ๐พ๐3
๐2 ๐ข (๐ฅ + ๐ฟ, ๐ฆ0 , ๐ง0 ). ๐๐ฅ2 0
1.2. Holomorphic functions
3
Similarly, there exist 0 < ๐, ๐ < ๐ so that the propagation of heat through ๐ in the ๐ฆ and ๐ง directions is given by ๐พ๐3
๐2 ๐ข (๐ฅ , ๐ฆ + ๐, ๐ง0 ) ๐๐ฆ2 0 0
and
๐พ๐3
๐2 ๐ข (๐ฅ , ๐ฆ , ๐ง + ๐), ๐๐ง2 0 0 0
respectively, and the total propagation is then given by ๐พ๐3 (
๐2 ๐ข ๐2 ๐ข ๐2 ๐ข (๐ฅ + ๐ฟ, ๐ฆ , ๐ง ) + (๐ฅ , ๐ฆ + ๐, ๐ง ) + (๐ฅ , ๐ฆ , ๐ง + ๐)). 0 0 0 0 0 0 ๐๐ฅ2 ๐๐ฆ2 ๐๐ง2 0 0 0
Since the system is in equilibrium, the total propagation must be equal to 0. As we are assuming that ๐ข is a smooth function, its partial derivatives are continuous, so we obtain, as ๐ โ 0, the equation ๐2 ๐ข ๐2 ๐ข ๐2 ๐ข + + 2 =0 ๐๐ฅ2 ๐๐ฆ2 ๐๐ง
(1.1) at the point (๐ฅ0 , ๐ฆ0 , ๐ง0 ).
Equation (1.1) is called the Laplace equation. We can also write it as ฮ๐ข = 0, where the differential operator ฮ is given by ฮ๐ข =
๐2 ๐ข ๐2 ๐ข ๐2 ๐ข + + 2. ๐๐ฅ2 ๐๐ฆ2 ๐๐ง
ฮ๐ข is called the Laplacian of ๐ข. The solutions of equation (1.1) are called harmonic functions.
1.2. Holomorphic functions In this section we observe that harmonic functions also appear in complex analysis. Recall that ๐ is holomorphic (or analytic) in an open set ๐ท โ โ if, for each ๐ง โ ๐ท, its derivative (1.2)
๐(๐ง + โ) โ ๐(๐ง) โ โโ0
๐โฒ (๐ง) = lim
exists. If we write the holomorphic function ๐ as ๐(๐ง) = ๐ข(๐ฅ, ๐ฆ) + ๐๐ฃ(๐ฅ, ๐ฆ), where ๐ง = ๐ฅ+๐๐ฆ and ๐ข and ๐ฃ are its real and imaginary parts, respectively, then ๐ข and ๐ฃ satisfy the CauchyโRiemann equations (1.3)
๐๐ข ๐๐ฃ = ๐๐ฅ ๐๐ฆ
and
๐๐ฃ ๐๐ข =โ . ๐๐ฅ ๐๐ฆ
4
1. Motivation and preliminaries
These equations follow directly from the differentiability of ๐. Indeed, if we take the limit in (1.2) by approaching โ โ 0 with real numbers, we obtain ๐๐ข ๐๐ฃ ๐โฒ (๐ง) = (๐ฅ, ๐ฆ) + ๐ (๐ฅ, ๐ฆ). ๐๐ฅ ๐๐ฅ Meanwhile, if we approach โ โ 0 with purely imaginary numbers, we get 1 ๐๐ข ๐๐ฃ ๐๐ฃ ๐๐ข ๐โฒ (๐ง) = (๐ฅ, ๐ฆ) + (๐ฅ, ๐ฆ) = (๐ฅ, ๐ฆ) โ ๐ (๐ฅ, ๐ฆ). ๐ ๐๐ฆ ๐๐ฆ ๐๐ฆ ๐๐ฆ As these two expressions for ๐โฒ (๐ง) must be equal, we obtain (1.3). Assuming ๐ข and ๐ฃ are smooth functions,1 we can differentiate the CauchyโRiemann equations and get ๐2 ๐ข ๐2 ๐ฃ = ๐๐ฅ๐๐ฆ ๐๐ฅ2
and
๐2 ๐ฃ ๐2 ๐ข = โ 2. ๐๐ฆ๐๐ฅ ๐๐ฆ
All mixed derivatives are continuous, so they must be equal and thus ๐2 ๐ข ๐2 ๐ข = โ . ๐๐ฅ2 ๐๐ฆ2 Therefore ๐ข is a harmonic function. We can similarly verify that ๐ฃ is harmonic. As ๐ฃ is the imaginary part of a holomorphic function of which ๐ข is the real part, we say that the function ๐ฃ is a conjugate harmonic function to ๐ข. Note that conjugate harmonic functions are not unique, because adding any constant to ๐ฃ will give another conjugate harmonic function. Also, observe that โ๐ข is the conjugate harmonic function to ๐ฃ. Under appropiate conditions on the set ๐ท, one can prove that every harmonic ๐ข has a conjugate harmonic function ๐ฃ. See Exercise (3) for the special case of the complex plane ๐ท = โ. We will dedicate Chapter 9 to the study of the properties of conjugate harmonic functions in the upper half-plane.
1.3. Know thy calculus Before moving on, letโs dedicate a section to set the notation used in this text, and review some of the results from advanced calculus that weโll need later on. This will just be a quick summary of these results, so we 1 It is a fact, proven in any basic complex analysis text (see [Gam01], for example), that both แต and ๐ฃ are smooth functions whenever ๐ is holomorphic.
1.3. Know thy calculus
5
invite the reader to consult advanced calculus texts, such as [Fle77] or [Spi65] for the details and proofs. We denote the ๐-dimensional Euclidean space by โ๐ . Thus โ๐ = {๐ฅ = (๐ฅ1 , ๐ฅ2 , . . . , ๐ฅ๐ ) โถ ๐ฅ๐ โ โ}. We will usually denote the points in the plane โ2 and in the space โ3 by (๐ฅ, ๐ฆ) and (๐ฅ, ๐ฆ, ๐ง), respectively. We denote the Euclidean norm of a vector ๐ฅ โ โ๐ by |๐ฅ|. Thus |๐ฅ| = โ๐ฅ12 + ๐ฅ22 + . . . + ๐ฅ๐2 . For ๐ฅ โ โ๐ , ๐ฅโฒ is the point in โ๐โ1 formed by the first ๐ โ 1 coordinates of ๐ฅ. We can thus write ๐ฅ = (๐ฅโฒ , ๐ฅ๐ ). If we need to explicitly distinguish the last coordinate, then we refer to โ๐+1 = {(๐ฅ, ๐ก) โถ ๐ฅ โ โ๐ , ๐ก โ โ}. We denote by โ๐+1 the upper half-space of points (๐ฅ, ๐ก) โ โ๐+1 with + ๐ก > 0. The open ball of radius ๐ > 0 centered at ๐ฅ0 is given by ๐ต๐ (๐ฅ0 ) = {๐ฅ โ โ๐ โถ |๐ฅ โ ๐ฅ0 | < ๐}. If ๐ฅ0 = 0, we simply denote it by ๐ต๐ . If, in addition, ๐ = 1, we denote it by ๐น. The sphere of radius ๐ > 0 centered at ๐ฅ0 is given by ๐๐ (๐ฅ0 ) = {๐ฅ โ โ๐ โถ |๐ฅ โ ๐ฅ0 | = ๐}, and we denote it by ๐๐ if ๐ฅ0 = 0, and by ๐ if we also have ๐ = 1. For an open set ฮฉ โ โ๐ , ๐ถ(ฮฉ) is the space of continuous functions ฬ the space of continuous functions on its closure. We in ฮฉ, and ๐ถ(ฮฉ) ๐ denote by ๐ถ (ฮฉ) the space of ๐-continuously differentiable functions in ฮฉ, and by ๐ถ โ (ฮฉ) the space of smooth functions. Note that ๐ถ โ (ฮฉ) =
โ
๐ถ ๐ (ฮฉ).
๐โฅ1
We denote by ๐ถ๐โ (ฮฉ) the space of smooth functions with compact support in ฮฉ. That is, ๐ โ ๐ถ๐โ (ฮฉ) if ๐ is smooth in ฮฉ and there exists a compact subset ๐พ โ ฮฉ such that ๐(๐ฅ) = 0 for all ๐ฅ โ ๐พ. In particular, we say that ๐ is zero โclose to the boundary.โ
6
1. Motivation and preliminaries
We denote the partial derivative of ๐ with respect to ๐ฅ๐ either by ๐๐ , by ๐๐ฅ๐ ๐ or simply by ๐๐ ๐, if there is no confusion. The gradient of a ๐๐ฅ๐ function ๐ is given by the vector โ๐ = (๐1 ๐, ๐2 ๐, . . . , ๐๐ ๐). Note that its norm is given by |โ๐| = (|๐1 ๐|2 + |๐2 ๐|2 + . . . + |๐๐ ๐|2 )
1/2
.
If ๐ผ = (๐ผ1 , ๐ผ2 , . . . , ๐ผ๐ ) is a multi-index, where each ๐ผ๐ โ โ, we define ๐ฅ๐ผ as the monomial ๐ผ ๐ผ ๐ผ ๐ฅ ๐ผ = ๐ฅ 1 1 ๐ฅ2 2 โฏ ๐ฅ ๐ ๐ and ๐๐ผ ๐ as the higher order derivative ๐ผ
๐ผ
๐ผ
๐๐ผ ๐ = ๐1 1 ๐2 2 โฏ ๐๐ ๐ ๐. The order of the multi-index ๐ผ is given by |๐ผ| = ๐ผ1 + ๐ผ2 + . . . + ๐ผ๐ . A hypersurface in โ๐ is a differentiable manifold ๐ of dimension ๐ โ 1. Locally, for each ๐ฅ0 โ ๐, there exists an open set ๐ that contains ๐ฅ0 such that ๐ โฉ ๐ is the solution set to the equation (1.4)
๐(๐ฅ) = 0,
for some continuously differentiable function ๐ in ๐ with โ๐ โ 0. By the implicit function theorem, and relabeling the coordinates if needed, we can assume ๐ is of the form ๐(๐ฅ) = ๐ฅ๐ โ ๐(๐ฅโฒ ), and thus ๐ โฉ ๐ is given by (1.5)
๐ฅ๐ = ๐(๐ฅโฒ ).
We say that ๐ is a ๐ถ ๐ -hypersurface if the function ๐ above is in ๐ถ ๐ (๐) (and hence ๐ is a ๐ถ ๐ function in its domain). A domain in โ๐ is an open and connected subset ฮฉ โ โ๐ . The domain ฮฉ is a ๐ถ ๐ -domain if its boundary ๐ฮฉ is a ๐ถ ๐ -hypersurface. If ฮฉ is a ๐ถ 1 -domain and ๐ฅ0 โ ๐ฮฉ, then the normal vector at ๐ฅ0 is the unit vector ๐(๐ฅ0 ) orthogonal to the hypersuface ๐ฮฉ pointing outwards of ฮฉ (Figure 1.2). Thus, โ๐ ๐=ยฑ , |โ๐| where ๐ is a function that describes ๐ฮฉ locally near ๐ฅ0 , as in (1.4).
1.3. Know thy calculus
7
Figure 1.2. The normal vector ๐ at a point in the boundary of ฮฉ.
Example 1.6. The open ball ๐ต๐
(๐ฅ0 ) of radius ๐
and centered at ๐ฅ0 โ โ๐ is a ๐ถ 1 -domain (in fact, a ๐ถ ๐ -domain for every ๐), with boundary equal to the sphere ๐ ๐
(๐ฅ0 ). Note that ๐ is the solution set to the equation ๐ฅ12 + ๐ฅ22 + . . . + ๐ฅ๐2 = 1. Hence, for ๐ฅ โ ๐, ๐(๐ฅ) = ๐ฅ. The surface measure on a hypersurface ๐ is denoted by ๐๐. Locally, if ๐ is as in (1.5), we have that ๐๐ = โ1 + |โ๐|2 ๐๐ฅโฒ . 1.7. One can integrate over โ๐ (or over a subset with rotational symmetry) by using spherical coordinates. If we write a point ๐ฅ โ 0 in โ๐ as ๐ฅ = ๐๐, where ๐ = |๐ฅ| > 0 and ๐ = ๐ฅ/|๐ฅ| โ ๐, then โ
โซ ๐(๐ฅ)๐๐ฅ = โซ โซ ๐(๐๐)๐๐(๐)๐๐โ1 ๐๐. โ๐
0
๐
1.8. The area of the unit sphere in โ๐ is denoted by ๐๐ . Thus, ๐๐ = โซ ๐๐. ๐
We leave it as an exercise (Exercise (7)) to prove that ๐๐ =
2๐๐/2 , ฮ(๐/2)
8
1. Motivation and preliminaries
where ฮ(๐ ) is the gamma function given by โ
ฮ(๐ ) = โซ ๐โ๐ก ๐ก๐ โ1 ๐๐ก, 0
for every ๐ > 0. We will also make use of Theorem 1.9. Theorem 1.9 (Divergence theorem). Let ฮฉ โ โ๐ be a bounded ๐ถ 1 domain and ๐น a continuously differentiable vector field defined in a neighborhood of ฮฉ.ฬ Then โซ โ โ
๐น๐๐ฅ = โซ ๐น โ
๐๐๐.
(1.10)
โฆ
๐โฆ
If ๐น = (๐น 1 , ๐น 2 , . . . , ๐น ๐ ), then โ โ
๐น = ๐1 ๐น 1 + ๐2 ๐น 2 + . . . + ๐ ๐ ๐น ๐ is called the divergence of ๐น, and is also denoted by div ๐น. Note that the divergence theorem is a multi-dimensional version of the fundamental theorem of calculus. Indeed, in the line โ, ฮฉ is just an open interval, say ฮฉ = (๐, ๐), the normal vector at its boundary is given by ๐(๐) = โ1, ๐(๐) = 1, and โ โ
๐น is the derivative of ๐น, so (1.10) is ๐
โซ ๐น โฒ (๐ฅ)๐๐ฅ = โ๐น(๐) + ๐น(๐). ๐
1.11. Taking ๐น = (0, 0, . . . , ๐ข๐ฃ, . . . , 0), where the nonzero component is the ๐th term, we obtain the formula for integration by parts: โซ โฆ
๐๐ข ๐๐ฃ ๐ฃ๐๐ฅ = โซ ๐ข๐ฃ๐ ๐ ๐๐ โ โซ ๐ข ๐๐ฅ. ๐๐ฅ๐ ๐๐ฅ ๐ ๐โฆ โฆ
Theorem 1.12 (Greenโs identities). Let ฮฉ be a bounded ๐ถ 1 -domain in โ๐ . (1) If ๐ข is continuously differentiable and ๐ฃ is twice continuously differentiable in a neighborhood of ฮฉ,ฬ then (1.13)
โซ (๐ขฮ๐ฃ + โ๐ข โ
โ๐ฃ)๐๐ฅ = โซ ๐ข๐๐ ๐ฃ๐๐, โฆ
๐โฆ
where ๐๐ ๐ฃ = โ๐ฃ โ
๐ is the normal derivative of ๐ฃ at the boundary of ฮฉ.
1.4. The Dirichlet principle
9
(2) If ๐ข and ๐ฃ are twice continuously differentiable in a neighborhood of ฮฉ,ฬ then (1.14)
โซ (๐ขฮ๐ฃ โ ๐ฃฮ๐ข)๐๐ฅ = โซ (๐ข๐๐ ๐ฃ โ ๐ฃ๐๐ ๐ข)๐๐. โฆ
๐โฆ
Theorem 1.12 follows almost immediately from the divergence theorem (or 1.11), and we leave it as an exercise (Exercise (9)).
1.4. The Dirichlet principle The Green identities provide us with another motivation for the study of harmonic functions: they are minimizers of energy. Let ฮฉ be a ๐ถ 1 domain. We define the energy form on ฮฉ as the bilinear form (1.15)
โฐ(๐ข, ๐ฃ) = โซ โ๐ข โ
โ๐ฃ๐๐ฅ, โฆ
for smooth functions ๐ข and ๐ฃ in a neighborhood of ฮฉฬ (we denote the ฬ The energy of the function ๐ข, denoted space of such functions as ๐ถ โ (ฮฉ)). simply as โฐ(๐ข), is then given by โฐ(๐ข) = โฐ(๐ข, ๐ข) = โซ |โ๐ข|2 ๐๐ฅ. โฆ
We now consider the following question: can we find the function ๐ข that minimizes โฐ(๐ข), given its values at the boundary ๐ฮฉ of ฮฉ? That ฬ such that ๐ข|๐โฆ = ๐ is, given a function ๐ defined on ๐ฮฉ, find ๐ข โ ๐ถ โ (ฮฉ) and ฬ and ๐ฃ|๐โฆ = ๐}. โฐ(๐ข) = min{โฐ(๐ฃ) โถ ๐ฃ โ ๐ถ โ (ฮฉ) It is clear that we cannot expect the above problem to always have a ฬ ๐ cannot be arbitrary because solution ๐ข. First, as we require ๐ข โ ๐ถ โ (ฮฉ), it must be the restriction of such a function to ๐ฮฉ. Moreover, although it is true that the set ฬ and ๐ฃ|๐โฆ = ๐} {โฐ(๐ฃ) โถ ๐ฃ โ ๐ถ โ (ฮฉ) is bounded from below, because โฐ(๐ฃ) โฅ 0 for any smooth function ๐ฃ, it is not clear whether it has a minimum or not. However, we have the following fact: in the case when โฐ takes its minimum at ๐ข, then ๐ข is a harmonic function in ฮฉ, that is a function that satisfies the equation ฮ๐ข = 0
10
1. Motivation and preliminaries
in ฮฉ. To prove this, suppose โฐ takes its minimum at ๐ข for given values at ๐ฮฉ. Now, for any function ๐ฃ โ ๐ถ๐โ (ฮฉ) and any ๐ก โ โ, the function ๐ข + ๐ก๐ฃ is smooth and has the same values as ๐ข at the boundary. Since โฐ(๐ข) is minimal, we have that โฐ(๐ข) โค โฐ(๐ข + ๐ก๐ฃ). Hence, as a function of ๐ก, the function ๐ผ(๐ก) = โฐ(๐ข+๐ก๐ฃ) takes its minimum value at ๐ก = 0, and thus ๐ผ โฒ (0) = 0.
(1.16) Now
๐ผ(๐ก) = โฐ(๐ข + ๐ก๐ฃ) = โซ โ(๐ข + ๐ก๐ฃ) โ
โ(๐ข + ๐ก๐ฃ)๐๐ฅ โฆ
= โซ |โ(๐ข)|2 ๐๐ฅ + 2๐ก โซ โ๐ข โ
โ๐ฃ๐๐ฅ + ๐ก2 โซ |โ๐ฃ|2 ๐๐ฅ, โฆ
โฆ
โฆ
so ๐ผ โฒ (๐ก) = 2 โซ โ๐ข โ
โ๐ฃ๐๐ฅ + 2๐ก โซ |โ๐ฃ|2 ๐๐ฅ โฆ
โฆ
and (1.16) implies โซ โ๐ข โ
โ๐ฃ๐๐ฅ = 0. โฆ
By the Green identity (1.13), we thus have (1.17)
โซ ๐ฃฮ๐ข๐๐ฅ = โซ ๐ฃ๐๐ ๐ข๐๐ = 0, โฆ
๐โฆ
because ๐ฃ is zero near the boundary. Moreover, since (1.17) holds for every ๐ฃ โ ๐ถ๐โ (ฮฉ), we conclude ฮ๐ข = 0, and thus ๐ข is harmonic. We have left many open questions in the discussion above. We have mentioned that we cannot expect to have a minimizer ๐ข of the energy form satisfying that ๐ข|๐โฆ = ๐ for any function ๐. However, as a minimizer is a harmonic function, this leads to the following problem: given a domain ฮฉ and a function ๐ defined on ๐ฮฉ, find a harmonic function ๐ข in ฮฉ such that it is equal to ๐ on the boundary. This is known as the Dirichlet problem, in honor of the french mathematician Lejeune Dirichlet. It opens a handful of questions, such as the following: โข For which domains ฮฉ can we solve the Dirichlet problem? For which functions on its boundary does the solution exist?
Exercises
11
โข If ๐ข is a harmonic function in ฮฉ, what can we say about its behavior at the boundary of ฮฉ? Does it extend continuously to ๐ฮฉ? Throughout this text we will be discussing results related to the previous questions. In particular, we will focus our attention to harmonic functions in the domains ฮฉ = ๐น, the unit ball, and ฮฉ = โ๐+1 + , the upper half-space, and the behavior of such harmonic functions at the boundaries of their domains. The fact that the minimizers of the energy form โฐ are harmonic functions is called the Dirichlet principle. The first to give it this name was Bernard Riemann in [Rie51], who used this fact to prove the result in complex analysis that we now know as the Riemann mapping theorem. See, for example, [Ull08] for a study of the Riemann mapping theorem and its relation to the Dirichlet problem.
Exercises (1) Let ๐
be a rotation in the plane. (a) Consider the change of variables (๐, ๐) = ๐
(๐ฅ, ๐ฆ). Then ๐2 ๐ข ๐2 ๐ข ๐2 ๐ข ๐2 ๐ข + = + . ๐๐2 ๐๐ฅ2 ๐๐ฆ2 ๐๐2 (b) If ๐ข is harmonic, then ๐ข โ ๐
is also harmonic. (2) Let (๐, ๐) be the polar coordinates of the plane. Then ฮ๐ข =
๐2 ๐ข 1 ๐๐ข 1 ๐2 ๐ข + + . ๐ ๐๐ ๐2 ๐๐2 ๐๐2
(3) Let ๐ข be a harmonic function in โ2 . Then there exists a conjugate harmonic function ๐ฃ to ๐ข. (Hint: Consider a line integral of the 1form ๐๐ข ๐๐ข โ ๐๐ฅ + ๐๐ฆ.) ๐๐ฆ ๐๐ฅ (4) If ๐ฃ 1 and ๐ฃ 2 are conjugate to ๐ข in the plane, then ๐ฃ 1 โ ๐ฃ 2 is constant. (5) (a) If 0 is conjugate to ๐ข in the plane, then ๐ข is constant. (b) If ๐ is holomorphic in โ and real valued, then ๐ is constant. (6) Let ฮ(๐ ) be the gamma function.
12
1. Motivation and preliminaries (a) Integrate by parts to verify the identity ฮ(๐ + 1) = ๐ ฮ(๐ ). (b) For every ๐ โ โค+ , ฮ(๐) = (๐ โ 1)!.
(7) (a) Use polar coordinates to verify the identity 2
โซ ๐โ๐|๐ฅ| ๐๐ฅ = 1. โ2
(b) For every dimension ๐, 2
โซ ๐โ๐|๐ฅ| ๐๐ฅ = 1. โ๐
(c) Use spherical coordinates to verify 2๐๐/2 . ฮ(๐/2) (8) Use integration in spherical coordinates, fact 1.7, to prove that the volume of the unit ball ๐น is given by ๐๐ =
โซ ๐๐ฅ = ๐น
๐๐ . ๐
(9) Prove Theorem 1.12. (10) Consider the unit interval and, for smooth functions in [0, 1], define the form 1
โฐ(๐) = โซ ๐โฒ (๐ฅ)2 ๐๐ฅ. 0
(a) The minimizers of this form, given the values of ๐ at ๐ฅ = 0 and ๐ฅ = 1, are the linear functions ๐(๐ฅ) = ๐๐ฅ + ๐. (b) If โฐ(๐) is a minimum, then โฐ(๐) = (๐(1) โ ๐(0))2 .
Chapter 2
Basic properties
2.1. The mean value property Let ฮฉ โ โ๐ be an open set. As discussed in Chapter 1, we say that a twice differentiable function ๐ข is harmonic in ฮฉ if it satisfies ฮ๐ข = ๐12 ๐ข + ๐22 ๐ข + . . . + ๐๐2 ๐ข = 0 in ฮฉ. Example 2.1. Any linear function ๐ข = ๐1 ๐ฅ1 + ๐2 ๐ฅ2 + . . . + ๐๐ ๐ฅ๐ is harmonic in โ๐ , as all of its second derivatives are zero. Observe that, in the case ๐ = 1, the linear functions ๐ข(๐ฅ) = ๐๐ฅ + ๐ are precisely the functions that satisfy ๐ขโณ (๐ฅ) = 0, so the only harmonic functions in โ (or in any interval in the real line) are the linear functions. Example 2.2. The quadratic polynomial ๐ข(๐ฅ, ๐ฆ) = ๐ฅ2 โ ๐ฆ2 is harmonic in โ2 , as its second derivatives are equal to ๐12 ๐ข = 2 and ๐22 ๐ข = โ2. Note that ๐ข is the real part of the holomorphic function ๐(๐ง) = ๐ง2 . As observed in Section 1.2, the real and imaginary parts of an analytic function are harmoinc. The imaginary part of ๐ง2 , and thus a conjugate harmonic to ๐ข, is ๐ฃ(๐ฅ, ๐ฆ) = 2๐ฅ๐ฆ. Similarly, the functions ๐ข(๐ฅ, ๐ฆ) = โ((๐ฅ + ๐๐ฆ)๐ )
and
๐ฃ(๐ฅ, ๐ฆ) = โ((๐ฅ + ๐๐ฆ)๐ ),
the real and imaginary parts of ๐ง๐ , are harmonic in โ2 for each ๐ โ โ. 13
14
2. Basic properties
Example 2.3. The function ๐ข(๐ฅ, ๐ฆ) = sin ๐ฅ sinh ๐ฆ is harmonic in โ2 , because and ๐22 ๐ข = ๐ข. ๐12 ๐ข = โ๐ข Note that ๐ข is the imaginary part of the holomorphic function โ cos ๐ง. It is easy to see that the harmonic functions in any open set in โ๐ form a vector space because, if ๐ข and ๐ฃ are harmonic, then any linear combination ๐ผ๐ข + ๐ฝ๐ฃ of ๐ข and ๐ฃ is also a harmonic function. Moreover, the space of harmonic functions in โ๐ is invariant under translations and orthogonal transformations (see Exercises (1) and (2)). We observed in Example 2.1 that the linear functions are harmonic functions and, in fact they are the only harmonic functions in โ. We now make a rather immediate observation: if ๐ข is linear, say in the interval [๐, ๐], then its value at the midpoint (๐ + ๐)/2 is the average of its values at ๐ and ๐, ๐+๐ ๐ข(๐) + ๐ข(๐) ๐ข( . )= 2 2 It turns out that this is true in every dimension. Theorem 2.4 (Mean value property). Let ๐ข be a harmonic function in a neighborhhod of the closed ball ๐ต๐ฬ (๐ฅ0 ). Then (2.5)
๐ข(๐ฅ0 ) =
1 โซ ๐ข(๐)๐๐(๐). |๐๐ (๐ฅ0 )| ๐ (๐ฅ ) ๐
0
In other words, the average of the values of ๐ข over any sphere around ๐ฅ0 is equal to ๐ข(๐ฅ0 ). In the identity (2.5), ๐๐ is the surface measure on the sphere ๐๐ (๐ฅ0 ), and |๐๐ (๐ฅ0 )| is its surface area, |๐๐ (๐ฅ0 )| = ๐๐ ๐๐โ1 . With an appropiate change of variables, we can also write (2.5) as (2.6)
๐ข(๐ฅ0 ) =
1 โซ ๐ข(๐ฅ0 + ๐๐)๐๐(๐), ๐๐ ๐
where ๐ is the unit sphere around the origin.
2.1. The mean value property
15
2.7. Integrating over the ball ๐ต๐ (๐ฅ0 ) in spherical coordinates, it follows from (2.5) that, under the same assumptions of Theorem 2.4, ๐ข(๐ฅ0 ) =
1 โซ |๐ต๐ (๐ฅ0 )| ๐ต
๐ข(๐ฅ)๐๐ฅ =
๐ (๐ฅ0 )
๐ โซ ๐ข(๐ฅ0 + ๐๐ฅ)๐๐ฅ, ๐๐ ๐น
where ๐น is the unit ball centered at the origin. We leave this as an exercise (Exercise (3)). Again, this identity is immediate in โ (Exercise (4)). Proof of Theorem 2.4. We only need to prove the case ๐ โฅ 2, by the observations made before the statement of the theorem. By translating by ๐ฅ0 , we can assume ๐ฅ0 = 0 (see Exercise (1)). For 0 < ๐ < ๐, let ฮฉ = ๐ต๐ โงต ๐ต๐ฬ , where ๐ต๐ = ๐ต๐ (0) and ๐ต๐ = ๐ต๐ (0), as in Figure 2.1. Define
Figure 2.1. The domain ฮฉ = ๐ต๐ โงต ๐ต๐ฬ . On a point in ๐ ๐ , the normal vector ๐ points away from the origin, while on a point in ๐ ๐ points towards the origin.
the function ๐ฃ in โ๐ โงต {0} by log |๐ฅ| ๐ฃ(๐ฅ) = { 2โ๐ |๐ฅ|
๐=2
๐ โฅ 3. ๐ฅ Note that, for any ๐ฅ โ ฮฉ, โ๐ฃ(๐ฅ) = ๐ ๐ ๐ , where ๐ ๐ = 1 if ๐ = 2 and |๐ฅ| ๐ ๐ = 2 โ ๐ if ๐ โฅ 3. Also, ๐ฮฉ = ๐๐ โ ๐๐ , where we have written ๐๐ and ๐๐ for ๐๐ (0) and ๐๐ (0), respectively (as oriented manifolds,1 see Figure 2.1), 1 We are only interested, at this moment, in the fact that the normal vectors on โ๐๐ point opposite to those at ๐๐ .
16
2. Basic properties
and
๐ฅ ๐(๐ฅ) = { ๐ ๐ฅ โ ๐
on ๐๐ on โ ๐๐ .
Thus
๐๐ on ๐๐ ๐โ1 ๐ ๐๐ ๐ฃ = { ๐ ๐ โ ๐โ1 on โ ๐๐ . ๐ We can also verify explicitly that ฮ๐ฃ = 0 (Exercise (5)), and hence โซ (๐ขฮ๐ฃ โ ๐ฃฮ๐ข)๐๐ฅ = 0. โฆ
Applying Greenโs identity (1.14), and the previous explicit calculations, we obtain 0 = โซ (๐ข๐๐ ๐ฃ โ ๐ฃ๐๐ ๐ข)๐๐ ๐โฆ
= โซ (๐ข ๐๐
๐๐ ๐๐ โ ๐๐ (๐)๐๐ ๐ข)๐๐ โ โซ (๐ข ๐โ1 โ ๐๐ (๐)๐๐ ๐ข)๐๐, ๐๐โ1 ๐ ๐๐
where log ๐ ๐ = 2 ๐๐ (๐ ) = { 1 ๐ โฅ 3, ๐ ๐โ2 for ๐ = ๐ or ๐ = ๐, which are constant over ๐๐ and ๐๐ . Thus, since the surface integral of ๐๐ ๐ข over a sphere is zero (Exercise (6)), we obtain ๐๐ ๐๐ โซ ๐ข๐๐ = ๐โ1 โซ ๐ข๐๐ ๐๐โ1 ๐๐ ๐ ๐๐ for any ๐ > 0. Since ๐ข is continuous we obtain, taking ๐ โ 0 (see Exercise (7)), 1 โซ ๐ข๐๐ = ๐ข(0). ๐๐ ๐๐โ1 ๐๐ โก If a continuous function ๐ข in โ satisfies the mean value property, that is, ๐ข(๐ฅ) + ๐ข(๐ฆ) ๐ฅ+๐ฆ ๐ข( )= 2 2
2.1. The mean value property
17
for all ๐ฅ, ๐ฆ โ โ, then ๐ข must be a linear function. Indeed, let ๐ = ๐ข(1) โ ๐ข(0) and ๐ = ๐ข(0), so we have ๐ข(1) = ๐ + ๐ and ๐ข(0) = ๐. Since ๐ข(1) = (๐ข(0) + ๐ข(2))/2, we see that ๐ข(2) = 2๐ข(1) โ ๐ข(0) = 2๐ + ๐, and we can verify, inductively, that (2.8)
๐ข(๐) = ๐๐ + ๐
for every ๐ โ โ. We can similarly prove that (2.8) holds for negative integers ๐. Now, for every ๐ โ โค, ๐ข(๐) + ๐ข(๐ + 1) ๐๐ + ๐ + ๐(๐ + 1) + ๐ 2๐ + 1 = )= 2 2 2 2๐ + 1 = ๐( ) + ๐, 2 and similarly for every number of the form ๐/2๐ , for every ๐ โ โค and every ๐ โ โ. Since such numbers are dense in โ and ๐ข is continuous, we conclude that ๐ข(๐ฅ) = ๐๐ฅ + ๐ ๐ข(
for every ๐ฅ โ โ. As we have stated above, the linear functions are the harmonic functions in โ so, therefore, the continuous functions that satisfy the mean value property are precisely the harmonic functions. This is also true in higher dimensions. Theorem 2.9 (Converse to the mean value property). Let ฮฉ โ โ๐ be open and ๐ข a continuous function on ฮฉ that satisfies that, whenever ๐ต๐ฬ (๐ฅ) โ ฮฉ, (2.10)
๐ข(๐ฅ) =
1 โซ ๐ข(๐ฅ + ๐๐)๐๐(๐). ๐๐ ๐
Then ๐ข โ ๐ถ โ (ฮฉ) and ๐ข is harmonic in ฮฉ. As harmonic functions must be twice differentiable, we must prove that a function ๐ข that satisfies (2.10) is at least twice differentiable before proving that ฮ๐ข = 0 in ฮฉ. However, the conclusion of Theorem 2.9 is much stronger: ๐ข is actually an infinitely differentiable function. We thus conclude Corollary 2.11, which follows by applying Theorems 2.4 and 2.9. Corollary 2.11. If ๐ข is harmonic in an open set ฮฉ, then it is infinitely differentiable in ฮฉ.
18
2. Basic properties In order to prove Theorem 2.9, we will make use of Lemma 2.12.
Lemma 2.12. There exists a smooth radial function ๐ on โ๐ such that it is supported in ๐น and โซ ๐ = 1. Proof. Consider the function ๐ on โ given by 1
๐ (4๐กโ1)(2๐กโ1) ๐(๐ฅ) = { 0
1/4 < ๐ก < 1/2 otherwise.
It is a standard calculus exercise to verify that ๐ is a smooth function in โ, supported in [1/4, 1/2]. Indeed, it is infinitely flat at the points 1/4 and 1/2 (see Figure 2.2). Now, we define on โ๐ the function ๐(๐ฅ) = ๐๐(|๐ฅ|),
1
1
4
2
Figure 2.2. The cut-off function ๐(๐ก). Note that it is supported in [1/4, 1/2], and infinitely flat at the points 1/4 and 1/2.
where ๐ is such that โซ ๐(๐ฅ)๐๐ฅ = 1. โ๐
Such ๐ exists because ๐ is nonnegative, and thus โซโ๐ ๐(|๐ฅ|)๐๐ฅ > 0. Now, ๐ is ๐ถ โ and supported away from zero, and hence ๐ โ ๐ถ โ (โ๐ ), because
2.1. The mean value property
19
๐ฅ โฆ |๐ฅ| is smooth away from zero. Finally, as ๐(๐ก) = 0 unless 1/4 < ๐ก < 1/2, then ๐(๐ฅ) = 0 unless 1/4 < |๐ฅ| < 1/2, and thus supp ๐ โ ๐น. โก Proof of Theorem 2.9. Let ๐ข be a continuous function that satisfies ฬ (๐ฅ0 ) โ ฮฉ. (2.10) for every ๐ต๐ฬ (๐ฅ) โ ฮฉ. Let ๐ฅ0 โ ฮฉ, and ๐ > 0 such that ๐ต2๐ ฬ Let ๐(๐ฅ) = ๐(|๐ฅ|) as in Lemma 2.12, and define the function ๐๐ (๐ฅ) = ๐โ๐ ๐(๐โ1 ๐ฅ). Note that ๐๐ โ ๐ถ โ (โ๐ ), it is supported in ๐ต๐ (0), and โซ ๐๐ (๐ฅ)๐๐ฅ = 1. โ๐
In particular, for any ๐ฅ โ ๐ต๐ (๐ฅ0 ), the function ๐ฆ โฆ ๐๐ (๐ฅ โ๐ฆ) is supported in ๐ต2๐ (๐ฅ0 ) โ ฮฉ. Hence, we observe that, for ๐ฅ โ ๐ต๐ (๐ฅ0 ), โซ ๐ข(๐ฆ)๐๐ (๐ฅ โ ๐ฆ)๐๐ฆ = โซ ๐ข(๐ฅ โ ๐ฆ)๐๐ (๐ฆ)๐๐ฆ = โซ โ๐
โ๐ ๐
๐ข(๐ฅ โ ๐ฆ)๐๐ (๐ฆ)๐๐ฆ
๐ต๐ (0)
ฬ โ1 ๐)๐๐โ1 ๐๐. = โซ โซ ๐ข(๐ฅ โ ๐๐)๐๐(๐) โ
๐โ๐ ๐(๐ 0
๐
Using (2.10) we obtain ๐
ฬ โ1 ๐)๐๐โ1 ๐๐ โซ ๐ข(๐ฆ)๐๐ (๐ฅ โ ๐ฆ)๐๐ฆ = ๐ข(๐ฅ) โ
๐๐ โซ ๐โ๐ ๐(๐ โ๐
0 ๐
ฬ โ1 ๐)๐๐(๐)๐๐โ1 ๐๐ = ๐ข(๐ฅ) โซ โซ ๐โ๐ ๐(๐ 0
๐
= ๐ข(๐ฅ) โซ
๐๐ (๐ฆ)๐๐ฆ = ๐ข(๐ฅ).
๐ต๐ (0)
Note that the function ๐ข(๐ฆ)๐๐ (๐ฅ โ ๐ฆ) is ๐ถ โ in ๐ฅ and continuous with compact support in ๐ฆ, so we can differentiate under the integral the function ๐ฅ โฆ โซ ๐ข(๐ฆ)๐๐ (๐ฅ โ ๐ฆ)๐๐ฆ โ๐
as many times as we want, and thus we conclude ๐ข โ ๐ถ โ (๐ต๐ (๐ฅ0 )). In particular, ฮ๐ข is a continuous function in ๐ต๐ (๐ฅ0 ). Now, for any ๐ฅ โ ๐ต๐ (๐ฅ0 ) and 0 < ๐ < ๐ such that ๐ต๐ฬ (๐ฅ) โ ๐ต๐ (๐ฅ0 ), ๐ข(๐ฅ) =
1 โซ ๐ข(๐ฅ + ๐๐)๐๐(๐), ๐๐ ๐
20
2. Basic properties
so if we differentiate with respect to ๐ we obtain 0= =
๐ โซ ๐ข(๐ฅ + ๐๐)๐๐(๐) = โซ โ๐ข(๐ฅ + ๐๐) โ
๐๐๐(๐) ๐๐ ๐ ๐ 1 ๐๐โ1
โซ
๐๐ ๐ข๐๐ =
๐๐ (๐ฅ)
1 ๐๐โ1
โซ
ฮ๐ข๐๐ฅ.
๐ต๐ (๐ฅ)
In the last equality we have used Greenโs identity (1.14) with ๐ฃ = โ1. Hence, the integral of ฮ๐ข over any ball in ๐ต๐ (๐ฅ0 ) is zero. By the continuity of ฮ๐ข, ฮ๐ข = 0 in ๐ต๐ (๐ฅ0 ). Since ๐ฅ0 โ ฮฉ is arbitrary, we conclude ๐ข is harmonic in ฮฉ.
โก
2.2. The maximum principle From the mean value property 2.4 we obtain another basic property of harmonic functions, the maximum principle. Corollary 2.13 (Maximum principle). If ฮฉ โ โ๐ is a domain and ๐ข is harmonic in ฮฉ, then ๐ข does not take a maximum nor a minumum in ฮฉ, unless ๐ข is constant. This is easy to see in the case ๐ = 1, where harmonic functions coincide with linear functions: if ๐ข is linear in the interval (๐, ๐), then it clearly does not take neither a maximum or a minimum, because ๐ข is either strictly increasing or strictly decreasing, unless it is constant. Proof of Corollary 2.13. Suppose that ๐ข takes its maximum ๐ at some ๐ฅ0 โ ฮฉ, so ๐ข(๐ฅ0 ) = ๐. Let ๐ = {๐ฅ โ ฮฉ โถ ๐ข(๐ฅ) = ๐} be the set of points in ฮฉ where ๐ข takes the value ๐. Note that ๐ โ โ
because ๐ฅ0 โ ๐. We prove that ๐ = ฮฉ. First, ๐ is closed in ฮฉ because ๐ = ๐ขโ1 ({๐}) and ๐ข is continuous, so it is the pre-image of a closed set under a continuous function.2 Now let ๐ฅ โ ๐. Since ฮฉ is open, there exists ๐ > 0 such that ๐ต๐ฬ (๐ฅ) โ ฮฉ. By the mean value property, ๐ข(๐ฅ) =
1 โซ |๐ต๐ (๐ฅ)| ๐ต
๐ข(๐ฆ)๐๐ฆ.
๐ (๐ฅ)
2
See Sections A.2 and A.4 for a summary of results from topology in Euclidean spaces.
2.2. The maximum principle
21
As we are assuming ๐ฅ โ ๐, this integral must equal ๐. Now ๐ข(๐ฆ) โค ๐ for all ๐ฆ โ ฮฉ, because ๐ is the maximum of ๐ข. Hence, if at some ๐ฆ โ ๐ต๐ (๐ฅ) we had ๐ข(๐ฆ) < ๐, this integral would be smaller than ๐, because ๐ข is continuous. Thus ๐ข(๐ฆ) = ๐ for all ๐ฆ โ ๐ต๐ (๐ฅ). Therefore ๐ต๐ (๐ฅ) โ ๐, and ๐ is also open in ฮฉ. Since ฮฉ is connected, we conclude ๐ = ฮฉ, and therefore ๐ข is the constant function ๐ข(๐ฅ) = ๐. By taking โ๐ข, we also see that ๐ข takes its minimum in ฮฉ only if it is constant. โก The maximum principle implies that if ฮฉ is bounded, so ฮฉฬ is compact, and ๐ข is harmonic in ฮฉ and continuous on ฮฉ,ฬ then ๐ข takes its maximum (and its minimum) at the boundary of ฮฉ (Exercise (8)). Again, this is clear in the case of a linear function in a closed interval. The maximum principle also implies uniqueness of harmonic functions in a bounded domain ฮฉ, given their values on the boundary. See Exercise (9) for details. The maximum principle states that harmonic functions, unless they are constant, do not take their maxima nor minima in their domains. If the domain ฮฉ is bounded, a nonconstant harmonic function in ฮฉ may be bounded, of course, and in that case it may be possible to extend it to the boundary of ฮฉ, and hence its extrema would be achieved in ๐ฮฉ. Even if the domain ฮฉ is unbounded, we may have bounded harmonic functions, as we will see later on. However, in the case where the domain of the harmonic function is all of the Euclidean space, we have the following result. Theorem 2.14 (Liouville). If ๐ข is harmonic and bounded in โ๐ , then it is constant. Proof. Suppose ๐ข is harmonic and |๐ข(๐ฅ)| โค ๐ for all ๐ฅ โ โ๐ . We prove that ๐ข(๐ฅ) = ๐ข(0) for all ๐ฅ โ โ๐ . Fix ๐ฅ โ โ๐ and let ๐
> |๐ฅ|. By the mean value property in balls, 2.7, we have that ๐ข(๐ฅ) โ ๐ข(0) =
1 โซ |๐ต๐
(๐ฅ)| ๐ต
๐
(๐ฅ)
๐ข(๐ฆ)๐๐ฆ โ
1 โซ ๐ข(๐ฆ)๐๐ฆ, |๐ต๐
| ๐ต ๐
22
2. Basic properties
where ๐ต๐
= ๐ต๐
(0). Since |๐ต๐
(๐ฅ)| = |๐ต๐
| = ๐๐ ๐
๐ /๐, we can write this difference as ๐ ๐ข(๐ฅ) โ ๐ข(0) = ๐ข(๐ฆ)๐๐ฆ โ โซ ๐ข(๐ฆ)๐๐ฆ). (โซ ๐๐ ๐
๐ ๐ต๐
(๐ฅ) ๐ต๐
Now, if ๐ด is the annulus ๐ด = {๐ฆ โ โ๐ โถ ๐
โ |๐ฅ| โค |๐ฆ| โค ๐
+ |๐ฅ|}, we see that the symmetric difference of the balls ๐ต๐
(๐ฅ) and ๐ต๐
satisfies ๐ต๐
(๐ฅ) โณ ๐ต๐
โ ๐ด. (See Figure 2.3) Indeed, if ๐ฆ โ ๐ต๐
(๐ฅ) โงต ๐ต๐
, then |๐ฆ โ ๐ฅ| < ๐
and |๐ฆ| โฅ
Figure 2.3. The annulus ๐ด containing the symmetric difference of the balls ๐ต๐
(๐ฅ) and ๐ต๐
.
๐
โฅ ๐
โ |๐ฅ|, and further |๐ฆ| โค |๐ฆ โ ๐ฅ| + |๐ฅ| < ๐
+ |๐ฅ|; similarly, if ๐ฆ โ ๐ต๐
โงต ๐ต๐
(๐ฅ), then |๐ฆ| < ๐
โค ๐
+ |๐ฅ| and |๐ฆ โ ๐ฅ| โฅ ๐
, so |๐ฆ| โฅ |๐ฆ โ ๐ฅ| โ |๐ฅ| โฅ ๐
โ |๐ฅ|. If we integrate using spherical coordinates, we obtain ๐
+|๐ฅ|
๐ ๐ โซ |๐ข(๐ฆ)|๐๐ฆ โค โซโซ ๐๐๐โ1 ๐๐๐๐ |๐ข(๐ฅ) โ ๐ข(0)| โค ๐ ๐ ๐
๐ ๐ด ๐๐ ๐
๐ ๐ ๐
โ|๐ฅ| =
(๐
+ |๐ฅ|)๐ โ (๐
โ |๐ฅ|)๐ ๐ถ ๐ ๐ โ
๐๐ โ
โค ๐ฅ , ๐ ๐ ๐
๐ ๐ ๐
๐
2.2. The maximum principle
23
where we have used the fact that |๐ข(๐ฆ)| โค ๐ for all ๐ฆ โ โ๐ , and the constant ๐ถ๐ฅ only depends on ๐ and |๐ฅ|. As ๐ฅ is fixed and ๐
is arbitrary, we conclude that |๐ข(๐ฅ) โ ๐ข(0)| = 0, and therefore ๐ข(๐ฅ) = ๐ข(0). โก We have seen above that, if ๐(๐ง) is holomorphic, then it is real and imaginary parts are harmonic functions. Therefore, Theorem 2.14 implies that, if ๐ is a holomorphic function in โ (such a function is called an entire function) and is bounded, then ๐ must be constant (this is also know as Liouvilleโs theorem). This fact provides a proof for the fundamental theorem of algebra: If ๐(๐ง) is polynomial over โ of degree at least 1, then it has a root in โ. Indeed, if ๐(๐ง) is a polynomial over โ with no roots, then 1/๐(๐ง) is an entire bounded function, and thus constant, so ๐(๐ง) is a constant polynomial. See Exercise (10) for the details. We can refine Theorem 2.14 to obtain the same conclusion even when ๐ข is only bounded from below or from above. Theorem 2.15. If ๐ข is harmonic and nonnegative in โ๐ , then it is constant. Proof. The proof of Theorem 2.15 follows similarly as the proof of Theorem 2.14, but we now have to be more careful when estimating the difference ๐ข(๐ฅ) โ ๐ข(0) =
๐ ๐ข(๐ฆ)๐๐ฆ โ โซ ๐ข(๐ฆ)๐๐ฆ), (โซ ๐๐ ๐
๐ ๐ต๐
(๐ฅ) ๐ต๐
for ๐ฅ โ โ๐ and ๐
> |๐ฅ|. This time, we use the fact that ๐ข(๐ฆ) โฅ 0 to observe that, if again ๐ด is the annulus ๐ด = {๐ฆ โ โ๐ โถ ๐
โ |๐ฅ| โค |๐ฆ| โค ๐
+ |๐ฅ|}, then |๐ข(๐ฅ) โ ๐ข(0)| โค =
๐ โซ ๐ข(๐ฆ)๐๐ฆ ๐ ๐ ๐
๐ ๐ด ๐ ๐ข(๐ฆ)๐๐ฆ โ โซ ๐ข(๐ฆ)๐๐ฆ) (โซ ๐๐ ๐
๐ ๐ต๐
+|๐ฅ| ๐ต๐
โ|๐ฅ|
where ๐ต๐
+|๐ฅ| and ๐ต๐
โ|๐ฅ| are the balls of radii ๐
+ |๐ฅ| and ๐
โ |๐ฅ| centered at 0, respectively. We use again the mean value property and, as before,
24
2. Basic properties
for some constant ๐ถ๐ฅ that depends only on ๐ and |๐ฅ|, |๐ข(๐ฅ) โ ๐ข(0)| โค
๐ |๐ข(0) โ |๐ต๐
โ|๐ฅ| |๐ข(0)) (|๐ต ๐๐ ๐
๐ ๐
+|๐ฅ|
(๐
+ |๐ฅ|)๐ โ (๐
โ |๐ฅ|)๐ โ
๐ข(0) ๐
๐ ๐ถ ๐ข(0) โค ๐ฅ . ๐
We can now conclude again that ๐ข(๐ฅ) = ๐ข(0). =
โก
It is clear that we obtain the same conclusion of Theorem 2.15 whenever ๐ข is a harmonic function in โ๐ and there exists a constant ๐ผ โ โ such that either ๐ข(๐ฅ) โฅ ๐ผ for all ๐ฅ โ โ๐ , or ๐ข(๐ฅ) โค ๐ผ for all ๐ฅ โ โ๐ .
2.3. Poisson kernel and Poisson integrals in the ball We have seen that the value of a harmonic function at the center of a sphere is equal to the average of its values over the sphere. One can ask naturally if the value at any other point in the interior of the sphere is similarly determined by the values over the sphere, and if this value corresponds to a perhaps weighted average over such values. This is certainly the case for a linear function in a close interval [๐, ๐]: if ๐ก โ (๐, ๐) and ๐ข is linear, then ๐โ๐ก ๐กโ๐ ๐ข(๐) + ๐ข(๐), ๐โ๐ ๐โ๐ which is a convex combination of ๐ข(๐) and ๐ข(๐). ๐ข(๐ก) =
We will prove that this is true for harmonic functions in โ๐ , for ๐ โฅ 2, as well. The weight function is called the Poisson kernel. For ๐ฅ โ ๐น and ๐ โ ๐, we define (2.16)
๐(๐ฅ, ๐) =
1 1 โ |๐ฅ|2 . ๐๐ |๐ฅ โ ๐|๐
The Poisson kernel satisfies the following facts. 2.17. ๐(๐ฅ, ๐) > 0 for any ๐ฅ โ ๐น and ๐ โ ๐, which is easily seen from (2.16) because |๐ฅ| < 1. 2.18. For each fixed ๐ โ ๐, the function ๐ฅ โฆ ๐(๐ฅ, ๐) is harmonic in ๐น. This is followed by explicit differentiation (Exercise (13)).
2.3. Poisson kernel and Poisson integrals in the ball
25
2.19. For each fixed ๐ฅ โ ๐น, โซ ๐(๐ฅ, ๐)๐๐(๐) = 1.
(2.20)
๐
This is clear if ๐ฅ = 0, since (2.21)
๐(0, ๐) =
1 ๐๐
for any ๐ โ ๐, and thus โซ ๐(0, ๐)๐๐(๐) = โซ ๐
๐
1 ๐๐(๐) = 1. ๐๐
For ๐ฅ โ 0, ๐ฅ/|๐ฅ| โ ๐ and, since ๐(โ
, ๐ฅ/|๐ฅ|) is harmonic in ๐น, the mean value property implies (2.22)
๐(0,
๐ฅ 1 ๐ฅ โซ ๐(|๐ฅ|๐, )= )๐๐(๐), |๐ฅ| ๐๐ ๐ |๐ฅ|
because the integral on the right side is the average over the sphere centered at the origin of radius |๐ฅ| < 1, contained in the ball ๐น. By the identity ||๐ฅ|๐ โ ๐ฅ | = |๐ฅ โ ๐|, | |๐ฅ| | known as the symmetry lemma (Exercise (14)), we have that 2
๐(|๐ฅ|๐,
1 1 โ ||๐ฅ|๐ | 1 1 โ |๐ฅ|2 ๐ฅ = = ๐(๐ฅ, ๐), )= |๐ฅ| ๐๐ | ๐๐ |๐ฅ โ ๐|๐ ๐ฅ |๐ ||๐ฅ|๐ โ |๐ฅ| |
and thus ๐(0,
๐ฅ 1 โซ ๐(๐ฅ, ๐)๐๐(๐). )= |๐ฅ| ๐๐ ๐
The identity (2.20) follows using (2.21). Observe that, if ๐ = 1, we simply have ๐1 = 2 and hence ๐(๐ฅ, โ1) =
1 1 โ |๐ฅ|2 1 = (1 โ ๐ฅ) 2 |๐ฅ โ (โ1)| 2
and ๐(๐ฅ, 1) =
1 1 โ |๐ฅ|2 1 = (1 + ๐ฅ) 2 |๐ฅ โ 1| 2
26
2. Basic properties
for every ๐ฅ โ (โ1, 1), since the boundary of the unit interval (โ1, 1) is the set {โ1, 1} of two points. Note that each function ๐ฅ โฆ ๐(๐ฅ, ยฑ1) is linear, so it is harmonic in (โ1, 1), and (2.20) is just ๐(๐ฅ, โ1) + ๐(๐ฅ, 1) = 1. Example 2.23. In the case when ๐ = 2, we can write the Poisson kernel ๐(๐ฅ, ๐) in polar coordinates. Indeed, if ๐ฅ = ๐๐๐๐ for some 0 โค ๐ < 1 and ๐ = ๐๐๐ , then ๐(๐๐๐๐ , ๐๐๐ ) =
1 1 โ ๐2 1 1 โ ๐2 = โ
. 2๐ |๐๐๐๐ โ ๐๐๐ |2 2๐ 1 + 2๐ cos(๐ โ ๐) + ๐2
Note that (2.20) is now the identity 2๐
1 โซ 2๐ 0
1 โ ๐2 ๐๐ = 1. 1 + 2๐ cos(๐ โ ๐) + ๐2
The following fact states that, as we approach a point in the boundary, the weight of the Poisson kernel concentrates on that point. 2.24. For any ๐ โ ๐ and ๐ > 0, โซ
๐(๐ฅ, ๐)๐๐(๐) โ 0
|๐โ๐|โฅ๐
as ๐ฅ โ ๐, where the integral is taken over the subset of ๐ of points ๐ โ ๐ that satisfy |๐ โ ๐| โฅ ๐. To verify this limit observe that, if |๐ โ ๐| โฅ ๐ and |๐ฅ โ ๐| < ๐/2, then ๐ ๐ |๐ฅ โ ๐| = |๐ โ ๐ + ๐ โ ๐ฅ| โฅ |๐ โ ๐| โ |๐ โ ๐ฅ| > ๐ โ = , 2 2 so we have 1 1 โ |๐ฅ|2 1 1 โ |๐ฅ|2 ๐(๐ฅ, ๐) = โค . ๐๐ |๐ฅ โ ๐|๐ ๐๐ (๐/2)๐ Therefore, if |๐ฅ โ ๐| < ๐/2, โซ
๐(๐ฅ, ๐)๐๐(๐) โค โซ
|๐โ๐|โฅ๐
|๐โ๐|โฅ๐
1 1 โ |๐ฅ|2 2 ๐ ๐๐(๐) โค ( ) (1 โ |๐ฅ|2 ), ๐ ๐๐ (๐/2) ๐
and hence โซ
๐(๐ฅ, ๐)๐๐(๐) โ 0
|๐โ๐|โฅ๐
as ๐ฅ โ ๐, because |๐| = 1 and thus |๐ฅ| โ 1.
2.3. Poisson kernel and Poisson integrals in the ball
27
The facts 2.17, 2.20 and 2.24 make the family {๐ โฆ ๐(๐ฅ, ๐) โถ ๐ฅ โ ๐น} of functions on ๐ resemble a family of good kernels as ๐ฅ โ ๐ โ ๐, as defined in [SS03]. We will study such families later in this text. Let ๐ โ ๐ถ(๐). The Poisson integral of ๐ is given by (2.25)
๐ซ๐(๐ฅ) = โซ ๐(๐ฅ, ๐)๐(๐)๐๐(๐), ๐
for each ๐ฅ โ ๐น. The function ๐ข(๐ฅ) = ๐ซ๐(๐ฅ) defined by the Poisson integral of ๐ is well defined in ๐น for any continuous function ๐ on ๐. This follows because ๐(๐ฅ, ๐) is continuous as well in ๐. In fact, it is not required for ๐ to be continuous on ๐ for the integral in (2.25) to be defined. It is sufficient for ๐ to be Riemann-integrable on ๐. 2.26. The Poisson integral ๐ข(๐ฅ) of ๐ โ ๐ถ(๐) is harmonic in ๐น. This is followed by differentiating inside the integral (2.25), and using the fact that ๐(๐ฅ, ๐) is harmonic in ๐ฅ. (This is true for a Riemann-integrable function ๐ on ๐, as well; see Exercise (15).) In the case ๐ = 1, the Poisson integral of ๐ โถ {โ1, 1} โ โ is the sum ๐ข(๐ฅ) = ๐(๐ฅ, โ1)๐(โ1) + ๐(๐ฅ, 1)๐(1) 1 1 = (1 โ ๐ฅ)๐(โ1) + (1 + ๐ฅ)๐(1). 2 2 This is a linear combination of linear functions, so it is linear and clearly harmonic in (โ1, 1). Note that ๐ข(๐ฅ) โ ๐(ยฑ1) as ๐ฅ โ ยฑ1. The Poisson integral solves the Dirichlet problem for the ball: given ๐ โ ๐ถ(๐), find a function ๐ข on ๐นฬ such that it is harmonic in the interior and coincides with ๐ on the boundary, that is (2.27)
ฮ๐ข = 0 { ๐ข=๐
in ๐น on ๐.
We prove Theorem 2.28. Theorem 2.28. Let ๐ โ ๐ถ(๐) and ๐ข = ๐ซ๐ its Poisson integral. Then ๐ข is harmonic in ๐น, extends continuously to ๐นฬ and ๐ข|๐ = ๐.
28
2. Basic properties
Proof. From fact 2.26, we know that ๐ข is harmonic in ๐น. It is thus sufficient to prove that, for each ๐ โ ๐, ๐ข(๐ฅ) โ ๐(๐) as ๐ฅ โ ๐. Since ๐ is continuous on the compact set ๐, it is bounded.3 Let ๐ > 0 be such that |๐(๐)| โค ๐ for all ๐ โ ๐. Given ๐ > 0, we can choose ๐ > 0 such that, if |๐ โ ๐| < ๐, then ๐ |๐(๐) โ ๐(๐)| < . 2 We write, using identity (2.20), |๐ข(๐ฅ) โ ๐(๐)| = || โซ ๐(๐ฅ, ๐)๐(๐)๐๐(๐) โ ๐(๐) โซ ๐(๐ฅ, ๐)๐๐(๐)|| ๐
๐
โค โซ ๐(๐ฅ, ๐)|๐(๐) โ ๐(๐)|๐๐(๐) ๐
=โซ
+โซ
|๐โ๐| 0, we define the function in ๐นฬ โ ๐ฃ ๐ (๐ฅ) = ๐ข(๐ฅ) โ ๐ซ(๐ข|๐ )(๐ฅ) + ๐(|๐ฅ|2โ๐ โ 1).
Exercises
31
As |๐ฅ|2โ๐ is harmonic in โ๐ โงต {0}, ๐ฃ ๐ is harmonic in ๐นโ and, if we set ๐ซ(๐ข|๐ )(๐) = ๐ข(๐) if ๐ โ ๐, ๐ฃ ๐ is continuous on ๐นฬ โ . We observe that, for ๐ โ ๐, ๐ฃ ๐ (๐) = 0. Moreover, as ๐ฅ โ 0, we have ๐ฃ ๐ (๐ฅ) โ โ because ๐ข is bounded near 0. Thus, by the maximum principle, ๐ฃ ๐ (๐ฅ) > 0 for all ๐ฅ โ ๐นโ , because otherwise ๐ข would take a negative minimum in ๐นโ , and that is not possible by the maximum principle. Since ๐ > 0 is arbitrary, we obtain that ๐ข(๐ฅ) โฅ ๐ซ(๐ข|๐ )(๐ฅ) for all ๐ฅ โ ๐นโ . If we repeat the argument for โ๐ข, we obtain ๐ข(๐ฅ) โค ๐ซ(๐ข|๐ )(๐ฅ) for all ๐ฅ โ ๐นโ . Thus ๐ข(๐ฅ) = ๐ซ(๐ข|๐ )(๐ฅ) in ๐นโ . Therefore, the Poisson integral ๐ซ(๐ข|๐ )(๐ฅ) is the harmonic extension of ๐ข to all of ๐น. โก Note that, in the proof of Theorem 2.33, we are extending ๐ข to ๐ฅ = 0 by its average over ๐. It is actually not necessary to assume that ๐ข is bounded near ๐ฅ0 to conclude that ๐ฅ0 is a removable singularity. See Exercise (21).
Exercises (1) A translation in โ๐ is a map ๐ โถ โ๐ โ โ๐ of the form ๐(๐ฅ) = ๐ฅ + โ, for some โ โ โ๐ . (a) If ๐ is a translation, ฮ(๐ข โ ๐) = (ฮ๐ข) โ ๐. (b) If ๐ข is harmonic in โ๐ and ๐ is a translation, then ๐ข โ ๐ is also harmonic in โ๐ . (2) An orthogonal transformation in โ๐ is a map ๐ โถ โ๐ โ โ๐ of the form ๐(๐ฅ) = ๐ด๐ฅ, for some orthogonal ๐ ร ๐ matrix ๐ด, that is, ๐ด satisfies that ๐ด๐ด๐ก = ๐ผ๐ , where ๐ด๐ก is the transpose of ๐ด and ๐ผ๐ is the ๐ ร ๐ identity matrix. (a) If ๐ is an orthogonal transformation, then ฮ(๐ข โ ๐) = (ฮ๐ข) โ ๐. (b) If ๐ข is harmonic in โ๐ and ๐ is orthogonal, then ๐ข โ ๐ is also harmonic in โ๐ . (3) Prove fact 2.7.
32
2. Basic properties
(4) Let ๐ข(๐ฅ) = ๐๐ฅ + ๐. Then ๐ฅ +๐
0 1 โซ ๐ข(๐ฅ0 ) = ๐ข(๐ฅ)๐๐ฅ. 2๐ ๐ฅ โ๐ 0
(5) Prove that the function ๐ฃ in the proof of Theorem 2.4 is harmonic. (6) Suppose ๐ข is harmonic in a neighborhood of ฮฉ,ฬ where ฮฉ is a ๐ถ 1 domain. Then โซ ๐๐ ๐ข ๐๐ = 0. ๐โฆ
(7) Let ๐ be Riemann-integrable on the rectangle ๐
, and continuous at the interior point ๐ฅ0 โ ๐
. As ๐ โ 0, 1 โซ ๐ โ ๐(๐ฅ0 ); and (a) |๐ต๐ (๐ฅ0 )| ๐ต (๐ฅ ) ๐ 0 1 โซ ๐ โ ๐(๐ฅ0 ). (b) |๐๐ (๐ฅ0 )| ๐ (๐ฅ ) ๐
0
(8) If ฮฉ โ โ๐ is a bounded domain and ๐ข is harmonic in ฮฉ and continuous on ฮฉ,ฬ then ๐ข takes its maximum and its minimum on ๐ฮฉ. (9) Let ฮฉ โ โ๐ be a bounded domain, ๐ข and ๐ฃ harmonic in ฮฉ and continuous on ฮฉ.ฬ If ๐ข = ๐ฃ on ๐ฮฉ, then ๐ข = ๐ฃ in ฮฉ. (10) The following exercises provide the details of the proof of the fundamental theorem of algebra. (a) If ๐ is a holomorphic function without zeroes in its domain ฮฉ, then 1/๐ is holomorphic in ฮฉ. (b) If ๐(๐ง) is a polynomial over โ, then either ๐(๐ง) is constant or |๐(๐ง)| โ โ as |๐ง| โ โ. (c) If ๐(๐ง) is polynomial over โ with no roots, then 1/๐(๐ง) is an entire bounded function. (11) If ๐ is an entire function and its real part is nonnegative, then ๐ is constant. (12) If ๐ข is a radial harmonic function in ๐น, then it is constant. (13) ๐ฅ โฆ ๐(๐ฅ, ๐) is harmonic in ๐น, for each ๐ โ ๐. (Hint: Write ๐(๐ฅ, ๐) = 2 โ๐ ๐โ1 and use the identity ฮ(๐ข๐ฃ) = (ฮ๐ข)๐ฃ + 2โ๐ข โ
๐ (1 โ |๐ฅ| )|๐ฅ โ ๐| โ๐ฃ + ๐ขฮ๐ฃ.) (14) Symmetry Lemma: If ๐ฅ โ ๐น and ๐ โ ๐, then ||๐ฅ|๐ โ ๐ฅ | = |๐ฅ โ ๐|. | |๐ฅ| |
Exercises
33
See Figure 2.4.
Figure 2.4. If ๐ฅ โ ๐น and ๐ โ ๐, the distance between the points ๐ฅ/|๐ฅ| and |๐ฅ|๐ is the same as the distance between ๐ฅ and ๐, as stated by the symmetry lemma.
(15) Let ๐ be Riemann-integrable on ๐. Then its Poisson integral ๐ข is harmonic in ๐น. (16) Hopf lemma: If ๐ข is a nonconstant harmonic function in ๐น, is continuous on ๐น,ฬ and attains its maximum at ๐ โ ๐, then there exists ๐ > 0 such that ๐ข(๐) โ ๐ข(๐๐) > ๐(1 โ ๐), for any 0 < ๐ < 1. (17) Harnack inequality: If ๐ข is a harmonic function in ๐น, is continuous on ๐น,ฬ and is positive, then 1 โ |๐ฅ| 1 + |๐ฅ| ๐ข(0) โค ๐ข(๐ฅ) โค ๐ข(0) ๐โ1 (1 + |๐ฅ|) (1 โ |๐ฅ|)๐โ1 for all ๐ฅ โ ๐น. (18) If ๐ข is harmonic in ฮฉ and ๐ต๐ฬ (๐ฅ0 ) โ ฮฉ, then the values of ๐ข in ๐ต๐ (๐ฅ0 ) are determined by its values on ๐๐ (๐ฅ0 ). (19) Let ๐ข๐ be a sequence of harmonic functions in ฮฉ such that ๐ข๐ โ ๐ข on any compact ๐พ โ ฮฉ. Then ๐ข is harmonic in ฮฉ. (20) Prove Theorem 2.33 for ๐ = 2.
34
2. Basic properties
(21) Let ๐ข be harmonic in a domain in โ๐ with an isolated singularity at ๐ฅ0 . If ๐ = 2 and lim ๐ข(๐ฅ) log |๐ฅ โ ๐ฅ0 | = 0,
๐ฅโ๐ฅ0
or ๐ > 2 and lim ๐ข(๐ฅ)|๐ฅ โ ๐ฅ0 |๐โ2 = 0,
๐ฅโ๐ฅ0
then ๐ฅ0 is a removable singularity.
Notes The results of this chapter are basic classical results, proven in every text in harmonic functions and partial differential equations. A classical reference for the theory of harmonic functions is [Kel67]. Theorem 2.4 is a result by Gauss [Gau40]. The proof presented here is the most popular and can be found, for instance, in [ABR01] or in [Fol95]. It can also be proven by differentiating the integral over a sphere with respect to its radius, as in [Eva10] or in [MS13]. The proof of Theorem 2.9 is also in [Fol95]. The proof of Theorem 2.14 is an elaboration of the proof by Nelson [Nel61]. Simรฉon Denis Poisson developed explicit expressions to solutions to the Laplace equation in terms of integrals over the sphere in [Poi20], and thus the Poisson kernel and integral are named after him. Theorem 2.33 is a result by Riemann [Rie51]. The proof presented here can be found in [ABR01].
Chapter 3
Fourier series
3.1. Separation of variables In Chapter 2 we solved the Dirichlet problem (2.27) using the Poisson integral, which provides an explicit integral form of the solution from its values on the boundary. We now attempt to solve the problem by decomposing the function ๐ on ๐ in fundamental pieces, which is the original Fourier approach in [Fou55]. In this chapter we consider the Dirichlet problem in the unit disk ๐ป in the plane โ2 . In order to find such fundamental pieces, we first recall that the Laplacian in polar coordinates (๐, ๐) is given by ฮ๐ข =
๐2 ๐ข 1 ๐๐ข 1 ๐2 ๐ข + + 2 2. 2 ๐ ๐๐ ๐ ๐๐ ๐๐
(See Exercise (2) of Chapter 1.) We now search for solutions of the form ๐ข(๐, ๐) = ๐ฃ(๐)๐(๐), where ๐ฃ(๐) is a twice differentiable function defined for 0 โค ๐ < 1 and ๐(๐) is a twice differentiable periodic function defined on โ, with period 2๐, as the pair (๐, ๐) denotes a point in the disk. Thus we want to solve the equation 1 1 ฮ๐ข = ๐ฃโณ (๐)๐(๐) + ๐ฃโฒ (๐)๐(๐) + 2 ๐ฃ(๐)๐โณ (๐) = 0, ๐ ๐ 35
36
3. Fourier series
which we can rewrite, when ๐ฃ(๐) โ 0 and ๐(๐) โ 0, as ๐โณ (๐) ๐2 ๐ฃโณ (๐) + ๐๐ฃโฒ (๐) =โ . ๐ฃ(๐) ๐(๐) The left hand side of this equation does not depend on ๐, and the right hand side does not depend on ๐, so we conclude that both sides are equal to a constant, say, ๐ โ โ. Hence we obtain the equations (3.1)
๐2 ๐ฃโณ (๐) + ๐๐ฃโฒ (๐) = ๐๐ฃ(๐)
and ๐โณ (๐) = โ๐๐(๐),
(3.2)
subject to the constrains stated above. Equation (3.2) has periodic solutions ๐(๐) = cos(๐๐)
and
๐(๐) = sin(๐๐),
2
with period 2๐, when ๐ = ๐ and ๐ โ โ. We have a pair of linearly independent solutions for each natural number ๐ โฅ 1. For ๐ = 0, we have the linearly independent solutions ๐(๐) = 1 and ๐(๐) = ๐, but the latter is not periodic. Now, with ๐ = ๐2 , equation (3.1) is ๐2 ๐ฃโณ (๐) + ๐๐ฃโฒ (๐) โ ๐2 ๐ฃ(๐) = 0 and has linearly independent solutions ๐ฃ(๐) = ๐๐
and
๐ฃ(๐) = ๐โ๐
for each ๐ โฅ 1, though only the former is well defined on [0, 1). Again, if ๐ = 0, we have the solutions ๐ฃ(๐) = 1 and ๐ฃ(๐) = log ๐, but only the former is defined on [0, 1). We thus obtain, for each ๐ โ โ, the harmonic functions ๐ข๐ (๐, ๐) = ๐๐ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)), where ๐๐ , ๐๐ โ โ (note that ๐ข0 is the constant function ๐ข0 (๐, ๐) = ๐0 ). Any linear combination of such functions, ๐
(3.3)
๐ข(๐, ๐) = โ ๐๐ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)), ๐=0
3.1. Separation of variables
37
is harmonic in the disk, and its limit at the boundary ๐ โ 1 is the trigonometric polynomial ๐
(3.4)
๐(๐) = โ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)). ๐=0
In fact, this limit is uniform as ๐ โ 1 on ๐. We have thus solved the problem ฮ๐ข = 0 in ๐ป { ๐ข=๐ on ๐, where ๐ is the trigonometric polynomial (3.4). It is clear that the trigonometric polynomial (3.4) can be seen either as a function on ๐ or as a periodic function on โ, through the map ๐ โฆ (cos ๐, sin ๐). In general, that is true for any 2๐-periodic function ๐ on โ: the function ๐น(cos ๐, sin ๐) = ๐(๐) is a well-defined function on ๐. We note that ๐ is continuous on โ if and only if ๐น is continuous on ๐, so we can identify the space ๐ถ(๐) of continuous functions on ๐ with the subspace of ๐ถ(โ) of periodic functions with period 2๐, or with the subspace of ๐ถ([0, 2๐]) of functions satisfying ๐(0) = ๐(2๐). Similarly, Riemann-integrable functions on ๐ can be identified with Riemann-integrable functions ๐ on [0, 2๐] satisfying ๐(0) = ๐(2๐), as well as with 2๐-periodic functions on โ that are Riemann-integrable on each closed interval. We will interchangeably use the terms โfunction on ๐โ, โfunction on [0, 2๐]โ and โ2๐-periodic function on โโ throughout this chapter. If we recall de Moivreโs formula for complex numbers, (cos ๐ + ๐ sin ๐)๐ = cos(๐๐) + ๐ sin(๐๐), we see that (3.4) is indeed a polynomial in cos ๐ and sin ๐ (that is why we call it a trigonometric polynomial), and thus a polynomial in the coordinate functions on ๐. We can ask whether we can use the linear combinations (3.3) to solve the Dirichlet problem (3.5)
ฮ๐ข = 0 { ๐ข=๐
in ๐ป on ๐,
given any continuous function ๐ on the boundary ๐, and not only a trigonometric polynomial. More precisely, we have:
38
3. Fourier series Question 1: For ๐ โ ๐ถ(๐), is there a sequence of trigonometric polynomials ๐๐ that converge to ๐ (pointwise or uniformly) such that the corresponding solutions ๐ข๐ as in (3.3) converge to a solution of the Dirichlet problem (3.5)? Question 2: For ๐ โ ๐ถ(๐), is there a series โ
โ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)) ๐=0
such that the series โ
๐ข(๐, ๐) = โ ๐๐ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)) ๐=0
converges to a solution of the Dirichlet problem (3.5)? Note that an affirmative answer to Question 2 would not necessarily give a positive answer to Question 1, as the partial sums of the series might not satisfy the requirement of the trigonometric polynomials ๐๐ and the functions ๐ข๐ . Note also that a sequence ๐๐ of trigonometric polynomials converging uniformly to ๐ is by all means not unique, and thus neither the sequence ๐ข๐ . These questions also make sense if ๐ is only Riemann-integrable, except for the fact that we cannot expect uniform convergence anymore because the uniform limit of continuous functions is continuous. In that case, we would restrict to convergence at certain points of ๐.
3.2. Fourier series Consider the problem of writing a function ๐ on ๐ as a series โ
(3.6)
โ (๐๐ cos ๐๐ + ๐๐ sin ๐๐) ๐=0
for ๐ โ [0, 2๐], where we agree that ๐0 = 0. By Eulerโs formula ๐๐๐ฅ = cos ๐ฅ + ๐ sin ๐ฅ, we can write, for ๐ โฅ 1, ๐๐ cos ๐๐ + ๐๐ sin ๐๐ = ๐๐ ๐๐๐๐ + ๐โ๐ ๐โ๐๐๐ , where ๐ ยฑ๐ =
๐๐ โ ๐๐๐ . 2
3.2. Fourier series
39
If we set ๐ 0 = ๐0 , the series (3.6) can be written as โ
(3.7)
โ
๐ 0 + โ (๐๐ ๐๐๐๐ + ๐โ๐ ๐โ๐๐๐ ) = โ ๐๐ ๐๐๐๐ , ๐=1
๐=โโ
where the double infinite series on the right of (3.7) is understood as the limit of the partial sums ๐
โ ๐๐ ๐๐๐๐ ๐=โ๐
as ๐ โ โ. Suppose the series (3.7) converges uniformly to ๐, so ๐
๐ ๐ (๐) = โ ๐๐ ๐๐๐๐ โ ๐(๐). ๐=โ๐
Thus, for any ๐ โ โค (see Section A.3), 2๐
2๐
โซ
๐ ๐ (๐)๐โ๐๐๐ ๐๐ โ โซ
0
0
๐(๐)๐โ๐๐๐ ๐๐.
Now for any ๐ โ โค, 2๐
(3.8)
โซ
๐๐๐๐ ๐โ๐๐๐ ๐๐ = {
0
2๐ ๐ = ๐ 0
๐โ 0
(Exercise (3)) and thus, for ๐ โฅ ๐, 2๐
โซ
๐ ๐ (๐)๐โ๐๐๐ ๐๐ = 2๐๐๐ .
0
Therefore, if the series (3.7) converges uniformly to ๐, the coefficients ๐๐ are given by ๐๐ =
1 โซ 2๐ 0
2๐
๐(๐)๐โ๐๐๐ ๐๐
for each ๐ โ โค. For any Riemann-integrable function ๐ on ๐, we define its Fourier coefficients, for ๐ โ โค, by 2๐
(3.9)
ฬ = 1 โซ ๐(๐) 2๐ 0
๐(๐)๐โ๐๐๐ ๐๐.
40
3. Fourier series
The series ๐๐๐ ฬ โ ๐(๐)๐
(3.10)
๐โโค
is called the Fourier series of ๐. Note that, since a Riemann-integrable function ๐ is bounded, its Fourier coefficients (3.9) are bounded. Indeed, they are bounded by any bound for ๐ because, if |๐(๐)| โค ๐ for all ๐, then 1 ฬ โซ |๐(๐)| โค || 2๐ 0
(3.11)
2๐
2๐
๐(๐)๐
โ๐๐๐
1 โซ ๐๐|| โค 2๐ 0
|๐(๐)|๐๐ โค ๐.
We also observe that in the definition (3.9) of the Fourier coefficients of ๐, as ๐ is periodic, we can choose any interval of length 2๐ in the integration, as convenient (see Exercise (4)). Example 3.12 (The sawtooth function). Consider the periodic function ๐, with period 2๐, given in [โ๐, ๐) by ๐(๐) = ๐ (Figure 3.1). Its Fourier ฯ
-2 ฯ
-ฯ
2ฯ
ฯ
3ฯ
-ฯ
Figure 3.1. The sawtooth function given by ๐(๐) = ๐, โ๐ โค ๐ < ๐. Note that ๐ is discontinuous at ๐.
coefficients are given, for ๐ = 0, by ๐
๐
ฬ = 1 โซ ๐(๐)๐๐ = 1 โซ ๐๐๐ = 0, ๐(0) 2๐ โ๐ 2๐ โ๐ and, for ๐ โ 0, by ๐
๐
๐ ฬ = 1 โซ ๐(๐)๐โ๐๐๐ ๐๐ = 1 โซ ๐๐โ๐๐๐ ๐๐ = ๐(โ1) , ๐(๐) 2๐ โ๐ 2๐ โ๐ ๐
3.2. Fourier series
41
and thus the Fourier series of ๐ is given by โ
(โ1)๐ ๐๐๐ (โ1)๐ ๐ = โ2 โ sin ๐๐. ๐ ๐ ๐โ 0 ๐=1
๐โ
Using Dirichletโs test,1 one can prove that this series converges for every ๐ (Exercise (5)). At ๐ = 0, the series clearly converges to 0, and at ๐ = ๐/2, the series โ
โ
(โ1)๐ ๐๐ (โ1)๐ 1 1 1 sin =2โ = 2(1 โ + โ + โฏ ) ๐ 2 2๐ + 1 3 5 7 ๐=1 ๐=0
โ2 โ
converges to ๐/2, as seen in a calculus course. In both cases, the series at each ๐ converges to ๐(๐). However, at ๐ = ๐, the series converges to 0, even though ๐(๐) = โ๐. Note that ๐ is discontinuous at ๐. Example 3.13 (The sharkteeth function). We now consider the periodic function ๐ given in [โ๐, ๐) by ๐(๐) = |๐| (Figure 3.2). This time the ฯ
-2 ฯ
-ฯ
ฯ
2ฯ
3ฯ
Figure 3.2. The sharkteeth function given by ๐(๐) = |๐|, โ๐ โค ๐ < ๐. Note that ๐ is continuous at every point.
function ๐ is continuous at every point. Its Fourier coefficients are given by ๐ ๐=0 โง โช2 even ๐ โ 0 ๐(๐) ฬ = 0 โจ 2 โชโ โฉ ๐๐2 odd ๐ (Exercise (6)), and thus its Fourier series is given by โ
๐๐๐๐ cos(2๐ + 1)๐ ๐ 2 ๐ 4 โ 2 = โ โ โ . 2 ๐ odd ๐ ๐ 2 ๐ ๐=0 (2๐ + 1)2 1
Theorem A.5 in Section A.1.
42
3. Fourier series
We see that, this time, the Fourier series of ๐ converges uniformly on [โ๐, ๐), by the Weierstrass ๐-test,2 as the coefficients decrease as 1/๐2 . For example, for ๐ = ๐/2, each ๐ cos(2๐ + 1) = 0, 2 so the series converges to ๐/2, the value of ๐ at ๐/2. We can ask then if the series converges to ๐(๐) at any other ๐. Example 3.14 (The flounces function). Consider now the periodic function โ given in [โ๐, ๐) by โ(๐) = (๐2 โ๐2 )2 , shown in Figure 3.3. โ is not
-ฯ
ฯ
Figure 3.3. The flounces function given by โ(๐) = (๐2 โ ๐2 )2 , โ๐ โค ๐ < ๐. Note that โ is continuous and differentiable at every point.
only continuous, but also differentiable at everypoint, with โโฒ (๐๐) = 0 for every ๐ โ โค. Its Fourier coefficients (Exercise (7)) are given by 8 4 ๐ 15 ฬ โ(๐) = { (โ1)๐ โ24 4 ๐ so its Fourier series is then
๐=0 ๐ โ 0, โ
8 4 (โ1)๐ ๐๐๐ 8 4 (โ1)๐ ๐ = cos ๐๐. ๐ โ 24 โ ๐ โ 48 โ 4 15 15 ๐ ๐4 ๐โ 0 ๐=1 As in example 3.13 above, the Fourier series converges absolutely and uniformly, though this time it is not clear what the limit of the series is at any point. However, we observe that the series converges more rapidly than in the previous example, and thus one questions if the regularity of โโthe fact that it is not only continuous but also differentiableโhas any effect in the decrease of the coefficients. We have now another set of questions. 2
Theorem A.6 in Section A.1.
3.3. Abel means and Poisson integrals
43
Question 3: Does the Fourier series (3.10) converge? In what sense? Pointwise, uniformly? In any other sense? Does the regularity of the function have any effect on the convergence? Question 4: If the Fourier series (3.10) of ๐ converges, does it converge to ๐? It turns out that Question 4 can be answered thanks to our previous analysis of Poisson integrals.
3.3. Abel means and Poisson integrals โ
Consider a series โ๐=0 ๐๐ . The Abel means of โ ๐๐ are given by โ
๐ด๐ = โ ๐๐ ๐๐ ,
(3.15)
๐=0
for 0 < ๐ < 1. We say that the series โ ๐๐ is Abel-summable to ๐ if lim ๐ด๐ = ๐ . ๐โ1
All convergent series are also Abel-summable, a result know as Abelโs theorem. Theorem 3.16. Suppose that the series โ ๐๐ converges to ๐ . Then โ ๐๐ is Abel-summable to ๐ . Proof. Let ๐ ๐ = ๐0 + ๐1 + . . . ๐๐ be the ๐th partial sum of the series, so we have ๐ ๐ โ ๐ . In particular, ๐ ๐ is bounded, so there exists ๐ > 0 such that |๐ ๐ | โค ๐ for all ๐. The boundedness of ๐ ๐ implies the convergence of โ ๐ ๐ ๐๐ , so for each 0 โค ๐ < 1, using the fact that ๐๐ = ๐ ๐ โ ๐ ๐โ1 for each ๐ โฅ 1, โ
โ
๐ด๐ = โ ๐๐ ๐๐ = ๐0 + โ (๐ ๐ โ ๐ ๐โ1 )๐๐ ๐=0
๐=1 โ
โ
= ๐ 0 + โ ๐ ๐ ๐๐ โ โ ๐ ๐โ1 ๐๐ ๐=1 โ
๐=1 โ
โ
= โ ๐ ๐ ๐๐ โ โ ๐ ๐ ๐๐+1 = (1 โ ๐) โ ๐ ๐ ๐๐ . ๐=0
๐=0
๐=0
44
3. Fourier series Given ๐ > 0 there exists ๐ such that |๐ ๐ โ ๐ |
0 we have |๐โฒ (๐)| โค ๐, so (3.38)
|๐โ (๐)| โค 2๐โ
52
3. Fourier series
for every ๐. Now, 2๐
๐ ห โ (๐) =
1 โซ (๐(๐ + โ) โ ๐(๐ โ โ))๐โ๐๐๐ ๐๐ 2๐ 0 2๐
2๐
=
1 โซ 2๐ 0
๐(๐ + โ)๐โ๐๐๐ ๐๐ โ
1 โซ 2๐ 0
=
1 โซ 2๐ 0
๐(๐)๐โ๐๐(๐โโ) ๐๐ โ
1 โซ 2๐ 0
=
๐๐๐โ โ ๐โ๐๐โ โซ 2๐ 0
2๐
๐(๐ โ โ)๐โ๐๐๐ ๐๐
2๐
๐(๐)๐โ๐๐(๐+โ) ๐๐
2๐
ฬ ๐(๐)๐โ๐๐๐ ๐๐ = 2๐ sin ๐โ๐(๐),
so we have ฬ 2. โ |ห ๐โ (๐)|2 = 4 โ | sin ๐โ|2 |๐(๐)|
(3.39)
๐โโค
๐โโค
Lemma 3.36, together with equations (3.38) and (3.39), implies ฬ 2โค โ | sin ๐โ|2 |๐(๐)| ๐โโค
1 1 โซ โ
4 2๐ 0
2๐
|๐โ (๐)|2 ๐๐ โค
(2๐โ)2 โ
2๐ = ๐ 2 โ2 . 8๐
In particular, for each ๐ โฅ 1, ฬ 2 โค ๐ 2 โ2 . | sin ๐โ|2 |๐(๐)|
โ
(3.40)
2๐โ1 โค|๐| 0 such that, if |๐| < ๐, then ๐ |๐(๐ โ ๐) โ ๐(๐)| < . 2 Now, as ๐ is Riemann-integrable, it is bounded, so there exists ๐ > 0 such that |๐(๐)| โค ๐ for all ๐. By (3.48), there exists ๐พ such that, if ๐ โฅ ๐พ, ๐ | 1 โซ ๐น๐ (๐)๐๐|| < . | 2๐ 4๐ ๐โค|๐|โค๐ Therefore, for ๐ โฅ ๐พ, using (3.47), ๐
1 โซ ๐น๐ (๐)๐(๐ โ ๐)๐๐ โ ๐(๐)|| |๐๐ (๐) โ ๐(๐)| = || 2๐ โ๐
๐
๐
= ||
1 1 โซ ๐น (๐)๐(๐ โ ๐)๐๐ โ ๐(๐) โ
โซ ๐น (๐)๐๐|| 2๐ โ๐ ๐ 2๐ โ๐ ๐
= ||
1 โซ ๐น (๐)(๐(๐ โ ๐) โ ๐(๐))๐๐|| 2๐ โ๐ ๐
๐
โค
1 โซ ๐น (๐)|๐(๐ โ ๐) โ ๐(๐)|๐๐ 2๐ |๐| 0, โซ
|๐พ๐ (๐)|๐๐ โ 0
๐โค|๐|โค๐
as ๐ โ โ. It is not hard to verify that the proof of Theorem 3.45 applies to any family of good kernels, so, if a function is continuous at ๐, then ๐
1 โซ ๐พ (๐ โ ๐)๐(๐)๐๐ โ ๐(๐) 2๐ โ๐ ๐ as ๐ โ โ. The fact that the Cesร ro sums of a convergent series converge to the same limit implies the same results of Corollary 3.21. However, we additionally have the following stronger result. Corollary 3.49. The space of trigonometric polynomials is dense in ๐ถ(๐). In other words, if ๐ โ ๐ถ(๐) and ๐ > 0, there exists a trigonometric polynomial ๐ such that |๐(๐) โ ๐(๐)| < ๐ for every ๐. Proof. By Fejรฉrโs theorem, the Cesร ro sums ๐๐ of the Fourier series of ๐ converge uniformly to ๐. In other words, given ๐ > 0, there exists ๐ such that |๐(๐) โ ๐๐ (๐)| < ๐
3.6. Mean-square convergence
59
for all ๐. The results follows from the fact that ๐๐ is a trigonometric polynomial. โก Corollary 3.49 gives an affirmative answer to Question 1 above.
3.6. Mean-square convergence We now come back to the orthogonality of the Fourier expansions to prove Theorem 3.50. Theorem 3.50. If ๐ is a 2๐-periodic Riemann-integrable function and, for each ๐ โ โ, ๐ ๐๐๐ ฬ ๐ ๐ (๐) = โ ๐(๐)๐ , ๐=โ๐
then (3.51)
lim โ๐ โ ๐ ๐ โ = 0.
๐โโ
In other words, the Fourier series of ๐ ๐๐๐ ฬ โ ๐(๐)๐ ๐โโค
converges to ๐ in the sense, 2๐
๐
1 โซ 2๐ 0
2
๐๐๐ | |๐(๐) โ โ ๐(๐)๐ ฬ | | ๐๐ โ 0 ๐=โ๐
as ๐ โ โ. This is called mean-square convergence. Proof. As ๐ is Riemann-integrable, it is bounded, so there exists ๐ > 0 such that |๐(๐)| โค ๐ for all ๐. Now, given ๐ > 0, by Theorem A.10 (Section A.3) we can choose a continuous function ๐ on ๐ such that |๐(๐)| โค ๐ and 2๐
โซ
|๐(๐) โ ๐(๐)|๐๐
0 small, let ๐๐ be the continuous function on [โ๐, ๐] that is equal to ๐๐ (๐) in each interval [๐๐ + ๐ฟ, ๐๐+1 โ ๐ฟ], and is equal to the linear function from ๐๐ (๐๐ โ ๐ฟ) to ๐๐ (๐๐ + ๐ฟ) around each ๐๐ (see Figure 3.4).4 We now observe that ๐๐ =
Figure 3.4. The functions ๐ท๐ , ๐๐ and ๐๐ for ๐ = 5. ๐
โซ |๐๐ (๐) โ ๐๐ (๐)|๐๐ = 2๐๐ฟ, โ๐
so, using the fact that |๐ท๐ (๐)| โค 2๐ + 1, we obtain ๐
1 1 1 โซ |๐ท (๐)||๐๐ (๐) โ ๐๐ (๐)|๐๐ โค (2๐ + 1)2๐๐ฟ < , 2๐ โ๐ ๐ 2๐ 4 if we choose ๐ฟ< 4
Cf. the proof of Theorem A.10.
๐ . 4๐(2๐ + 1)
3.7. Convergence for continuous functions
65
Hence ๐
1 โซ ๐ท (๐)๐๐ (๐)๐๐ 2๐ โ๐ ๐ ๐
๐
1 1 โซ ๐ท (๐)๐๐ (๐)๐๐ โ โซ |๐ท (๐)||๐๐ (๐) โ ๐๐ (๐)|๐๐ โฅ 2๐ โ๐ ๐ 2๐ โ๐ ๐ 4 โฅ 2 log ๐. ๐ Set ๐ large enough so that 4 log ๐ โฅ ๐ + 1. ๐2 By Corollary 3.49, there exists a trigonometric polynomial ๐(๐) such that |๐๐ (๐) โ ๐(๐)| < Hence
1 . 2๐ + 1
๐
1 โซ |๐ท (๐)||๐๐ (๐) โ ๐(๐)|๐๐ < 1 2๐ โ๐ ๐ and, as above ๐
1 โซ ๐ท (๐)๐(๐)๐๐ 2๐ โ๐ ๐ ๐
โฅ
๐
1 1 โซ ๐ท (๐)๐๐ (๐)๐๐ โ โซ |๐ท (๐)||๐๐ (๐) โ ๐(๐)|๐๐ 2๐ โ๐ ๐ 2๐ โ๐ ๐
โฅ ๐. Since |๐๐ (๐)| โค 1, we clearly have |๐(๐)| โค 2.
โก
Proof of Theorem 3.56. For each ๐ โฅ 0, by Lemma 3.57 we can find a trigonometric polynomial ๐ ๐ and an integer ๐ ๐ such that |๐ ๐ (๐)| โค 2 and ๐
(3.58)
1 โซ ๐ท (๐)๐ ๐ (๐)๐๐ โฅ 2๐ . 2๐ โ๐ ๐๐
Let ๐๐ be a sequence of integers such that ๐๐+1 > 3๐๐ and such that ๐๐ is at least as large as the degree of ๐ ๐ and as ๐ ๐ , so that we can write ๐๐
(3.59)
๐ ๐ (๐) = โ ๐ ห๐ (๐)๐๐๐๐ , ๐=โ๐๐
66
3. Fourier series
and estimate (3.58) implies ๐๐
โ ๐ ห๐ (๐) โฅ 2๐ .
(3.60)
๐=โ๐๐
Now define ๐๐ (๐) = ๐2๐๐๐ ๐ ๐ ๐ (๐), so by (3.59) we have 3๐๐
๐๐
ห๐ (๐)๐ ๐๐ (๐) = โ ๐
๐(๐+2๐๐ )๐
๐=โ๐๐
= โ ๐ ห๐ (๐ โ 2๐๐ )๐๐๐๐ , ๐=๐๐
so (3.60) is now 2๐๐ +๐๐
๐ห๐ (๐) โฅ 2๐ .
โ
(3.61)
๐=2๐๐ โ๐๐
Note that |๐๐ (๐)| โค 2 and, for each ๐, ๐ห๐ (๐) โ 0 for at most one ๐ because ๐๐ > 3๐๐โ1 . We can now define โ
๐(๐) = โ 2โ๐ ๐๐ (๐). ๐=0
By the Weierstrass ๐-test, the series above converges uniformly and, since each ๐๐ is continuous (itโs a trigonometric polynomial), then ๐ is continuous. Moreover, by the observations above and the uniform conฬ = 2โ๐ ๐ห๐ (๐) for ๐๐ โค ๐ โค vergence of the series, for each ๐ we have ๐(๐) 3๐๐ . Therefore, if ๐ ๐ (0) is the ๐th partial sum of the Fourier series of ๐ at 0, 2๐๐ +๐๐
๐ 2๐๐ +๐๐ (0) โ ๐ 2๐๐ โ๐๐ โ1 (0) = 2โ๐
โ
๐ห๐ (๐) โฅ 1,
๐=2๐๐ โ๐๐
by (3.61). Therefore, the sequence ๐ ๐ (0) is not a Cauchy sequence, and cannot converge. โก By an appropiate translation of ๐, we can verify the existence of a continuous function with divergent Fourier series at any ๐0 โ ๐. By adding such functions, we conclude that there exist continuous functions with divergent Fourier series at any finite, or even countable infinite, number of points (Exercise (18)). Theorem 3.56 implies that being continuous is not enough for a function to have convergent Fourier series. However, if the function is โregular enoughโ, then we can guarantee its convergence, as in Theorem
3.7. Convergence for continuous functions
67
3.32, where we have ๐ โ ๐ถ 1 (๐). Moreover, it is enough for a function to be differentiable at a point to conclude that its Fourier series converges at that point. Theorem 3.62. Let ๐ be Riemann-integrable on ๐ and differentiable at ๐0 . Then its Fourier series converges at ๐0 . Proof. Let ๐ be differentiable at ๐0 . We want to prove that ๐
(3.63)
๐ ๐ (๐0 ) โ ๐(๐0 ) =
1 โซ ๐ท (๐)(๐(๐0 โ ๐) โ ๐(๐0 ))๐๐ 2๐ โ๐ ๐
converges to 0 as ๐ โ โ. Since ๐ท๐ (๐) =
sin ๐๐ cos ๐/2 + cos ๐๐ sin ๐/2 sin(๐ + 1/2)๐ = , sin ๐/2 sin ๐/2
we can write (3.63) as ๐
(3.64)
๐ ๐ (๐0 ) โ ๐(๐0 ) =
1 sin ๐๐ cos ๐/2 โซ (๐(๐0 โ ๐) โ ๐(๐0 ))๐๐ 2๐ โ๐ sin ๐/2 ๐
+
1 โซ cos ๐๐(๐(๐0 โ ๐) โ ๐(๐0 ))๐๐. 2๐ โ๐
Note that the second integral in (3.64) is the Fourier coefficient ๐๐ of the Riemann-integrable function ๐ โฆ ๐(๐0 โ ๐) โ ๐(๐0 ), and thus converges to 0 as ๐ โ โ, by the RiemannโLebesgue lemma 3.43. The first integral in (3.64) is the Fourier coefficient ๐๐ of the function ๐ โฆ cos ๐/2
๐(๐0 โ ๐) โ ๐(๐0 ) , sin ๐/2
which is also Riemann-integrable as the limit lim
๐โ0
๐(๐0 โ ๐) โ ๐(๐0 ) sin ๐/2
exists because ๐ is differentiable at ๐0 . Thus, it also converges to 0 when ๐ โ โ. โก
68
3. Fourier series
Exercises (1) Calculate the solutions of equations (3.1) and (3.2) when ๐ < 0. (2) If the sequences ๐๐ and ๐๐ are bounded, then โ
๐ข(๐, ๐) = โ ๐๐ (๐๐ cos(๐๐) + ๐๐ sin(๐๐)) ๐=0
is harmonic in ๐ป. (3) For ๐, ๐ โ โค, 2๐
โซ 0
2๐ ๐ = ๐ ๐๐๐๐ ๐โ๐๐๐ ๐๐ = { 0 ๐ โ ๐.
(4) If ๐ is Riemann-integrable and periodic with period ๐, then ๐+๐
โซ
๐
๐(๐ฅ)๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ
๐
0
for any ๐ โ โ. (5) Use Dirichletโs test to prove that the Fourier series of Example 3.12 converges for every ๐. (6) Complete the details of Example 3.13. (7) Complete the details of Example 3.14. (8) If โ ๐๐ is Abel-summable to ๐ and ๐๐๐ โ 0, then โ ๐๐ converges to ๐ . (9) Use Example 3.14 to obtain the identity โ
1 ๐4 = . 4 90 ๐ ๐=1 โ
(10) Suppose ๐ is a Riemann-integrable function that has left and right limits at ๐0 , say lim ๐(๐) = ๐(๐0 โ)
๐โ๐0โ
and
lim ๐(๐) = ๐(๐0 +),
๐โ๐0+
then ๐(๐0 โ) + ๐(๐0 +) , 2 as ๐ โ 1, where ๐๐ ๐(๐) are the Abel means of its Fourier series. ๐๐ ๐(๐0 ) โ
Exercises
69
(11) The following exercises prove inequality (3.35) for the inner product (3.33) and the partial sums (3.34). All functions are Riemannintegrable periodic functions with period 2๐. (a) For a function ๐, if ๐ ๐ is given by (3.34) and ๐ โ ๐ฏ๐ , then ๐ โ๐ ๐ and ๐ are orthogonal, โจ๐ โ ๐ ๐ , ๐โฉ = 0. (b) (Pythagorasโs theorem) If ๐, ๐ are orthogonal, then โ๐ + ๐โ2 = โ๐โ2 + โ๐โ2 . (c) Conclude (3.35) by noting that ๐ โ ๐ = ๐ โ ๐ ๐ + ๐ ๐ โ ๐, and that ๐ โ ๐ ๐ is orthogonal to ๐ ๐ โ ๐. (12) Let โ(๐) be the space of Riemann-integrable functions on ๐, and let ๐
= โ(๐)/ โผ, where โผ is the equivalence relation so that ๐ โผ ๐ if, and only if, ๐(๐ฅ) = ๐(๐ฅ) for all ๐ฅ except at a set of measure 0. (a) The subspace of continuous functions ๐ถ(๐) identifies with itself in ๐
. (b) The bilinear form โจ๐, ๐โฉ =
1 โซ 2๐ 0
2๐
๐(๐)๐(๐)๐๐
induces an inner product on ๐
. (c) The quadratic form โ๐โ = โโจ๐, ๐โฉ induces a norm in ๐
. (13) We say that ๐ is Hรถlder continuous with exponent ๐ผ, and write ๐ โ ๐ถ ๐ผ (๐), for some 0 < ๐ผ โค 1, if there exists ๐ > 0 such that |๐(๐ฅ) โ ๐(๐ฆ)| โค ๐|๐ฅ โ ๐ฆ|๐ผ . Prove Bernsteinโs theorem: If ๐ โ ๐ถ ๐ผ (๐) for some ๐ผ > 1/2, then ฬ < โ. (Hint: Proceed as in the proof of Theorem 3.32). โ |๐(๐)| (14) If โ ๐๐ is Cesร ro-summable to ๐ , then is it Abel-summable to ๐ . (15) For any ๐ โ โ and ๐ โฅ 1, ๐โ1
๐
1 1 sin2 (๐๐/2) โ โ ๐๐๐๐ = . ๐ ๐=0 ๐=โ๐ ๐ sin2 (๐/2) (16) Suppose ๐ is a Riemann-integrable function that has left and right limits at ๐0 , say lim ๐(๐) = ๐(๐0 โ)
๐โ๐0โ
and
lim ๐(๐) = ๐(๐0 +),
๐โ๐0+
70
3. Fourier series then
๐(๐0 โ) + ๐(๐0 +) , 2 where ๐๐ (๐) are the Cesร ro sums of its Fourier series. ๐๐ (๐0 ) โ
(17) The Dirichlet kernel ๐
๐ท๐ (๐) = โ ๐๐๐๐ ๐=โ๐
is given explicitly by sin(๐ + 1/2)๐ . sin ๐/2 (18) (a) For each ๐ โ ๐, there exists ๐ โ ๐ถ(๐) with divergent Fourier series at ๐. (b) If ๐1 , ๐2 , . . . , ๐๐ โ ๐, there exists ๐ โ ๐ถ(๐) with divergent Fourier series at ๐1 , ๐2 , . . . , ๐๐ . (c) If ๐ด โ ๐ is countable, there exists ๐ โ ๐ถ(๐) with divergent Fourier series at each point in ๐ด. ๐ท๐ (๐) =
Notes The idea of solving Laplaceโs equation, and the heat equation, by decomposing its solutions in trigonometric series was introduced by Fourier in [Fou55]. Parsevalโs identity was stated by Marc-Antoine Parseval in [PdC06], and it was proven by Michel Plancherel in [Pla10]. The Riemannโ Lebesgueโs lemma was first proven by Riemann for the case of Riemannintegrable functions, and then proven for general measurable functions by Lebesgue in [Leb03]. Theorem 3.45 was proven by Lipรณt Fejรฉr in [Fej00]. An extensive discussion of summability methods can be found in [Zyg02]. The proof of Theorem 3.62 is taken from [SS03]. Bernsteinโs theorem was proven in [Ber14]. The steps to prove it in Exercise (13) are also taken from [SS03], which are elaborated from its proof in [Zyg02]. The first to provide an explicit example of a continuous function with divergent Fourier series was Paul du Bois-Reymond in [du 76]. The construction discussed here follows [K8ฬ8]. Another construction can be found in [SS03].
Chapter 4
Poisson kernel in the half-space
4.1. The Poisson kernel in the half-space We now study harmonic functions in the upper half-space โ๐+1 = {(๐ฅ, ๐ก) โถ ๐ฅ โ โ๐ , ๐ก > 0}. + As in the case of the ball in โ๐ , we will study explicit formulas for harmonic functions in โ๐+1 + , given their values at its boundary โ๐ ร {0} = {(๐ฅ, 0) โถ ๐ฅ โ โ๐ } Note that, while the ball is a bounded set, and hence its boundary is compact, the upper half-space is unbounded, and so is its boundary โ๐ ร {0}. This will force us to be more careful when defining objects analogous to the Poisson integrals studied before. The Poisson kernel for the upper half-space is given by the function (4.1)
๐๐ก (๐ฅ) =
๐๐+1
(|๐ฅ|2
2๐ก , + ๐ก2 )(๐+1)/2
๐
defined for ๐ฅ โ โ and ๐ก > 0. Compare the function ๐๐ก (๐ฅ) with the Poisson kernel for the ball, given by 1 โ |๐ฅ|2 ๐(๐ฅ, ๐) = ๐๐ |๐ฅ โ ๐|๐ 71
72
4. Poisson kernel in the half-space
for ๐ฅ โ ๐น and ๐ โ ๐. Both functions have a multiple of the distance to the boundary (2๐ก and (1 + |๐ฅ|)(1 โ |๐ฅ|), respectively), and both have the distance to a boundary point powered to the dimension: in the case of the upper half-space, this is the distance between (๐ฅ, ๐ก) to the origin, so it is |(๐ฅ, ๐ก)|๐+1 , and in the case of the ball, the distance between ๐ฅ and ๐, for each ๐ โ ๐, so it is |๐ฅ โ ๐|๐ . Example 4.2. For ๐ = 1, since ๐2 = 2๐, the circunference of the unit circle, the Poisson kernel for the upper half-plane is given by ๐๐ก (๐ฅ) =
1 ๐ก . 2 ๐ ๐ฅ + ๐ก2
๐๐ก (๐ฅ) is an even function, and since we can write ๐๐ก (๐ฅ) =
1 1 1 โ
, ๐ก ๐ (๐ฅ/๐ก)2 + 1
we see that it has a bump at the origin that increases as ๐ก โ 0, as seen in Figure 4.1.
Figure 4.1. The Poisson kernel for the upper half-plane. Note that it is an even function, with a bump at the origin.
4.3. The function (๐ฅ, ๐ก) โฆ ๐๐ก (๐ฅ) is harmonic in โ๐+1 + . This can be verified explicitly by differentiating (Exercise (1)). 4.4. For each ๐ก > 0, ๐๐ก (๐ฅ) =
๐ฅ 1 ๐1 ( ). ๐ ๐ก ๐ก
4.1. The Poisson kernel in the half-space
73
Indeed, we have 2๐ก 2๐ก = ๐๐+1 (|๐ฅ|2 + ๐ก2 )(๐+1)/2 ๐๐+1 ๐ก๐+1 (|๐ฅ/๐ก|2 + 1)(๐+1)/2 1 2 ๐ฅ 1 = ๐ = ๐ ๐1 ( ). (๐+1)/2 2 ๐ก ๐ก ๐๐+1 (|๐ฅ/๐ก| + 1) ๐ก
๐๐ก (๐ฅ) =
Thus, ๐๐ก is the dilation of ๐1 , which motivates the notation of ๐ก as a subindex parameter, rather than as another variable of ๐. For any ๐ก > 0, โซ ๐๐ก (๐ฅ)๐๐ฅ = 1.
(4.5)
โ๐
The integral in (4.5) must be understood in the improper sense, i.e. โซ ๐๐ก (๐ฅ)๐๐ฅ = lim โซ ๐โโ
โ๐
๐๐ก (๐ฅ)๐๐ฅ.
|๐ฅ|โค๐
The fact the this limit exists for ๐ก > 0 follows from the estimate 2๐ก 2๐ก โค , ๐๐ก (๐ฅ) = (๐+1)/2 2 2 ๐๐+1 |๐ฅ|๐+1 ๐๐+1 (|๐ฅ| + ๐ก ) so, for ๐ โฅ ๐, using spherical coordinates we obtain |โซ |
๐๐ก (๐ฅ)๐๐ฅ โ โซ
|๐ฅ|โค๐
|๐ฅ|โค๐
๐๐ก (๐ฅ)๐๐ฅ|| โค โค
๐๐ฅ 2๐ก โซ ๐๐+1 ๐ 0, โซ
๐๐ก (๐ฅ)๐๐ฅ โ 0
|๐ฅ|โฅ๐ฟ
as ๐ก โ 0. Thus, the weight of the Poisson kernel gets concentrated at the origin as ๐ก โ 0. To verify this limit, observe first that, for any ๐ก > 0, ๐๐ก (๐ฅ) โค
2๐ก , ๐๐+1 |๐ฅ|๐+1
74
4. Poisson kernel in the half-space
and thus, using spherical coordinates as above โ
โซ
๐๐ก (๐ฅ)๐๐ฅ โค ๐๐ก โซ
|๐ฅ|โฅ๐ฟ
|๐ฅ|โฅ๐ฟ
1 ๐ก ๐๐ฅ = ๐๐ก โซ โซ ๐+1 ๐๐๐๐โ1 ๐๐ = ๐โฒ , ๐ฟ |๐ฅ|๐+1 ๐ ๐ฟ ๐
โฒ
where ๐, ๐ are the same constants as above, and all integrals at infinity are improper integrals. Therefore, the integral clearly goes to 0 as ๐ก โ 0, for each ๐ฟ > 0.
4.2. Poisson integrals in the half-space In this section we study the Poisson integrals in the half-space. Again, as in the previous section, we understand all integrals in โ๐ in the improper sense. For this, let โ(โ๐ ) be the space of bounded, locally Riemannintegrable functions ๐ on โ๐ (๐ is Riemann-integrable on any rectangle in โ๐ ) such that the sequence โซ
(4.7)
|๐(๐ฅ)|๐๐ฅ
|๐ฅ|โค๐
is bounded. For ๐ โ โ(โ๐ ), we define its improper integral as โซ ๐(๐ฅ)๐๐ฅ = lim โซ
(4.8)
๐โโ
โ๐
๐(๐ฅ)๐๐ฅ.
|๐ฅ|โค๐
To see that this limit exists, note that, since the sequence (4.7) is increasing and bounded, it converges, so for any ๐ > 0 there exists ๐พ such that, if ๐ โฅ ๐ โฅ ๐พ, then โซ ๐โค|๐ฅ|โค๐
|๐(๐ฅ)|๐๐ฅ = || โซ
|๐(๐ฅ)|๐๐ฅ โ โซ
|๐ฅ|โค๐
|๐ฅ|โค๐
|๐(๐ฅ)|๐๐ฅ|| < ๐.
Thus, for ๐ โฅ ๐ โฅ ๐พ, |โซ |
๐(๐ฅ)๐๐ฅ โ โซ
|๐ฅ|โค๐
|๐ฅ|โค๐
๐(๐ฅ)๐๐ฅ|| โค โซ
|๐(๐ฅ)|๐๐ฅ < ๐,
๐โค|๐ฅ|โค๐
and therefore the limit in (4.8) exists. Improper integrals on โ๐ are translation invariant, that is, (4.9)
โซ ๐(๐ฅ)๐๐ฅ = โซ ๐(๐ฅ โ โ)๐๐ฅ, โ๐
โ๐
4.2. Poisson integrals in the half-space
75
for any ๐ โ โ(โ๐ ) and any โ โ โ๐ . They also satisfy, for any ๐ > 0, โซ ๐(๐ฅ)๐๐ฅ =
(4.10)
โ๐
๐ฅ 1 โซ ๐( )๐๐ฅ. ๐๐ โ๐ ๐
These two properties follow from the boundedness of the sequence (4.7), and are left as an exercise (Exercise (3)). For a bounded locally Riemann-integrable function ๐ in โ๐ , we define its Poisson integral to be the function ๐ข(๐ฅ, ๐ก) given by ๐ข(๐ฅ, ๐ก) = ๐ซ๐ก ๐(๐ฅ) = โซ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ,
(4.11)
โ๐
for each (๐ฅ, ๐ก) โ โ๐+1 + . The integral in (4.11) is well defined because ๐ is bounded and thus ๐ฆ โฆ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ) is in โ(โ๐ ). Note that we can also write the Poisson integral as ๐ข(๐ฅ, ๐ก) = โซ ๐๐ก (๐ฆ)๐(๐ฅ โ ๐ฆ)๐๐ฆ, โ๐ 1
using (4.9).
The Poisson integral ๐ข(๐ฅ, ๐ก) of a bounded function ๐ is a bounded continuous function. Indeed, if |๐(๐ฅ)| โค ๐, |๐ข(๐ฅ, ๐ก)| โค โซ ๐๐ก (๐ฅ โ ๐ฆ)|๐(๐ฆ)|๐๐ฆ โค ๐ โซ ๐๐ก (๐ฅ โ ๐ฆ)๐๐ฆ = ๐, โ๐
โ๐
by (4.5) and (4.9). The continuity follows from the continuity of ๐๐ก (๐ฅ), its integrability, and the boundedness of ๐. Note that, for |๐ง| > 2|๐ฅ|, we have |๐ฅ โ ๐ง| > |๐ง|/2 and thus ๐๐ก (๐ฅ โ ๐ง) โค
2๐ก 2๐+2 ๐ก < . ๐๐+1 |๐ฅ โ ๐ง|๐+1 ๐๐+1 |๐ง|๐+1
Hence, if ๐ > 2|๐ฅ|, 2๐+2 ๐๐ ๐ก๐ ๐๐ก (๐ฅ โ ๐ง)๐(๐ง)๐๐ง|| โค . ๐๐+1 ๐ |๐ง|>๐
|โซ | 1
The operator ๐ โฆ ๐ซ๐ก ๐ is also called the Poisson semigroup operator. It satisfies ๐ซ๐ก ๐ซ๐ ๐ = ๐ซ๐ก+๐ ๐ ๐
for any ๐ โ โ(โ ) and ๐ก, ๐ > 0. This follows from the identity โซ ๐๐ก (๐ฅ โ ๐ง)๐๐ (๐ง โ ๐ฆ)๐๐ง = ๐๐ก+๐ (๐ฅ โ ๐ฆ), โ๐
which will be proved later using the Fourier transform (Chapter 10).
76
4. Poisson kernel in the half-space
Given (๐ฅ, ๐ก) โ โ๐+1 and ๐ > 0, choose ๐ > 2(|๐ฅ| + 1) such that + 2๐+2 ๐๐ (๐ก + 1)๐ ๐ < . ๐๐+1 ๐ 3 Now, choose ๐ฟ > 0 such that ๐ฟ < 1 and, if (|๐ฅ โ ๐ฆ|2 + (๐ก โ ๐ )2 )1/2 < ๐ฟ, then ๐๐ |๐๐ก (๐ฅ โ ๐ง) โ ๐๐ (๐ฆ โ ๐ง)| < 3๐๐ ๐๐ ๐ for any |๐ง| โค ๐. Such ๐ฟ exists since the closed ball of radius ๐ + 1 is compact and, hence, ๐๐ก (๐ฅ โ ๐ง) is uniformly continuous. Therefore, using the fact that |๐ฅ โ ๐ฆ| < 1 and |๐ก โ ๐ | < 1 if (|๐ฅ โ ๐ฆ|2 + (๐ก โ ๐ )2 )1/2 < ๐ฟ, so that we have ๐ > 2|๐ฆ| and ๐ < ๐ก + 1, | | | โซ ๐๐ก (๐ฅ โ ๐ง)๐(๐ง)๐๐ง โ โซ ๐๐ (๐ฆ โ ๐ง)๐(๐ง)๐๐ง| โ๐
โ๐
โค โซ |๐๐ก (๐ฅ โ ๐ง) โ ๐๐ (๐ฆ โ ๐ง)||๐(๐ง)|๐๐ง โ๐
2๐+2 ๐๐ ๐ก๐ 2๐+2 ๐๐ ๐ ๐ ๐๐ ๐๐๐ง + + ๐ ๐๐+1 ๐ ๐๐+1 ๐ |๐ง|โค๐ 3๐ ๐ ๐๐ ๐ ๐ ๐ < + + = ๐. 3 3 3 โคโซ
As above, we can also verify that ๐ข(๐ฅ, ๐ก) is differentiable, using the fact that ๐๐ก (๐ฅ) is continuously differentiable and in โ(โ๐ ) for each ๐ก. Indeed, the partial derivatives of ๐๐ก (๐ฅ) are given by 2(๐ + 1)๐ก๐ฅ๐ ๐๐๐ก =โ ๐๐ฅ๐ ๐๐+1 (|๐ฅ|2 + ๐ก2 )(๐+3)/2 for ๐ = 1, . . . , ๐, and 2(|๐ฅ|2 โ ๐๐ก2 ) ๐๐๐ก , = ๐๐ก ๐๐+1 (|๐ฅ|2 + ๐ก2 )(๐+3)/2 and thus, as above, for |๐ง| > 2|๐ฅ|, | ๐๐๐ก (๐ฅ โ ๐ง)| โค ๐ถ๐ก | ๐๐ฅ | |๐ง|๐+2 ๐
and
| ๐๐๐ก (๐ฅ โ ๐ง)| โค ๐ถ , | ๐๐ก | |๐ง|๐+1
4.3. Boundary limits
77
for some constant ๐ถ that depends only on ๐. Moreover, note that for โ โ โ, ๐ข(๐ฅ + โ๐๐ , ๐ก) โ ๐ข(๐ฅ, ๐ก) ๐๐๐ก โโซ (๐ฅ โ ๐ง)๐(๐ง)๐๐ง โ ๐๐ฅ ๐ โ๐ =โซ ( โ๐
๐๐ก (๐ฅ + โ๐๐ โ ๐ง) โ ๐๐ก (๐ฅ โ ๐ง) ๐๐ โ ๐ก (๐ฅ โ ๐ง))๐(๐ง)๐๐ง โ ๐๐ฅ๐ =โซ ( โ๐
๐๐๐ก ๐๐ (๐ฆ โ ๐ง) โ ๐ก (๐ฅ โ ๐ง))๐(๐ง)๐๐ง, ๐๐ฅ๐ ๐๐ฅ๐
๐
where ๐ฆ โ โ is of the form ๐ฆ = ๐ฅ + ๐ ๐๐ where ๐ is a number between 0 and โ that depends on ๐ฅ, ๐ก and ๐ง. Since ๐๐๐ก ๐๐ (๐ฆ โ ๐ง) โ ๐ก (๐ฅ โ ๐ง) ๐๐ฅ๐ ๐๐ฅ๐ as โ โ 0 for each ๐ง, we can proceed as above and conclude that ๐๐๐ก ๐๐ข =โซ (๐ฅ โ ๐ง)๐(๐ง)๐๐ง, ๐๐ฅ๐ ๐๐ฅ ๐ โ๐ so we can โdifferentiate under the integralโ. Similarly for the partial derivative of ๐ข with respect to ๐ก. Hence, we have the following result. Proposition 4.12. If ๐ is a bounded locally Riemann-integrable function in โ๐ , its Poisson integral ๐ข(๐ฅ, ๐ก) is harmonic in โ๐+1 + . Proof. Similarly, as in the above cases, we can differentiate inside the Poisson integral to verify that ๐
๐2 ๐๐ก ๐2 ๐๐ก (๐ฅ โ ๐ง) + (๐ฅ โ ๐ง))๐(๐ง)๐๐ง. 2 ๐๐ก2 ๐=1 ๐๐ฅ๐
ฮ๐ข(๐ฅ, ๐ก) = โซ ( โ โ๐
The theorem then follows from the fact that (๐ฅ, ๐ก) โฆ ๐๐ก (๐ฅ) is harmonic. We leave the details as an exercise (Exercise (4)). โก
4.3. Boundary limits We now study the behavior of a Poisson integrals ๐ข(๐ฅ, ๐ก) as the point (๐ฅ, ๐ก) approaches a boundary point. We start with Theorem 4.13, analogous to Theorem 2.28.
78
4. Poisson kernel in the half-space
Theorem 4.13. Let ๐ be a bounded locally Riemann-integrable function in โ๐ , continuous at ๐ฅ0 โ โ๐ , and let ๐ข(๐ฅ, ๐ก) be its Poisson integral. Then lim
(๐ฅ,๐ก)โ(๐ฅ0 ,0)
๐ข(๐ฅ, ๐ก) = ๐(๐ฅ0 ).
In particular, if ๐ is continuous at every point of โ๐ , its Poisson integral ๐ข(๐ฅ, ๐ก) in โ๐+1 extends continuously to the boundary โ๐ ร {0}, with + the value ๐(๐ฅ) at each (๐ฅ, 0). Unsurprinsingly, the proof is very similar to the proofs of Theorems 2.28 and 3.45. Proof. Given ๐ > 0, since ๐ is continuous at ๐ฅ0 there exists ๐ > 0 such that, if |๐ฅ โ ๐ฅ0 | < ๐, then |๐(๐ฅ) โ ๐(๐ฅ0 )| < ๐/2. Now, write ๐ข(๐ฅ, ๐ก) โ ๐(๐ฅ0 ) = โซ ๐๐ก (๐ง)๐(๐ฅ โ ๐ง)๐๐ง โ ๐(๐ฅ0 ) โ๐
= โซ ๐๐ก (๐ง)(๐(๐ฅ โ ๐ง) โ ๐(๐ฅ0 ))๐๐ง, โ๐
where we have used (4.5). Now, if |๐ฅ โ ๐ฅ0 | < ๐/2 and |๐ง| < ๐/2, |๐ฅ โ ๐ง โ ๐ฅ0 | < ๐ and thus |โซ |
|๐ง| 0, โซ
|ฮฆ๐ก (๐ฅ)|๐๐ฅ โ 0 as ๐ก โ 0.
|๐ฅ|โฅ๐ฟ
These operators are called convolution operators, and are denoted by ฮฆ๐ก โ ๐. Note that the Poisson integral of ๐ is the convolution ๐๐ก โ ๐. The collection of functions {ฮฆ๐ก }๐ก>0 form a family of good kernels. See Exercise (9).
80
4. Poisson kernel in the half-space
Exercises (1) The function (๐ฅ, ๐ก) โฆ ๐๐ก (๐ฅ) is harmonic in โ๐+1 + . (2) For any dimension ๐ โฅ 1, โซ โ๐
(|๐ฅ|2
๐๐ฅ ๐(๐+1)/2 = , (๐+1)/2 ฮ((๐ + 1)/2) + 1)
and verify (4.5). (Hint: Use spherical coordinates and the identity โ
โซ ๐ก๐ผ ๐โ๐ก๐ 0
ฮ(๐ผ) ๐๐ก = ๐ผ ๐ก ๐
for any ๐ผ, ๐ > 0.) (3) Let ๐ โ โ(โ๐ ). (a) For any โ โ โ๐ , โซ ๐(๐ฅ)๐๐ฅ = โซ ๐(๐ฅ โ โ)๐๐ฅ. โ๐
โ๐
(b) For any ๐ > 0, โซ ๐(๐ฅ)๐๐ฅ = โ๐
1 ๐ฅ โซ ๐( )๐๐ฅ. ๐๐ โ๐ ๐
(Hint: The integrals โซ|๐ฅ|โค๐ ๐(๐ฅ)๐๐ฅ form a Cauchy sequence.) (4) Complete the details of the proof of Proposition 4.12. (5) If ๐ โ ๐ถ0 (โ๐ ), then ๐ข(๐ฅ, ๐ก) โ ๐(๐ฅ) as ๐ก โ 0, uniformly in ๐ฅ โ โ๐ . (6) Let ๐ โ ๐ถ๐ (โ๐ ). Then โซ |๐(๐ฅ โ โ) โ ๐(๐ฅ)|๐๐ฅ โ 0 โ๐ ๐
as โ โ 0 in โ . (Hint: ๐ is uniformly continuous on its compact support.) (7) Prove the following version of Fubiniโs theorem: Let ๐(๐ฅ, ๐ฆ) โ ๐ถ(โ๐+๐ ) such that (a) there exists ๐ด > 0 such that, for all ๐ฆ โ โ๐ , |๐(๐ฅ, ๐ฆ)| โค
๐ด ; (|๐ฅ|๐+1 + 1)
Exercises
81
(b) there exists a compact ๐พ โ โ๐ such that, for all ๐ฅ โ โ๐ , ๐(๐ฅ, โ
) is supported in ๐พ. Then ๐(๐ฅ, ๐ฆ) is integrable in โ2๐ and โซ ๐(๐ฅ, ๐ฆ)๐๐ฅ๐๐ฆ = โซ ( โซ ๐(๐ฅ, ๐ฆ)๐๐ฅ)๐๐ฆ = โซ ( โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ)๐๐ฅ. โ2๐
โ๐
โ๐
โ๐
โ๐
(8) If ๐ โ ๐ถ๐ (โ๐ ), โซ |๐ข(๐ฅ, ๐ก) โ ๐(๐ฅ)|๐๐ฅ โ 0 โ๐
as ๐ก โ 0. (Hint: Use Exercises (6) and (7).) (9) Let {๐พ๐ก }๐ก>0 be a family of functions in โ(โ๐ ). We say that it is a family of good kernels if โข โซ ๐พ๐ก (๐ฅ)๐๐ฅ = 1 for all ๐ก > 0; โ๐
โข there exists ๐ > 0 such that โซ |๐พ๐ก (๐ฅ)|๐๐ฅ โค ๐ for all ๐ก > 0; โ๐
and โข for ๐ฟ > 0, โซ
|๐พ๐ก (๐ฅ)|๐๐ฅ โ 0 as ๐ก โ 0.
|๐ฅ|โฅ๐ฟ ๐
(a) If ฮฆ โ โ(โ ) and โซ ฮฆ = 1, then its dilations {ฮฆ๐ก }๐ก>0 form a family of good kernels. (b) If {๐พ๐ก }๐ก>0 is a family of good kernels and ๐ โ ๐ถ(โ๐ ) is bounded, then, for each ๐ฅ โ โ๐ , lim
(๐ฆ,๐ก)โ(๐ฅ,0)
๐พ๐ก โ ๐(๐ฆ) = ๐(๐ฅ).
(c) If {๐พ๐ก }๐ก>0 is a family of good kernels and ๐ โ ๐ถ0 (โ๐ ), then ๐พ๐ก โ ๐ โ ๐ as ๐ก โ 0. (10) (Principle of subordination) Let ๐ป๐ก (๐ฅ) be the heat kernel, ๐ป๐ก (๐ฅ) =
1 2 ๐โ|๐ฅ| /4๐ก . (4๐๐ก)๐/2
Then ๐๐ก (๐ฅ) =
๐ก 2โ๐
โ 2 /4๐
โซ ๐โ๐ก 0
๐ป๐ (๐ฅ)
๐๐ . ๐ 3/2
(Hint: use the identity of the gamma function as in Exercise (2).)
82
4. Poisson kernel in the half-space
Notes The results of this chapter are classical, and can be found in the references cited previously, as [ABR01]. An extended treatment on Poisson integrals can be found in [Ste70].
Chapter 5
Measure theory in Euclidean space
5.1. The need for an integration theory We have seen in Chapter 4 that, when dealing with Poisson integrals in the upper half-space, that improper integrals add a further difficulty when dealing with boundary limits. However, the main problem with Riemann integration is the fact that the Riemann integral is not compatible with pointwise limits. Indeed, consider the set ๐ = [0, 1] โฉ โ of rational numbers in the interval [0, 1]. This set is countable, so we can write ๐ = {๐๐ โถ ๐ โ โ}. Now, for each ๐, let ๐๐ be the function on [0, 1] given by 1 ๐ฅ = ๐๐ for some ๐ โค ๐ ๐๐ (๐ฅ) = { 0 otherwise. Then, we see that ๐๐ (๐ฅ) โ ๐(๐ฅ) for every ๐ฅ โ [0, 1], where ๐ is the function 1 ๐(๐ฅ) = { 0
๐ฅโ๐ ๐ฅ โ ๐.
However, while each ๐๐ is Riemann-integrable on [0, 1], ๐ is not. Given any partition ๐ซ = {๐ฅ0 = 0 < ๐ฅ1 < ๐ฅ2 < . . . < ๐ก๐ = 1} 83
84
5. Measure theory in Euclidean space
of [0, 1], we see that ๐๐ = sup{๐(๐ฅ) โถ ๐ฅ โ [๐ฅ๐โ1 , ๐ฅ๐ ]} = 1 for all ๐ = 1, . . . , ๐, because every interval in โ contains a rational number, so the upper Riemann sum of ๐ is ๐
๐
๐(๐, ๐ซ) = โ ๐๐ (๐ฅ๐ โ ๐ฅ๐โ1 ) = โ (๐ฅ๐ โ ๐ฅ๐โ1 ) = 1. ๐=1
๐=1
On the other hand, ๐๐ = inf{๐(๐ฅ) โถ ๐ฅ โ [๐ฅ๐โ1 , ๐ฅ๐ ]} = 0 for all ๐ = 1, . . . , ๐, because every interval in โ also contains an irrational number, so the lower Riemann sum is ๐
๐ฟ(๐, ๐ซ) = โ ๐๐ (๐ฅ๐ โ ๐ฅ๐โ1 ) = 0. ๐=1
As it is impossible to find a partition to get ๐(๐, ๐ซ) โ ๐ฟ(๐, ๐ซ) < 1, we conclude ๐ is not Riemann integrable. We now ask if it is possible to extend the definition of the integral to satisfy the following requirements: (1) every Riemann-integrable function is also integrable in the new definition; (2) if each ๐๐ is integrable and ๐๐ โ ๐ pointwise, then ๐ is also integrable; (3) under appropiate conditions, if ๐๐ โ ๐ pointwise, then we have โซ ๐๐ โ โซ ๐ in the extended definition. Property (1) is necessary because we want to extend the definition of the integral, so that the results on integration we already have must still hold in the new definition. The โappropiate conditionsโ above should not be too restrictive, of course. For example, the Riemann integral is consistent with uniform convergence, which is an hypothesis too strong to be useful in most situations. The purpose of this chapter is to give an introduction to measure and integration theory. Lebesgueโs theory provides an integral that satisfies the above conditions, and will allow us to both extend the results of Chapter 4 and deal with further related problems.
5.2. Outer measure in Euclidean space
85
5.2. Outer measure in Euclidean space The first ingredient in Lebesgueโs theory is the idea of measure of a set. Recall that if ๐
is a rectangle in โ๐ , say ๐
= ๐ผ 1 ร ๐ผ2 ร โฏ ร ๐ผ ๐ , where each ๐ผ๐ is a bounded interval, then its volume is given by vol(๐
) = |๐ผ1 | โฏ |๐ผ๐ |, the product of the lengths of the intervals ๐ผ1 , . . . , ๐ผ๐ . A cube ๐ is a rectangle with all intervals ๐ผ๐ above of the same length, say ๐ฟ, and in that case vol(๐) = ๐ฟ๐ . ๐ is a dyadic cube if ๐ฟ = 2๐ for some ๐ โ โค, and the limits of each ๐ผ๐ are of the form ๐ โ
2๐ for some ๐ โ โค. The intervals that define the rectangle ๐
may be open, closed, or neither. Note that ๐
is an open set if all the intervals ๐ผ๐ are open, and ๐
is closed if all of them are closed. For any rectangle ๐
, its interior ๐
0 and its closure ๐
ฬ satisfy vol(๐
0 ) = vol(๐
)ฬ = vol(๐
). For any ๐ > 0, we can find an open rectangle ๐ โ ๐
such that vol(๐) < vol(๐
) + ๐, by widening (if necessary) each interval ๐ผ๐ , and a closed rectangle ๐ โ ๐
such that vol(๐) > vol(๐
) โ ๐, by shrinking (if necessary) each interval ๐ผ๐ (Exercise (1)). In order to extend the concept of volume to general sets ๐ด โ โ๐ , we first define the outer measure of ๐ด by |๐ด|โ = inf { โ vol(๐
๐ ) โถ each ๐
๐ is a rectangle and ๐ด โ ๐โฅ1
โ
๐
๐ }.
๐โฅ1
Thus, |๐ด|โ is the infimum of the set of all possible sums of volumes of rectangles covering ๐ด, where each cover may have a finite or a countable infinite number of rectangles. If no such sum is finite, then we say |๐ด|โ = โ. Thus, for each ๐ > 0, there exists a cover of rectangles ๐
1 , ๐
2 , . . . for ๐ด such that โ vol(๐
๐ ) < |๐ด|โ + ๐, ๐โฅ1
86
5. Measure theory in Euclidean space
and, by the observations above, we can choose all of these rectangles to be open, or all of them to be closed.1 The outer measure is also usually called exterior measure. 5.1. If ๐ด = โ
, then clearly |๐ด|โ = 0. Also if ๐ด contains only one point, or if ๐ด is finite. In fact, if ๐ด is countable, |๐ด|โ = 0 (Exercise (2)). As seen in Appendix A, if |๐ด|โ = 0, we say that ๐ด has measure zero, or that itโs a measure zero set. One can show that the countable union of measure zero sets is also of measure zero (Exercise (2)). The Cantor ternary set is also a set of measure zero (Exercise (3)). 5.2. If ๐
is a closed rectangle, then |๐
|โ = vol(๐
). That is, its outer measure coincides with its volume. Indeed, first note that |๐
|โ โค vol(๐
), because ๐
is a cover for itself. For the reverse inequality, suppose ๐ > 0 is given and ๐
1 , ๐
2 , . . . are open rectangles that cover ๐
such that โ vol(๐
๐ ) < |๐
|โ + ๐. ๐โฅ1
Since ๐
is closed, it is compact, and thus we can assume we have a finite number of the ๐
๐ , say, we have ๐ of them. Now, the edges of these rectangles can be extended to form a grid of subrectangles ๐ ๐ of ๐
, as in
Figure 5.1. Given a cover of rectangles for a rectangle, the edges can be extended to form a grid of subrectangles.
Figure 5.1, each one of them also a subrectangle of the ๐
๐ , such that ๐
=
โ
๐๐
๐
1
We use the convention that ๐ฅ < โ for all ๐ฅ โ โ, and later on we will also use โ + โ = โ.
5.2. Outer measure in Euclidean space
87
and the ๐ ๐ are almost disjoint, that is, ๐ ๐ โฉ ๐ ๐ is either empty or just part of their boundary. Therefore ๐
vol(๐
) = โ vol(๐ ๐ ) โค โ vol(๐
๐ ) < |๐
|โ + ๐ ๐
๐=1
(Exercise (4)) and, since ๐ > 0 is arbitrary, vol(๐
) โค |๐
|โ . 5.3. If ๐
is any rectangle, then we also have that |๐
|โ = vol(๐
). Indeed, |๐
|โ โค vol(๐
) because again ๐
forms a cover for itself. Also, for any closed rectangle ๐ โ ๐
, |๐|โ โค |๐
|โ , because any cover of rectangles for ๐
also a cover for ๐. Now, |๐|โ = vol(๐) and, for any ๐ > 0, we can choose ๐ such that vol(๐) > vol(๐
) โ ๐. Thus vol(๐
) โ ๐ < |๐
|โ , so we again have vol(๐
) โค |๐
|โ because we can make ๐ arbitrarily small. 5.4. The fact for rectangles used above is true for general subsets of โ๐ : if ๐ด โ ๐ต, then |๐ด|โ โค |๐ต|โ , as any cover of rectangles for ๐ต is also a cover for ๐ด. This property is called monotonicity. 5.5. If ๐ด = โ๐โฅ1 ๐ด๐ , then |๐ด|โ โค โ |๐ด๐ |โ . ๐โฅ1 ๐
Indeed, let ๐ > 0 be given and, for each ๐ด๐ , let {๐
๐ } be a cover of rectangles for ๐ด๐ such that ๐ ๐ โ vol(๐
๐ ) < |๐ด๐ |โ + ๐ . 2 ๐โฅ1 ๐
Thus โ๐,๐โฅ1 {๐
๐ } is a cover of rectangles for ๐ด, and ๐
|๐ด|โ โค โ vol(๐
๐ ) < โ (|๐ด๐ |โ + ๐,๐โฅ1
๐โฅ1
๐ ) = โ |๐ด๐ |โ + ๐. 2๐ ๐โฅ1
We obtain the result because ๐ > 0 is arbitrary. Property 5.5 is called countable subadditivity. A natural question to ask is whether we have equality in 5.5 when the ๐ด๐ are disjoint. In general we donโt,2 but we have the following special cases. 2
This is beyond the goal of this text, but you can find an example in [Fol99, Section 1.1].
88
5. Measure theory in Euclidean space
5.6. If ๐ด, ๐ต โ โ๐ and dist(๐ด, ๐ต) > 0, then |๐ด โช ๐ต|โ = |๐ด|โ + |๐ต|โ . Note that ๐ด and ๐ต are not only disjoint, but separated by a positive distance from each other, that is, there is some ๐ฟ > 0 such that, for any ๐ฅ โ ๐ด, ๐ฆ โ ๐ต, then |๐ฅ โ ๐ฆ| > ๐ฟ. By 5.5 we have |๐ด โช ๐ต|โ โค |๐ด|โ + |๐ต|โ , so we have to prove the reverse inequality. Let ๐ > 0 be given and {๐
๐ } be a cover of rectangles for ๐ด โช ๐ต such that โ vol(๐
๐ ) < |๐ด โช ๐ต|โ + ๐ ๐
and, by subdividing them if necessary, the diameter of each ๐
๐ is smaller than ๐ฟ. Thus, if ๐
๐ โฉ ๐ด โ โ
, then ๐
๐ โฉ ๐ต = โ
, and vice versa. Hence, if ๐๐ด = {๐ โถ ๐
๐ โฉ ๐ด โ โ
} and ๐๐ต = {๐ โถ ๐
๐ โฉ ๐ต โ โ
}, then ๐๐ด โฉ ๐๐ต = โ
and each of {๐
๐ }๐โ๐๐ด and {๐
๐ }๐โ๐๐ต is a cover for ๐ด and ๐ต, respectively. Thus |๐ด|โ + |๐ต|โ โค โ vol(๐
๐ ) + โ vol(๐
๐ ) โค โ vol(๐
๐ ) < |๐ด โช ๐ต|โ + ๐, ๐โ๐๐ด
๐โ๐๐ต
๐
and the result follows because ๐ > 0 is arbitrary. Note that, inductively, we can extend 5.6 for any finite number of sets at positive distance. 5.7. If ๐ด = โ๐ ๐๐ where the ๐๐ are almost disjoint cubes, then |๐ด|โ = โ |๐|โ = โ vol(๐). ๐
๐
To verify this, let ๐ > 0 be given and consider strictly thinner cubes ๐ฬ๐ โ ๐๐ such that ๐ vol(๐ฬ๐ ) > vol(๐๐ ) โ ๐ . 2 With strictly thinner we mean that the closure of each ๐ฬ๐ is contained in the interior of ๐๐ . Thus, for any ๐ โ ๐, dist(๐ฬ ๐ , ๐ฬ๐ ) > 0 and, by 5.6, for each ๐ we have ๐
๐
๐
๐
๐ | | | โ ๐ฬ๐ |โ = โ |๐ฬ๐ |โ > โ (|๐๐ |โ โ 2๐ ) > โ |๐๐ |โ โ ๐. ๐=1 ๐=1 ๐=1 ๐=1
5.3. Measurable sets and measure
89
๐
Since ๐ด โ โ๐=1 ๐ฬ๐ , we have, by 5.4, ๐
|๐ด|โ > โ |๐๐ |โ โ ๐. ๐=1
As ๐ is arbitrary, we have |๐ด|โ โฅ โ |๐๐ |โ โ ๐, ๐
and we obtain the result because ๐ > 0 is also arbitrary. Fact 5.7 is particularly useful when ๐ด is an open set, since every nonempty open set in โ๐ is an almost disjoint union of closed cubes which can even be chosen to be dyadic (Exercise (5)). As we have mentioned above, we donโt have equality in 5.5, even if we have a finite number of sets. Thus, we need to restrict ourselves to those sets on which the equality would be true.
5.3. Measurable sets and measure We say that ๐ด โ โ๐ is measurable if, for each ๐ต โ โ๐ , |๐ต|โ = |๐ต โฉ ๐ด|โ + |๐ต โงต ๐ด|โ .
(5.8)
Note that ๐ต โฉ ๐ด and ๐ต โงต ๐ด are disjoint and ๐ต = (๐ต โฉ ๐ด) โช (๐ต โงต ๐ด), so ๐ด is measurable when it splits any other set ๐ต in parts whose outer measures add to the outer measure of ๐ต. Recall that we always have, by 5.5, the inequality |๐ต|โ โค |๐ต โฉ ๐ด|โ + |๐ต โงต ๐ด|โ , so ๐ด is measurable if the reverse inequality |๐ต|โ โฅ |๐ต โฉ ๐ด|โ + |๐ต โงต ๐ด|โ is true for all ๐ต โ โ๐ . 5.9. โ
and โ๐ are measurable. This follows because ๐ตโฉโ
=โ
and
๐ต โงต โ
= ๐ต,
๐ต โฉ โ๐ = ๐ต
and
๐ต โงต โ๐ = โ
.
as well as 5.10. If ๐ด is measurable, then its complement โ๐ โงต๐ด is measurable. This also follows immediately because ๐ต โฉ (โ๐ โงต ๐ด) = ๐ต โงต ๐ด
and
๐ต โงต (โ๐ โงต ๐ด) = ๐ต โฉ ๐ด.
90
5. Measure theory in Euclidean space
5.11. If ๐ด and ๐ต are measurable, then ๐ด โช ๐ต is measurable. To prove this note that, for any ๐ถ โ โ๐ , |๐ถ|โ = |๐ถ โฉ ๐ด|โ + |๐ถ โงต ๐ด|โ = |๐ถ โฉ ๐ด โฉ ๐ต|โ + |(๐ถ โฉ ๐ด) โงต ๐ต|โ + |(๐ถ โงต ๐ด) โฉ ๐ต|โ + |(๐ถ โงต ๐ด) โงต ๐ต|โ , using (5.8) for each of ๐ด and ๐ต, which are both measurable. Now ๐ด โช ๐ต = (๐ด โฉ ๐ต) โช (๐ด โงต ๐ต) โช (๐ต โงต ๐ด), ๐ถ โฉ (๐ด โงต ๐ต) = (๐ถ โฉ ๐ด) โงต ๐ต,
and
๐ถ โฉ (๐ต โงต ๐ด) = (๐ถ โงต ๐ด) โฉ ๐ต,
so we have ๐ถ โฉ (๐ด โช ๐ต) = (๐ถ โฉ ๐ด โฉ ๐ต) โช (๐ถ โฉ (๐ด โงต ๐ต)) โช (๐ถ โฉ (๐ต โงต ๐ด)) = (๐ถ โฉ ๐ด โฉ ๐ต) โช ((๐ถ โฉ ๐ด) โงต ๐ต) โช ((๐ถ โงต ๐ด) โฉ ๐ต), and thus |๐ถ โฉ ๐ด โฉ ๐ต|โ + |(๐ถ โฉ ๐ด) โงต ๐ต|โ + |(๐ถ โงต ๐ด) โฉ ๐ต|โ โฅ |๐ถ โฉ (๐ด โช ๐ต)|โ . Also (๐ถ โงต ๐ด) โงต ๐ต = ๐ถ โงต (๐ด โช ๐ต), and hence |๐ถ|โ โฅ |๐ถ โฉ (๐ด โช ๐ต)|โ + |๐ถ โงต (๐ด โช ๐ต)|โ , so ๐ด โช ๐ต is measurable. You can visualize the decomposition of ๐ถ used in the proof above with the Venn diagram shown in Figure 5.2. Inductively, for any ๐, the union of ๐ measurable sets ๐ด1 , . . . , ๐ด๐ is measurable. 5.12. If ๐ด1 , ๐ด2 , . . . are measurable, then their union ๐ด = โ๐ ๐ด๐ is also measurable. To prove this, let ๐ต1 = ๐ด1 and, for each ๐ โฅ 2, define ๐ต๐ = ๐โ1
๐
๐
๐ด๐ โงต โ๐=1 ๐ด๐ . Thus the ๐ต๐ are disjoint, โ๐=1 ๐ต๐ = โ๐=1 ๐ด๐ for all ๐ and ๐
โ๐ ๐ต๐ = ๐ด. Also, each โ๐=1 ๐ต๐ is measurable. Now, for ๐ถ โ โ๐ and any ๐, ๐
|๐ถ โฉ
โ ๐=1
๐
๐ต๐ |โ = |(๐ถ โฉ
โ
๐
๐ต๐ ) โฉ ๐ต๐ |โ + |(๐ถ โฉ
๐=1
๐=1
๐โ1
= |๐ถ โฉ ๐ต๐ |โ + |๐ถ โฉ
โ
โ ๐=1
๐ต๐ |โ ,
๐ต๐ ) โงต ๐ต๐ |โ
5.3. Measurable sets and measure
91
Figure 5.2. Venn diagram of the decomposition of ๐ถ โฉ (๐ด โช ๐ต) as (๐ถ โฉ ๐ด โฉ ๐ต) โช ((๐ถ โฉ ๐ด) โงต ๐ต) โช ((๐ถ โงต ๐ด) โฉ ๐ต).
and inductively ๐
|๐ถ โฉ
โ
๐
๐ต๐ |โ = โ |๐ถ โฉ ๐ต๐ |โ . ๐=1
๐=1
Thus ๐
๐
|๐ถ|โ = |๐ถ โฉ
โ ๐=1
๐ต๐ |โ + |๐ถ โงต
โ
๐
๐ต๐ |โ = โ |๐ถ โฉ ๐ต๐ |โ + |๐ถ โงต
๐=1
๐=1
๐
โ
๐ต๐ |โ
๐=1
๐
โฅ โ |๐ถ โฉ ๐ต๐ |โ + |๐ถ โงต ๐ด|โ , ๐=1 ๐
because โ๐=1 ๐ต๐ โ ๐ด. As ๐ is arbitrary, we obtain |๐ถ|โ โฅ โ |๐ถ โฉ ๐ต๐ |โ + |๐ถ โงต ๐ด|โ โฅ || (๐ถ โฉ ๐ต๐ )|| + |๐ถ โงต ๐ด|โ โ โ ๐โฅ1
= |๐ถ โฉ ๐ด| + |๐ถ โงต ๐ด|โ , and we conclude ๐ด is measurable.
๐โฅ1
92
5. Measure theory in Euclidean space
Note that we can conclude, by 5.10 and 5.12, that the intersection of a countable number of measurable sets is measurable, as we can write โ
๐ด๐ = โ๐ โงต
๐
โ
(โ๐ โงต ๐ด๐ ).
๐
5.13. A closed cube ๐ is measurable. Let ๐ด โ โ๐ , and we want to prove |๐ด|โ โฅ |๐ด โฉ ๐|โ + |๐ด โงต ๐|โ . This inequality is obvious if |๐ด|โ = โ, so we assume |๐ด|โ < โ. For each ๐, define ๐๐ = {๐ฅ โ โ๐ โถ dist(๐ฅ, ๐)
0 be given. (1) There exists an open set ๐ โ ๐ด such that |๐ โงต ๐ด| < ๐. (2) There exists a closed set ๐ธ โ ๐ด such that |๐ด โงต ๐ธ| < ๐. (3) If |๐ด| < โ, there exists a compact set ๐พ โ ๐ด such that |๐ดโงต๐พ| < ๐. (4) If |๐ด| < โ, there exist finitely many closed cubes ๐1 , ๐2 , . . ., ๐๐ ๐ such that, if ๐น = โ๐=1 ๐๐ , then |๐ดโณ๐น| < ๐. The symbol โณ above denotes the symmetric difference of the sets, ๐ดโณ๐น = (๐ด โงต ๐น) โช (๐น โงต ๐ด). 5.18. Moreover, statement (1) of Corollary 5.17 is equivalent to the statement that ๐ด is measurable. Indeed, suppose ๐ด โ โ๐ such that, for any ๐ > 0, there exists an open ๐ โ ๐ด such that |๐ โงต ๐ด|โ < ๐.
96
5. Measure theory in Euclidean space
Let ๐ต โ โ๐ . Given ๐ > 0, choose ๐ โ ๐ด such that |๐ โงต ๐ด|โ < ๐. Now ๐ต โฉ ๐ด โ ๐ต โฉ ๐ and ๐ต โงต ๐ด โ (๐ต โงต ๐) โช (๐ โงต ๐ด), so |๐ต โฉ ๐ด|โ + |๐ต โงต ๐ด|โ โค |๐ต โฉ ๐|โ + |๐ต โงต ๐|โ + |๐ โงต ๐ด|โ < |๐ต|โ + ๐, because ๐ is measurable. Since ๐ > 0 is arbitrary, |๐ต โฉ ๐ด|โ + |๐ต โงต ๐ด|โ โค |๐ต|โ , so ๐ด is measurable.3 5.19. Lebesgue measure is translation and dilation invariant: For ๐ด โ โ๐ , ๐ฅ0 โ โ๐ and ๐ฟ > 0, we define ๐ฅ0 + ๐ด = {๐ฅ0 + ๐ฅ โถ ๐ฅ โ ๐ด}
and
๐ฟ๐ด = {๐ฟ๐ฅ โถ ๐ฅ โ ๐ด}.
These sets are measurable, and we have |๐ฅ0 + ๐ด| = |๐ด| and |๐ฟ๐ด| = ๐ฟ๐ |๐ด| (Exercises (12) and (13)).
5.4. Measurable functions In this section we discuss the functions that are to be integrated with respect to the Lebesgue measure discussed in Section 5.3. Such functions are called measurable. For convenience, we will allow functions to have infinite values at some points, so we consider the extended real line [โโ, โ] with the conventions ๐ฅ+โ=โ+๐ฅ=โ
for all ๐ฅ โ โ,
๐ฅ โ โ = โโ + ๐ฅ = โโ
for all ๐ฅ โ โ,
๐ฅโ
โ=โโ
๐ฅ=โ
for all ๐ฅ โ โ,
๐ฅ > 0,
๐ฅ โ
โ = โ โ
๐ฅ = โโ
for all ๐ฅ โ โ,
๐ฅ < 0,
and the corresponding products with โโ. We also agree that โ+โ=โ
and
โ โ โ โ = โโ,
or ยฑโ โ
โ = ยฑโ, but we do not define โ โ โ nor 0 โ
โ. We also say that โโ < ๐ฅ and ๐ฅ < โ for all ๐ฅ โ โ. Let ๐ โถ โ๐ โ [โโ, โ] be an extended real valued function. We say that ๐ is measurable if, for all ๐ โ โ, the set ๐โ1 ([โโ, ๐)) = {๐ฅ โ โ๐ โถ ๐(๐ฅ) < ๐} 3
Statement (1) is indeed used as the definition of a measurable set in some texts, as [SS05].
5.4. Measurable functions
97
is measurable. The definition of a measurable function is equivalent to saying that ๐โ1 ([โโ, ๐]) is measurable for all ๐ โ โ. Indeed, since 1 1 [โโ, ๐] = [ โ โ, ๐ + ) and [โโ, ๐) = [ โ โ, ๐ โ ], โ โ ๐ ๐ ๐
๐
โ1
we see that the sets ๐ ([โโ, ๐)) are measurable for all ๐ โ โ if and only if the sets ๐โ1 ([โโ, ๐]) are measurable for all ๐ โ โ. We could also have used the sets ๐โ1 ((๐, โ]) or ๐โ1 ((๐, โ]), or even the pre-images of bounded intervals ๐โ1 ((๐, ๐]), ๐โ1 ((๐, ๐)), etc. together with ๐โ1 ({โ}) and ๐โ1 ({โโ}), to define measurability. We usually denote the set ๐โ1 ([โโ, ๐)) by {๐ < ๐}, and the set ๐ ([โโ, ๐]) by {๐ โค ๐}. Similarly, ๐โ1 ((๐, โ]) by {๐ > ๐}, ๐โ1 ([๐, โ]) by {๐ โฅ ๐}, etc. โ1
The fact that ๐ is measurable is also equivalent to the fact that ๐โ1 (๐) is measurable for any open ๐ โ โ, or ๐โ1 (๐ธ) is measurable for any closed ๐ธ โ โ (Exercise (14)). The previous equivalence implies that all continuous functions ๐ are measurable. Indeed, since every open set is measurable and, for any open ๐ โ โ, ๐โ1 (๐) is open, then ๐ is measurable, as stated above. The most important property of measurability of functions is its stability under pointwise limits. This is a consequence of the following result. Theorem 5.20. Let ๐1 , ๐2 , . . . be measurable extended real valued functions. Then the functions sup ๐๐ ,
inf ๐๐ ,
lim sup ๐๐ ,
and
lim inf ๐๐
are measurable. In the theorem, sup ๐๐ denotes the function sup ๐๐ (๐ฅ) = sup{๐๐ (๐ฅ) โถ ๐ โฅ 1}, which we define as โ in the case when the set {๐๐ (๐ฅ) โถ ๐ โฅ 1} is not bounded. Similarly for inf ๐๐ . lim sup ๐๐ and lim inf ๐๐ are defined by lim sup ๐๐ (๐ฅ) = inf sup ๐๐ (๐ฅ) = inf { sup{๐๐ (๐ฅ) โถ ๐ โฅ ๐} โถ ๐ โฅ 1}, ๐โฅ1 ๐โฅ๐
lim inf ๐๐ (๐ฅ) = sup inf ๐๐ (๐ฅ) = sup { inf{๐๐ (๐ฅ) โถ ๐ โฅ ๐} โถ ๐ โฅ 1}. ๐โฅ1 ๐โฅ๐
98
5. Measure theory in Euclidean space
Proof. To see that sup ๐๐ is measurable, observe that {sup ๐๐ > ๐} =
โ
{๐๐ > ๐}
๐
and each set {๐๐ > ๐} is measurable. Similarly, inf ๐๐ is measurable because {inf ๐๐ < ๐} = {๐ < ๐} โ ๐ ๐
and also each {๐๐ < ๐} is measurable. Now, lim sup ๐๐ is measurable because, for each ๐, sup๐โฅ๐ ๐๐ is measurable, and therefore inf๐โฅ1 sup๐โฅ๐ ๐๐ is measurable. Similarly for lim inf ๐๐ . โก Corollary 5.21. If ๐1 , ๐2 , . . . is a sequence of extended real valued measurable functions and ๐๐ โ ๐ pointwise, then ๐ is measurable. โก
Proof. If ๐๐ โ ๐, then lim sup ๐๐ = lim inf ๐๐ = ๐. ๐
5.22. If ๐ is measurable and ๐(๐ฅ) = ๐(๐ฅ) for all ๐ฅ โ โ except at a set of measure zero, then ๐ is also measurable. Indeed, for each ๐ โ โ, whenever ๐(๐ฅ) < ๐ we have either ๐(๐ฅ) < ๐ or ๐(๐ฅ) โฅ ๐, so {๐ < ๐} is contained in the union of {๐ < ๐} and {๐ โฅ ๐} โฉ {๐ < ๐}. However, if ๐(๐ฅ) < ๐, it could happen that ๐(๐ฅ) โฅ ๐, so we need to remove the set {๐ < ๐} โฉ {๐ โฅ ๐}. Thus, we have {๐ < ๐} = {๐ < ๐} โช ({๐ โฅ ๐} โฉ {๐ < ๐}) โงต ({๐ < ๐} โฉ {๐ โฅ ๐}). As the two sets {๐ โฅ ๐} โฉ {๐ < ๐} and {๐ < ๐} โฉ {๐ โฅ ๐} are contained in the set where ๐(๐ฅ) โ ๐(๐ฅ), they are sets of measure zero and thus measurable, so we conclude {๐ < ๐} is measurable. If ๐(๐ฅ) = ๐(๐ฅ) for all ๐ฅ โ โ๐ except at a set of measure zero, we say that ๐ = ๐ almost everywhere, or at almost every point, and we denote it by a.e. In general, we say that a property ๐(๐ฅ) holds a.e. if the set where ๐(๐ฅ) is false is of measure zero. We can then refine Corollary 5.21 as the following statement. Corollary 5.23. If ๐1 , ๐2 , . . . is a sequence of extended real valued measurable functions and ๐๐ โ ๐ a.e., then ๐ is measurable. 5.24. If ๐ is measurable and ๐ โ โค+ , then (๐)๐ is measurable. This follows because, if ๐ is odd, {(๐)๐ < ๐} = {๐ < ๐1/๐ }
5.4. Measurable functions
99
and, if ๐ is even, {(๐)๐ < ๐} = {โ๐1/๐ < ๐ < ๐1/๐ } if ๐ > 0, or empty if ๐ โค 0, and both of the sets on right side are measurable because ๐ is measurable. 5.25. If ๐ and ๐ are measurable and real valued, then ๐ + ๐ and ๐๐ are measurable. For the sum, observe that (5.26)
{๐ + ๐ < ๐} =
โ
({๐ < ๐ โ ๐} โฉ {๐ < ๐})
๐โโ
and that each set {๐ < ๐ โ ๐} โฉ {๐ < ๐} is measurable. To prove the identity (5.26), first note that, if ๐(๐ฅ) < ๐ โ ๐ and ๐(๐ฅ) < ๐, we clearly have ๐(๐ฅ) + ๐(๐ฅ) < ๐, so the union of the right side of (5.26) is contained in the left side. For the reverse inclusion, assume ๐(๐ฅ)+๐(๐ฅ) < ๐, so ๐(๐ฅ) < ๐โ๐(๐ฅ). Let ๐ โ โ such that ๐(๐ฅ) < ๐ < ๐โ๐(๐ฅ). Hence ๐(๐ฅ) < ๐ and ๐(๐ฅ) < ๐โ๐, so ๐ฅ โ {๐ < ๐ โ ๐} โฉ {๐ < ๐}, which implies (5.26). Now, the multiplication ๐๐ is a measurable function because we can write 1 ๐๐ = ((๐ + ๐)2 โ (๐ โ ๐)2 ), 4 and each of the functions (๐ ยฑ ๐)2 is measurable by the previous results. 5.27. If ๐ด is measurable set, its characteristic function 1 ๐๐ด (๐ฅ) = { 0
๐ฅโ๐ด ๐ฅ โ ๐ด,
is a measurable function. Indeed, for each ๐ โ โ, ๐
โงโ {๐๐ด < ๐} = โ๐ โงต ๐ด โจ โฉโ
๐>1 0 0}.
๐=0
We have thus partitioned the set where ๐ is positive in the 22๐ + 1 sets 2๐ ๐ด0๐ , ๐ด1๐ , . . . , ๐ด2๐ โ1 and ๐ต๐ . Now define the simple function 22๐ โ1
๐๐ = โ ๐=0
๐ ๐ ๐ + 2๐ ๐๐ต๐ . 2๐ ๐ด๐
Note that 0 โค ๐๐ โค 2๐ , and ๐๐ splits the values of ๐, up to 2๐ , in small jumps of size 2โ๐ (see Figure 5.4). The ๐๐ clearly satisfy ๐๐ โค ๐๐+1 and
2n
Figure 5.4. The approximation of a function with simple functions.
๐๐ โ ๐. Moreover, given ๐ > 0, if ๐ด = {๐ฅ โ โ๐ โถ ๐(๐ฅ) โค ๐}, then, for ๐ such that 2๐ > ๐, |๐(๐ฅ) โ ๐๐ (๐ฅ)| < 2โ๐ โก
for all ๐ฅ โ ๐ด. Therefore ๐๐ โ ๐ on ๐ด. f (x)
f + (x)
f - (x)
Figure 5.5. The positive and negative parts of a function ๐(๐ฅ).
5.30. We can apply Theorem 5.29 to approximate a general measurable function ๐ with with simple functions ๐๐ such that |๐๐ | โค |๐๐+1 |, |๐๐ | โค |๐| and ๐๐ โ ๐, and the convergence to be uniform on any set where ๐
102
5. Measure theory in Euclidean space
is bounded. We just need to write ๐ = ๐+ โ ๐โ , where ๐+ and ๐โ are the positive and negative parts of ๐, respectively, given by ๐+ = ๐ โ
๐{๐โฅ0}
๐โ = โ๐ โ
๐{๐โค0} .
and
(See Figure 5.5.) We leave the details as an exercise (Exercise (16)). Note that we also have |๐| = ๐+ + ๐โ .
Exercises (1) Let ๐
be a rectangle and ๐ > 0. Then there exist an open rectangle ๐ โ ๐
and a closed rectangle ๐ โ ๐
such that vol(๐) < vol(๐
) + ๐
and
vol(๐) > vol(๐
) โ ๐.
(2) (a) If ๐ด is countable, then |๐ด|โ = 0. (b) If ๐ด1 , ๐ด2 , . . . are sets of measure zero, then their union has measure zero. (3) Let ๐ be the Cantor ternary set ๐ถ, constructed by the removal of middle third intervals starting from ๐ถ0 = [0, 1]. Then ๐ถ is of measure zero. (4) Let ๐
= [๐1 , ๐1 ] ร [๐2 , ๐2 ] ร โฏ ร [๐๐ , ๐๐ ] be a rectangle and, for each ๐ = 1, 2, . . . , ๐, ๐
๐
๐
๐
๐ซ๐ = {๐ฅ0 = ๐๐ < ๐ฅ1 < ๐ฅ2 < . . . < ๐ฅ๐๐ = ๐๐ } a partition of [๐๐ , ๐๐ ]. If ๐ฎ is the grid of subrectangles ๐
of the form ๐
๐
๐ = ๐ผ1 ร ๐ผ2 ร โฏ ร ๐ผ๐ , where each ๐ผ๐ = [๐ฅ๐โ1 , ๐ฅ๐ ], then vol(๐
) = โ vol(๐). ๐โ๐ฎ
(5) If ๐ โ โ๐ is open and nonempty, then ๐ = โ๐ ๐๐ where ๐๐ are dyadic almost disjoint closed cubes. For the proof, follow the next steps: (a) For ๐ โฅ 0, let ๐๐ be the collection of closed dyadic cubes ๐ with sides of length 2โ๐ such that ๐ โ ๐. (b) Show that, for ๐ โค ๐, ๐ โ ๐๐ and ๐โฒ โ ๐๐ , then either ๐ โ ๐โฒ or they are almost disjoint.
Notes
103
(c) Define ๐0 = ๐0 and, for each ๐ > 0, let ๐๐ be collection of ๐ โ ๐๐ such that ๐ is almost disjoint to every ๐โฒ โ ๐๐ , for ๐ = 0, . . . , ๐ โ 1. (d) Let ๐ = โช๐โฅ0 ๐๐ . (e) Prove that ๐= ๐. โ ๐โ๐
๐
(6) For any ๐ด โ โ , |๐ด|โ = inf{|๐|โ โถ ๐ is open and ๐ด โ ๐}. (7) If ๐ด1 , ๐ด2 , . . . are measurable, then โ๐ ๐ด๐ is measurable. (8) A set of measure zero is measurable. (9) Let ๐ด โ โ๐ . The following are equivalent. (a) ๐ด is measurable. (b) ๐ด = ๐ โงต ๐, where ๐ is a ๐บ ๐ฟ set and |๐| = 0. (c) ๐ด = ๐ โช ๐, where ๐ is an ๐น๐ set and |๐| = 0. (10) Part (2) of Corollary 5.16 is false if all ๐ด๐ have infinite measure. (11) Prove Corollary 5.17. (12) For ๐ด โ โ๐ measurable and ๐ฅ0 โ โ๐ , |๐ฅ0 + ๐ด| = |๐ด|. (13) Let ๐ด โ โ๐ be a measurable set. (a) For ๐ฟ > 0, |๐ฟ๐ด| = ๐ฟ๐ |๐ด|. (b) For a ๐-tuple ๐ฟ ฬ = (๐ฟ1 , . . . , ๐ฟ ๐ ) with each ๐ฟ๐ > 0, ๐ = 1, . . . , ๐, define ฬ = {(๐ฟ1 ๐ฅ1 , . . . , ๐ฟ ๐ ๐ฅ๐ ) โถ (๐ฅ1 , . . . , ๐ฅ๐ ) โ ๐ด}. ๐ฟ๐ด ฬ = ๐ฟ1 โฏ ๐ฟ ๐ |๐ด|. Then |๐ฟ๐ด| (14) Let ๐ โถ โ๐ โ [โโ, โ] be an extended valued function. Then the following are equivalent. (a) ๐ is measurable. (b) For any open ๐ โ โ, ๐โ1 (๐) is measurable. (c) For any closed ๐ธ โ โ, ๐โ1 (๐ธ) is measurable. (15) If ๐ โถ โ โ [โโ, โ] is monotone, then it is measurable. (16) Write the details for Fact 5.30.
Notes Measure theory has a long history with a long list of motivations, as discussed in detail in the texts [Bre07] and [Bre08] by David M. Bressoud.
104
5. Measure theory in Euclidean space
Henri Lebesgueโs contribution was to consider countable covers for sets, as in the definition of outer measure presented here, and the approximation of a function by decomposition of its image, as it was done in 5.29. Lebesgueโs theory is discussed in his papers [Leb98], [Leb99a], and [Leb99b], as well as his text [Leb04]. More detailed introductions to measure theory can be found in [Fol99] and [SS05].
Chapter 6
Lebesgue integral and Lebesgue spaces
6.1. Integration of measurable functions We are now ready to define the Lebesgue integral of a measurable function, and we first consider nonnegative functions. By Theorem 5.29, we can approximate any nonnegative measurable function by simple functions. Thus, we start by defining the integral of a nonnegative simple function. 6.1.1. Nonnegative simple functions. Let ๐ be a nonnegative simple function on โ๐ and โ๐ ๐๐ ๐๐ด๐ its reduced form, so each ๐๐ > 0 and ๐ด๐ โฉ ๐ด๐ = โ
if ๐ โ ๐. We define the integral of ๐ by (6.1)
โซ ๐ = โ ๐๐ |๐ด๐ |. ๐
The integral of ๐ is also written โซ ๐(๐ฅ)๐๐ฅ if we need to explicitly make reference to its argument ๐ฅ. We require no special conditions on the sets ๐ด๐ that define the simple function ๐, except to be measurable. Thus, the integral (6.1) may 105
106
6. Lebesgue integral and Lebesgue spaces
be infinite, for example, if some |๐ด๐ | = โ. Note that assuming ๐๐ > 0 avoids the conflicting operations โ โ โ or 0 โ
โ in (6.1). 6.2. For any ๐ > 0, โซ ๐๐ = ๐ โซ ๐. This follows clearly from the fact that ๐ โ ๐๐ ๐๐ด๐ = โ ๐๐๐ ๐๐ด๐ , ๐
๐
if ๐ > 0. 6.3. For nonnegative simple functions ๐ and ๐, โซ(๐ + ๐) = โซ ๐ + โซ ๐. Indeed, if ๐ = โ๐ ๐๐ ๐๐ด๐ and ๐ = โ๐ ๐๐ ๐๐ต๐ , then we can write the reduced form of ๐ + ๐ as ๐ + ๐ = โ ๐ ๐ ๐๐ถ๐ = โ(๐โฒ๐ + ๐โณ๐ )๐๐ถ๐ , ๐
๐
where each ๐ด๐ =
๐ถ๐ ,
โ
๐ต๐ =
๐ ๐โฒ๐ = { ๐ 0
if ๐ถ๐ โ ๐ด๐ otherwise
โ
๐ถ๐ ,
๐๐
if ๐ถ๐ โ ๐ต๐
0
otherwise.
๐ถ๐ โฉ๐ต๐ โ โ
๐ถ๐ โฉ๐ด๐ โ โ
and
๐โณ๐ = {
Note that each ๐ถ๐ is either of the form ๐ด๐ โฉ ๐ต๐ for some ๐, ๐, of the form ๐ด๐ โงต โ๐ ๐ต๐ for some ๐, or ๐ต๐ โงต โ๐ ๐ด๐ for some ๐. Thus โซ(๐ + ๐) = โ ๐ ๐ |๐ถ๐ | = โ(๐โฒ๐ + ๐โณ๐ )|๐ถ๐ | = โ ๐โฒ๐ |๐ถ๐ | + โ ๐โณ๐ |๐ถ๐ | ๐
=โ
๐;๐โฒ๐ โ 0
๐
โ
๐โฒ๐ |๐ถ๐ | + โ
๐ ๐ถ๐ โฉ๐ด๐ โ โ
๐โณ๐ |๐ถ๐ |
โ
๐ ๐ถ๐ โฉ๐ต๐ โ โ
= โ ๐๐
โ
๐
๐ถ๐ โฉ๐ด๐ โ โ
|๐ถ๐ | + โ ๐๐ ๐
โ
|๐ถ๐ |
๐ถ๐ โฉ๐ต๐ โ โ
= โ ๐๐ |๐ด๐ | + โ ๐๐ |๐ต๐ | = โซ ๐ + โซ ๐, ๐
๐
because the sets ๐ถ๐ are pairwise disjoint and measurable.
๐;๐โณ ๐ โ 0
6.1. Integration of measurable functions
107
6.4. If ๐ โฅ 0 is simple, ๐ฅ0 โ โ๐ and ๐ is the translation of ๐ by ๐ฅ0 , ๐(๐ฅ) = ๐(๐ฅ โ ๐ฅ0 ), then โซ ๐ = โซ ๐. This follows from the fact that |๐ฅ0 + ๐ด| = |๐ด| for any measurable set ๐ด โ โ๐ (Exercise (12), Chapter 5), because this implies that โซ ๐๐ด (๐ฅ โ ๐ฅ0 )๐๐ฅ = โซ ๐๐ฅ0 +๐ด = |๐ฅ0 + ๐ด| = |๐ด| = โซ ๐๐ด , and thus the result follows for any linear combination of ๐๐ด , by 6.2 and 6.3. 6.5. If ๐ โฅ 0 is simple, ๐ฟ > 0 and ๐ is the dilation of ๐ by ๐ฟ, ๐(๐ฅ) = ๐ฟโ๐ ๐(๐ฅ/๐ฟ), then โซ ๐ = โซ ๐. This follows from the fact that |๐ฟ๐ด| = ๐ฟ๐ |๐ด| for any measurable set ๐ด โ โ๐ (Exercise (13), Chapter 5), because this implies that, using 6.2, โซ ๐ฟโ๐ ๐๐ด (๐ฅ/๐ฟ)๐๐ฅ = ๐ฟโ๐ โซ ๐๐ฟ๐ด = ๐ฟโ๐ |๐ฟ๐ด| = ๐ฟโ๐ โ
๐ฟ๐ |๐ด| = โซ ๐๐ด , and thus, as above, the result follows for any linear combination of ๐๐ด . 6.6. If ๐ โค ๐, โซ ๐ โค โซ ๐. Using the same decomposition as in 6.3, we have โซ ๐ = โ ๐๐ |๐ด๐ | = โ ๐โฒ๐ |๐ถ๐ | โค โ ๐โณ๐ |๐ถ๐ | = โ ๐๐ |๐ต๐ | = โซ ๐, ๐
since each
๐โฒ๐
โค
๐
๐
๐
๐โณ๐ .
If ๐ด โ โ๐ is measurable, we define the integral of ๐ over ๐ด as โซ ๐ = โซ ๐๐๐ด .
(6.7)
๐ด
Note that, if ๐ = โ๐ ๐๐ ๐๐ด๐ , then โซ ๐ = โ ๐๐ |๐ด๐ โฉ ๐ด|. ๐ด
๐
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6.8. If ๐ด, ๐ต โ โ๐ are measurable and ๐ด โฉ ๐ต = โ
, then โซ
๐ = โซ ๐ + โซ ๐.
๐ดโช๐ต
๐ด
๐ต
Follows from the fact that, since ๐ด โฉ ๐ต = โ
, ๐๐ถโฉ(๐ดโช๐ต) = ๐๐ถโฉ๐ด + ๐๐ถโฉ๐ต for any set ๐ถ. 6.1.2. Nonnegative measurable functions. For any measurable extended real valued function ๐ โฅ 0, we can approximate ๐ pointwise with simple functions ๐, by Theorem 5.29. This allows us to define the integral of ๐ by (6.9)
โซ ๐ = sup { โซ ๐ โถ ๐ is simple and 0 โค ๐ โค ๐}.
Again, it is possible to have โซ ๐ = โ. The integral of ๐ is also usually denoted by โซ ๐, โ๐
if one wants to make explicit the fact that ๐ is a function on โ๐ , or โซ ๐(๐ฅ)๐๐ฅ, โ๐
making the argument ๐ฅ explicit. As before, โซ๐ด ๐ means โซ ๐ = โซ ๐๐๐ด , ๐ด ๐
for any measurable ๐ด โ โ . The definition of the integral implies the following properties, which follow from their versions for simple functions. 6.10. For any measurable function ๐ โฅ 0 and any ๐ > 0, โซ ๐๐ = ๐ โซ ๐. This follows directly from 6.2 and the definition (6.9).
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109
6.11. The translation and dilation invariance of the integral for simple functions, 6.4 and 6.5, also imply the invariance of the integral for measurable ๐ โฅ 0, by (6.9). That is, if ๐ โฅ 0 is measurable, ๐ฅ0 โ โ๐ and ๐ฟ > 0, then โซ ๐(๐ฅ โ ๐ฅ0 )๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ and โซ ๐ฟโ๐ ๐(๐ฅ/๐ฟ)๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ. 6.12. If ๐, ๐ are measurable and 0 โค ๐ โค ๐, then โซ ๐ โค โซ ๐. Again, this follows from the definition (6.9), and the fact that, if ๐ is a simple function that satisfies 0 โค ๐ โค ๐, then 0 โค ๐ โค ๐ and thus { โซ ๐ โถ ๐ is simple and 0 โค ๐ โค ๐} โ { โซ ๐ โถ ๐ is simple and 0 โค ๐ โค ๐}. The supremum of the set on the left side cannot be larger than that of the set on the right side. We are ready to state and prove the first of the Lebesgue integration theorems, called the monotone convergence theorem. Theorem 6.13 (Monotone convergence). If ๐๐ โฅ 0 are measurable and ๐๐ โ ๐, then โซ ๐๐ โ โซ ๐. Proof. The limit ๐ is measurable by Theorem 5.20. By 6.12, the sequence โซ ๐๐ is increasing, so the limit exists (possibly โ). Also, by 6.12, โซ ๐๐ โค โซ ๐ for all ๐, so lim โซ ๐๐ โค โซ ๐. To prove the reverse inequality, we verify that the limit of the integrals is at least as large as the integral of any simple function ๐ that
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6. Lebesgue integral and Lebesgue spaces
satisfies 0 โค ๐ โค ๐, so this limit is an upper bound for the set of all such integrals. By the definition (6.9) of the integral, this will imply lim โซ ๐๐ โฅ โซ ๐. So let ๐ be a simple function such that 0 โค ๐ โค ๐, and let 0 < ๐ < 1. Define ๐ธ๐ = {๐ฅ โถ ๐๐ (๐ฅ) โฅ (1 โ ๐)๐(๐ฅ)}. Since ๐๐ is increasing, the sequence ๐ธ๐ of sets is increasing because, if ๐ฅ โ ๐ธ๐ , then ๐๐+1 (๐ฅ) โฅ ๐๐ (๐ฅ) โฅ (1 โ ๐)๐(๐ฅ) and thus ๐ฅ โ ๐ธ๐+1 . We also have โ๐ ๐ธ๐ = โ๐ because ๐๐ โ ๐ and ๐ โฅ ๐. Suppose ๐ = โ ๐๐ ๐๐ด๐ . Then, for each ๐, โซ ๐๐ โฅ โซ ๐๐ ๐๐ธ๐ โฅ (1 โ ๐) โซ ๐๐๐ธ๐ = (1 โ ๐) โ ๐๐ |๐ด๐ โฉ ๐ธ๐ |. ๐
For each ๐, the sequence of sets ๐ด๐ โฉ ๐ธ๐ is increasing in ๐ and โ
๐ด๐ โฉ ๐ธ๐ = ๐ด๐ ,
๐
so we have, by the monotone continuity of the measure (Corollary 5.16), |๐ด๐ โฉ ๐ธ๐ | โ |๐ด๐ | and thus lim โซ ๐๐ โฅ (1 โ ๐) โซ ๐. Since 0 < ๐ < 1 is arbitrary, we obtain lim โซ ๐๐ โฅ โซ ๐, โก
as desired.
The monotone convergence theorem can be used to prove the linear and analytic properties of the Lebesgue integral, as in the following two facts. 6.14. If ๐, ๐ โฅ are measurable, then โซ(๐ + ๐) = โซ ๐ + โซ ๐.
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111
This is the analogue of fact 6.3 for nonnegative functions. For the proof, let ๐๐ and ๐๐ be sequences of simple functions with 0 โค ๐๐ โค ๐ and 0 โค ๐๐ โค ๐ such that ๐๐ โ ๐ and ๐๐ โ ๐, which exist by Theorem 5.29. Thus ๐๐ + ๐๐ โ ๐ + ๐ and, by the monotone convergence theorem, โซ(๐ + ๐) = lim โซ(๐๐ + ๐๐ ) = lim โซ ๐๐ + lim โซ ๐๐ = โซ ๐ + โซ ๐. Inductively, we can extend 6.14 to any finite number of nonnegative measurable functions ๐1 , ๐2 , . . . , ๐๐ , so ๐
(6.15)
๐
โซ โ ๐๐ = โ โซ ๐๐ . ๐=1
๐=1
The monotone convergence theorem also implies, in fact, that the integral of a series of nonnegative functions is equal to the series of the integrals. 6.16. If ๐๐ is a sequence of nonnegative measurable functions and ๐ = โ๐ ๐๐ , then โซ ๐ = โ โซ ๐๐ . ๐
Indeed, the partial sums ๐ ๐ =
๐ โ๐=1 ๐๐
satisfy
๐
โซ ๐ ๐ = โ โซ ๐๐ , ๐=1
by (6.15). Since each ๐๐ is nonnegative, ๐ ๐ โ ๐, so by the monotone convergence theorem we obtain ๐
โ โซ ๐๐ โ โซ ๐. ๐=1
The monotone convergence theorem is true even if we only assume ๐๐ โ ๐ a.e. (Exercise (4)). If the sequence ๐๐ is not monotone, nor converges to ๐, we can still say something about the sequence of integrals. The following result, our second of the Lebesgue theorems, is known as Fatouโs lemma.
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6. Lebesgue integral and Lebesgue spaces
Theorem 6.17 (Fatouโs lemma). For any sequence ๐๐ of nonnegative measurable functions, โซ lim inf ๐๐ โค lim inf โซ ๐๐ . Proof. For each ๐, consider the function inf๐โฅ๐ ๐๐ given by inf ๐๐ (๐ฅ) = inf{๐๐ (๐ฅ) โถ ๐ โฅ ๐},
๐โฅ๐
the largest lower bound of the set of values ๐๐ (๐ฅ) for ๐ โฅ ๐. Thus, for each ๐ โฅ ๐, we have inf๐โฅ๐ ๐๐ โค ๐๐ and thus โซ inf ๐๐ โค โซ ๐๐ ,
(6.18)
๐โฅ๐
by 6.12. Now, since ๐ โฅ ๐ is arbitrary, the number on the left of (6.18) is then a lower bound for the set { โซ ๐๐ โถ ๐ โฅ ๐}, so we have that โซ inf ๐๐ โค inf โซ ๐๐ .
(6.19)
๐โฅ๐
๐โฅ๐
Since inf๐โฅ๐ ๐๐ โ lim inf ๐๐ as ๐ โ โ, the monotone convergence theorem implies โซ lim inf ๐๐ = lim โซ inf ๐๐ . ๐
๐โฅ๐
Therefore, by (6.19), โซ lim inf ๐๐ โค lim inf โซ ๐๐ = lim inf โซ ๐๐ . ๐ ๐โฅ๐
โก 6.20. In the case when ๐๐ โ ๐, Fatouโs lemma implies that โซ ๐ โค lim inf โซ ๐๐ . Thus, the integral of the limit of a sequence of functions can never be larger that the limit of the integrals, if it exists.
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113
Example 6.21. In general, we donโt have equality in 6.20. Consider the sequence ๐๐ = ๐(๐,๐+1] . Clearly ๐๐ โ 0, but each โซ ๐๐ = 1, so โซ ๐๐ โ 1 > 0. 6.1.3. Extended real valued functions. Now we consider the integral for general measurable extended real valued functions ๐. We say the ๐ is integrable if โซ ๐+ < โ
โซ ๐โ < โ,
and
where ๐+ and ๐โ are the positive and negative parts of ๐, defined above by ๐+ = ๐ โ
๐{๐โฅ0} and ๐โ = โ๐ โ
๐{๐โค0} . + โ See Figure 5.5. As |๐| = ๐ + ๐ , the integrability of ๐ is equivalent to โซ |๐| < โ. Note that ๐ = ๐+ โ ๐โ . Thus, if ๐ is integrable, its integral is defined by โซ ๐ = โซ ๐+ โ โซ ๐โ . Example 6.22. Let ๐ โถ โ โ โ be given by sin ๐ฅ ๐ฅ>0 ๐(๐ฅ) = { ๐ฅ 0 ๐ฅ โค 0. ๐ is not integrable. We have that (2๐+1)๐
โ
โซ ๐+ = โ โซ ๐=0 2๐๐
sin ๐ฅ ๐๐ฅ ๐ฅ (2๐+1)๐
โ
1 โซ (2๐ + 1)๐ 2๐๐ ๐=0
โฅ โ
sin ๐ฅ๐๐ฅ = โ,
because each integral in the series is equal to 2.1 1
Recall, however, that the improper integral of ๐ indeed exists: โ
โซ 0
๐
sin ๐ฅ ๐ sin ๐ฅ ๐๐ฅ = lim โซ ๐๐ฅ = . ๐ฅ ๐ฅ 2 ๐
โโ 0
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It is not hard to verify that the integral of extended real valued functions, by the definition above, satisfies the linear properties โซ ๐๐ = ๐ โซ ๐
(6.23) and (6.24)
โซ ๐ + ๐ = โซ ๐ + โซ ๐.
Indeed, for the proof of (6.23), consider first the case ๐ > 0. We have ๐๐ = (๐๐)+ โ (๐๐)โ = ๐๐+ โ ๐๐โ , so โซ ๐๐ = โซ ๐๐+ โ โซ ๐๐โ = ๐ โซ ๐+ โ ๐ โซ ๐โ = ๐ inf ๐, where we have used 6.10. The case ๐ = 0 is trivial, as we have both sides of (6.23) equal to 0. If ๐ < 0, we write ๐๐ = (โ๐)(โ๐) and thus โซ ๐๐ = โซ(โ๐)(โ๐) = (โ๐) โซ(โ๐) = โ๐( โซ ๐โ โ โซ ๐+ ) = ๐ โซ ๐, because (โ๐)+ = ๐โ and (โ๐)โ = ๐+ . Since we are allowing the functions to be extended real valued, the undefined operation 0โ
โ will occur in the case ๐ = 0 at the points where ๐ is equal to โ. However this may only occur in a set of measure zero (Exercise (3)) for an integrable function ๐, so the scalar multiplication ๐๐ is zero a.e. and โซ ๐๐ = 0 (Exercise (1)). To prove (6.24), first note that, in the case when the measurable functions ๐, ๐ satisfy ๐, ๐ โฅ 0, ๐ โ ๐ โฅ 0 and โซ ๐ < โ, then โซ(๐ โ ๐) = โซ ๐ โ โซ ๐. Indeed, by 6.14, we have โซ ๐ = โซ(๐ โ ๐ + ๐) = โซ(๐ โ ๐) + โซ ๐, and we can substract โซ ๐ from both sides because โซ ๐ < โ. Now, to verify (6.24), for extended real valued functions ๐, ๐ we have (๐ + ๐)+ = ๐+ ๐๐ด โ ๐โ ๐๐ด + ๐+ ๐๐ด โ ๐โ ๐๐ด ,
6.1. Integration of measurable functions
115
where ๐ด = {๐ฅ โถ ๐(๐ฅ) + ๐(๐ฅ) โฅ 0}, and similarly for (๐ + ๐)โ (with opposite signs). Thus, by the previous observation and 6.14, โซ(๐ + ๐)+ = โซ(๐+ ๐๐ด โ ๐โ ๐๐ด + ๐+ ๐๐ด โ ๐โ ๐๐ด ) = โซ(๐+ ๐๐ด + ๐+ ๐๐ด ) โ โซ(๐โ ๐๐ด + ๐โ ๐๐ด ) = โซ ๐+ ๐๐ด + โซ ๐+ ๐๐ด โ โซ ๐โ ๐๐ด โ โซ ๐โ ๐๐ด = โซ ๐๐๐ด + โซ ๐๐๐ด . Combining with the corresponding expresion for โซ(๐ + ๐)โ we obtain (6.24). We leave the rest of the details as an exercise (Exercise (7)). Again, as we are allowing the functions to be extended valued, we may have the operation โ โ โ. As before, this may only occur in a set of measure zero, which doesnโt modify the integrals (Exercise (2)). 6.1.4. Complex valued functions. Now, let ๐ โถ โ๐ โ โ. We say that ๐ is integrable if both its real and imaginary parts are measurable and โซ |๐| < โ. In this case, the integral of ๐ is defined by โซ ๐ = โซ โ๐ + ๐ โซ โ๐, where โ๐ and โ๐ are the real and imaginary parts of ๐, respectively. Note that the complex valued function ๐ is integrable if and only if its real and imaginary parts are integrable. As above, the integral of ๐ is denoted by โซ ๐ โ๐
when we want to make the Euclidean space โ๐ explicit, or by โซ ๐(๐ฅ)๐๐ฅ, โ๐
if we need to show explicitly the variable ๐ฅ of ๐. We denote the set of complex valued integrable functions by ๐ฟ1 (โ๐ ). We sometimes denote it simply by ๐ฟ1 , if there is no confusion. It is not hard to see that ๐ฟ1 is a complex vector space and ๐ โฆ โซ ๐ is a linear
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functional on ๐ฟ1 (Exercise (8)). This time we donโt have to worry about operations with infinity, but one still needs to verify carefully the identity โซ ๐ผ๐ = ๐ผ โซ ๐ for complex scalars ๐ผ. 6.25. If ๐ โ ๐ฟ1 (โ๐ ), ๐ฅ0 โ โ๐ and ๐ฟ > 0, let ๐(๐ฅ) = ๐(๐ฅ โ ๐ฅ0 ) be the translation of ๐ by ๐ฅ0 , and โ(๐ฅ) = ๐ฟโ๐ ๐(๐ฅ/๐ฟ) the dilation of ๐ by ๐ฟ. Then, by applying 6.11 to (โ๐)ยฑ , (โ๐)ยฑ we obtain โซ ๐ = โซ โ = โซ ๐. 6.26. For ๐ โ ๐ฟ1 (โ๐ ), | โซ ๐| โค โซ |๐|. | | This inequality is obvious if โซ ๐ = 0. If ๐ is real valued, | โซ ๐| = | โซ ๐+ โ โซ ๐โ | โค โซ ๐+ + โซ ๐โ = โซ |๐|. | | | | If ๐ is complex valued and โซ ๐ โ 0, write โซ ๐ = ๐๐๐๐ , its polar form. Then | โซ ๐| = ๐ = ๐โ๐๐ โซ ๐ = โซ ๐โ๐๐ ๐, | | so โซ ๐โ๐๐ ๐ is a real number, and hence โซ ๐โ๐๐ ๐ = โซ โ(๐โ๐๐ ๐) + ๐ โซ โ(๐โ๐๐ ๐) = โซ โ(๐โ๐๐ ๐). Therefore, by 6.12, | โซ ๐| = โซ ๐โ๐๐ ๐ = โซ โ(๐โ๐๐ ๐) | | โค โซ |โ(๐โ๐๐ ๐)| โค โซ |๐โ๐๐ ๐| = โซ |๐|. For a measurable ๐ด โ โ๐ , we say that ๐ is integrable on ๐ด if โซ |๐| = โซ |๐|๐๐ด < โ. ๐ด
If ๐ is integrable on ๐ด, we have โซ ๐ = โซ ๐๐๐ด . ๐ด
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117
We denote the set of integrable functions on ๐ด by ๐ฟ1 (๐ด). Recall that we say that ๐ = ๐ almost everywhere (and we write a.e.) if the set {๐ฅ โถ ๐(๐ฅ) โ ๐(๐ฅ)} has measure zero. Proposition 6.27. Let ๐, ๐ โ ๐ฟ1 (โ๐ ). The following are equivalent. (1) ๐ = ๐ a.e. (2) โซ |๐ โ ๐| = 0. (3) For all measurable ๐ด โ โ๐ , โซ ๐ = โซ ๐. ๐ด
๐ด
Proof. We prove the implications (1) โ (2) โ (3) โ (1). (1) โ (2): If ๐ = ๐ a.e., then |๐ โ ๐| = 0 a.e. Hence โซ |๐ โ ๐| = 0. (2) โ (3): For any measurable ๐ด โ โ๐ , by 6.26, | โซ ๐ โ โซ ๐| = | โซ(๐ โ ๐)๐ | โค โซ |๐ โ ๐|๐ โค โซ |๐ โ ๐| = 0. ๐ด| ๐ด | | | ๐ด
๐ด
(3) โ (1): We prove the contrapositive, so assume ๐ โ ๐ in a set of positive measure. Then at least one of the functions โ(๐ โ ๐)ยฑ or โ(๐ โ ๐)ยฑ is positive in a set of positive measure. Assume ๐ด = {๐ฅ โ โ๐ โถ โ(๐ โ ๐)+ > 0} has positive measure. Note that, if ๐ฅ โ ๐ด, โ(๐ โ ๐)โ (๐ฅ) = 0. Thus โซ ๐ โ โซ ๐ = โซ (๐ โ ๐) = โซ โ(๐ โ ๐) + ๐ โซ โ(๐ โ ๐) ๐ด
๐ด
๐ด
๐ด
๐ด
and โซ โ(๐ โ ๐) = โซ โ(๐ โ ๐)+ > 0, ๐ด
so โซ๐ด ๐ โ โซ๐ด ๐.
๐ด
โก
We are ready for our third Lebesgue convergence theorem for integrals, known as the dominated convergence theorem.
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6. Lebesgue integral and Lebesgue spaces
Theorem 6.28 (Dominated convergence). Let ๐๐ โ ๐ฟ1 (โ๐ ) such that ๐๐ โ ๐ and there exists ๐ โ ๐ฟ1 (โ๐ ) with |๐๐ | โค ๐ for all ๐. Then ๐ โ ๐ฟ1 (โ๐ ) and โซ ๐๐ โ โซ ๐. Proof. ๐ is measurable by Corollary 5.21 and is integrable by 6.12, because |๐| โค ๐. By taking real and imaginary parts, we can assume all functions are real valued. Since |๐๐ | โค ๐, we have ๐ ยฑ ๐๐ โฅ 0 for all ๐, and we can apply Fatouโs lemma (Theorem 6.17) to the sequences of nonnegative functions ๐ ยฑ ๐๐ . Hence we have โซ(๐ + ๐) โค lim inf โซ(๐ + ๐๐ ) = โซ ๐ + lim inf โซ ๐๐ and โซ(๐ โ ๐) โค lim inf โซ(๐ โ ๐๐ ) = โซ ๐ โ lim sup โซ ๐๐ . As โซ ๐ < โ and โซ(๐ ยฑ ๐) = โซ ๐ ยฑ โซ ๐, we have lim sup โซ ๐๐ โค โซ ๐ โค lim inf โซ ๐๐ , โก
and therefore โซ ๐๐ โ โซ ๐.
As in the case of the monotone convergence theorem, Theorem 6.28 also holds if we only assume ๐๐ โ ๐ a.e. As a corollary, we can extend 6.16 to complex valued integrable functions. Corollary 6.29. Let ๐๐ โ ๐ฟ1 (โ๐ ) such that โ๐ โซ |๐๐ | < โ. Then the series โ๐ ๐๐ converges almost everywhere to an integrable function ๐ and โซ ๐ = โ โซ ๐๐ . ๐
Proof. Consider the function ๐ = โ๐ |๐๐ |. Then ๐ is the limit of an increasing sequence, because each |๐๐ | โฅ 0. By the monotone convergence
6.1. Integration of measurable functions
119
theorem we have โซ ๐ = โ โซ |๐๐ | < โ, ๐ 1
๐
so ๐ โ ๐ฟ (โ ). In particular, ๐ is finite almost everywhere (Exercise (3)) and hence โ๐ ๐๐ converges almost everywhere. Also, for every ๐, ๐
๐
๐=1
๐=1
| โ ๐ | โค โ |๐ | โค ๐. ๐| ๐ | By the dominated convergence theorem, if โ๐ ๐๐ โ ๐, we have that ๐ โ ๐ฟ1 (โ๐ ) and โซ ๐ = โ โซ ๐๐ . ๐
โก 6.30. If ๐ โ ๐ฟ1 (โ๐ ) and, for each ๐ โฅ 1, ๐๐ = ๐๐๐ต๐ , where ๐ต๐ is the ball around the origin of radius ๐, then clearly |๐๐ | โค |๐|, ๐๐ โ ๐ and thus, by the dominated convergence theorem, โซ ๐๐ โ โซ ๐. This can be written as โซ ๐ โ โซ ๐, ๐ต๐
โ๐
or, explicitly, lim โซ
๐โโ
|๐ฅ|โค๐
๐(๐ฅ)๐๐ฅ = โซ ๐(๐ฅ)๐๐ฅ, โ๐
a fact that will later be useful to calculate integrals explicitly. There is nothing special about the balls ๐ต๐ : we can use any increasing sequence of measurable sets ๐ด๐ such that โ๐ ๐ด๐ = โ๐ and obtain the same result. Corollary 6.31 states the conditions for the continuity and differentiability of integrals, also implied by the dominated convergence theorem. We leave its proof as an exercise (Exercise (9)). Corollary 6.31. Let ๐(๐ฅ, ๐ก) be a function on โ๐ ร [๐, ๐] such that ๐(โ
, ๐ก) โ ๐ฟ1 (โ๐ ) for each ๐ก โ [๐, ๐]. (1) If ๐(๐ฅ, โ
) is continuous on [๐, ๐] for each ๐ฅ โ โ๐ and there exists ๐ โ ๐ฟ1 (โ๐ ) such that |๐(๐ฅ, ๐ก)| โค ๐(๐ฅ) for all ๐ฅ โ โ๐ and ๐ก โ [๐, ๐], then ๐ก โฆ โซโ๐ ๐(๐ฅ, ๐ก)๐๐ฅ is continuous on [๐, ๐].
120
6. Lebesgue integral and Lebesgue spaces (2) If ๐(๐ฅ, โ
) is differentiable in (๐, ๐) for each ๐ฅ โ โ๐ and there exists โ โ ๐ฟ1 (โ๐ ) such that | ๐(๐ฅ, ๐ก) โ ๐(๐ฅ, ๐ ) | โค โ(๐ฅ) | | ๐กโ๐ for all ๐ฅ โ โ๐ and ๐ก, ๐ โ (๐, ๐), ๐ก โ ๐ , then ๐ก โฆ โซโ๐ ๐(๐ฅ, ๐ก)๐๐ฅ is differentiable in (๐, ๐) and ๐ ๐ โซ ๐(๐ฅ, ๐ก)๐๐ฅ = โซ ๐(๐ฅ, ๐ก)๐๐ฅ. ๐๐ก โ๐ ๐๐ก โ๐
6.2. Fubiniโs theorem In this section we prove Fubiniโs theorem on iterated integrals. For ๐1 , ๐2 โ โค+ , we write the elements of the space โ๐1 +๐2 as (๐ฅ, ๐ฆ), where ๐ฅ โ โ๐1 and ๐ฆ โ โ๐2 . Now consider a measurable set ๐ด โ โ๐1 +๐2 . For each ๐ฅ โ โ๐1 , the ๐ฅ-section of ๐ด is defined as ๐ด๐ฅ = {๐ฆ โ โ๐2 โถ (๐ฅ, ๐ฆ) โ ๐ด}, while the ๐ฆ-section of ๐ด is defined by ๐ด ๐ฆ = {๐ฅ โ โ๐1 โถ (๐ฅ, ๐ฆ) โ ๐ด}. ๐ด๐ฅ and ๐ด ๐ฆ are the projections of the cross sections of ๐ด onto โ๐2 and โ๐1 , respectively (see Figure 6.1). It may happen that ๐ด๐ฅ or ๐ด ๐ฆ are nonmeasurable sets. However, we have the following result. Lemma 6.32. The sets ๐ด๐ฅ and ๐ด ๐ฆ are measurable for a.e. ๐ฅ โ โ๐1 and a.e. ๐ฆ โ โ๐2 , respectively, the functions ๐ฅ โฆ |๐ด๐ฅ | and ๐ฆ โฆ |๐ด ๐ฆ | are measurable, and |๐ด| = โซ |๐ด๐ฅ |๐๐ฅ = โซ |๐ด ๐ฆ |๐๐ฆ.
(6.33)
โ๐ 2
โ๐ 1
Proof. The lemma follows immediately if ๐ด = ๐
ร ๐, where ๐
โ โ๐1 and ๐ โ โ๐2 are rectangles, because then ๐ฅโ๐
๐ฅโ๐
and
๐
๐ด๐ฆ = { โ
|๐ด๐ฅ | = |๐|๐๐
(๐ฅ)
and
|๐ด ๐ฆ | = |๐
|๐๐ (๐ฆ),
๐ด๐ฅ = {
๐ โ
๐ฆโ๐ ๐ฆ โ ๐,
so
6.2. Fubiniโs theorem
121
Figure 6.1. The ๐ฅ-section and the ๐ฆ-section of ๐ด. Note that they are the projections of the cross sections of ๐ด onto โ๐2 and โ๐1 .
and |๐ด| = |๐
| โ
|๐| = โซ |๐|๐๐
(๐ฅ)๐๐ฅ = โซ |๐
|๐๐ (๐ฆ)๐๐ฆ. โ๐2
โ๐ 1
If ๐ด is a finite almost disjoint union of closed rectangles, then each ๐ด๐ฅ and ๐ด ๐ฆ is a finite disjoint union of closed rectangles (except for a finite number of points ๐ฅ and ๐ฆ corresponding to intersecting boundaries), so they are measurable, and (6.33) follows by the linearity of the integral. ๐ Indeed, if ๐ด = โ๐=1 ๐
๐ where the ๐
๐ are almost disjoint, then ๐ด๐ฅ = ๐
๐
โ๐=1 (๐
๐ )๐ฅ and ๐ด ๐ฆ = โ๐=1 (๐
๐ )๐ฆ are also almost disjoint unions, so ๐
|๐ด๐ฅ | = โ |(๐
๐ )๐ฅ | ๐=1
๐
and
|๐ด ๐ฆ | = โ |(๐
๐ )๐ฆ | ๐=1
122
6. Lebesgue integral and Lebesgue spaces
except at most a finite number of ๐ฅ and ๐ฆ. Hence, ๐
๐
|๐ด| = โ |๐
๐ | = โ โซ |(๐
๐ )๐ฅ |๐๐ฅ ๐=1 โ๐1
๐=1 ๐
=โซ
โ |(๐
๐ )๐ฅ |๐๐ฅ = โซ |๐ด๐ฅ |๐๐ฅ,
โ๐1 ๐=1
โ๐1
and similarly for the integrals of the ๐ฆ-sections. By Exercise (5) of Chapter 5, if ๐ด is open then it is the countable union of almost disjoint closed cubes, so the lemma follows by the monotone convergence theorem. Indeed, if we write ๐ด = โ๐ ๐
๐ , where the ๐
๐ are almost disjoint closed rectangles, then we also have ๐ด = โ๐ ๐ด๐ , where the ๐ด๐ is an increasing sequence of sets which are a finite union of almost disjoint rectangles, so by monotone continuity |๐ด| = lim |๐ด๐ | = lim โซ |(๐ด๐ )๐ฅ |๐๐ฅ = โซ |๐ด๐ฅ |๐๐ฅ ๐
๐
โ๐ 1
โ๐1
as |(๐ด๐ )๐ฅ | โ |๐ด๐ฅ | for each ๐ฅ. The result for the ๐ฆ-sections follows in the same way. If ๐ด is a bounded ๐บ ๐ฟ set, then ๐ด = โ๐ ๐๐ , where ๐๐ is a decreasing sequence of bounded open sets. Then ๐ด๐ฅ =
โ
(๐๐ )๐ฅ
and
๐
๐ด๐ฆ =
โ
(๐๐ )๐ฆ
๐
๐ฆ
are measurable. Since |(๐1 )๐ฅ |, |(๐1 ) | < โ because ๐1 is bounded, the sequences |(๐๐ )๐ฅ | and |(๐๐ )๐ฆ | converge to |๐ด๐ฅ | and |๐ด ๐ฆ |, respectively, satisfy that |(๐๐ )๐ฅ | โค |(๐1 )๐ฅ |, |(๐๐ )๐ฆ | โค |(๐1 )๐ฆ |, and โซ |(๐1 )๐ฅ |๐๐ฅ = โซ |(๐1 )๐ฆ |๐๐ฆ = |๐1 | < โ, โ๐ 2
โ๐1
๐ฆ
so |(๐1 )๐ฅ | and |(๐1 ) | are integrable. Thus (6.33) follows by the dominated convergence theorem, following similar lines as in the previous cases. For a nonbounded ๐บ ๐ฟ set ๐ด, we can write ๐ด = โ๐ (๐ด โฉ ๐ต๐ ), where each ๐ต๐ is the open ball of radius ๐ around the origin, and thus ๐ด is the increasing union of bounded ๐บ ๐ฟ sets, so the lemma again follows by the monotone convergence theorem.
6.2. Fubiniโs theorem
123
Let ๐ด be a measure zero set. By Corollary 5.17(1), for each ๐ there exists an open set ๐๐ โ ๐ด such that |๐๐ | < 1/๐. If ๐ = โ๐ ๐๐ , then ๐ is a ๐บ ๐ฟ set of measure zero, ๐๐ฅ and ๐ ๐ฆ are measurable, and โซ |๐๐ฅ |๐๐ฅ = โซ |๐ ๐ฆ |๐๐ฆ = 0. โ๐ 2
โ๐1
Thus |๐๐ฅ | = 0 for a.e. ๐ฅ โ โ๐1 and |๐ ๐ฆ | = 0 for a.e. ๐ฆ โ โ๐2 . Since ๐ด โ ๐, each ๐ด๐ฅ โ ๐๐ฅ and ๐ด ๐ฆ โ ๐ ๐ฆ , so ๐ด๐ฅ is a measure zero set for a.e. ๐ฅ โ โ๐1 and ๐ด ๐ฆ is a measure zero set for a.e. ๐ฆ โ โ๐2 , and thus measurable. Therefore (6.33) is true for the measure zero set ๐ด. For a general measurable set ๐ด, we can use Corollary 5.17(1) as above to write ๐ด = ๐ โงต ๐, where ๐ is a ๐บ ๐ฟ set and ๐ is a measure zero set. Since ๐ด๐ฅ = ๐๐ฅ โงต ๐๐ฅ and ๐ด ๐ฆ = ๐ ๐ฆ โงต ๐ ๐ฆ , ๐ด๐ฅ and ๐ด ๐ฆ are measurable for a.e. ๐ฅ โ โ๐1 and for a.e. ๐ฆ โ โ๐2 , respectively, and (6.33) follows by the previous cases. โก Indeed, the monotone convergence theorem is the protagonist in the proof of Lemma 6.32. Do not be surprised, as it will be in many of the results further in this book. It is, of course, in the proof of Theorem 6.34. Given a function ๐ โถ โ๐1 +๐2 โ โ we define, for each ๐ฅ โ โ๐1 , the function ๐๐ฅ on โ๐2 by ๐๐ฅ (๐ฆ) = ๐(๐ฅ, ๐ฆ) and, for each ๐ฆ โ โ๐2 , the function ๐๐ฆ on โ๐1 by ๐๐ฆ (๐ฅ) = ๐(๐ฅ, ๐ฆ). Theorem 6.34 (Fubini). If ๐ โ ๐ฟ1 (โ๐1 +๐2 ), for a.e. ๐ฅ โ โ๐1 and ๐ฆ โ โ๐2 we have ๐๐ฅ โ ๐ฟ1 (โ๐2 ) and ๐๐ฆ โ ๐ฟ1 (โ๐1 ) and โซ
(6.35)
โ๐1 +๐2
๐ = โซ ( โซ ๐๐ฅ )๐๐ฅ = โซ ( โซ ๐๐ฆ )๐๐ฆ. โ๐1
โ๐ 2
โ๐2
โ๐ 1
Proof. By taking (โ๐)ยฑ and (โ๐)ยฑ , we can assume ๐ โฅ 0. If ๐ = ๐๐ด for some measurable set ๐ด โ โ๐1 +๐2 , the identity (6.35) is the same as (6.33). By linearity of the integral, (6.35) follows if ๐ is a simple function. For general ๐ โฅ 0, take a sequence of nonnegative simple functions ๐๐ โ ๐, as in Theorem 5.29. Then (๐๐ )๐ฅ โ ๐๐ฅ and (๐๐ )๐ฆ โ ๐๐ฆ as well, for any ๐ฅ โ โ๐1 and ๐ฆ โ โ๐2 . Thus ๐๐ฅ and ๐๐ฆ are measurable for a.e. ๐ฅ โ โ๐1 , ๐ฆ โ โ๐2 and by the monotone convergence theorem we have the two limits โซ (๐๐ )๐ฅ โ โซ ๐๐ฅ โ๐ 2
โ๐2
and
โซ (๐๐ )๐ฆ โ โซ ๐๐ฆ , โ๐1
โ๐1
124
6. Lebesgue integral and Lebesgue spaces
and a second application of the monotone convergence theorem gives us โซ ( โซ (๐๐ )๐ฅ )๐๐ฅ โ โซ ( โซ ๐๐ฅ )๐๐ฅ โ๐2
โ๐ 1
โ๐ 1
and
โ๐2
โซ ( โซ (๐๐ )๐ฆ )๐๐ฆ โ โซ ( โซ ๐๐ฆ )๐๐ฆ. โ๐ 2
โ๐2
โ๐ 1
โ๐1
Equation (6.35) follows because, once more, the monotone convergence theorem implies โซ ๐๐ โ โซ ๐ on โ๐1 +๐2 . Since we are assuming โซ ๐ < โ, we see that all integrals in (6.35) are finite, so the functions ๐ฅ โฆ โซ ๐๐ฅ and ๐ฆ โฆ โซ ๐๐ฆ are finite almost everywhere, so ๐๐ฅ and ๐๐ฆ are integrable for a.e. ๐ฅ โ โ๐1 and ๐ฆ โ โ๐2 . โก We usually write (6.35) as ๐ = โซ โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ๐๐ฅ = โซ โซ ๐(๐ฅ, ๐ฆ)๐๐ฅ๐๐ฆ,
โซ โ๐1 +๐2
โ๐1
โ๐ 2
โ๐2
โ๐1
and we call the second and third integrals as the iterated integrals of โซ ๐. From the proof of Fubiniโs theorem, we can see that (6.35) is true if we only assume ๐ โฅ 0, as the integrability of ๐ was only used to conclude the integrability of ๐๐ฅ and ๐๐ฆ , needed to extend the result to general integrable functions. The result for the case of nonnegative functions is known as Tonelliโs theorem. We usually use Tonelliโs theorem to verify the integrability of ๐, as we can estimate โซ |๐| using its iterated integrals. 6.36. An immediate consequence of Fubiniโs theorem is following inequality, known as Minkowskiโs inequality, โซ || โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ||๐๐ฅ โค โซ โซ |๐(๐ฅ, ๐ฆ)|๐๐ฅ๐๐ฆ. โ๐ 1
โ๐ 2
โ๐2
โ๐ 1
6.3. The Lebesgue space ๐ฟ1 We observed above that ๐ฟ1 (โ๐ ) is a vector space under the usual operations (๐ + ๐)(๐ฅ) = ๐(๐ฅ) + ๐(๐ฅ)
and
(๐ผ๐)(๐ฅ) = ๐ผ ๐(๐ฅ),
which follow from the basic properties of the Lebesgue integral. Now we can define โ๐โ๐ฟ1 = โซ |๐|,
6.3. The Lebesgue space ๐ฟ1
125
and see that it satisfies: (1) โ๐โ๐ฟ1 โฅ 0 for all ๐ โ ๐ฟ1 (โ๐ ); (2) โ๐ผ๐โ๐ฟ1 = |๐ผ| โ๐โ๐ฟ1 for all ๐ โ ๐ฟ1 (โ๐ ) and ๐ผ โ โ; and (3) โ๐ + ๐โ๐ฟ1 โค โ๐โ๐ฟ1 + โ๐โ๐ฟ1 , which also follow from the basic properties of the integral. For instance, inequality (3), called the triangle inequality, follows from 6.12, โ๐ + ๐โ๐ฟ1 = โซ |๐ + ๐| โค โซ(|๐| + |๐|) = โซ |๐| + โซ |๐| = โ๐โ๐ฟ1 + โ๐โ๐ฟ1 , because |๐(๐ฅ) + ๐(๐ฅ)| โค |๐(๐ฅ)| + |๐(๐ฅ)| for all ๐ฅ. So we can almost say that โ โ
โ๐ฟ1 defines a norm, except for the fact that the condition โโ๐โ๐ฟ1 = 0 if and only if ๐ = 0โ is not true. For instance, take ๐ด = โ (or any other subset of โ of measure zero) and ๐ = ๐๐ด . Then ๐ โ 0, but โ๐โ๐ฟ1 = 0. However, by Proposition 6.27, ๐ = ๐ a.e. if and only if โซ |๐โ๐| = 0, so ๐ = ๐ a.e. if and only if โ๐ โ ๐โ๐ฟ1 = 0. Thus, if we define the equivalence relation ๐ โผ ๐ if and only if ๐ = ๐ a.e., we see that โ โ
โ๐ฟ1 defines a norm on the space โ = ๐ฟ1 (โ๐ )/ โผ of equivalence classes. Indeed, first note that the sum and scalar multiplication are well defined on โ, because if ๐1 โผ ๐2 and ๐1 โผ ๐2 , then {๐ฅ โ โ๐ โถ ๐1 (๐ฅ) + ๐1 (๐ฅ) โ ๐2 (๐ฅ) + ๐2 (๐ฅ)} โ {๐ฅ โ โ๐ โถ ๐1 (๐ฅ) โ ๐2 (๐ฅ)} โช {๐ฅ โ โ๐ โถ ๐1 (๐ฅ) โ ๐2 (๐ฅ)}, so ๐1 + ๐1 = ๐2 + ๐2 a.e., and similarly for the scalar multiplication. For โ โ
โ๐ฟ1 , note that if ๐ โผ ๐ then |๐| = |๐| a.e., so โซ |๐| = โซ |๐| and hence โ๐โ๐ฟ1 = โ๐โ๐ฟ1 . Thus, โ โ
โ๐ฟ1 is well defined on โ, and it is a norm, by (1), (2) and (3). Therefore, โ is a normed space. As there is no reason for confusion, we will also denote the space โ of equivalence classes of integrable functions by ๐ฟ1 (โ๐ ). 6.37. The norm โ โ
โ๐ฟ1 is invariant under translations and dilations, which follows immediately by 6.25. Theorem 6.38. ๐ฟ1 (โ๐ ) is a complete normed space.
126
6. Lebesgue integral and Lebesgue spaces Any norm โ โ
โ on a vector space ๐ induces the metric ๐(๐, ๐) = โ๐ โ ๐โ,
so it makes ๐ a metric space. Recall that a complete metric space is a metric space where its Cauchy sequences converge. If ๐ is a complete normed vector space, we call it a Banach space. See Appendix A.5. Proof of Theorem 6.38. We use Theorem A.13: assume that we have a sequence ๐๐ โ ๐ฟ1 (โ๐ ) such that โ๐ โ๐๐ โ๐ฟ1 < โ, and we have to prove that the series โ๐ ๐๐ converges in ๐ฟ1 (โ๐ ), that is, there exists ๐ โ ๐ฟ1 (โ๐ ) such that ๐
|| โ ๐ โ ๐|| โ 0 ๐ || ||๐ฟ1 ๐=1
as ๐ โ โ. By Corollary 6.29, there exists ๐ โ ๐ฟ1 (โ๐ ) such that โ๐ ๐๐ = ๐ a.e. If we set ๐ = โ๐ |๐๐ | as in the proof of 6.29, then ๐
| โ ๐ โ ๐| โค 2๐ ๐ | | ๐=1
for all ๐. Since ๐
| โ ๐ โ ๐| โ 0 ๐ | |
a.e.,
๐=1
the dominated convergence theorem implies that ๐
โซ || โ ๐๐ โ ๐|| โ 0. ๐=1
โก 6.39. Simple functions are dense in ๐ฟ1 (โ๐ ). This follows by 5.30, which states that we can find simple functions ๐๐ such that |๐๐ | โค |๐๐+1 |, |๐๐ | โค |๐| and ๐๐ โ ๐, and the dominated convergence theorem. Note that, since each ๐๐ is integrable, if ๐๐ = โ ๐๐ ๐๐ด๐ , ๐
is the reduced form of ๐๐ , then the sets ๐ด๐ have finite measure.
6.3. The Lebesgue space ๐ฟ1
127
6.40. The space ๐ถ๐ (โ๐ ) of continuous functions of compact support is also dense in ๐ฟ1 (โ๐ ). By 6.30, we can approximate any ๐ โ ๐ฟ1 (โ๐ ) with integrable functions with compact support, and thus we can assume that the simple functions in 6.39 also have compact support, so each ๐ด๐ in their reduced form is bounded. Now, given ๐ > 0, we can find a compact set ๐น๐ โ ๐ด๐ and a bounded open set ๐ ๐ โ ๐ด๐ such that |๐ด๐ โงต๐น๐ |, |๐ ๐ โงต๐ด๐ | < ๐, and by Theorem A.11 in Appendix A there exists a continuous function ๐๐ , with 0 โค ๐๐ โค 1, such that ๐๐ is supported in ๐ ๐ and ๐ = 1 on ๐น๐ . Thus โซ |๐๐ด๐ โ ๐๐ | โค |๐ ๐ โงต ๐ด๐ | < 2๐, so โซ || โ ๐๐ ๐๐ด๐ โ โ ๐๐ ๐๐ || โค 2๐ โ |๐๐ |, ๐
๐
๐
which implies the result because ๐ > 0 is arbitrary. Note that, explicitly, 6.39 and 6.40 imply that, for every ๐ > 0, we can find a simple ๐ and ๐ โ ๐ถ๐ (โ๐ ) such that โ๐ โ ๐โ๐ฟ1 < ๐
and
โ๐ โ ๐โ๐ฟ1 < ๐.
6.41. In particular, 6.40 implies that translations are continuous under the norm in ๐ฟ1 , in the following sense. Let ๐ โ ๐ฟ1 (โ๐ ) and, for each ๐ฆ โ โ๐ , let ๐๐ฆ (๐ฅ) = ๐(๐ฅ โ ๐ฆ) be the translation of ๐ by ๐ฆ. Then lim โ๐๐ฆ โ ๐โ๐ฟ1 = 0,
๐ฆโ0
so ๐๐ฆ โ ๐ in ๐ฟ1 as ๐ฆ โ 0. Note first that the result is true for ๐ โ ๐ถ๐ (โ๐ ), because every continuous function of compact support is uniformly continuous. Indeed, asume ๐ is supported in the ball ๐ต๐ . Given ๐ > 0, there exists ๐ฟ > 0 such that ๐ฟ < 1 and, if |๐ฆ| < ๐ฟ then ๐ |๐(๐ฅ โ ๐ฆ) โ ๐(๐ฅ)| < . |๐ต๐+1 | Hence, if |๐ฆ| < ๐ฟ, โ๐๐ฆ โ ๐โ๐ฟ1 = โซ ๐ต๐+1 ๐ฆ
|๐(๐ฅ โ ๐ฆ) โ ๐(๐ฅ)|๐๐ฅ โค โซ ๐ต๐+1
๐ ๐๐ฅ = ๐. |๐ต๐+1 |
1
Thus ๐ โ ๐ in ๐ฟ as ๐ฆ โ 0. For any other ๐ โ ๐ฟ1 (โ๐ ) and given ๐ > 0, choose ๐ โ ๐ถ๐ (โ๐ ) such that โ๐ โ ๐โ๐ฟ1 < ๐. Thus, by the triangle inequality and the invariance of the norm under translations, โ๐๐ฆ โ ๐โ๐ฟ1 โค โ๐๐ฆ โ ๐๐ฆ โ๐ฟ1 + โ๐๐ฆ โ ๐โ๐ฟ1 + โ๐ โ ๐โ๐ฟ1 < 2๐ + โ๐๐ฆ โ ๐โ๐ฟ1 .
128
6. Lebesgue integral and Lebesgue spaces
Since โ๐๐ฆ โ ๐โ๐ฟ1 โ 0 and ๐ > 0 is arbitrary, we conclude that ๐๐ฆ โ ๐ in ๐ฟ1 as ๐ฆ โ 0. 6.42. For ๐, ๐ โ ๐ฟ1 (โ๐ ), their convolution is defined by ๐ โ ๐(๐ฅ) = โซ ๐(๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ. โ๐
By Minkowskiโs inequality and the translation invariance of the integral, โซ |๐ โ ๐(๐ฅ)|๐๐ฅ = โซ || โซ ๐(๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ||๐๐ฅ โ๐
โ๐
โ๐
โค โซ โซ |๐(๐ฅ โ ๐ฆ)๐(๐ฆ)|๐๐ฅ๐๐ฆ โ๐
โ๐
= โซ |๐(๐ฆ)| โซ |๐(๐ฅ โ ๐ฆ)|๐๐ฅ๐๐ฆ โ๐
โ๐
= โซ |๐(๐ฆ)| โซ |๐(๐ฅ)|๐๐ฅ๐๐ฆ = โ๐โ๐ฟ1 โ๐โ๐ฟ1 . โ๐
โ๐
Therefore ๐ โ ๐ โ ๐ฟ1 (โ๐ ) and its norm satisfies (6.43)
โ๐ โ ๐โ๐ฟ1 โค โ๐โ๐ฟ1 โ๐โ๐ฟ1 .
Note that we have ๐ โ ๐ = ๐ โ ๐ for any ๐, ๐ โ ๐ฟ1 (โ๐ ) (Exercise (10)). 6.44. Recall, as an example, the convolution operators 4.14 defined by ๐ โฆ ฮฆ๐ก โ ๐, where the dilations ฮฆ๐ก (๐ฅ) =
1 ๐ฅ ฮฆ( ), ๐ก ๐ก๐
form a family of good kernels. Note that this is true for any ฮฆ โ ๐ฟ1 (โ๐ ) with โซ ฮฆ = 1, because we donโt use the continuity of ฮฆ in Exercise (9) of Chapter 4. Also (6.43) and 6.37 imply โฮฆ๐ก โ ๐โ๐ฟ1 โค โฮฆ๐ก โ๐ฟ1 โ๐โ๐ฟ1 = โฮฆโ๐ฟ1 โ๐โ๐ฟ1 , so the convolutions ฮฆ๐ก โ ๐ are uniformly bounded in ๐ฟ1 .
6.3. The Lebesgue space ๐ฟ1
129
Now, by Minkwoskiโs inequality and invariance under dilations, โฮฆ๐ก โ ๐ โ ๐โ๐ฟ1 = โซ ||๐ โ ฮฆ๐ก (๐ฅ) โ ๐(๐ฅ) โซ ฮฆ๐ก ||๐๐ฅ โ๐
โ๐
= โซ || โซ (๐(๐ฅ โ ๐ฆ) โ ๐(๐ฅ))ฮฆ๐ก (๐ฆ)๐๐ฆ||๐๐ฅ โ๐
โ๐
โค โซ โซ |๐(๐ฅ โ ๐ฆ) โ ๐(๐ฅ)| |ฮฆ๐ก (๐ฆ)|๐๐ฅ๐๐ฆ โ๐
โ๐
= โซ โซ |๐(๐ฅ โ ๐ก๐ฆ) โ ๐(๐ฅ)| |ฮฆ(๐ฆ)|๐๐ฅ๐๐ฆ โ๐
โ๐
= โซ โ๐๐ก๐ฆ โ ๐โ๐ฟ1 |ฮฆ(๐ฆ)|๐๐ฆ. โ๐
Now, by 6.41, โ๐๐ก๐ฆ โ ๐โ๐ฟ1 โ 0 for each ๐ฆ โ โ๐ as ๐ก โ 0. Since โ๐๐ก๐ฆ โ ๐โ๐ฟ1 โค 2โ๐โ๐ฟ1 and ฮฆ is integrable, the dominated convergence theorem implies that โฮฆ๐ก โ ๐ โ ๐โ๐ฟ1 โ 0
as ๐ก โ 0.
6.45. As a consequence of 6.44, we also see that the space ๐ถ๐โ (โ๐ ) of smooth functions of compact support are dense in ๐ฟ1 (โ๐ ). Indeed, by Corollary 6.31, if ๐ โ ๐ถ๐โ (โ๐ ) and ๐ โ ๐ฟ1 (โ๐ ), ๐โ๐ is a smooth function because we can differentiate ๐ โ ๐(๐ฅ) inside the integral. Now, if ๐ โ ๐ถ๐ (โ๐ ), then ๐๐ก โ ๐ has compact support for any ๐ก > 0, and ๐๐ก โ ๐ โ ๐ in ๐ฟ1 as ๐ก โ 0, if โซ ๐ = 1. Therefore, given ๐ โ ๐ฟ1 (โ๐ ) and ๐ > 0, we can choose ๐ โ ๐ถ๐ (โ๐ ) such that โ๐โ๐โ๐ฟ1 < ๐/2, and for a fixed ๐ โ ๐ถ๐โ (โ๐ ) with โซ ๐ = 1 we can choose ๐ก > 0 such that โ๐๐ก โ ๐ โ ๐โ๐ฟ1 < ๐/2, and hence โ๐๐ก โ ๐ โ ๐โ๐ฟ1 โค โ๐๐ก โ ๐ โ ๐โ๐ฟ1 + โ๐ โ ๐โ๐ฟ1 < ๐. 6.46. Observe that, if {๐พ๐ก } is a family of good kernels, ๐ โ ๐ฟ1 (โ๐ ) is bounded and continuous at ๐ฅ, then ๐พ๐ก โ ๐(๐ฅ) โ ๐(๐ฅ) as ๐ก โ 0, as the same proof as in the ones the case of the Poisson integral in Chapter 4 will apply to this case.
130
6. Lebesgue integral and Lebesgue spaces
6.4. The Lebesgue space ๐ฟ2 We now define ๐ฟ2 (โ๐ ) as the set of square integrable measurable functions on โ๐ , that is, functions that satisfy โซ |๐|2 < โ. As in the case of ๐ฟ1 (โ๐ ), we in fact identify ๐ฟ2 (โ๐ ) to be the set of equivalence classes with respect to the relation ๐ โผ ๐ if and only if ๐(๐ฅ) = ๐(๐ฅ) for almost every ๐ฅ. ๐ฟ2 (โ๐ ) is a vector space (Exercise (11)) with inner product (6.47)
โจ๐, ๐โฉ = โซ ๐(๐ฅ)๐(๐ฅ)๐๐ฅ. โ๐
The integral in (6.47) converges, as Cauchyโs inequality |๐(๐ฅ)๐(๐ฅ)| โค
1 (|๐(๐ฅ)|2 + |๐(๐ฅ)|2 ) 2
implies ๐๐ฬ is integrable whenever |๐|2 and |๐|2 are integrable. The inner product (6.47) induces the ๐ฟ2 norm โ๐โ๐ฟ2 =
โซ |๐|2 . โ
Recall the CauchyโSchwarz inequality | โซ ๐(๐ฅ)๐(๐ฅ)๐๐ฅ| โค โ๐โ 2 โ๐โ 2 , ๐ฟ ๐ฟ | | which is the main ingredient when verifying that โ โ
โ๐ฟ2 satisfies the triangle inequality. The ๐ฟ2 norm is also invariant by translations, as the ๐ฟ1 norm above, although it is not invariant by dilations. Theorem 6.48. ๐ฟ2 (โ๐ ) is a complete inner product space. As in the proof of Theorem 6.38, the proof of Theorem 6.48 consists on verifying that every absolutely convergent series of functions in ๐ฟ2 (โ๐ ) converges in ๐ฟ2 , and we leave the proof as an exercise (Exercise (12)). Moreover, we can also prove that the simple functions, as well as the space ๐ถ๐ (โ๐ ) of continuous functions of compact support, are dense in
6.4. The Lebesgue space ๐ฟ2
131
๐ฟ2 (โ๐ ), with similar proofs to the ๐ฟ1 (โ๐ ) case. The density of ๐ถ๐ (โ๐ ) in ๐ฟ2 (โ๐ ) also implies the continuity of translations in the ๐ฟ2 norm, that is (6.49)
โ๐๐ฆ โ ๐โ๐ฟ2 โ 0
as ๐ฆ โ 0.
To discuss the analogous results to convolutions in ๐ฟ2 (โ๐ ), we need the following result. Theorem 6.50. Let ๐ be a measurable function such that, for any ๐ โ ๐ฟ2 (โ๐ ), ๐๐ โ ๐ฟ1 (โ๐ ) and ๐ = sup {|| โซ ๐๐|| โถ ๐ โ ๐ฟ2 (โ๐ ) and โ๐โ๐ฟ2 = 1} < โ. Then ๐ โ ๐ฟ2 (โ๐ ) and โ๐โ๐ฟ2 = ๐. Theorem 6.50 can be seen as the converse to the CauchyโSchwarz inequality. Proof. The result is trivial if ๐ = 0 a.e., so we assume it is not. Let ๐๐ be a sequence of simple funtions such that |๐๐ | โค |๐๐+1 |, |๐๐ | โค |๐| and ๐๐ โ ๐, as in 5.30. If we set ๐๐ = ๐๐ ๐๐ต๐ , then ๐๐ โ ๐ with |๐๐ | โ |๐| and each ๐๐ is supported in the ball ๐ต๐ , so ๐๐ โ ๐ฟ2 (โ๐ ). Since we are assuming ๐ is not zero almost everywhere, we can also assume each โ๐๐ โ๐ฟ2 > 0. We write ๐ in polar coordinates as ๐(๐ฅ) = |๐(๐ฅ)|๐๐๐(๐ฅ) , and define ๐๐ =
|๐๐ |๐โ๐๐ . โ๐๐ โ๐ฟ2
The sequence ๐๐ has the following properties. โข ๐๐ โ ๐ฟ2 (โ๐ ) and โ๐๐ โ๐ฟ2 = 1. Indeed, โซ |๐๐ |2 =
1 โซ |๐๐ |2 = 1. โ๐๐ โ2๐ฟ2
โข โซ |๐๐ | |๐๐ | = โ๐๐ โ๐ฟ2 . This follows directly by the calculation โซ |๐๐ | |๐๐ | = โซ |๐๐ |
|๐๐ | 1 โซ |๐๐ |2 = โ๐๐ โ๐ฟ2 . = โ๐๐ โ๐ฟ2 โ๐๐ โ๐ฟ2
132
6. Lebesgue integral and Lebesgue spaces โข โซ ๐๐๐ = โซ |๐| |๐๐ |. It follows from the fact that ๐๐โ๐๐ = |๐|, because ๐๐๐ = ๐
|๐๐ |๐โ๐๐ |๐๐ | = |๐| = |๐| |๐๐ |. โ๐โ๐ฟ2 โ๐โ๐ฟ2
Thus, by Fatouโs lemma and the above properties โ๐โ๐ฟ2 โค lim inf โ๐๐ โ๐ฟ2 = lim inf โซ |๐๐ | |๐๐ | โค lim inf โซ |๐| |๐๐ | = lim inf โซ ๐๐๐ โค ๐.
Hence ๐ โ ๐ฟ2 (โ๐ ) and โ๐โ๐ฟ2 โค ๐. By the CauchyโSchwarz inequality, for any ๐ โ ๐ฟ2 (โ๐ ) such that โ๐โ๐ฟ2 = 1, | โซ ๐๐| โค โ๐โ 2 โ๐โ 2 = โ๐โ 2 . ๐ฟ ๐ฟ ๐ฟ | | โก
Therefore ๐ โค โ๐โ๐ฟ2 , as required.
6.51. As a consequence of Theorem 6.50, we have the Minkowskiโs inequality for ๐ฟ2 , 2
โ
โซ || โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ|| ๐๐ฅ โค โซ โ๐ 1
โ๐2
โ๐2
โซ |๐(๐ฅ, ๐ฆ)|2 ๐๐ฅ๐๐ฆ. โ โ๐1
Note that, if we write ๐๐ฆ = ๐(๐ฅ, ๐ฆ), this can be written as || โซ ๐ ๐๐ฆ|| โค โซ โ๐ โ 2 ๐๐ฆ, ๐ฆ ๐ฆ ๐ฟ || ||๐ฟ2 so it can be seen as an extension of the triangle inequality for integrals of functions. This follows by Theorem 6.50 because, for any ๐ โ ๐ฟ2 (โ๐1 ) such that โ๐โ๐ฟ2 = 1, using Tonelliโs theorem and the CauchyโSchwarz
6.4. The Lebesgue space ๐ฟ2
133
inequality, โซ || โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ|| |๐(๐ฅ)|๐๐ฅ โค โซ โซ |๐(๐ฅ, ๐ฆ)|๐๐ฆ |๐(๐ฅ)|๐๐ฅ โ๐ 1
โ๐ 2
โ๐ 1
โ๐ 2
= โซ โซ |๐(๐ฅ, ๐ฆ)| |๐(๐ฅ)|๐๐ฅ๐๐ฆ โ๐2
โคโซ โ๐ 2
=โซ โ๐2
โ๐ 1
โ
โซ |๐(๐ฅ, ๐ฆ)|2 ๐๐ฅ โ
โ๐โ๐ฟ2 ๐๐ฆ โ๐ 1
โซ |๐(๐ฅ, ๐ฆ)|2 ๐๐ฅ๐๐ฆ. โ โ๐1
Therefore, if the last integral is finite, ๐ฅ โฆ ( โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ)๐(๐ฅ) โ๐2
is integrable and satisfies | โซ ( โซ ๐(๐ฅ, ๐ฆ)๐๐ฆ)๐(๐ฅ)๐๐ฅ| โค โซ | | โ๐ 1
โ๐ 2
โ๐2
โซ |๐(๐ฅ, ๐ฆ)|2 ๐๐ฅ๐๐ฆ, โ โ๐ 1
so we obtain the result. 6.52. We can obtain the following inequality for convolutions. If ๐ โ ๐ฟ1 (โ๐ ) and ๐ โ ๐ฟ2 (โ๐ ), then ๐ โ ๐ โ ๐ฟ2 (โ๐ ) and โ๐ โ ๐โ๐ฟ2 โค โ๐โ๐ฟ1 โ๐โ๐ฟ2 . This follows by Minkowskiโs inequality because 2
โซ |๐ โ ๐(๐ฅ)|2 ๐๐ฅ = โซ || โซ ๐(๐ฆ)๐(๐ฅ โ ๐ฆ)๐๐ฆ|| ๐๐ฅ โ โ โคโซ
โซ |๐(๐ฆ)|2 |๐(๐ฅ โ ๐ฆ)|2 ๐๐ฅ๐๐ฆ โ
โซ |๐(๐ฅ โ ๐ฆ)|2 ๐๐ฅ๐๐ฆ โ = โ๐โ๐ฟ1 โ๐โ๐ฟ2 , = โซ |๐(๐ฆ)|
where have used the invariance of the ๐ฟ2 norm under translations.
134
6. Lebesgue integral and Lebesgue spaces
We can also prove analogous result in ๐ฟ2 to the convergence of convolution operators with good kernels discussed in 6.44, in ๐ฟ1 . We leave it as an exercise.
Exercises (1) Let ๐ โฅ 0 be measurable. Then โซ ๐ = 0 if and only if ๐ = 0 a.e. (2) Let ๐ โฅ 0 be measurable and ๐ด a set of measure 0. Then โซ๐ด ๐ = 0. (3) Let ๐ โฅ 0 be measurable. If โซ ๐ < โ, then ๐ < โ a.e. (4) If ๐๐ โฅ 0 are measurable and ๐๐ โ ๐ a.e., then โซ ๐๐ โ โซ ๐. (5) If ๐๐ โฅ 0 are measurable, ๐๐ โ ๐ and โซ ๐ = lim โซ ๐๐ < โ, then โซ ๐๐ โ โซ ๐ ๐ด
๐ด
for all measurable ๐ด โ โ๐ . The statement is false if โซ ๐ = lim โซ ๐๐ = โ. (6) Fatouโs lemma implies the monotone convergence theorem. (7) Complete the details of the proof of (6.24). (8) The set ๐ฟ1 (โ๐ ) of integrable functions is a vector space with the usual pointwise operations, and ๐ โฆ โซ ๐ defines a linear functional on ๐ฟ1 (โ๐ ). (9) Prove Corollary 6.31. (10) If ๐, ๐ โ ๐ฟ1 (โ๐ ), then ๐ โ ๐ = ๐ โ ๐. (11) The set of equivalence clases ๐ฟ2 (โ๐ ) of square integrable functions forms a vector space, and (6.47) defines an inner product. (12) Prove Theorem 6.48. (13) Let ฮฆ โ ๐ฟ1 (โ๐ ) with โซ ฮฆ = 1. If {ฮฆ๐ก }๐ก>0 are the dilations of ฮฆ, then lim โฮฆ๐ก โ ๐ โ ๐โ๐ฟ2 = 0 ๐กโ0
for any ๐ โ ๐ฟ2 (โ๐ ).
Notes
135
Notes Lebesgue developed what is now known as the Lebesgue integral in his papers cited in Chapter 5, as well as [Leb01a], [Leb01b], [Leb01c], his thesis [Leb02] and his text [Leb04], where he also proved the dominated convergence theorem for the special case when the functions are dominated on [0, 1] by a constant. The general case was proved in [Leb10]. Fatouโs lemma was stated and proved by Pierre Fatou in [Fat06]. Beppo Levi stated and proved the monotone convergence theorem in [Lev06]. The spaces ๐ฟ๐ (for 1 โค ๐ โค โ, although we only discussed ๐ = 1, 2) were introduced by Frigyes Riesz in [Rie10], where he used the letter โ๐ฟโ to denote them in honor to Lebesgue. Hermann Minkowski proved the inequality now known as Minkowskiโs inequality for sums and series in [Min96], and the integral version is due to Riesz [Rie13].
Chapter 7
Maximal functions
7.1. Indefinite integrals and averages A well known result from calculus states that, if ๐ is Riemann-integrable on the interval [๐, ๐], then 1 โซ โโ0 โ ๐ฅ
๐ฅ+โ
lim
๐ = ๐(๐ฅ)
at each point ๐ฅ โ (๐, ๐) where ๐ is continuous. This says that, if ๐ฅ
๐น(๐ฅ) = โซ ๐ ๐
is the indefinite integral of ๐, then ๐น is differentiable at each point ๐ฅ where ๐ is continuous, and ๐น โฒ (๐ฅ) = ๐(๐ฅ). This result is true if ๐ is Lebesgue integrable, of course, and, in fact, can be extended to functions on โ๐ if we replace difference quotients by averages. Proposition 7.1. Let ๐ โ ๐ฟ1 (โ๐ ) and continuous at ๐ฅ โ โ๐ . Then 1 โซ |๐ต ๐โ0 ๐ (๐ฅ)| ๐ต
lim
๐ = ๐(๐ฅ).
๐ (๐ฅ)
Proof. Given ๐ > 0, let ๐ฟ > 0 such that, if |๐ฆ โ ๐ฅ| < ๐ฟ, then |๐(๐ฆ) โ ๐(๐ฅ)| < ๐. 137
138
7. Maximal functions
Then, if ๐ < ๐ฟ, | 1 โซ | |๐ต (๐ฅ)| ๐
๐ต๐ (๐ฅ)
๐ โ ๐(๐ฅ)|| โค
1 โซ |๐ต๐ (๐ฅ)| ๐ต
|๐(๐ฆ) โ ๐(๐ฅ)|๐๐ฆ
0
1 โซ |๐ต๐ (๐ฅ)| ๐ต
๐ (๐ฅ)
|๐|,
7.2. The HardyโLittlewood maximal function
139
for each ๐ฅ โ โ๐ . Note that we are taking the supremum of the averages of |๐| over all balls centered at ๐ฅ, of all radii. If the set of averages is not bounded, then ๐๐(๐ฅ) = โ. If ๐ is bounded, say, |๐(๐ฅ)| โค ๐ด for all ๐ฅ โ โ๐ (or almost everywhere), then ๐๐(๐ฅ) is finite at every ๐ฅ, of course, and |๐๐(๐ฅ)| โค ๐ด. Example 7.3. A priori, nothing guarantees that ๐๐ is finite in a positive measure set, or anywhere (see Exercises (2) and (3)). It is easy to see that, in general, ๐๐ โ ๐ฟ1 (โ๐ ), even if ๐ โ ๐ฟ1 (โ๐ ). For example, consider ๐ = ๐[0,1] on โ. For ๐ฅ > 1, ๐๐(๐ฅ) โฅ
1 โซ 2๐ฅ 0
2๐ฅ
๐[0,1] =
1 , 2๐ฅ
and thus ๐๐ โ ๐ฟ1 (โ). However, Hardy and Littlewood proved that, indeed, ๐๐ is finite almost everywhere if ๐ โ ๐ฟ1 (โ๐ ) and, even though it might not be integrable, Example 7.3 is essentially the worst case scenario. Theorem 7.4 (HardyโLittlewood). There exists ๐ด > 0 such that, for any ๐ โ ๐ฟ1 (โ๐ ) and ๐ผ > 0, |{๐ฅ โ โ๐ โถ ๐๐(๐ฅ) > ๐ผ}| โค
๐ด โ๐โ๐ฟ1 . ๐ผ
We say then that the maximal function is an operator of weak type (1, 1) . We first observe that Theorem 7.4 states that the situation of Example 7.3 is pretty much the โworst caseโ we would get for ๐๐, as the function 1/๐ฅ is the natural example of a measurable function ๐ that satisfies ๐ด |{๐ฅ โ โ โถ |๐(๐ฅ)| > ๐ผ}| = ๐ผ for some constant ๐ด. Indeed, we clearly have 1 2 |{๐ฅ โ โ โถ | | > ๐ผ}| = ๐ฅ ๐ผ for any ๐ผ > 0, as the set on the left hand side is the interval (โ๐ผ, ๐ผ).
140
7. Maximal functions In fact, any integrable function satisfies the above inequality, as |{๐ฅ โ โ๐ โถ |๐(๐ฅ)| > ๐ผ}| = โซ
1๐๐ฅ
{๐ฅโโ๐ โถ|๐(๐ฅ)|>๐ผ}
โคโซ {๐ฅโโ๐ โถ|๐(๐ฅ)|>๐ผ}
|๐(๐ฅ)| ๐๐ฅ ๐ผ
1 (7.5) โค โ๐โ๐ฟ1 . ๐ผ The inequality (7.5) is called Chebyshevโs inequality. In the proof of Theorem 7.4, we will use Lemma 7.6, due to Vitali. Lemma 7.6 (Vitali). Let ๐ต1 , ๐ต2 , . . . , ๐ต๐ be a finite collection of balls in โ๐ . Then there exist disjoint ๐ต๐1 , ๐ต๐2 , . . . , ๐ต๐๐ among them such that ๐
๐
๐ | | | โ ๐ต๐ | โค 3 โ |๐ต๐๐ |. ๐=1
๐=1
In other words, from any finite collection of balls we can obtain a disjoint subcollection such that, even as its union might be a smaller set than the union of the original balls, the volume of the new union is at least as large as a fixed proportion of the original total volume. Proof. Given ๐ต1 , ๐ต2 , . . . , ๐ต๐ , choose a ball of maximal radius, say ๐ต๐1 . Once ๐ต๐1 , ๐ต๐2 , . . . , ๐ต๐๐ are chosen, choose ๐ต๐๐+1 as a ball of maximal radius among all remaining balls disjoint to ๐ต๐1 , ๐ต๐2 , . . . , ๐ต๐๐ , until exhausting the collection. For each ๐ต๐ in the collection, there exists ๐๐ such that โข ๐ต๐๐ intersects ๐ต๐ ; and โข the radius of ๐ต๐๐ is at least as large as the radius of ๐ต๐ . Otherwise, ๐ต๐ would have been chosen in the construction. Thus, if 3๐ต๐๐ is the ball with the same center as ๐ต๐๐ and three times its radius, then ๐ต๐ โ 3๐ต๐๐ (see Figure 7.1) and, thus, ๐
โ ๐=1
๐
๐ต๐ โ
โ
3๐ต๐๐ .
๐=1
The lemma follows from the fact that |3๐ต๐๐ | = 3๐ |๐ต๐๐ |.
โก
7.2. The HardyโLittlewood maximal function
141
Figure 7.1. The ball 3๐ต๐๐ with the same center as ๐ต๐๐ and three times its radius contains all balls intersecting it with smaller or equal radii.
Proof of Theorem 7.4. As the measure of any measurable set is the supremum of the measures of its compact subsets, we let ๐พ โ {๐ฅ โ โ๐ โถ ๐๐(๐ฅ) > ๐ผ} be compact. Hence, it is enough to estimate the measure of ๐พ. Now, by definition, for each ๐ฅ โ ๐พ there exists a ball ๐ต๐ฅ centered at ๐ฅ such that 1 โซ |๐| > ๐ผ. |๐ต๐ฅ | ๐ต ๐ฅ
The balls ๐ต๐ฅ , for ๐ฅ โ ๐พ, cover the compact set ๐พ, so there exists a finite collection ๐ต1 , ๐ต2 , . . . , ๐ต๐ of them such that ๐
๐พโ
โ
๐ต๐ .
๐=1
By Lemma 7.6, there exist disjoint ๐ต๐1 , ๐ต๐2 , . . . , ๐ต๐๐ among them such that ๐
๐
๐ | | | โ ๐ต๐ | โค 3 โ |๐ต๐๐ |. ๐=1
๐=1
142
7. Maximal functions
Thus ๐
๐
|๐พ| โค || ๐ต | โค 3๐ โ |๐ต๐๐ |. โ ๐| ๐=1
๐=1
Now, for each of those balls, 1 โซ |๐| > ๐ผ, |๐ต๐๐ | ๐ต ๐๐
so |๐ต๐๐ |
๐ผ, for any fixed positive ๐ผ, is finite. 7.7. We can also consider, for ๐ โ ๐ฟ1loc (โ๐ ), the maximal function 1 ฬ โซ |๐| โถ ๐ต is a ball with ๐ฅ โ ๐ต}. ๐๐(๐ฅ) = sup { |๐ต| ๐ต ฬ is the supremum of the averages of |๐| over all balls that That is, ๐๐ contain ๐ฅ, and not only those centered at ๐ฅ. If ๐ฅ โ ๐ต and ๐ต has radius ๐, then ๐ต โ ๐ต2๐ (๐ฅ) and |๐ต2๐ (๐ฅ)| = 2๐ |๐ต|, and thus 1 2๐ โซ |๐| โค โซ |๐ต| ๐ต |๐ต2๐ (๐ฅ)| ๐ต
|๐|,
2๐ (๐ฅ)
ฬ ฬ also satisfies the concluand hence ๐๐(๐ฅ) โค 2๐ ๐๐(๐ฅ). Therefore, ๐๐ sion of the HardyโLittlewood maximal theorem.
7.3. The Lebesgue differentiation theorem
143
7.3. The Lebesgue differentiation theorem We are now ready to discuss the question posed in Section 7.1, on whether lim ๐โ0
1 โซ |๐ต๐ (๐ฅ)| ๐ต
๐
๐ (๐ฅ)
exists for any ๐ฅ โ โ๐ , if ๐ โ ๐ฟ1 (โ๐ ). In fact, it turns out that the limit exists almost everywhere, as stated by Lebesgueโs theorem. Theorem 7.8 (Lebesgue). If ๐ โ ๐ฟ1 (โ๐ ), then, for almost every ๐ฅ โ โ๐ , lim ๐โ0
1 โซ |๐ต๐ (๐ฅ)| ๐ต
๐ = ๐(๐ฅ).
๐ (๐ฅ)
Lebesgueโs theorem 7.8 states that the limit not only exists, but that it is actually equal to ๐ almost everywhere. This might seem remarkable when compared to Proposition 7.1, because ๐ could be discontinuous everywhere.1 Proof of Theorem 7.8. Let ๐น โ โ๐ be the set where either the limit does not exist or is not equal to ๐. Thus, ๐น = {๐ฅ โ โ๐ โถ lim sup |๐ผ๐ ๐(๐ฅ) โ ๐(๐ฅ)| > 0}. ๐โ0
where we have written ๐ผ๐ ๐(๐ฅ) for the average of ๐ on the ball of radius ๐ around ๐ฅ, 1 โซ ๐ผ๐ ๐(๐ฅ) = ๐. |๐ต๐ (๐ฅ)| ๐ต (๐ฅ) ๐
We want to prove that |๐น| = 0. As ๐น = โ๐ ๐น1/๐ , where ๐น๐ผ = {๐ฅ โ โ๐ โถ lim sup |๐ผ๐ ๐(๐ฅ) โ ๐(๐ฅ)| > ๐ผ}, ๐โ0
it is sufficient to prove that |๐น๐ผ | = 0 for any ๐ผ > 0. Given ๐ > 0, let ๐ โ ๐ถ๐ (โ๐ ) such that โ๐ โ ๐โ๐ฟ1 < ๐. By Proposition 7.1, lim ๐ผ๐ ๐(๐ฅ) = ๐(๐ฅ) ๐โ0
1 Recall, however, that Riemann-integrable functions are indeed continuous almost everywhere (Theorem A.9).
144
7. Maximal functions
for every ๐ฅ โ โ๐ , and thus lim sup|๐ผ๐ ๐(๐ฅ) โ ๐(๐ฅ)| ๐โ0
= lim sup |๐ผ๐ (๐ โ ๐)(๐ฅ) + ๐ผ๐ ๐(๐ฅ) โ ๐(๐ฅ) + ๐(๐ฅ) โ ๐(๐ฅ)| ๐โ0
= lim sup |๐ผ๐ (๐ โ ๐)(๐ฅ)| + |๐(๐ฅ) โ ๐(๐ฅ)| ๐โ0
โค ๐(๐ โ ๐)(๐ฅ) + |๐(๐ฅ) โ ๐(๐ฅ)|, where, in the last inequality, we have used the fact |๐ผ๐ (๐ โ ๐)(๐ฅ)| โค
1 โซ |๐ต๐ (๐ฅ)| ๐ต
|๐ โ ๐| โค ๐(๐ โ ๐)(๐ฅ).
๐ (๐ฅ)
Then |๐น๐ผ | โค |{๐ฅ โถ ๐(๐ โ ๐)(๐ฅ) >
๐ผ ๐ผ }| + |{๐ฅ โถ |๐(๐ฅ) โ ๐(๐ฅ)| > }|. 2 2
The first term is estimated by the HardyโLittlewood theorem, |{๐ฅ โถ ๐(๐ โ ๐)(๐ฅ) >
๐ผ ๐ด 2๐ด }| โค โ๐ โ ๐โ๐ฟ1 < ๐, 2 ๐ผ/2 ๐ผ
and the second term by Chebyshevโs inequality (7.5), |{๐ฅ โถ |๐(๐ฅ) โ ๐(๐ฅ)| >
๐ผ 2 1 }| โค โ๐ โ ๐โ๐ฟ1 < ๐ 2 ๐ผ/2 ๐ผ
Therefore 2(๐ด + 1) ๐, ๐ผ and the theorem follows because ๐ > 0 is arbitrary. |๐น๐ผ |
0 such that, for every ๐ โ ๐ฟ1 (โ๐ ) and (๐ฅ, ๐ก) โ โ๐+1 + , |๐ข(๐ฅ, ๐ก)| โค ๐ด๐๐(๐ฅ). Hence Poisson integrals are uniformly estimated from above by the maximal function. Proof. We estimate the Poisson integral ๐ข(๐ฅ, ๐ก) by integrating over dyadic annuli around the point ๐ฅ. Indeed, since ๐ โ ๐ฟ1 (โ๐ ), for each (๐ฅ, ๐ก) โ โ๐+1 we can write + โ
๐ข(๐ฅ, ๐ก) = โซ
๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ + โ โซ
๐ต๐ก (๐ฅ)
๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ,
๐=0 ๐ด๐ (๐ฅ)
where each ๐ด๐ (๐ฅ) is the annulus ๐ด๐ (๐ฅ) = {๐ฆ โ โ๐ โถ 2๐ ๐ก โค |๐ฅ โ ๐ฆ| < 2๐+1 ๐ก}. Now ๐๐ก (๐ฅ โ ๐ฆ) =
2 2๐ก โค , (๐+1)/2 2 2 ๐๐+1 ๐ก๐ ๐๐+1 (|๐ฅ โ ๐ฆ| + ๐ก )
so the first integral in the sum above is estimated by |โซ |
๐ต๐ก (๐ฅ)
๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ|| โค
2 ๐๐+1 ๐ก๐
โซ
|๐(๐ฆ)|๐๐ฆ โค ๐๐๐(๐ฅ),
๐ต๐ก (๐ฅ)
where ๐ > 0 is a constant that depends only on the dimension ๐. Similarly, if |๐ฅ โ ๐ฆ| โฅ 2๐ ๐ก, we have ๐๐ก (๐ฅ โ ๐ฆ) โค
2๐ก 2 โค , ๐๐+1 |๐ฅ โ ๐ฆ|๐+1 ๐๐+1 2๐(๐+1) ๐ก๐
146
7. Maximal functions
and thus |โซ |
๐ด๐ (๐ฅ)
๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ|| โค
2 โซ ๐๐+1 2๐(๐+1) ๐ก๐ ๐ต ๐+1 2
|๐(๐ฆ)|๐๐ฆ (๐ฅ) ๐ก
๐โฒ โค ๐ ๐๐(๐ฅ), 2 โฒ where ๐ > 0 is another constant that depends only on ๐. Therefore โ
๐โฒ ๐๐(๐ฅ) = ๐ด๐๐(๐ฅ), 2๐ ๐=0
|๐ข(๐ฅ, ๐ก)| โค ๐๐๐(๐ฅ) + โ
โก
where ๐ด = ๐ + 2๐โฒ .
We can now state the following result on pointwise boundary limits of Poisson integrals. Theorem 7.12. Lef ๐ โ ๐ฟ1 (โ๐ ) and ๐ข(๐ฅ, ๐ก) its Poisson integral. Then lim ๐ข(๐ฅ, ๐ก) = ๐(๐ฅ) ๐กโ0
๐
for almost every ๐ฅ โ โ . The proof of Theorem 7.12 follows as the proof of Theorem 7.8, by showing that the set {๐ฅ โ โ๐ โถ lim sup |๐ข(๐ฅ, ๐ก) โ ๐(๐ฅ)| > ๐ผ} ๐กโ0
has measure 0 for any ๐ผ > 0, by first approximating with a compactly supported continuous function and then comparing with the maximal function, using Proposition 7.11. We leave the details as an exercise (Exercise (5)). If we compare Theorem 7.12 with Theorem 4.13, we see that this time we are only approaching the boundary point (๐ฅ, 0) vertically, as we only consider ๐ข(๐ฅ, ๐ก) as ๐ก > 0, while in Theorem 4.13 we approach (๐ฅ, 0) from any direction in the upper half-space. We can extend Theorem 7.12 if we consider nontangential limits. For ๐ฅ โ โ๐ , the cone of aperture ๐ > 0 over ๐ฅ is defined as the set ฮ๐ (๐ฅ) = {(๐ฆ, ๐ก) โ โ๐+1 โถ |๐ฅ โ ๐ฆ| < ๐๐ก}. + See Figure 7.2. Thus, we consider the limit of ๐ข(๐ฆ, ๐ก) when we approach (๐ฅ, 0) within this cone. We first prove Theorem 7.13.
7.4. Boundary limits of harmonic functions
147
Figure 7.2. The cone ฮ๐ (๐ฅ) over the point ๐ฅ โ โ๐ .
Theorem 7.13. For any ๐ > 0, there exists ๐ด๐ > 0 such that, for any ๐ โ ๐ฟ1 (โ๐ ) and ๐ฅ โ โ๐ , |๐ข(๐ฆ, ๐ก)| โค ๐ด๐ ๐๐(๐ฅ) for all (๐ฆ, ๐ก) โ ฮ๐ (๐ฅ). Proof. By Proposition 7.11, the theorem follows once we prove that there exists ๐ ๐ > 0 such that, for all ๐ง โ โ๐ , (7.14)
๐๐ก (๐ฆ โ ๐ง) โค ๐ ๐ ๐๐ก (๐ฅ โ ๐ง),
whenever (๐ฆ, ๐ก) โ ฮ๐ (๐ฅ). For this, fix ๐ฅ, ๐ง โ โ๐ and (๐ฆ, ๐ก) โ ฮ๐ (๐ฅ). If |๐ฅ โ ๐ง| โฅ 2|๐ฅ โ ๐ฆ|, then 1 1 |๐ฆ โ ๐ง| = |๐ฅ โ ๐ง + ๐ฆ โ ๐ฅ| โฅ |๐ฅ โ ๐ง| โ |๐ฆ โ ๐ฅ| โฅ |๐ฅ โ ๐ง| โ |๐ฅ โ ๐ง| = |๐ฅ โ ๐ง|, 2 2 and hence 1 1 |๐ฆ โ ๐ง|2 + ๐ก2 โฅ |๐ฅ โ ๐ง|2 + ๐ก2 โฅ (|๐ฅ โ ๐ง|2 + ๐ก2 ), 4 4 so 2๐ก ๐๐ก (๐ฆ โ ๐ง) = ๐๐+1 (|๐ฆ โ ๐ง|2 + ๐ก2 )(๐+1)/2 2๐ก = 2๐+1 ๐๐ก (๐ฅ โ ๐ง). โค (๐+1)/2 2 2 ๐๐+1 ((|๐ฅ โ ๐ง| + ๐ก )/4)
148
7. Maximal functions
If |๐ฅ โ ๐ง| < 2|๐ฅ โ ๐ฆ|, we have |๐ฅ โ ๐ง| < 2๐๐ก beacuse (๐ฆ, ๐ก) โ ฮ๐ (๐ฅ), and thus 1 1 1 1 |๐ฆ โ ๐ง|2 + ๐ก2 โฅ ๐ก2 = ๐ก2 + ๐ก2 โฅ 2 |๐ฅ โ ๐ง|2 + ๐ก2 โฅ ๐(|๐ฅ โ ๐ง|2 + ๐ก2 ), 2 2 2 8๐ where ๐ = min{1/8๐2 , 1/2}, and thus, as above, ๐๐ก (๐ฆ โ ๐ง) โค
1
๐๐ก (๐ฅ ๐(๐+1)/2
โ ๐ง).
We obtain the inequality (7.14) with ๐ ๐ = max{2๐+1 , 1/๐(๐+1)/2 }.
โก
We then have the following result on the existence of nontangential limits. Corollary 7.15. Lef ๐ โ ๐ฟ1 (โ๐ ) and ๐ข(๐ฅ, ๐ก) its Poisson integral. Then, for any ๐ > 0, lim ๐ข(๐ฅ, ๐ก) = ๐(๐ฅ) (๐ฆ,๐ก)โ(๐ฅ,0) (๐ฆ,๐ก)โฮ๐ (๐ฅ)
for almost every ๐ฅ โ โ๐ . We leave its proof as an exercise (Exercise (10)).
Exercises (1) The set ๐ฟ1loc (โ๐ ) of locally integrable functions is a vector space. (2) If ๐ โ ๐ฟ1loc (โ๐ ), then ๐๐ might be infinite at every point. (3) If ๐ โ ๐ฟ1 (โ๐ ), then ๐๐ might be infinite at some points. (4) Define on โ the function 1 ๐(๐ฅ) = { |๐ฅ|(log |๐ฅ|)2 0
if |๐ฅ| โค 1/2 otherwise.
Then ๐ โ ๐ฟ1 (โ), but ๐๐ โ ๐ฟ1loc (โ). (5) Prove Theorem 7.12. (6) We say that {๐พ๐ก }๐ก>0 is a family of better kernels if it satisfies โข โซ ๐พ๐ก (๐ฅ)๐๐ฅ = 1 for all ๐ก > 0; โ๐
Exercises
149
โข there exists ๐ด > 0 such that |๐พ๐ก (๐ฅ)| โค ๐ก > 0; and โข there exists ๐ดโฒ > 0 such that |๐พ๐ก (๐ฅ)| โค
๐ด for all ๐ฅ โ โ๐ and ๐ก๐ ๐ดโฒ ๐ก for all ๐ฅ โ โ๐ |๐ฅ|๐+1
and ๐ก > 0. (a) If ฮฆ โ ๐ฟ1 (โ๐ ), โซ ฮฆ = 1 and |ฮฆ(๐ฅ)| โค ๐ด/(1 + |๐ฅ|)๐+1 , then its dilations {ฮฆ๐ก }๐ก>0 form a family of better kernels. (b) If {๐พ๐ก }๐ก>0 is a family of better kernels, then it is a family of good kernels. (c) If {๐พ๐ก }๐ก>0 is a family of better kernels, then there exists a constant ๐ > 0 such that, if ๐ โ ๐ฟ1 (โ๐ ), |๐พ๐ก โ ๐(๐ฅ)| โค ๐๐๐(๐ฅ) for all ๐ฅ โ โ๐ and ๐ก > 0. (d) If {๐พ๐ก }๐ก>0 is a family of better kernels and ๐ โ ๐ฟ1 (โ๐ ), then lim ๐พ๐ก โ ๐(๐ฅ) = ๐(๐ฅ) ๐กโ0
for almost every ๐ฅ โ โ๐ . (7) The results of the previous exercise are still true if we change the third hypothesis by |๐พ๐ก (๐ฅ)| โค
๐ดโฒ ๐ก๐ , |๐ฅ|๐+๐
for some ๐ > 0. (8) If {๐พ๐ก }๐ก>0 is a family of better kernels and ๐ โ ๐ฟ1 (โ๐ ) is continuous at ๐ฅ, then lim ๐พ๐ก โ ๐(๐ฅ) = ๐(๐ฅ). ๐กโ0
(Hint: Write ๐ = ๐ โ
๐๐ต๐ฟ (๐ฅ) + ๐ โ
(1 โ ๐๐ต๐ฟ (๐ฅ) ), where ๐ต๐ฟ (๐ฅ) is a ball on which ๐ is bounded.) (9) Let {๐พ๐ก }๐ก>0 be a family that satisfies the hypothesis of a family of better kernels except that, instead of the first hypothesis in Exercise (6), it satisfies, for some ๐ โ โ, โซ ๐พ๐ก (๐ฅ)๐๐ฅ = ๐ โ๐
for all ๐ก > 0. Then lim ๐พ๐ก โ ๐(๐ฅ) = ๐๐(๐ฅ) ๐กโ0
150
7. Maximal functions for almost every ๐ฅ โ โ๐ .
(10) Prove Corollary 7.15. (11) Let {๐พ๐ก }๐ก>0 be a family of better kernels and ๐ > 0. (a) There exists a constant ๐ ๐ > 0 such that, for ๐ โ ๐ฟ1 (โ๐ ) and ๐ฅ โ โ๐ , |๐พ๐ก โ ๐(๐ฆ)| โค ๐ ๐ ๐๐(๐ฅ) for all (๐ฆ, ๐ก) โ ฮ๐ (๐ฅ). (b) If ๐ โ ๐ฟ1 (โ๐ ), then lim
(๐ฆ,๐ก)โ(๐ฅ,0) (๐ฆ,๐ก)โฮ๐ (๐ฅ)
๐พ๐ก โ ๐(๐ฆ) = ๐(๐ฅ)
for almost every ๐ฅ โ โ๐ .
Notes The maximal function was introduced by Hardy and Littlewood in [HL30], where they proved Theorem 7.4 for the 1-dimensional case. They also proved Theorem 7.11 in the same paper, for Poisson integrals in the circle. Lemma 7.6 appeared in the paper [Vit08] by Giuseppe Vitali, in the 1-dimensional case. The ๐-dimensional case, and the proof presented here, is due to Stephan Banach [Ban24]. Lebesgueโs differentiation theorem 7.8 appeared in the one variable case in his book [Leb04]. A proof using Vitaliโs lemma appeared in [Leb10], and the proof using the HardyโLittlewood maximal theorem is due to Riesz [Rie32]. Our discussion on nontangential limits can be found in [Ste70].
Chapter 8
Fourier transform
8.1. Integrable functions In this chapter we discuss the representation of a function on โ๐ in terms of its Fourier transform, analogously to the Fourier series representation of a function in a circle discussed in previous chapters. The Fourier transform of a function ๐ โ ๐ฟ1 (โ๐ ) is defined as ฬ = โซ ๐(๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ, ๐(๐)
(8.1)
โ๐
for each ๐ โ โ๐ . The integral above converges since ๐, and thus the function ๐ฅ โฆ ๐(๐ฅ)๐โ2๐๐๐ฅโ
๐ , is integrable for each ๐. 8.2. For any ๐ โ ๐ฟ1 (โ๐ ), ๐ ฬ is a bounded continuous function. Indeed, ฬ |๐(๐)| โค โซ |๐(๐ฅ)๐โ2๐๐๐ฅโ
๐ |๐๐ฅ = โ๐โ๐ฟ1 , โ๐
and, if ๐ โ ๐0 , then ฬ โ ๐(๐ ฬ 0 )| โค โซ |๐(๐ฅ)| |๐โ2๐๐๐ฅโ
๐ โ ๐โ2๐๐๐ฅโ
๐0 |๐๐ฅ โ 0, |๐(๐) โ๐
by the dominated convergence theorem. 8.3. The Fourier transform defines on ๐ฟ1 (โ๐ ) a linear operator, i.e. 151
152
8. Fourier transform ห ฬ โข for each ๐ โ ๐ฟ1 (โ๐ ) and ๐ผ โ โ, ๐ผ๐(๐) = ๐ผ๐(๐); and 1 ๐ ฬ + ๐(๐). โข for ๐, ๐ โ ๐ฟ (โ ), ๐ห + ๐(๐) = ๐(๐) ฬ
By 8.2, this operator continuously maps ๐ฟ1 (โ๐ ) into the space ๐ถ๐ต (โ๐ ) of bounded continuous functions with the uniform norm. Example 8.4. If ๐ = ๐[โ1,1] in โ, then, for ๐ โ 0, 1
2๐๐๐ sin(2๐๐) โ ๐โ2๐๐๐ ฬ = โซ ๐โ2๐๐๐ฅ๐ ๐๐ฅ = ๐ ๐(๐) = . 2๐๐๐ ๐๐ โ1
In particular, ๐ ฬ โ ๐ฟ1 (โ๐ ). Although the Fourier transform of a function ๐ โ ๐ฟ1 (โ๐ ) is not necessarily an integrable function, as in Example 8.4, we observe that, in this ฬ example, ๐(๐) โ 0 as |๐| โ โ. This property is true for all functions ๐ โ ๐ฟ1 (โ๐ ). ฬ Proposition 8.5 (RiemannโLebesgue lemma). If ๐ โ ๐ฟ1 (โ๐ ), then ๐(๐) โ 0 as |๐| โ โ. Proof. One can check explicitly that, if ๐
is a rectangle in โ๐ , then ๐ ห๐
(๐) โ 0
as |๐| โ โ.
(Exercise (2)). If ๐ด โ โ๐ is a finite measure set, by Corollary 5.17(4), for any ๐ > 0 there exist cubes ๐1 , . . . , ๐๐ such that |๐ดโณ โ ๐๐ | < ๐, so โ๐๐ด โ ๐โ ๐๐ โ๐ฟ1 < ๐ and thus |ห ๐๐ด (๐) โ ๐ห โ ๐๐ (๐)| < ๐. Since we can write ๐
๐โ ๐๐ = โ ๐๐
๐ , ๐=1
where each ๐
๐ is a rectangle, we have that ๐ห โ ๐๐ (๐) โ 0 as |๐| โ โ, and thus ๐ ห ๐ด (๐) โ 0 as |๐| โ โ beacuse ๐ > 0 above is arbitrary. We ฬ thus have that, for a simple ๐ โ ๐ฟ1 (โ๐ ), ๐(๐) โ 0 as |๐| โ โ, because an integrable simple function is the linear combination of characteristic functions of finite measure sets. We thus get the proposition for every ๐ โ ๐ฟ1 (โ๐ ), because the simple functions are dense in ๐ฟ1 (โ๐ ). โก The RiemannโLebesgue lemma implies that the Fourier transform defines a linear operator on ๐ฟ1 (โ๐ ) into ๐ถ0 (โ๐ ).
8.1. Integrable functions
153
8.6. If ๐ โ ๐ฟ1 (โ๐ ), the Fourier transform of its dilation ๐๐ก (๐ฅ) = ๐กโ๐ ๐(๐ฅ/๐ก), for ๐ก > 0, is given by ฬ ๐ห๐ก (๐) = ๐(๐ก๐). This follows from the dilation property of Lebesgue measure. Indeed, 1 ๐ห๐ก (๐) = โซ ๐๐ก (๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ = ๐ โซ ๐(๐ฅ/๐ก)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ ๐ก โ๐ โ๐ ฬ = โซ ๐(๐ฆ)๐โ2๐๐๐ก๐ฆโ
๐ ๐๐ฆ = ๐(๐ก๐), โ๐
where we have applied the dilation ๐ฅ โฆ ๐ก๐ฆ. 8.7. The translation invariance of Lebesgue measure on โ๐ further implies, for ๐ โ ๐ฟ1 (โ๐ ), โข if ๐โ (๐ฅ) = ๐(๐ฅ โ โ) for some โ โ โ๐ , then หโ (๐) = ๐โ2๐๐๐ฅโ
โ ๐(๐); ฬ ๐ and โข if ๐(๐ฅ) = ๐2๐๐๐ฅโ
โ ๐(๐ฅ) for some โ โ โ๐ , then ฬ โ โ). ๐(๐) ฬ = ๐(๐ We leave the proof of these properties as an exercise (Exercise (3)). 2
Example 8.8. Consider the function ๐บ(๐ฅ) = ๐โ๐|๐ฅ| in โ๐ . Then 2 ฬ ๐บ(๐) = ๐โ๐|๐| = ๐บ(๐),
so ๐บ is equal to its own Fourier transform. To prove this by Fubiniโs theorem it is enough to show that โ 2
โซ ๐โ๐๐ฅ ๐โ2๐๐๐ฅ๐ ๐๐ฅ = ๐โ๐๐
2
โโ 2
2
2
2
for each ๐ โ โ, because ๐โ๐|๐ฅ| = ๐โ๐๐ฅ1 ๐โ๐๐ฅ2 โฏ ๐โ๐๐ฅ๐ . Since โ
๐ 2
2
โซ ๐โ๐๐ฅ ๐โ2๐๐๐ฅ๐ ๐๐ฅ = lim โซ ๐โ๐๐ฅ ๐โ2๐๐๐ฅ๐ ๐๐ฅ ๐โโ
โโ
โ๐ ๐
2
2
= ๐โ๐๐ lim โซ ๐โ๐(๐ฅ+๐๐) ๐๐ฅ, ๐โโ
โ๐
it remains to verify that ๐
๐ 2
2
lim โซ ๐โ๐(๐ฅ+๐๐) ๐๐ฅ = lim โซ ๐โ๐๐ฅ ๐๐ฅ,
๐โโ
โ๐
๐โโ
โ๐
154
8. Fourier transform
as we know that the integral on the right hand side is equal to 1. The above identity is achieved by considering the contour integral 2
โซ ๐โ๐๐ง ๐๐ง = 0 ๐พ
over the rectangle ๐พ with vertices ๐, ๐ + ๐๐, โ๐ + ๐๐ and โ๐. See Figure 8.1. We leave the details as an exercise (Exercise (4)).
Figure 8.1. The contour ๐พ to obtain the Fourier transform of the function ๐บ(๐ฅ).
8.9. The function ๐บ(๐ฅ) of Example 8.8 is called the Gaussian kernel, and is related to the heat kernel introduced in Exercise (10) of Chapter 4, ๐ป๐ก (๐ฅ) =
1 2 ๐โ|๐ฅ| /4๐ก . ๐/2 (4๐๐ก)
Note that ๐ป๐ก correspond to the dilation ๐บโ4๐๐ก (๐ฅ) of the Gaussian. By the dilation property 8.6 of the Fourier transform we have that ห๐ก (๐) = ๐โ4๐2 ๐ก|๐|2 . ๐ป Example 8.10. Consider now, for a fixed ๐ก > 0, the Poisson kernel ๐๐ก (๐ฅ). Then ห๐ก (๐) = ๐โ2๐๐ก|๐| . ๐ To show this, recall that ๐๐ก is the dilation of ๐1 , so it is sufficient to prove that ๐ห1 (๐) = ๐โ2๐|๐| . We calculate ฮ((๐ + 1)/2) ๐โ2๐๐๐ฅโ
๐ โซ ๐ห1 (๐) = โซ ๐1 (๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ = ๐๐ฅ, (๐+1)/2 (๐+1)/2 2 ๐ โ๐ โ๐ (|๐ฅ| + 1)
8.1. Integrable functions
155
where we have used the fact 2๐(๐+1)/2 . ฮ((๐ + 1)/2)
๐๐+1 = Now, by the identity
โ
1 1 ๐๐ 2 โซ ๐ (๐+1)/2 ๐โ(|๐ฅ| +1)๐ , = (๐+1)/2 2 ๐ ฮ((๐ + 1)/2) 0 (|๐ฅ| + 1) (see Exercise (2) of Chapter 4) and Fubiniโs theorem we obtain ๐ห1 (๐ฅ) = =
โ
1 ๐(๐+1)/2
โซ ๐โ2๐๐๐ฅโ
๐ ( โซ ๐ (๐+1)/2 ๐โ(|๐ฅ| โ๐ โ
2 +1)๐
0
๐๐ )๐๐ฅ ๐
1 2 โซ ๐ (๐โ1)/2 ๐โ๐ ( โซ ๐โ2๐๐๐ฅโ
๐ ๐โ|๐ฅ| ๐ ๐๐ฅ)๐๐ . ๐(๐+1)/2 0 ๐ โ
From 8.8 and the dilation ๐ฅ โฆ (โ๐/๐ )๐ฆ, the integral inside is 2
โซ ๐โ2๐๐๐ฅโ
๐ ๐โ|๐ฅ| ๐ ๐๐ฅ = โ๐
and hence
โ
1 โซ ๐ห1 (๐ฅ) = โ๐ 0
1 โ๐
๐๐/2 โ๐2 |๐|2 /๐ ๐ , ๐ ๐/2
๐โ๐ ๐โ๐
2 |๐|2 /๐
๐๐ .
We calculate the last integral in two steps: (I) For any ๐ข > 0, โ
โ
๐โ2๐๐แต๐ฃ 1 1 โซ โซ ๐๐ฃ = ๐ โโ 1 + ๐ฃ2 โ๐ 0
1 โ๐
๐โ๐ ๐โ๐
2 แต2 /๐
(II) For any ๐ข > 0, โ
1 ๐โ2๐๐แต๐ฃ โซ ๐๐ฃ = ๐โ2๐แต . ๐ โโ 1 + ๐ฃ2 Step (I) follows from the identities โ
1 2 = โซ ๐โ(1+๐ฃ )๐ ๐๐ 1 + ๐ฃ2 0 and
โ 2
โซ ๐โ2๐๐แต๐ฃ ๐โ๐ฃ ๐ ๐๐ฃ = โโ
๐ โ๐2 แต2 /๐ ๐ . โ๐
๐๐ ;
156
8. Fourier transform
The first is a straightforward calculation, while the second follows from Example 8.8 in the 1-dimensional case and the dilation property 8.6 of the Fourier transform. For Step (II), we calculate the contour integral โซ ๐พ
๐โ2๐๐แต๐ง ๐๐ง 1 + ๐ง2
over the lower semicircle around the origin of radius ๐, and we let ๐ โ โ. See Figure 8.2. We leave the details of Steps (I) and (II) as an exercise (Exercise (5)).
Figure 8.2. The contour ๐พ to obtain the Fourier transform of the Poisson kernel ๐1 (๐ฅ). Note that the function ๐ง โฆ ๐โ2๐๐แต๐ง /(1 + ๐ง2 ) has a pole at โ๐.
8.2. The Fourier inversion formula A natural question to ask is whether one can recover a function from its Fourier transform. Recall that the expansion formula ๐๐๐ ฬ ๐(๐) = โ ๐(๐)๐ ๐โโค
of a function on the circle in terms of its Fourier series holds for appropiate continuous functions. In fact, it holds whenever the series ฬ โ |๐(๐)| ๐โโค
converges, so the Fourier series of ๐ is absolutely convergent.
8.2. The Fourier inversion formula
157
Example 8.8, as well as the convergence of the Fourier series above, suggests that, if ๐ ฬ is integrable, then we should have the following Fourier inversion formula, 2๐๐๐ฅโ
๐ ฬ ๐(๐ฅ) = โซ ๐(๐)๐ ๐๐.
(8.11)
โ๐
It is indeed the case. Theorem 8.12. If ๐ โ ๐ฟ1 (โ๐ ) as well as ๐ ฬ โ ๐ฟ1 (โ๐ ), then the Fourier inversion formula (8.11) holds for almost every ๐ฅ โ โ๐ . In the proof of Theorem 8.12 we will use Lemma 8.13. Lemma 8.13. If ๐, ๐ โ ๐ฟ1 (โ๐ ), then โซ ๐๐ฬ = โซ ๐๐.ฬ โ๐
โ๐
Proof. We first observe that both integrals converge because ๐ and ๐ are integrable, and both ๐ ฬ and ๐ฬ are continuous and bounded. Also, if we define ๐น(๐ฅ, ๐ฆ) = ๐(๐ฅ)๐(๐ฆ)๐โ2๐๐๐ฅโ
๐ฆ ๐ ๐ on โ ร โ , then ๐น is integrable, because โซ
|๐น| โค โซ |๐(๐ฅ)|๐๐ฅ โซ |๐(๐ฆ)|๐๐ฆ < โ.
โ๐ รโ๐
โ๐
โ๐
Thus, by Fubiniโs theorem, โซ โ๐ รโ๐
๐น = โซ ( โซ ๐น(๐ฅ, ๐ฆ)๐๐ฅ)๐๐ฆ โ๐
โ๐
= โซ ( โซ ๐(๐ฅ)๐โ2๐๐๐ฅโ
๐ฆ ๐๐ฅ)๐(๐ฆ)๐๐ฆ โ๐
โ๐
ฬ = โซ ๐(๐ฆ)๐(๐ฆ)๐๐ฆ, โ๐
and โซ โ๐ รโ๐
๐น = โซ ( โซ ๐น(๐ฅ, ๐ฆ)๐๐ฆ)๐๐ฅ โ๐
โ๐
= โซ ๐(๐ฅ)( โซ ๐(๐ฆ)๐โ2๐๐๐ฅโ
๐ฆ ๐๐ฅ)๐๐ฅ โ๐
โ๐
= โซ ๐(๐ฅ)๐(๐ฅ)๐๐ฅ. ฬ โ๐
158
8. Fourier transform โก
Proof of Theorem 8.12. For each ๐ฅ โ โ๐ and ๐ก > 0, consider the function 2 ๐(๐) = ๐โ๐๐ก|๐| ๐2๐๐๐ฅโ
๐ . By 8.6, 8.7 and 8.8, its Fourier transform is the Gaussian kernel ๐(๐ฆ) ฬ =
1 ๐ก๐/2
2
๐โ๐|๐ฅโ๐ฆ| /๐ก .
The collection {๐พ๐ก }๐ก>0 given by 1 โ๐|๐ฅ|2 /๐ก ๐ ๐ก๐/2 is a collection of better kernels (see Exercise (6) of Chapter 7; see also Exercise (6) of this chapter), because it is the collection of dilations of the Gaussian kernel seen in 8.8. By Lemma 8.13 with ๐ and ๐, we obtain ๐พ๐ก (๐ฅ) =
(8.14)
โ๐๐ก|๐|2 2๐๐๐ฅโ
๐ ฬ โซ ๐(๐)๐ ๐ ๐๐ = โซ ๐(๐ฆ)๐พ๐ก (๐ฅ โ ๐ฆ)๐๐ฆ, โ๐
โ๐
where we have again used 8.7. Since ๐ ฬ โ ๐ฟ1 (โ๐ ), the left side of (8.14) converges to 2๐๐๐ฅโ
๐ ฬ โซ ๐(๐)๐ ๐๐ โ๐
as ๐ก โ 0, by the dominated convergence theorem. Since the collection {๐พ๐ก }๐ก>0 is a collection of better kernels, โซ ๐(๐ฆ)๐พ๐ก (๐ฅ โ ๐ฆ)๐๐ฆ = ๐ โ ๐พ๐ก (๐ฅ) โ ๐(๐ฅ) โ๐
for almost every ๐ฅ โ โ๐ , by Exercise (6) of Chapter 7.
โก
8.15. Note that, in particular, if ๐ and ๐ ฬ are integrable and ๐ is continuous at ๐ฅ โ โ๐ , then 2๐๐๐ฅโ
๐ ฬ ๐(๐ฅ) = โซ ๐(๐)๐ ๐๐. โ๐
(Exercise (8) of Chapter 7.) Example 8.16. Since the Fourier transform of the Poisson kernel ๐๐ก (๐ฅ) is given by ห๐ก (๐) = ๐โ2๐๐ก|๐| , ๐
8.3. Mean-square convergence
159
ห๐ก is integrable, we have that by 8.10, and clearly ๐ โซ ๐โ2๐๐ก|๐| ๐2๐๐๐ฅโ
๐ ๐๐ = ๐๐ก (๐ฅ), โ๐
for every ๐ฅ โ โ๐ . Moreover, since both ๐๐ก (๐ฅ) = ๐๐ก (โ๐ฅ) and ๐โ2๐๐ก|๐ฅ| = ๐โ2๐๐ก|โ๐ฅ| , we see that the Fourier transform of ๐โ2๐๐ก|๐ฅ| is ๐๐ก (๐), as well. Theorem 8.12 leads to the question of whether the Fourier transform of a given function is integrable. In general, we have seen that the Fourier transform is continuous and has limit 0 at infinity but, in general, itโs not a function in ๐ฟ1 (โ๐ ), as the simple example of a characteristic function shows (Example 8.4). However, an application of the limit results above and the monotone convergence theorem give us the following test. ฬ Proposition 8.17. Let ๐ โ ๐ฟ1 (โ๐ ) be continuous at 0. If ๐(๐) โฅ 0 for ๐ 1 ๐ ฬ every ๐ โ โ , then ๐ โ ๐ฟ (โ ). Proof. From equation (8.14) (which is true for all ๐ โ ๐ฟ1 (โ๐ )) and the hypothesis that ๐ is continuous at 0, we have that โ๐๐ก|๐|2 ฬ lim โซ ๐(๐)๐ ๐๐ = ๐(0). ๐กโ0
โ๐
ฬ โฅ 0, we clearly have that Also, since ๐(๐) โ๐๐ก|๐|2 ฬ ฬ ๐(๐)๐ โ ๐(๐)
as ๐ก โ 0 so, by the monotone convergence theorem, โ๐๐ก|๐|2 ฬ ฬ โซ ๐(๐)๐๐ = lim โซ ๐(๐)๐ ๐๐. โ๐
๐กโ0
โ๐
Therefore, ๐ ฬ โ ๐ฟ1 (โ๐ ) and, in fact, โ๐โฬ ๐ฟ1 = ๐(0).
โก
8.3. Mean-square convergence The Fourier transform is only defined for ๐ โ ๐ฟ1 (โ๐ ), as we require that the integral (8.1) converges. However, it is possible to extend the definition of the Fourier transform to functions in ๐ฟ2 (โ๐ ), and even obtain its inverse.
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8. Fourier transform
For this, we will use the fact that the set ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) is dense in ๐ฟ (โ๐ ): for any ๐ โ ๐ฟ2 (โ๐ ), there exists a sequence ๐๐ โ ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) that converges to ๐ in ๐ฟ2 (โ๐ ), that is 2
โซ |๐ โ ๐๐ |2 โ 0. โ๐
This follows from the fact that both ๐ฟ1 (โ๐ ) and ๐ฟ2 (โ๐ ) contain the space ๐ถ๐ (โ๐ ) of continuous functions of compact support, and that ๐ถ๐ (โ๐ ) is dense in both ๐ฟ1 (โ๐ ) and ๐ฟ2 (โ๐ ), as we discussed in Chapter 6. We have Theorem 8.18. Theorem 8.18. If ๐ โ ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ), then โซ |๐|2 = โซ |๐|ฬ 2 . โ๐
โ๐
That is, if โจโ
, โ
โฉ is the inner product in ๐ฟ2 (โ๐ ), then the operator ๐ โฆ ๐ฬ is an isometry on the subset ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) of ๐ฟ2 (โ๐ ), so โ๐โ๐ฟ2 = โ๐โฬ ๐ฟ2 . By the polarization identity, we also have โจ๐, ๐โฉ = โจ๐,ฬ ๐โฉฬ
(8.19)
for any ๐, ๐ โ ๐ฟ1 (โ๐ )โฉ๐ฟ2 (โ๐ ) (Exercise (7)). See Section A.5 for a review of results of inner product spaces. Proof. For the proof of Theorem 8.18, we will use the identity ห ฬ ๐(๐), ๐ โ ๐(๐) = ๐(๐) ฬ for any ๐, ๐ โ ๐ฟ1 (โ๐ ), which follows from Fubiniโs theorem applied to the function ๐น(๐ฅ, ๐ฆ) = ๐(๐ฅ โ ๐ฆ)๐(๐ฆ)๐โ2๐๐๐ฅโ
๐ (Exercise (8)). In particular, if we take ๐(๐ฅ) = ๐(โ๐ฅ) and โ = ๐ โ ๐, then ฬ ๐(๐) ฬ 2, ฬ = ๐(๐) โ(๐) ฬ = |๐(๐)| because ฬ ๐(๐) ฬ = โซ ๐(โ๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ = โซ ๐(๐ฅ)๐2๐๐๐ฅโ
๐ ๐๐ฅ = ๐(๐). โ๐
โ๐
8.3. Mean-square convergence
161
Now, โ is a continuous function because, by the CauchyโSchwarz inequality, |โ(๐ฅ) โ โ(๐ฆ)| = |๐ โ ๐(๐ฅ) โ ๐ โ ๐(๐ฆ)| = || โซ (๐(๐ฅ โ ๐ง) โ ๐(๐ฆ โ ๐ง))๐(๐ง)๐๐ง|| โ๐
โค โ๐(๐ฅ โ โ
) โ ๐(๐ฆ โ โ
)โ๐ฟ2 โ๐โ๐ฟ2 goes to 0 as ๐ฆ โ ๐ฅ, by the continuity of translations in the ๐ฟ2 norm (6.49). Then, โ is continuous at 0 and โ ฬ โฅ 0, so by Proposition 8.17 we have that โ ฬ โ ๐ฟ1 (โ๐ ) and โ(0) = โซ โ.ฬ โ๐
Therefore โซ |๐|ฬ 2 = โซ โ ฬ = โ(0) = ๐ โ ๐(0) = โซ ๐(โ๐ฅ)๐(๐ฅ)๐๐ฅ โ๐
โ๐
โ๐
= โซ ๐(โ๐ฅ)๐(โ๐ฅ)๐๐ฅ = โซ |๐|2 . โ๐
โ๐
โก Theorem 8.18 allows us to extend the definition of the Fourier transform to any function in ๐ฟ2 (โ๐ ). Indeed, if ๐ โ ๐ฟ2 (โ๐ ), as we have seen above, there exists a sequence ๐๐ โ ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) such that ๐๐ โ ๐ in ๐ฟ2 (โ๐ ). Now the sequence of Fourier transforms ๐๐ฬ is a Cauchy sequence in ๐ฟ2 (โ๐ ), as โ๐๐ฬ โ ๐๐ฬ โ๐ฟ2 = โ๐๐ โ ๐๐ โ๐ฟ2 and the fact that ๐๐ converges in ๐ฟ2 (โ๐ ). ๐ฟ2 (โ๐ ) is complete, so ๐๐ฬ converges in ๐ฟ2 (โ๐ ) and we can define โฑ๐ = lim ๐๐ฬ .
(8.20)
The operator โฑ โถ ๐ฟ2 (โ๐ ) โ ๐ฟ2 (โ๐ ) is well defined because, if ๐๐ โ ๐ and ๐๐ โ ๐ in ๐ฟ2 (โ๐ ), then โ๐๐ฬ โ ๐๐ฬ โ๐ฟ2 = โ๐๐ โ ๐๐ โ๐ฟ2 โ 0, so lim ๐๐ฬ = lim ๐๐ฬ 2
๐
in ๐ฟ (โ ). It is also an isometry, as clearly (8.21)
โโฑ๐โ๐ฟ2 = โ๐โ๐ฟ2
162
8. Fourier transform
for all ๐ โ ๐ฟ2 (โ๐ ). Theorem 8.22. The operator โฑ โถ ๐ฟ2 (โ๐ ) โ ๐ฟ2 (โ๐ ) is a unitary operator in ๐ฟ2 (โ๐ ) with inverse โฑ โ1 ๐(๐ฅ) = โฑ๐(โ๐ฅ). A unitary operator is a surjective isometry. Theorem 8.22 is commonly known as Plancherelโs theorem. Proof. We first observe that โฑ has closed range. If โฑ๐๐ โ ๐ in ๐ฟ2 (โ๐ ), then โ๐๐ โ ๐๐ โ๐ฟ2 = โโฑ๐๐ โ โฑ๐๐ โ๐ฟ2 because โฑ is an isometry. Thus ๐๐ is Cauchy in ๐ฟ2 (โ๐ ), so it converges, say ๐๐ โ ๐ in ๐ฟ2 (โ๐ ). Hence โฑ๐๐ โ โฑ๐, and ๐ = โฑ๐. Let โณ be the range of โฑ. If โณ โ ๐ฟ2 (โ๐ ), the orthogonal complement to โณ would be nontrivial (see Appendix A.5), so there would be a nonzero ๐ โ ๐ฟ2 (โ๐ ) so that โจโฑ๐, ๐โฉ = 0 2
๐
for every ๐ โ ๐ฟ (โ ), which implies โซ(โฑ๐)๐ = 0 for all ๐ โ ๐ฟ2 (โ๐ ). Now, the density of ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) and Lemma 8.13 imply that โซ โฑ๐ ๐ = โซ ๐ โฑ๐ โ๐
โ๐
(Exercise (10)), so โซ ๐ โฑ๐ = 0 โ๐ 2
๐
for every ๐ โ ๐ฟ (โ ), and thus, choosing ๐ = โฑ๐, we obtain โซ |โฑ๐|2 = 0. โ๐
But then โ๐โ๐ฟ2 = โโฑ๐โ๐ฟ2 = 0, contradicting the fact that ๐ โ 0 in ๐ฟ2 (โ๐ ). Thus โณ = ๐ฟ2 (โ๐ ) and โฑ is surjective. In order to show that โฑ โ1 ๐(๐ฅ) = โฑ๐(โ๐ฅ), we first note that, for any ๐ โ ๐ฟ2 (โ๐ ), ฬ โฑ๐(๐) = โฑ ๐(โ๐)
8.3. Mean-square convergence
163
ฬ = ๐(๐ฅ). Indeed, if ๐๐ โ ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) and where ๐ ฬ is the function ๐(๐ฅ) 2 ๐ ๐๐ โ ๐ in ๐ฟ (โ ), then ๐๐ฬ (๐) = โซ ๐๐ (๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ = โซ ๐๐ (๐ฅ)๐2๐๐๐ฅโ
๐ ๐๐ฅ = ๐๐ฬฬ (โ๐), โ๐
โ๐
ฬ so ๐๐ฬ (๐) โ โฑ ๐(โ๐) in ๐ฟ2 (โ๐ ). Then, for any ๐ โ ๐ฟ2 (โ๐ ), if we set โ(๐ฅ) = โฑ๐(โ๐ฅ), ฬ โจโฑโ, ๐โฉ = โซ โฑโ(๐ฅ)๐(๐ฅ)๐๐ฅ = โซ โ(๐ฅ)โฑ ๐(๐ฅ)๐๐ฅ โ๐
โ๐
ฬ ฬ = โซ โฑ๐(๐ฅ)โฑ ๐(โ๐ฅ)๐๐ฅ = โซ โฑ๐(โ๐ฅ)โฑ ๐(๐ฅ)๐๐ฅ โ๐
โ๐
= โซ โฑ๐(๐ฅ)โฑ๐(๐ฅ)๐๐ฅ = โจโฑ๐, โฑ๐โฉ = โจ๐, ๐โฉ, โ๐
where in the last identiy we have used the fact that โฑ is an isometry. Since ๐ โ ๐ฟ2 (โ๐ ) is arbitrary, we have shown that โ = โฑ โ1 ๐. โก For a function ๐ โ ๐ฟ2 (โ๐ ), we will write its Fourier transform โฑ๐ simply as ๐.ฬ 8.23. If ๐ โ ๐ฟ2 (โ๐ ), then ๐๐ = ๐๐๐ต๐ โ ๐ฟ1 (โ๐ ), where ๐๐ต๐ is the characteristic function of the ball ๐ต๐ of radius ๐ around the origin. Indeed, โซ |๐๐ | = โซ |๐|๐๐ต๐ โค โ๐โ๐ฟ2 โ
โ|๐ต๐ | < โ โ๐
โ๐
for each ๐. Moreover, ๐๐ โ ๐ in ๐ฟ2 (โ๐ ) as ๐ โ โ, because โ๐ โ ๐๐ โ2๐ฟ2 = โซ
|๐(๐ฅ)|2 ๐๐ฅ โ 0.
|๐ฅ|โฅ๐
Thus ฬ = lim ๐๐ฬ (๐) = lim โซ ๐(๐ฅ)๐โ2๐๐๐ฅโ
๐ ๐๐ฅ ๐(๐) ๐โโ
2
๐โโ
๐ต๐
๐
in ๐ฟ (โ ) (i.e. the limit is taken in the ๐ฟ2 sense). By Theorem 8.22, 2๐๐๐ฅโ
๐ ฬ ๐๐ ๐(๐ฅ) = lim โซ ๐(๐)๐ ๐โโ
2
๐ต๐
๐
in ๐ฟ (โ ), that is 2
2๐๐๐ฅโ
๐ ฬ โซ ||๐(๐ฅ) โ โซ ๐(๐)๐ ๐๐ || ๐๐ฅ โ 0 โ๐
๐ต๐
164
8. Fourier transform
as ๐ โ โ. Hence we have an analog of Theorem 3.50, on the meansquare convergence of Fourier series, for the Fourier transform. 8.24. If ๐ โ ๐ฟ1 (โ๐ ) and ๐ โ ๐ฟ2 (โ๐ ), we have seen above that their convolution ๐ โ ๐ โ ๐ฟ2 (โ๐ ) and โ๐ โ ๐โ๐ฟ2 โค โ๐โ๐ฟ1 โ๐โ๐ฟ2 . Thus, if ๐๐ โ ๐ฟ1 (โ๐ ) โฉ ๐ฟ2 (โ๐ ) and ๐๐ โ ๐ in ๐ฟ2 (โ๐ ), then โ๐ โ ๐๐ โ ๐ โ ๐โ๐ฟ2 โค โ๐โ๐ฟ1 โ๐๐ โ ๐โ๐ฟ2 , so ๐ โ ๐๐ โ ๐ โ ๐ in ๐ฟ2 (โ๐ ). Note that we also have ๐ โ ๐๐ โ ๐ฟ1 (โ๐ ), and thus ๐ห โ ๐๐ = ๐๐ฬ ๐ฬ for each ๐. Therefore we obtain ห ๐ โ ๐ = ๐๐ฬ ฬ for any ๐ โ ๐ฟ1 (โ๐ ) and ๐ โ ๐ฟ2 (โ๐ ). Example 8.25. Let ๐ โ ๐ฟ2 (โ๐ ) and let ๐ซ๐ก ๐(๐ฅ) be its Poisson integral. Then, since ๐ซ๐ก ๐ = ๐๐ก โ ๐, we have โ2๐๐ก|๐| ฬ ฬ ห ๐(๐). ๐ซห ๐ก ๐(๐) = ๐๐ก (๐)๐(๐) = ๐
Exercises (1) The Fourier transform is a linear operator: (a) If ๐ผ โ โ and ๐ โ ๐ฟ1 (โ๐ ), then ห ฬ ๐ผ๐(๐) = ๐ผ๐(๐). (b) If ๐, ๐ โ ๐ฟ1 (โ๐ ), then ฬ + ๐(๐). ๐ห + ๐(๐) = ๐(๐) ฬ (2) If ๐
is a rectangle in โ๐ , then ๐ ห๐
(๐) โ 0 as |๐| โ โ. (3) Let ๐ โ ๐ฟ1 (โ๐ ). (a) If ๐โ (๐ฅ) = ๐(๐ฅ + โ) is the translation of ๐ with โ โ โ๐ , then หโ (๐) = ๐2๐๐โโ
๐ ๐(๐). ฬ ๐ (b) If ๐(๐ฅ) = ๐2๐๐๐ฅโ
โ with โ โ โ๐ , then ฬ โ โ). ๐(๐) ฬ = ๐(๐
Notes
165
(4) For any ๐ โ โ, ๐
๐ 2
2
lim โซ ๐โ๐(๐ฅ+๐๐) ๐๐ฅ = lim โซ ๐โ๐๐ฅ ๐๐ฅ = 1.
๐โโ
๐โโ
โ๐
โ๐ 2
(Hint: Consider the contour integral โซ๐พ ๐โ๐๐ง ๐๐ง = 0 over the rectangle ๐พ with vertices ๐, ๐ + ๐๐, โ๐ + ๐๐ and โ๐.) (5) Let ๐ข > 0. Then โ โ2๐๐แต๐ฃ โ 1 ๐ 1 1 โ๐ โ๐2 แต2 /๐ โซ โซ (a) ๐๐ฃ = ๐ ๐ ๐๐ ; ๐ โโ 1 + ๐ฃ2 โ๐ 0 โ๐ โ โ2๐๐แต๐ฃ 1 ๐ โซ (b) ๐๐ฃ = ๐โ2๐แต . ๐ โโ 1 + ๐ฃ2 2
(6) If ฮฆ(๐ฅ) = ๐โ๐|๐ฅ| , the collection {ฮฆ๐ก (๐ฅ)}๐ก>0 of its dilations is a collection of better kernels (see Exercise (6) of Chapter 9). (7) Use the polarization identity for complex inner products spaces to show 8.19. (8) If ๐, ๐ โ ๐ฟ1 (โ๐ ), then ห ฬ ๐(๐). ๐ โ ๐(๐) = ๐(๐) ฬ (Hint: Apply Fubiniโs theorem to the function ๐น(๐ฅ, ๐ฆ) = ๐(๐ฅ โ ๐ฆ)๐(๐ฆ)๐โ2๐๐๐ฅโ
๐ on โ๐ ร โ๐ .) (9) Let ๐ โ ๐ฟ1 (โ๐ ) and ๐
a rotation on โ๐ . (a) If ๐ = ๐ โ ๐
, ฬ ๐(๐) ฬ = ๐(๐
๐). (b) If ๐ is a radial function, then ๐ ฬ is also radial. (10) For every ๐, ๐ โ ๐ฟ2 (โ๐ ), โซ โฑ๐๐ = โซ ๐โฑ๐. โ๐
โ๐
Notes The results in this chapter, and a deeper study of the Fourier transform, can be found in [SW71]. As we mentioned above, Theorem 8.22 is usually known as Plancherelโs theorem, and the collective results of Section 8.3 are commonly described as Plancherelโs theory due to Michel Plancherelโs work in [Pla10].
Chapter 9
Hilbert transform
9.1. The conjugate function Let ๐ โ ๐ฟ1 (โ) and ๐ข(๐ฅ, ๐ก) be its Poisson integral, ๐ข(๐ฅ, ๐ก) = ๐ซ๐ก ๐(๐ฅ) = โซ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ, โ
where ๐๐ก is the Poisson kernel on โ, ๐๐ก (๐ฅ) =
1 ๐ก . 2 ๐ ๐ฅ + ๐ก2
We have seen above, in 4.3, that ๐ข(๐ฅ, ๐ก) is harmonic in the upper half plane โ2+ . Consider now the function (9.1)
๐๐ก (๐ฅ) =
1 ๐ฅ . ๐ ๐ฅ2 + ๐ก 2
The function ๐๐ก is not integrable, as it only decays as 1/|๐ฅ| when |๐ฅ| โ โ. However, it is bounded for each ๐ก > 0, so we can calculate (9.2)
๐ฃ(๐ฅ, ๐ก) = โซ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ โ
for any ๐ โ ๐ฟ1 (โ). ๐ฃ(๐ฅ, ๐ก) is a conjugate harmonic function to ๐ข(๐ฅ, ๐ก) in โ2+ , so the function ๐(๐ฅ, ๐ก) = ๐ข(๐ฅ, ๐ก) + ๐๐ฃ(๐ฅ, ๐ก) 167
168
9. Hilbert transform
is holomorphic in โ2+ . To prove this, it is sufficient to verify explicitly that (๐ฅ, ๐ก) โฆ ๐๐ก (๐ฅ) is harmonic, and so is ๐ฃ(๐ฅ, ๐ก), in โ2+ , and that ๐ข and ๐ฃ satisfy the Cauchy-Riemann equations (Exercise (1)). In fact, we note that ๐ ๐๐ก (๐ฅ) + ๐๐๐ก (๐ฅ) = , ๐(๐ฅ + ๐๐ก) so ๐๐ฆ (๐ฅ) and ๐๐ฆ (๐ฅ), for ๐ฆ > 0, are the real and imaginary parts of ๐/๐๐ง, where ๐ง = ๐ฅ + ๐๐ฆ. We also observe that ๐๐ก (๐ฅ) =
1 1 ๐ฅ/๐ก 1 ๐ฅ ( ) = ๐1 ( ), ๐ก ๐ (๐ฅ/๐ก)2 + 1 ๐ก ๐ก
so {๐๐ก }๐ก>0 is the family of dilations of ๐1 . However, since ๐1 โ ๐ฟ1 (โ), {๐๐ก }๐ก>0 is not a family of good kernels, so we cannot yet determine the existence of (9.3)
lim โซ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ. ๐กโ0
โ
Note that the limit (9.3) represents the limit at the boundary of the conjugate harmonic function ๐ฃ to the Poisson integral of ๐. We have the following questions. Question 1: If ๐ โ ๐ฟ1 (โ), does the limit (9.3) exist for any ๐ฅ โ โ? Question 2: Under what conditions on ๐ can we guarantee, and in what sense, that the limit (9.3) exists?
9.2. Mean-square convergence We first consider Question 2 by asking if the limit (9.3) exists in the mean-square sense, that is, in ๐ฟ2 (โ). The first problem we encounter is that, since ๐๐ก is not integrable, we cannot guarantee that the convolution ๐๐ก โ ๐ exists if ๐ โ ๐ฟ2 (โ). However, since indeed ๐๐ก โ ๐ฟ2 (โ), the convolution ๐๐ก โ ๐ โ ๐ฟ2 (โ) if ๐ โ ๐ฟ1 (โ) and, in particular, if ๐ โ ๐ฟ1 (โ) โฉ ๐ฟ2 (โ). Thus, before we consider the limit as ๐ก โ 0, we have to ask Question 3. Question 3: Can we extend the operator ๐ โฆ ๐๐ก โ ๐ from ๐ฟ1 (โ) โฉ ๐ฟ2 (โ) to ๐ฟ2 (โ), for any ๐ก > 0? In that case, does the limit (9.3) exist in the ๐ฟ2 sense?
9.2. Mean-square convergence
169
To answer Question 3 we need to find out if the family of operators ๐ โฆ ๐๐ก โ ๐ is uniformly bounded in ๐ฟ2 (โ), that is, if there exists a constant ๐ด > 0, independent of ๐ก, such that โ๐๐ก โ ๐โ๐ฟ2 โค ๐ดโ๐โ๐ฟ2 1
2
for every ๐ โ ๐ฟ (โ) โฉ ๐ฟ (โ) and all ๐ก > 0. This will allow us to extend ๐๐ก โ ๐ to all of ๐ฟ2 (โ), as we did in the case of the Fourier transform in the previous chapter, for each ๐ก > 0, and then take the limit as ๐ก โ 0. Lemma 9.4. For any ๐ก > 0, the Fourier transform of ๐๐ก in ๐ฟ2 (โ) is given by ห๐ก (๐) = โ๐ sgn(๐)๐โ2๐๐ก|๐| , where sgn(๐) is the sign function the function ๐ of ๐. Proof. As we noted in 8.23, the Fourier transform of the ๐ฟ2 function ๐๐ก is given by ๐
๐
๐ฅ ห๐ก (๐) = lim โซ ๐๐ก (๐ฅ)๐โ2๐๐๐ฅ๐ ๐๐ฅ = 1 lim โซ ๐ ๐โ2๐๐๐ฅ๐ ๐๐ฅ. 2 + ๐ก2 ๐ ๐ฅ ๐โโ ๐โโ โ๐ โ๐ As ๐๐ก is odd, the above limit is 0 if ๐ = 0. For ๐ โ 0, the limit on the right can be obtained by using the residue theorem [Gam01]. Indeed, if ๐ < 0, consider the contour ๐พ given by the upper semicircle of radius ๐
Figure 9.1. The contour ๐พ in the proof of Lemma 9.4. For ๐ sufficiently large, the pole ๐๐ก of the function ๐(๐ง) is inside ๐พ.
with center at the origin (as in Figure 9.1), and 1 ๐ง ๐(๐ง) = ๐โ2๐๐๐ง๐ . ๐ ๐ง 2 + ๐ก2
170
9. Hilbert transform
Then, by the residue theorem, โซ ๐(๐ง)๐๐ง = 2๐๐ Res๐ง=๐๐ก ๐(๐ง) = ๐๐2๐๐ก๐ . ๐พ
We also have โซ ๐(๐ง)๐๐ง = ๐พ ๐
โซ โ๐
๐
1 ๐ฅ 1 ๐๐๐๐ ๐๐ ๐โ2๐๐๐ฅ๐ ๐๐ฅ + โซ ๐โ2๐๐๐๐ ๐ ๐๐๐๐๐ ๐๐, 2 2 ๐๐ )2 + ๐ก2 ๐๐ฅ +๐ก ๐ (๐๐ 0
where the integrals on the right hand side correspond to the integrals over the segment [โ๐, ๐] and the semicircle ๐๐๐๐ , 0 โค ๐ โค ๐, of ๐พ, respectively. The first integral, as ๐ โ โ, gives our desired limit, while the second satisfies ๐
๐
๐๐๐๐ ๐๐ ๐๐ |โซ 1 ๐โ2๐๐๐๐ ๐ ๐๐๐๐๐ ๐๐|| โค ๐ถ โซ |๐โ2๐๐๐๐ ๐ |๐๐ | ๐๐ )2 + ๐ก2 ๐ (๐๐ 0 0 ๐
= ๐ถ โซ ๐2๐๐๐ sin ๐ ๐๐ 0 ๐/2
โค 2๐ถ โซ
๐โ4๐|๐|๐ ๐๐ โ 0,
0
where we have used the fact that ๐ < 0 and the estimate 2 sin ๐ โฅ ๐ ๐ on [0, ๐/2], along with the symmetry of sin ๐ around ๐ = ๐/2. Thus, we obtain ห๐ก (๐) = ๐๐2๐๐ก๐ ๐ for ๐ < 0. We leave as an exercise the details for ๐ > 0 (Exercise (2)), where we obtain ห๐ก (๐) = โ๐๐โ2๐๐ก๐ . ๐ Combining the above we conclude ห๐ก (๐) = โ๐ sgn(๐)๐โ2๐๐ก|๐| . ๐ โก 1
2
Corollary 9.5. If ๐ โ ๐ฟ (โ) โฉ ๐ฟ (โ), then, for any ๐ก > 0, โ๐๐ก โ ๐โ๐ฟ2 โค โ๐โ๐ฟ2 .
9.2. Mean-square convergence
171
Proof. The Fourier transform is an isometry in ๐ฟ2 (โ), so, using Lemma 9.4, โ2๐๐ก|๐| ฬ ห โ๐๐ก โ ๐โ๐ฟ2 = โ๐ ๐โ๐ฟ2 โค โ๐โฬ ๐ฟ2 = โ๐โ๐ฟ2 . ๐ก โ ๐โ๐ฟ2 = โ โ ๐ sgn(๐)๐
โก Corollary 9.5 allows us to define the convolution ๐๐ก โ ๐ for any ๐ โ ๐ฟ2 (โ), and we can do it in two different ways. The first way is analogous to the way we extended the Fourier transform to ๐ฟ2 (โ๐ ): using the fact that ๐ฟ1 (โ) โฉ ๐ฟ2 (โ) is dense in ๐ฟ2 (โ), we can find, for any ๐ โ ๐ฟ2 (โ), a sequence ๐๐ โ ๐ฟ1 (โ) โฉ ๐ฟ2 (โ) such that ๐๐ โ ๐ in ๐ฟ2 (โ), and then define ๐๐ก โ ๐ as the limit in ๐ฟ2 (โ) of the sequence ๐๐ก โ ๐๐ . This limit exists because ๐๐ก โ ๐๐ is a Cauchy sequence in ๐ฟ2 (โ), by Corollary 9.5, and ๐ฟ2 (โ) is complete. The other way is through the Fourier transform of ๐๐ก โ ๐. Since the Fourier transform is a unitary operator, by Theorem 8.18, we can define, for ๐ โ ๐ฟ2 (โ), ๐๐ก โ ๐ โ ๐ฟ2 (โ) as the function whose Fourier transform is equal to (9.6)
โ2๐๐ก|๐| ฬ ห ๐ ๐(๐). ๐ก โ ๐(๐) = โ๐ sgn(๐)๐
That is, we define, for ๐ โ ๐ฟ2 (โ), ๐๐ก โ ๐ as the inverse Fourier transform of (9.6). Moreover, it is clear that, as ๐ก โ 0, the limit ห lim ๐ ๐ก โ ๐(๐) ๐กโ0
ฬ exists for every ๐ โ โ, and is equal to โ๐ sgn(๐)๐(๐). This limit also converges in ๐ฟ2 (โ) because 2
ฬ โ ( โ ๐ sgn(๐)๐(๐)) ฬ | ๐๐ โซ | โ ๐ sgn(๐)๐โ2๐๐ก|๐| ๐(๐) โ 2
ฬ | ๐๐ โ 0, = โซ |(๐โ2๐๐ก|๐| โ 1)๐(๐) โ
by the dominated convergence theorem. We can therefore make Definition 9.7. Definition 9.7. Let ๐ โ ๐ฟ2 (โ). The Hilbert transform of ๐ is the function ๐ป๐ โ ๐ฟ2 (โ) equal to the inverse Fourier transform of ห ฬ ๐ป๐(๐) = โ๐ sgn(๐)๐(๐).
172
9. Hilbert transform
9.8. The Hilbert transform ๐ป is an isometry in ๐ฟ2 (โ), because ฬ ฬ | โ ๐ sgn(๐)๐(๐)| = |๐(๐)| and hence ห ๐ฟ2 (โ) = โ๐โฬ ๐ฟ2 (โ) = โ๐โ๐ฟ2 (โ) . โ๐ป๐โ๐ฟ2 (โ) = โ๐ป๐โ With the observations above, and Definition 9.7, we have answered Questions 2 and 3 of this and Section 9.1. However, to answer Question 1, we need to understand the local nature of the convolution ๐๐ก โ ๐ as ๐ก โ 0, which we discuss in Section 9.3.
9.3. The Hilbert transform of integrable functions In this section we study the Hilbert transform of integrable functions. As noted above, the kernels ๐๐ก are not integrable, and thus we cannot discuss the behavior of ๐๐ก โ ๐ as ๐ก โ 0 pointwise, nor in ๐ฟ1 (โ), directly. However, we have Lemma 9.9. Lemma 9.9. For ๐ โ ๐ฟ1 (โ), lim ( โซ ๐(๐ฅ โ ๐ฆ) ๐กโ0
โ
๐(๐ฅ โ ๐ฆ) ๐ฆ ๐๐ฆ โ โซ ๐๐ฆ) = 0 ๐ฆ ๐ฆ2 + ๐ก 2 |๐ฆ|โฅ๐ก
for almost every ๐ฅ โ โ. Proof. Define
๐ฆ 1 โ 2+1 ๐ฆ ๐ฆ ฮฆ(๐ฆ) = { ๐ฆ ๐ฆ2 + 1
Since
|๐ฆ| โฅ 1 |๐ฆ| < 1.
๐ฆ 1 1 โ =โ 2 +1 ๐ฆ ๐ฆ(๐ฆ + 1) for |๐ฆ| > 1, ฮฆ is integrable. Moreover, the collection of its dilations ๐ฆ 1 โ |๐ฆ| โฅ ๐ก 2 + ๐ก2 1 ๐ฆ ๐ฆ ๐ฆ ฮฆ๐ก (๐ฆ) = ฮฆ( ) = { ๐ฆ ๐ก ๐ก |๐ฆ| < ๐ก ๐ฆ2 + ๐ก 2 ฮฆ(๐ฆ) =
๐ฆ2
satisfies (1) โซ ฮฆ๐ก = 0 for all ๐ก > 0; โ
9.3. The Hilbert transform of integrable functions
173
1 for all ๐ก > 0 and ๐ฆ โ โ; and ๐ก ๐ก (3) |ฮฆ๐ก (๐ฆ)| โค 2 for all ๐ก > 0 and ๐ฆ โ โ, ๐ฆ โ 0. ๐ฆ (2) |ฮฆ๐ก (๐ฆ)| โค
The first one follows because ฮฆ is odd, and the other two from the explicit form of ฮฆ๐ก (๐ฆ) (Exercise (4)). Then โซ ๐(๐ฅ โ ๐ฆ) โ
๐(๐ฅ โ ๐ฆ) ๐ฆ ๐๐ฆ โ โซ ๐๐ฆ = ฮฆ๐ก โ ๐(๐ฅ) ๐ฆ ๐ฆ2 + ๐ก 2 |๐ฆ|โฅ๐ก
and {ฮฆ๐ก }๐ก>0 is a collection of better kernels, as introduced in Exercises (6) and (9) of Chapter 7, with constant ๐ = 0 in the latter. Therefore, lim ฮฆ๐ก โ ๐(๐ฅ) = 0 ๐กโ0
โก
for almost every ๐ฅ โ โ. Lemma 9.9 implies that, for almost every ๐ฅ โ โ, the limit lim โซ ๐(๐ฅ โ ๐ฆ) ๐กโ0
โ
๐ฆ ๐๐ฆ ๐ฆ2 + ๐ก 2
exists if and only if lim โซ ๐กโ0
|๐ฆ|โฅ๐ก
๐(๐ฅ โ ๐ฆ) ๐๐ฆ ๐ฆ
exists. Note that the latter limit exists at ๐ฅ โ โ if ๐ โ ๐ฟ1 (โ) and ๐ฅ is not in the support of ๐, because lim ๐กโ0
๐(๐ฅ โ ๐ฆ) ๐(๐ฆ) ๐(๐ฆ) 1 1 1 โซ ๐๐ฆ = lim โซ ๐๐ฆ = โซ ๐๐ฆ, ๐ |๐ฆ|โฅ๐ก ๐ฆ ๐ฅ โ ๐ฆ ๐ ๐ฅ โ๐ฆ ๐กโ0 ๐ |๐ฅโ๐ฆ|โฅ๐ก โ
as there exist some ๐ฟ > 0 such that, if |๐ฅ โ ๐ฆ| < ๐ฟ, then ๐ฆ โ supp ๐, so ๐(๐ฆ) = 0. The limit also exists if ๐ is differentiable at ๐ฅ, because in that case the integrals ๐(๐ฅ โ ๐ฆ) โซ ๐๐ฆ ๐ฆ ๐กโค|๐ฆ| 0, โ๐ป๐โ๐ฟ1 โค ๐ดโ๐โ๐ฟ1 ๐ถ๐โ (โ).
for every ๐ โ However, this is far from true, because ๐ป๐ might not even be integrable. Example 9.11. Let ๐ โ ๐ถ๐โ (โ) such that ๐ โฅ 0, ๐(๐ฅ) = 1 if 0 โค ๐ฅ โค 1, and ๐(๐ฅ) = 0 if ๐ฅ โฅ 2 or ๐ฅ โค โ1 (as in Figure 9.2). Then, for ๐ฅ > 2,
Figure 9.2. A function ๐ โ ๐ถ๐โ (โ) such that ๐ โฅ 0, ๐(๐ฅ) = 1 if 0 โค ๐ฅ โค 1, and ๐(๐ฅ) = 0 if ๐ฅ โฅ 2 or ๐ฅ โค โ1. 2
๐ป๐(๐ฅ) =
1
๐(๐ฆ) 1 1 1 1 โซ ๐๐ฆ โฅ โซ ๐๐ฆ โฅ , ๐ โ1 ๐ฅ โ ๐ฆ ๐ 0 ๐ฅโ๐ฆ ๐๐ฅ
so ๐ป๐ is not an integrable function. We see then that the Hilbert transform cannot be bounded in ๐ฟ1 (โ). However we have Theorem 9.12. Theorem 9.12. There exists a constant ๐ด > 0 such that, for any ๐ โ ๐ถ๐โ (โ) and ๐ผ > 0, ๐ด |{๐ฅ โ โ โถ |๐ป๐(๐ฅ)| > ๐ผ}| โค โ๐โ๐ฟ1 . ๐ผ As in the case of the maximal function, the Hilbert transform is of weak type (1, 1) (although, so far, it is only defined on ๐ถ๐โ (โ)).
9.3. The Hilbert transform of integrable functions
175
Proof. Fix ๐ผ > 0. We will write the function ๐ as ๐ + ๐, the sum of a โgoodโ and a โbadโ part. For this, we construct a collection โ of dyadic intervals (intervals of the form [๐ โ
2๐ , (๐ + 1) โ
2๐ ], with ๐, ๐ โ โค) in the following way. Let ๐ be large enough so that 1 โซ|๐| โค ๐ผ |๐ผ| ๐ผ for every dyadic interval ๐ผ of length |๐ผ| = 2๐ . Such ๐ exists because ๐ is integrable. Now, subdivide each ๐ผ in two subintervals ๐ผ โฒ of half the length of ๐ผ. For each one, we have either 1 โซ |๐| โค ๐ผ |๐ผ โฒ | ๐ผ โฒ
1 โซ |๐| > ๐ผ. |๐ผ โฒ | ๐ผ โฒ
or
In the second case, we add ๐ผ โฒ to the collection โ. Note that we have ๐ผ
๐ผ/2}| โค โซ โ
|๐ป๐|2 4 20 โค 2 โ
5๐ผโ๐โ๐ฟ1 = โ๐โ๐ฟ1 . ๐ผ ๐ผ (๐ผ/2)2
Define ๐ = ๐ โ ๐. Since |{๐ฅ โ โ โถ |๐ป๐(๐ฅ)| > ๐ผ}| โค |{๐ฅ โ โ โถ |๐ป๐(๐ฅ)| > ๐ผ/2}| + |{๐ฅ โ โ โถ |๐ป๐(๐ฅ)| > ๐ผ/2}|, it remains to estimate |{๐ฅ โ โ โถ |๐ป๐(๐ฅ)| > ๐ผ/2}|. ๐ is what we refer as the bad part of ๐, because we have no control over the size of |๐(๐ฅ)|. However, ๐ = 0 outside of ฮฉ and, for each ๐ผ โ โ, โซ๐ = โซ๐ โ โซ๐ = โซ๐ โ โซ( ๐ผ
๐ผ
๐ผ
๐ผ
๐ผ
1 โซ๐) = 0. |๐ผ| ๐ผ
Write ๐ = โ ๐๐ผ , ๐ผโโ
where each ๐๐ผ = ๐ โ
๐๐ผ , for ๐ผ โ โ. Then ๐ป๐ = โ ๐ป๐๐ผ . ๐ผโโ
Note that each ๐ป๐๐ผ is defined almost everywhere, since ๐๐ผ is either 0, outside of ๐ผ, or 1 โซ๐ ๐๐ผ (๐ฅ) = ๐(๐ฅ) โ |๐ผ| ๐ผ in the interior of ๐ผ, so it is differentiable, as we are assuming that ๐ is differentiable.
9.3. The Hilbert transform of integrable functions
177
Write 3๐ผ for the interval with the same center as ๐ผ, but 3 times its length (see Figure 9.3).
Figure 9.3. The interval 3๐ผ has the same center as ๐ผ and 3 times its length.
If ๐ฅ โ 3๐ผ, and ๐ฆ0 is the center of ๐ผ,
๐ป๐๐ผ (๐ฅ) =
๐ (๐ฆ) ๐ (๐ฆ) 1 1 โซ ๐ผ ๐๐ฆ โ โซ ๐ผ ๐๐ฆ ๐ ๐ผ ๐ฅโ๐ฆ ๐ ๐ผ ๐ฅ โ ๐ฆ0
=
1 1 1 โซ๐ (๐ฆ)( โ )๐๐ฆ ๐ ๐ผ ๐ผ ๐ฅ โ ๐ฆ ๐ฅ โ ๐ฆ0
=
๐ฆ โ ๐ฆ0 1 โซ๐ (๐ฆ) ๐๐ฆ, ๐ ๐ผ ๐ผ (๐ฅ โ ๐ฆ)(๐ฅ โ ๐ฆ0 )
where, in the first equality, we have used the fact that โซ๐ผ ๐๐ผ = 0. As ๐ฅ โ 3๐ผ and ๐ฆ โ ๐ผ in the integral above, we see that
|๐ฆ โ ๐ฆ0 | โค
1 1 |๐ผ| โค |๐ฅ โ ๐ฆ|, 2 2
so |๐ฅ โ ๐ฆ0 | โฅ |๐ฅ โ ๐ฆ| โ |๐ฆ โ ๐ฆ0 | โฅ
1 |๐ฅ โ ๐ฆ|. 2
Thus
|๐ป๐๐ผ (๐ฅ)| โค
|๐ฆ โ ๐ฆ0 | |๐ (๐ฆ)| 2 1 โซ|๐ (๐ฆ)| ๐๐ฆ โค |๐ผ| โซ ๐ผ 2 ๐๐ฆ, ๐ ๐ผ ๐ผ ๐ |๐ฅ โ ๐ฆ|2 |๐ฅ โ ๐ฆ| ๐ผ
178
9. Hilbert transform
and therefore โซ
|๐ป๐๐ผ (๐ฅ)|๐๐ฅ โค
โโงต3๐ผ
|๐ (๐ฆ)| 1 |๐ผ| โซ โซ ๐ผ 2 ๐๐ฆ๐๐ฅ ๐ |๐ฅ โ ๐ฆ| โโงต3๐ผ ๐ผ
โค
1 1 |๐ผ| โซ|๐๐ผ (๐ฆ)| โซ ๐๐ฅ๐๐ฆ ๐ |๐ฅ โ ๐ฆ|2 ๐ผ |๐ฅโ๐ฆ|โฅ|๐ผ|
โค
2 โซ|๐ (๐ฆ)|๐๐ฆ ๐ ๐ผ ๐ผ
โค
1 2 | โซ ๐(๐ฆ) โ โซ๐|๐๐ฆ ๐ ๐ผ| |๐ผ| ๐ผ |
โค
2 1 โซ( โซ|๐|)) ( โซ|๐| + ๐ ๐ผ |๐ผ| ๐ผ ๐ผ
=
4 8 โซ|๐| โค ๐ผ|๐ผ|, ๐ ๐ผ ๐
because ๐ผ โ โ. Hence, if we define ฮฉโ =
โ
3๐ผ,
๐ผโโ
we obtain โซ
|๐ป๐(๐ฅ)|๐๐ฅ โค โ โซ
โโงตโฆโ
|๐ป๐๐ผ (๐ฅ)|๐๐ฅ โค
๐ผโโ โโงต3๐ผ
=
8๐ผ โ |๐ผ| ๐ ๐ผโโ
8๐ผ 8 |ฮฉ| โค โ๐โ๐ฟ1 . ๐ ๐
Thus, by Chebyshevโs inequality, |{๐ฅ โ โ โงต ฮฉโ โถ |๐ป๐(๐ฅ)| > ๐ผ/2}| โค
16/๐ 2 โซ |๐ป๐(๐ฅ)|๐๐ฅ โค โ๐โ๐ฟ1 . ๐ผ โโงตโฆโ ๐ผ
It just remains to estimate |{๐ฅ โ ฮฉโ โถ |๐ป๐(๐ฅ)| > ๐ผ/2}| โค |ฮฉโ | โค โ |3๐ผ| = 3|ฮฉ| โค ๐ผโโ
3 โ๐โ๐ฟ1 . ๐ผ โก
Note that, from the proof above, we can take the constant ๐ด in Theorem 9.12 as 16 ๐ด = 20 + + 3 < 29. ๐
9.4. Convergence in measure
179
9.4. Convergence in measure Theorem 9.12 doesnโt guarantee that the limit lim ๐กโ0
๐(๐ฅ โ ๐ฆ) 1 โซ ๐๐ฆ ๐ |๐ฆ|โฅ๐ก ๐ฆ
converges for a function in ๐ฟ1 . However, the Hilbert transform can be defined โin measureโ. We say that the sequence ๐๐ converges in measure to ๐ if, for every ๐ > 0, |{๐ฅ โ โ๐ โถ |๐๐ (๐ฅ) โ ๐(๐ฅ)| โฅ ๐}| โ 0 as ๐ โ โ. The function ๐ is unique almost everywhere (Exercise (6)). If a sequence ๐๐ โ ๐ in ๐ฟ1 , then it converges in measure to ๐, but almost everywhere convergence does not imply convergence in measure (Exercise (7)). A sequence ๐๐ of measurable functions on โ๐ is Cauchy in measure if, for every ๐ > 0, |{๐ฅ โ โ๐ โถ |๐๐ (๐ฅ) โ ๐๐ (๐ฅ)| โฅ ๐}| โ 0 as ๐, ๐ โ โ. If ๐๐ converges in measure to ๐, then ๐๐ is Cauchy in measure because {๐ฅ โ โ๐ โถ |๐๐ (๐ฅ) โ ๐๐ (๐ฅ)| โฅ ๐} โ {๐ฅ โ โ๐ โถ |๐๐ (๐ฅ) โ ๐(๐ฅ)| โฅ ๐/2} โช {๐ฅ โ โ๐ โถ |๐๐ (๐ฅ) โ ๐(๐ฅ)| โฅ ๐/2}, by the triangle inequality. If ๐๐ is Cauchy in measure, there exists a sequence ๐๐ such that |{๐ฅ โ โ๐ โถ |๐๐๐ (๐ฅ) โ ๐๐๐+1 (๐ฅ)| โฅ
1 1 }| < ๐ . ๐ 2 2
If we define ๐ด๐ =
โ
{๐ฅ โ โ๐ โถ |๐๐๐ (๐ฅ) โ ๐๐๐+1 (๐ฅ)| โฅ
๐โฅ๐
then โ
1 1 = ๐โ1 . ๐ 2 2 ๐=๐
|๐ด๐ | โค โ
1 }, 2๐
180
9. Hilbert transform
In particular, |๐ด๐ | โ 0 and | โ๐ ๐ด๐ | = 0. If ๐ฅ โ ๐ด๐ and ๐ โฅ ๐ โฅ ๐, ๐
๐
(9.13)
|๐๐๐ (๐ฅ) โ ๐๐๐ (๐ฅ)| โค โ |๐๐๐ (๐ฅ) โ ๐๐๐+1 (๐ฅ)| < โ ๐=๐
๐=๐
1 1 < ๐โ1 . ๐ 2 2
Thus the sequence ๐๐๐ (๐ฅ) is Cauchy for each ๐ฅ โ ๐ด๐ , and thus converges. We can then define ๐(๐ฅ) = lim ๐๐๐ (๐ฅ) for each ๐ฅโ
(โ๐ โงต ๐ด๐ ) = โ๐ โงต ๐ด . โ โ ๐ ๐
๐
Therefore the subsequence ๐๐๐ โ ๐ almost everywhere. By (9.13), |๐๐๐ (๐ฅ) โ ๐(๐ฅ)|
0, |{๐ฅ โ โ โถ |๐ป(๐๐ โ ๐๐ )(๐ฅ)| > ๐}| โค
๐ด โ๐ โ ๐๐ โ๐ฟ1 โ 0, ๐ ๐
so the sequence ๐ป๐๐ is Cauchy in measure. Therefore, there exists a function, which we may call ๐ป๐, such that ๐ป๐๐ โ ๐ป๐ in measure. As we have seen before, ๐ป๐ is well defined and there exists a subsequence ๐๐๐ such that ๐ป๐๐๐ โ ๐ป๐ almost everywhere (Exercise (8)). Theorem 9.12 implies that ๐ป๐ thus defined is finite almost everywhere.
Exercises
181
Exercises (1) (a) The function ๐(๐ฅ, ๐ก) = ๐๐ก (๐ฅ) =
1 ๐ฅ ๐ ๐ฅ2 + ๐ก 2
is harmonic in โ2+ . (b) If ๐ โ ๐ฟ1 (โ), the function ๐ฃ(๐ฅ, ๐ก) = โซ ๐๐ก (๐ฅ โ ๐ฆ)๐(๐ฆ)๐๐ฆ โ
is harmonic in โ2+ . (c) If ๐ โ ๐ฟ1 (โ), the functions ๐ฃ(๐ฅ, ๐ก) and ๐ข(๐ฅ, ๐ก), the Poisson integral of ๐, are conjugate harmonic. (2) Let ๐พ be the lower semicircle of radius ๐ around the origin, and ๐ > 0. Then โซ ๐(๐ง)๐๐ง = 2๐๐ Res๐ง=โ๐๐ก ๐(๐ง) = โ๐๐โ2๐๐ก๐ , ๐พ
where ๐(๐ง) is the function defined in the proof of Lemma 9.4. (3) For ๐ โ ๐ถ๐ (โ), consider the Cauchy integral ๐น(๐ง) =
๐(๐ก) 1 โซ ๐๐ก, ๐๐ โ ๐ก โ ๐ง
for ๐ง = ๐ฅ + ๐๐ฆ โ โ2+ . (a) There exists a constant ๐ด > 0 such that |๐น(๐ง)| โค ๐ด/|๐ง|. (b) โซ ๐น(๐ฅ)2 ๐๐ฅ = 0. โ
(c) Calculate โ(๐น(๐ฅ)2 ) and conclude โ๐ป๐โ๐ฟ2 = โ๐โ๐ฟ2 . (4) Let ๐ฆ 1 โ |๐ฆ| โฅ 1 2+1 ๐ฆ ๐ฆ ฮฆ(๐ฆ) = { ๐ฆ |๐ฆ| < 1. ๐ฆ2 + 1 Then its collection {ฮฆ๐ก }๐ก>0 of dilations satisfies (a) โซ ฮฆ๐ก = 0 for all ๐ก > 0; โ
(b) |ฮฆ๐ก (๐ฆ)| โค
1 for all ๐ก > 0 and ๐ฆ โ โ; and ๐ก
182
9. Hilbert transform
๐ก for all ๐ก > 0 and ๐ฆ โ โ, ๐ฆ โ 0. ๐ฆ2 (5) If ๐ โ ๐ฟ1 (โ) is differentiable at ๐ฅ โ โ, then the limit (c) |ฮฆ๐ก (๐ฆ)| โค
lim โซ ๐กโ0
|๐ฆ|โฅ๐ก
๐(๐ฅ โ ๐ฆ) ๐๐ฆ ๐ฆ
exists. (Hint: Use the identity, for any ๐ฟ๐ > 0, โซ ๐กโค|๐ฆ| 0 such that โซ |๐๐|๐ โค ๐ด โซ |๐|๐ โ
โ
Notes
183 for some ๐ > 1.
Notes The study of the conjugate function started in the work of several authors in the early 20th century, using complex variable methods (as in Exercise (3)). See the references in [Zyg02] for a list of such authors. The decomposition method in the proof of Theorem 9.12 is due to Alberto P. Calderรณn and Antoni Zygmund [CZ52], where they generalize to other kernels (see Exercises (9), (10) and (11)) and other dimensions. See [Ste70] for a further discussion of these methods, and a more extensive study of singular integrals.
Chapter 10
Mathematics of fractals
10.1. Hausdorff dimension The purpose of this chapter is to introduce the basic ideas in the study of fractals. We start with the Hausdorff dimension, which provides a means to quantify the complexity of a fractal set. For a set ๐ด โ โ๐ , we denote its diameter by diam ๐ด, diam ๐ด = sup{|๐ฅ โ ๐ฆ| โถ ๐ฅ, ๐ฆ โ ๐ด}, the supremum over all distances between points in ๐ด. We observe, for example, that diam ๐ต๐ (๐ฅ) = 2๐, and that, if ๐ฅ โ ๐ด and ๐ = diam ๐ด, ๐ด โ ๐ต๐ฬ (๐ฅ). For ๐ฟ > 0, a ๐ฟ-cover for ๐ด โ โ๐ is a collection ๐1 , ๐2 , . . . of subsets of โ๐ such that diam ๐ ๐ โค ๐ฟ,
๐ = 1, 2, . . . ,
and
๐ดโ
โ
๐๐.
๐
The collection {๐ ๐ } may be finite or infinite (as long as it is countable), and the sets ๐ ๐ are arbitrary, as long as they cover ๐ด and have diameter not larger than ๐ฟ. 185
186
10. Mathematics of fractals For ๐ โฅ 0, consider the number, for each ๐ด โ โ๐ ,
(10.1)
๐ป๐ฟ๐ (๐ด) = inf { โ(diam ๐ ๐ )๐ โถ {๐ ๐ } is a ๐ฟ-cover for ๐ด}. ๐
In other words, we take all ๐ฟ-covers {๐ ๐ } for ๐ด, calculate for each one โ๐ (diam ๐ ๐ )๐ , which is clearly nonnegative, and then take the infimum of all such sums. It is possible that all of them are divergent, and in that case we have ๐ป๐ฟ๐ (๐ด) = โ. It is also possible that ๐ป๐ฟ๐ (๐ด) = 0. For example, if ๐ด = โ
, or if ๐ด is finite. Indeed, if ๐ด = {๐ฅ1 , ๐ฅ2 , . . . , ๐ฅ๐ } โ โ๐ , consider the ๐ฟ-cover for ๐ด given by the balls ๐ต๐ (๐ฅ1 ), ๐ต๐ (๐ฅ2 ), . . . , ๐ต๐ (๐ฅ๐ ), with ๐ > 0. Then, for ๐ > 0, ๐
โ (diam ๐ต๐ (๐ฅ๐ ))๐ = (2๐)๐ ๐, ๐=1
and thus ๐ป๐ฟ๐ (๐ด) โค (2๐)๐ ๐. As ๐ > 0 is arbitrary, we have ๐ป๐ฟ๐ (๐ด) = 0. It is not hard to see that, if ๐ด โ ๐ต, then ๐ป๐ฟ๐ (๐ด) โค ๐ป๐ฟ๐ (๐ต), and, if ๐ด = โ๐ ๐ด๐ , then ๐ป๐ฟ๐ (๐ด) โค โ ๐ป๐ฟ๐ (๐ด๐ ). ๐
See Exercise (1). These are the same properties which are also satisfied by the outer measure | โ
|โ . Now, observe that, if ๐ฟ < ๐, then every ๐ฟ-cover for a set ๐ด is also an ๐-cover for ๐ด. Hence, by the definition (10.1), we have ๐ป๐ฟ๐ (๐ด) โฅ ๐ป๐๐ (๐ด). We see that ๐ป๐ฟ๐ (๐ด) increases as ๐ฟ decreases, so the limit exists as ๐ฟ โ 0 (it might be infinite). We thus define (10.2)
โ ๐ (๐ด) = lim ๐ป๐ฟ๐ (๐ด) = sup{๐ป๐ฟ๐ (๐ด) โถ ๐ฟ > 0}. ๐ฟโ0
โ (๐ด) is called the Hausdorff measure with exponent ๐ of ๐ด. As ๐ป๐ฟ๐ , the Hausdorff measure also satisfies the properties of the outer measure | โ
|โ (Exercise (2)). ๐
If {๐ ๐ } is a ๐ฟ-cover for ๐ด โ โ๐ , and ๐ > ๐ก โฅ 0, โ(diam ๐ ๐ )๐ = โ(diam ๐ ๐ )๐ โ๐ก (diam ๐ ๐ )๐ก โค ๐ฟ๐ โ๐ก โ(diam ๐ ๐ )๐ก , ๐
๐
๐
10.1. Hausdorff dimension
187
and, by the definition (10.1), (10.3)
๐ป๐ฟ๐ (๐ด) โค ๐ฟ๐ โ๐ก ๐ป๐ฟ๐ก (๐ด).
We thus have Theorem 10.4. Theorem 10.4. Let ๐ด โ โ๐ and ๐ > ๐ก โฅ 0. (1) If โ ๐ก (๐ด) < โ, then โ ๐ (๐ด) = 0. (2) If โ ๐ (๐ด) > 0, then โ ๐ก (๐ด) = โ. Proof. To prove (1), assume that โ ๐ก (๐ด) < โ. Thus ๐ป๐ฟ๐ก (๐ด) is bounded in ๐ฟ and, by (10.3), we obtain โ ๐ (๐ด) = lim ๐ป๐ฟ๐ (๐ด) = 0. ๐ฟโ0
(2) is the contrapositive of (1).
โก
Theorem 10.4 allows us to conclude that there exists some number ๐ท such that โ ๐ (๐ด) = โ if ๐ < ๐ท, and โ ๐ (๐ด) = 0 if ๐ > ๐ท (see Figure H s (A) โ
D Figure 10.1. The number ๐ท, the Hausdorff dimension of ๐ด, satisfies that โ ๐ (๐ด) = โ if ๐ < ๐ท, and โ ๐ (๐ด) = 0 if ๐ > ๐ท.
10.1). Indeed, we see that ๐ท = inf{๐ โถ โ ๐ (๐ด) = 0}.
s
188
10. Mathematics of fractals
The value of โ ๐ท (๐ด) may be zero or infinity, or even a number 0 < โ ๐ท (๐ด) < โ. ๐ท is called the Hausdorff dimension of ๐ด, and is denoted by dim ๐ด. Example 10.5. For ๐ = 0, โ 0 (๐ด) = #๐ด, the number of elements of ๐ด, if ๐ด is finite, and thus dim ๐ด = 0. If ๐ด is infinite, โ 0 (๐ด) = โ. However, if ๐ด is countably infinite, its Hausdorff dimension is also zero (Exercise (3)). 10.6. We observe that, if ๐ด โ โ, โ 1 (๐ด) is the outer Lebesgue measure of ๐ด, |๐ด|โ . Thus, for any interval [๐, ๐], ๐ > ๐, โ 1 ([๐, ๐]) = ๐ โ ๐, and thus intervals have Hausdorff dimension 1. 10.7. If ๐ > ๐ and ๐ด โ โ๐ , โ ๐ (๐ด) = 0. Indeed, consider first a unit cube ๐ in โ๐ and, given ๐ฟ > 0, subdivide it in 2๐๐ subcubes ๐๐ of sides 2โ๐ < ๐ฟ/๐. Each subcube has diameter diam(๐๐ ) = 2โ๐ โ๐ < ๐ฟ, and hence {๐๐ } is a ๐ฟ-cover for ๐. Thus โ(diam ๐๐ )๐ = (2โ๐ โ๐)๐ โ
2๐๐ = ๐ ๐ /2 2(๐โ๐ )๐ . ๐
As ๐ is an arbitrary positive integer (as long as 2โ๐ < ๐ฟ/๐) and ๐ > ๐, then ๐ป๐ฟ๐ (๐) = 0. Hence โ ๐ (๐) = 0, and therefore โ ๐ (๐ด) = 0 for any ๐ด โ โ๐ , because any ๐ด is contained in the union of countably many unit cubes. Therefore dim ๐ด โค ๐ for any ๐ด โ โ๐ . Example 10.8. Consider the Cantor ternary set, constructed by the removal of middle intervals starting from ๐ถ0 = [0, 1]. Thus ๐ถ1 = [0, 1/3] โช [2/3, 1], ๐ถ2 = [0, 1/9] โช [2/9, 1/3] โช [2/3, 7/9] โช [8/9, 1],
...
as seen in Figure 10.2. The Cantor set is
โ
๐ถ=
โ
๐ถ๐ .
๐=0
The Cantor set is compact, perfect (does not have any isolated points), uncountable and of measure 0. Thus โ 1 (๐ถ) = 0. We show that its
10.1. Hausdorff dimension
189
Figure 10.2. The construction of the Cantor ternary set.
Hausdorff dimension is ๐ท=
log 2 โ 0.63. log 3
To see that dim ๐ถ โค ๐ท, we prove โ ๐ท (๐ถ) < โ. Given ๐ฟ > 0, let ๐ so that 3โ๐ < ๐ฟ. Thus, ๐ถ๐ is the union of 2๐ intervals ๐ผ๐ of length 3โ๐ < ๐ฟ, so they form a ๐ฟ-cover for ๐ถ. Hence 2๐
๐ป๐ฟ๐ท (๐ถ)
โค โ (diam ๐ผ๐ )๐ท = (3โ๐ )๐ท 2๐ = ( ๐=1
2 ๐ ) = 1, 3๐ท
as ๐ท is the number such that 3๐ท = 2. Since ๐ฟ > 0 is arbitrary, we have that โ ๐ท (๐ถ) โค 1, and therefore dim ๐ถ โค ๐ท. To show that dim ๐ถ โฅ ๐ท (and thus conclude they are equal), we verify that โ ๐ท (๐ถ) > 0. In fact, we prove that โ ๐ท (๐ถ) โฅ 1, and we will do it by contradiction. Assume โ ๐ท (๐ถ) < 1, so there exist intervals ๐ผ๐ so that ๐ถโ
โ
๐ผ๐
and
โ(diam ๐ผ๐ )๐ท < 1. ๐
๐
By widening the intervals a little bit so that their sum is still smaller than 1, we can assume all ๐ผ๐ are open, and thus, by the compactness of ๐ถ, we can choose a finite number of them that still cover ๐ถ. Hence we have ๐
(10.9)
๐ถโ
โ ๐=1
๐
๐ผ๐
and
โ (diam ๐ผ๐ )๐ท < 1. ๐=1
Now, again by compactness of ๐ถ, we can find ๐ large enough so that each interval of ๐ถ๐ is completely contained in one of the ๐ผ๐ . We can now shorten each ๐ผ๐ to a closed interval in such a way that its extreme points coincide with extreme points of the intervals of ๐ถ๐ , and they still cover
190
10. Mathematics of fractals
๐ถ. That is, if ๐ผ๐ contains the intervals ๐ท1 , ๐ท2 , . . . , ๐ท๐ in ๐ถ๐ , we replace ๐ผ๐ with the smallest closed interval that contains ๐ท1 โช ๐ท2 โช . . . โช ๐ท๐ . Fix ๐ผ๐ . Let ๐ธ0 be the gap of largest length between the ๐ท๐ . By the construction of ๐ถ๐ , such gap must exist. We write ๐ผ๐ โงต๐ธ0 as the union ๐ธ1 โช ๐ธ2 of closed intervals, as in Figure 10.3. We see that, by the construction, diam ๐ธ1 , diam ๐ธ2 โค diam ๐ธ0 .
Figure 10.3. The largest gap inside ๐ผ๐ , for the special case when ๐ผ๐ contains the intervals ๐ท1 , ๐ท2 , ๐ท3 . By the construction of ๐ถ ๐ , such largest gap must exist and is unique.
Thus diam ๐ธ0 โฅ
diam ๐ธ1 + diam ๐ธ2 2
and diam ๐ผ๐ = diam ๐ธ0 + diam ๐ธ1 + diam ๐ธ2 3 3 1 1 โฅ diam ๐ธ1 + diam ๐ธ2 = 3( diam ๐ธ1 + diam ๐ธ2 ). 2 2 2 2 Now, since ๐ท < 1, the function ๐ฅ โฆ ๐ฅ๐ท is concave (Exercise (6)), and thus ๐ท 1 1 (diam ๐ผ๐ )๐ท โฅ 3๐ท ( diam ๐ธ1 + diam ๐ธ2 ) 2 2 1 1 โฅ 3๐ท ( (diam ๐ธ1 )๐ท + (diam ๐ธ2 )๐ท ) 2 2 = (diam ๐ธ1 )๐ท + (diam ๐ธ2 )๐ท ,
where we have again used the fact that 3๐ท = 2. Therefore, we can replace ๐ผ๐ with the intervals ๐ธ1 and ๐ธ2 , and the new set of intervals still satisfies (10.9). We can continue replacing the intervals ๐ผ๐ until all gaps are removed, so we arrive to the cover formed precisely by the 2๐ intervals of ๐ถ๐ ,
10.2. Self-similar sets
191
which of course satisfies 2๐
โ (diam ๐ผ๐ )๐ท = 2๐ (3โ๐ )๐ท = 1, ๐=1
a contradiction with (10.9).
10.2. Self-similar sets Cantorโs example shows us the difficulty of calculating the Hausdorff dimension of a given set ๐ด โ โ๐ . However, under certain conditions, it is possible to calculate the Hausdorff dimension a self-similar set. A function ๐น โถ โ๐ โ โ๐ is called a contraction if there exists a constant ๐ผ < 1 such that |๐น(๐ฅ) โ ๐น(๐ฆ)| โค ๐ผ|๐ฅ โ ๐ฆ| for all ๐ฅ, ๐ฆ โ โ๐ . The number ๐ผ is called the contraction constant of ๐น. A nonempty compact set ๐พ โ โ๐ is self-similar if there exist contractions ๐1 , ๐2 , . . . , ๐๐ โถ โ๐ โ โ๐ such that (10.10)
๐พ = ๐1 (๐พ) โช ๐2 (๐พ) โช . . . โช ๐๐ (๐พ).
One can show that ๐พ is determined by the contractions ๐1 , ๐2 , . . . , ๐๐ , that is, there exists only one nonempty compact set ๐พ that satisfies (10.10) [YHK97]. Let ๐ผ1 , ๐ผ2 , . . . , ๐ผ๐ be the contraction constants of the functions ๐1 , ๐2 , . . . , ๐๐ in (10.10) and let ๐ท be the unique positive number such that (10.11)
๐ท ๐ท ๐ผ๐ท 1 + ๐ผ2 + . . . + ๐ผ๐ = 1.
Such number exists because the function ๐ฅ โฆ ๐ผ๐ฅ1 + ๐ผ๐ฅ2 + . . . + ๐ผ๐ฅ๐ is strictly decreasing on [0, โ), equal to ๐ at ๐ฅ = 0 and goes to 0 as ๐ฅ โ โ. We have the following result. Proposition 10.12. If ๐พ is the self-similar set determined by (10.10) and ๐ท satisfies (10.11), then dim ๐พ โค ๐ท. Proof. We prove that โ ๐ท (๐พ) < โ. For this, we will use the following notation. For a finite sequence ๐ค = ๐ค 1 ๐ค 2 โฏ ๐ค ๐ of length ๐ (we will call such sequences words), where each ๐ค๐ โ {1, 2, . . . , ๐}, let ๐๐ค denote
192
10. Mathematics of fractals
the composition ๐๐ค = ๐๐ค1 โ ๐๐ค2 โ โฏ โ ๐๐ค๐ , and ๐พ๐ค the set ๐๐ค (๐พ). Each ๐พ๐ค is called a cell of level ๐, and, by (10.10), ๐พ=
โ
๐พ๐ค ,
๐คโ๐๐
where ๐๐ is the set of words of length ๐. Also, since each ๐๐ is a contraction with constant ๐ผ๐ , we have diam ๐พ๐ค โค ๐ผ๐ค1 ๐ผ๐ค2 โฏ ๐ผ๐ค๐ diam ๐พ, and thus diam ๐พ๐ค โค ๐ด๐ diam ๐พ, where ๐ด = max{๐ผ1 , ๐ผ2 , . . . , ๐ผ๐ }. Since ๐ด < 1, given ๐ฟ > 0 we can choose ๐ large enough so that the collection of cells of level ๐, {๐พ๐ค }๐คโ๐๐ , is a ๐ฟ-cover for ๐พ. Also, ๐ท ๐ท ๐ท โ (diam๐พ๐ค )๐ท โค โ ๐ผ๐ท ๐ค1 ๐ผ๐ค2 โฏ ๐ผ๐ค๐ (diam ๐พ) ๐คโ๐๐
๐คโ๐๐ ๐
๐
๐
๐ท ๐ท ๐ท = ( โ ๐ผ๐ท ๐ค1 )( โ ๐ผ๐ค2 ) โฏ ( โ ๐ผ๐ค๐ )(diam ๐พ) ๐ค1 =1
๐ค2 =1
๐ค๐ =1
๐ท
= (diam ๐พ) , by (10.11). Hence ๐ป๐ฟ๐ท (๐พ) โค (diam ๐พ)๐ท and, as ๐ฟ > 0 is arbitrary, we have that โ ๐ท (๐พ) โค (diam ๐พ)๐ท < โ. Therefore dim ๐พ โค ๐ท. โก Consider the Cantor set ๐ถ discussed in Example 10.8. ๐ถ is selfsimilar with respect to the contractions ๐1 , ๐2 โถ โ โ โ given by ๐1 (๐ฅ) =
1 ๐ฅ 3
and
Thus, in this case, ๐ผ1 = ๐ผ2 =
๐2 (๐ฅ) =
1 2 ๐ฅ+ . 3 3
1 and the number ๐ท that satisfies 3
1 ๐ท 1 ๐ท ( ) +( ) =1 3 3 is precisely ๐ท=
log 2 , log 3
which we proved to be the Hausdorff dimension of ๐ถ. However, (10.11) does not guarantee that ๐ท is the Hausdorff dimension of ๐พ for any self-similar set, as Example 10.13 shows.
10.2. Self-similar sets
193
Example 10.13. Consider the functions ๐1 , ๐2 โถ โ โ โ given by 2 2 1 ๐ฅ and ๐2 (๐ฅ) = ๐ฅ + . 3 3 3 ๐1 and ๐2 are contractions and, if ๐ผ = [0, 1], then ๐1 (๐ฅ) =
๐ผ = ๐1 (๐ผ) โช ๐2 (๐ผ), so ๐ผ is self-similar with respect to ๐1 and ๐2 . Both functions have contraction constant 2/3, and the number ๐ท that satisfies 2 ๐ท 2 ๐ท ( ) +( ) =1 3 3 is
log 2 . log 3/2 Note that ๐ท > 1, so it is not equal to dim ๐ผ. ๐ท=
The problem with Example 10.13 is the overlap of the images ๐1 (๐ผ) = [0, 2/3]
and
๐2 (๐ผ) = [1/3, 1]
of ๐ผ under the contractions ๐1 and ๐2 , so we need conditions to avoid this problem. We say that a function ๐ โถ โ๐ โ โ๐ is a similitude if there exists a constant ๐ผ such that |๐(๐ฅ) โ ๐(๐ฆ)| = ๐ผ|๐ฅ โ ๐ฆ|. It can be verified that, if ๐ is a similitude, then itโs a composition of a dilation, a rotation and a translation (Exercise (7)). Thus, for any ball ๐ต๐ (๐ฅ) โ โ๐ , (10.14)
๐(๐ต๐ (๐ฅ)) = ๐ต๐ผ๐ (๐(๐ฅ)).
Let ๐1 , ๐2 , . . . , ๐๐ be contractive similitudes. They satisfy the open set condition if there exists a bounded open set ๐ โ โ๐ such that (1) ๐๐ (๐) โ ๐ for each ๐ = 1, 2, . . . , ๐; (2) ๐๐ (๐) โฉ ๐๐ (๐) = โ
, for ๐, ๐ = 1, 2, . . . , ๐ such that ๐ โ ๐. One can verify that, since ๐1 (๐) โช ๐2 (๐) โช . . . โช ๐๐ (๐) โ ๐, then ๐พ โ ๐,ฬ where ๐ฬ is the closure of ๐ (Exercise (8)).
194
10. Mathematics of fractals
Condition (2) guarantees that there are no interior overlaps between the images ๐๐ (๐พ) of the contractions. Theorem 10.15. Let ๐๐ โถ โ๐ โ โ๐ , ๐ = 1, 2, . . . , ๐, be contractive similitudes with constants ๐ผ๐ < 1, respectively, that satisfy the open set condition. If ๐พ is the self-similar set with respect to the ๐๐ , then dim ๐พ = ๐ท, where ๐ท is the unique solution to (10.11). Proof. We already know that dim ๐พ โค ๐ท, by Proposition 10.12. To prove that dim ๐พ โฅ ๐ท, we verify that โ ๐ท (๐พ) > 0. Let ๐ โ โ๐ be a bounded open set as in the open set condition, and let ๐ผ, ๐ฝ > 0 and ๐ฅ0 , ๐ฆ0 โ โ๐ such that (10.16)
๐ต๐ผฬ (๐ฅ0 ) โ ๐ โ ๐ต๐ฝ (๐ฆ0 ).
Since the ๐๐ are similitudes, for each word ๐ค โ ๐๐ we have, by (10.14), (10.17)
๐ต๐ผฬ ๐ค ๐ผ (๐๐ค (๐ฅ0 )) โ ๐๐ค โ ๐ต๐ผ๐ค ๐ฝ (๐๐ค (๐ฆ0 )),
where ๐ผ๐ค = ๐ผ๐ค1 ๐ผ๐ค2 โฏ ๐ผ๐ค๐ and ๐๐ค = ๐๐ค (๐). Let ๐พ = min{๐ผ1 , ๐ผ2 , . . . , ๐ผ๐ } and ๐=(
๐ผ๐พ ๐ ) . 2๐ฝ + 1
We will prove that โ ๐ท (๐พ) โฅ ๐. In order to arrive to a contradiction, assume โ ๐ท (๐พ) < ๐. Let ๐ฟ > 0 be small enough (๐ฟ < ๐ is sufficient) and {๐ ๐ } a ๐ฟ-cover for ๐พ such that (10.18)
โ diam(๐ ๐ )๐ท < ๐. ๐
By widening the ๐ ๐ up we can assume they are open, and so we can also assume they are finite because ๐พ is compact. For each ๐, consider the set of words ฮ๐ = {๐ค โถ ๐ผ๐ค โค diam ๐ ๐ < ๐ผ๐คโฒ } where, if ๐ค = ๐ค 1 ๐ค 2 . . . ๐ค ๐ , then ๐คโฒ = ๐ค 1 ๐ค 2 . . . ๐ค ๐โ1 . That is, ฮ๐ is the set of words ๐ค of sequences ๐ค 1 ๐ค 2 . . . ๐ค ๐ which make the product ๐ผ๐ค1 ๐ผ๐ค2 โฏ ๐ผ๐ค๐ precisely smaller or equal to diam ๐ ๐ , and larger if we miss the last factor ๐ผ๐ค๐ . The words in ฮ๐ satisfy the following two properties.
10.2. Self-similar sets
195
โข If ๐ข, ๐ค โ ฮ๐ are different, then the letters of ๐ข cannot be the first letters of ๐ค, nor the other way around. This guarantees that, by the open set condition, ๐แต โฉ ๐๐ค = โ
if ๐ข, ๐ค โ ฮ๐ , ๐ข โ ๐ค.
โข If ๐ค โ ฮ๐ , ๐พ diam ๐ ๐ < ๐ผ๐ค โค diam ๐ ๐ , so all the products ๐ผ๐ค , for ๐ค โ ฮ๐ , are essentially of the same size. Let ๐ฅ โ ๐ ๐ . Then ๐ ๐ โ ๐ตdiam ๐๐ (๐ฅ). Using (10.17), we have ๐๐ค โ ๐ต๐ผ๐ค ๐ฝ (๐๐ค (๐ฆ0 )) โ ๐ต2๐ผ๐ค ๐ฝ+diam ๐๐ (๐ฅ) ฬ โฉ ๐ ๐ โ โ
. As ๐ผ๐ค ๐ฝ โค ๐ฝ diam ๐ ๐ , we have for each ๐ค โ ฮ๐ such that ๐๐ค (10.19)
๐๐ค โ ๐ต(2๐ฝ+1) diam ๐๐ (๐ฅ).
Thus, the ball ๐ต(2๐ฝ+1) diam ๐๐ (๐ฅ) contains all sets ๐๐ค such that ๐ค โ ฮ๐ ฬ โฉ ๐ ๐ โ โ
. Moreover, each set ๐๐ค contains the ball ๐ต๐ผ ๐ผ (๐๐ค (๐ฅ0 )), and ๐๐ค ๐ค by (10.17), and all of them are disjoint, so the ball ๐ต(2๐ฝ+1) diam ๐๐ (๐ฅ) contains, say, ๐ balls of radius ๐ผ๐ค ๐ผ = ๐ผ๐คโฒ ๐ผ๐ค๐ ๐ผ > ๐พ๐ผ diam ๐ ๐ , where ฬ โฉ ๐ ๐ โ โ
}. ๐ = #{๐ค โ ฮ๐ โถ ๐๐ค Since the measure |๐ต๐ (๐ฅ)| of a ball of radius ๐ is given by ๐๐ ๐๐ /๐, we have the inequality ๐
๐
((2๐ฝ + 1) diam ๐ ๐ ) โฅ ๐(๐พ๐ผ diam ๐ ๐ ) , from the fact that we have a ball of radius (2๐ฝ + 1) diam ๐ ๐ that contains ๐ disjoint balls of radius ๐พ๐ผ diam ๐ ๐ . Therefore ๐โค(
2๐ฝ + 1 ๐ 1 ) = . ๐พ๐ผ ๐
Now, for each ๐ and ๐, we consider the sum ๐ด๐ (๐) =
โ ๐ค=๐ค1 ๐ค2 . . .๐ค๐ ๐ฬ ๐ค โฉ๐๐ โ 0
๐ผ๐ท ๐ค.
196
10. Mathematics of fractals
๐ด๐ (๐) is decreasing in ๐ because ๐๐ค๐ โ ๐๐ค for each ๐ = 1, 2, . . . , ๐, and ๐ โ๐=1 ๐ผ๐ท ๐ = 1. Hence, if ๐ฟ = max{๐ โถ ๐ค 1 ๐ค 2 . . . ๐ค ๐ โ ฮ๐ }, we have, as each ๐ผ๐ < 1, โ
๐ด๐ฟ (๐) โค
๐ผ๐ท ๐ค โค
๐คโฮ๐ ๐ฬ ๐ค โฉ๐๐ โ โ
โ
(diam ๐ ๐ )๐ท = ๐(diam ๐ ๐ )๐ท
๐คโฮ๐ ๐ฬ ๐ค โฉ๐๐ โ โ
1 (diam ๐ ๐ )๐ท . ๐ However, as ๐พ โ ๐,ฬ ๐พ โ โ๐ ๐ ๐ , we conclude โค
โ
1=
๐ค=๐ค1 ๐ค2 . . .๐ค๐ฟ ๐ฬ ๐ค โฉ๐พโ โ
0. (a) If ๐ด โ ๐ต, then ๐ป๐ฟ๐ (๐ด) โค ๐ป๐ฟ๐ (๐ต). (b) If ๐ด = โ๐ ๐ด๐ , then ๐ป๐ฟ๐ (๐ด) โค โ ๐ป๐ฟ๐ (๐ด๐ ). ๐
(2) Let ๐ โฅ 0. (a) If ๐ด โ ๐ต, then โ ๐ (๐ด) โค โ ๐ (๐ต). (b) If ๐ด = โ๐ ๐ด๐ , then โ ๐ (๐ด) โค โ โ ๐ (๐ด๐ ). ๐ ๐
(c) If dist(๐ด, ๐ต) > 0, then โ (๐ด โช ๐ต) = โ ๐ (๐ด) + โ ๐ (๐ต). (3) If ๐ด is countable, then dim ๐ด = 0. (4) There exist constants ๐, ๐ถ > 0 such that, for any measurable ๐ด โ โ๐ , ๐|๐ด| โค โ ๐ (๐ด) โค ๐ถ|๐ด|, and thus โ ๐ is comparable to Lebesgue measure on โ๐ . (5) If ๐ด โ โ๐ is open, then dim ๐ด = ๐. (6) If 0 < ๐ < 1, the function ๐ฅ โฆ ๐ฅ๐ is concave, that is, for ๐ฅ, ๐ฆ > 0 and 0 โค ๐ก โค 1, (๐ก๐ฅ + (1 โ ๐ก)๐ฆ)๐ โฅ ๐ก๐ฅ๐ + (1 โ ๐ก)๐ฆ๐ . (7) Let ๐ โถ โ๐ โ โ๐ be a similitude with coefficient ๐ผ > 0: for every ๐ฅ, ๐ฆ โ โ๐ , |๐(๐ฅ) โ ๐(๐ฆ)| = ๐ผ|๐ฅ โ ๐ฆ| 1 Let ๐(๐ฅ) = (๐(๐ฅ) โ ๐(0)). ๐ผ (a) For all ๐ฅ, ๐ฆ โ โ๐ , ๐(๐ฅ) โ
๐(๐ฆ) = ๐ฅ โ
๐ฆ. (b) If ๐ 1 , ๐ 2 , . . . , ๐ ๐ is the standard basis for โ๐ , then ๐(๐ 1 ), ๐(๐ 2 ), . . ., ๐(๐ ๐ ) is an orthonormal basis for โ๐ . (c) For ๐ฅ, ๐ฆ โ โ๐ and ๐ = 1, 2, . . . , ๐, ๐(๐ฅ + ๐ฆ) โ
๐(๐๐ ) = (๐(๐ฅ) + ๐(๐ฆ)) โ
๐(๐๐ ). (d) For ๐ฅ โ โ๐ , ๐ โ โ and ๐ = 1, 2, . . . , ๐, ๐(๐๐ฅ) โ
๐(๐๐ ) = ๐๐(๐ฅ) โ
๐(๐๐ ).
202
10. Mathematics of fractals (e) ๐ is an orthogonal linear transformation. (f) Conclude that every similitude in โ๐ is of the form ๐(๐ฅ) = ๐ผ๐๐ฅ + ๐ฅ0 , where ๐ผ > 0, ๐ โ ๐(๐) and ๐ฅ0 โ โ๐ .
(8) Let ๐พ be the self-similar set with respect to the contractions ๐1 , ๐2 , . . . , ๐๐ โถ โ๐ โ โ๐ , and suppose ๐ด โ โ๐ is nonempty and satisfies ๐1 (๐ด) โช ๐2 (๐ด) โช . . . โช ๐๐ (๐ด) โ ๐ด. Then ๐พ โ ๐ด,ฬ where ๐ดฬ is the closure of ๐ด. (Hint: Prove that, if ๐ด๐ = {๐ฅ โ โ๐ โถ there exists ๐ฆ โ ๐ด such that |๐ฅ โ ๐ฆ| < ๐}, then ๐พ โ ๐ด๐ for any ๐ > 0.) (9) Describe explicitly the convex hull of the Hata tree set of Example 10.22, and prove that its interior is an open set under which the contractions that define the Hata set satisfy the open set conditions. (10) Describe explicitly an open set, under which the open set condition is satisfied, for the golden fractal of Example 10.23. (11) The polynomial ๐ฅ3 + 2๐ฅ โ 1 has only one real root, and it belongs to the interval (0, ๐พ), where ๐พ = 1/๐ is the reciprocal of the golden ratio.
Notes The Hausdorff dimension was introduced by Felix Hausdorff in [Hau18]. Theorem 10.15 was proven by John Hutchinson in [Hut81]. It is sometimes known as Moranโs theorem due to Patrick Alfred Pierce Moranโs related work in [Mor46]. The proof presented here is from [YHK97]. The Sierpiลski gasket was studied by Wacลaw Sierpiลski in [Sie15]. The snowflake set is presented in [Kig01]. The Hata set was introduced by Masayoshi Hata in [Hat85], and the golden fractal by Marc Frantz in [Fra09].
Chapter 11
The Laplacian on the Sierpiลski gasket
In the previous chapters, we studied harmonic funcions on regions given by open subsets of the Euclidean space โ๐ . However, in the study of diffusion in disordered media, fractals sets, as the ones we studied in Chapter 10, are better suited as models to those systems. The purpose of this chapter is to construct a Laplacianโand study the harmonic functions, on the Sierpiลski gasketโthe set ๐ described in Example 10.20. We will do this by working on discrete approximations to ๐. To motivate our methods, we start by discussing a discrete approach to harmonic analysis on the interval [0, 1].
11.1. Discrete energy on the interval We start with the following model. Consider a spring placed between two nodes at points (0, ๐) and (1, ๐) in the plane, as in Figure 11.1 (left). If we assume that it is in equilibrium when ๐ = ๐, then its energy is given by a constant multiple of (11.1)
โฐ0 = (๐ โ ๐)2 .
Now, assume we put a node at the point (1/2, ๐ฅ), as in Figure 11.1 (right). We now have a pair of springs whose total energy is given by (11.2)
โฐ1 = ๐ผ((๐ โ ๐ฅ)2 + (๐ฅ โ ๐)2 ), 203
204
11. The Laplacian on the Sierpiลski gasket
Figure 11.1. The spring on the left has energy โฐ0 = (๐ โ ๐)2 , while the pair of springs on the right have total energy โฐ1 = ๐ผ((๐ โ ๐ฅ)2 + (๐ฅ โ ๐)2 ).
for an appropriate constant ๐ผ. It is not hard to see that the value of ๐ฅ that minimizes (11.2) is ๐+๐ ๐ฅโ = , 2 and at this value the energy โฐ1 in (11.2) is equal to ๐ผ โฐ1 = . 2 Note that if the node is placed at (1/2, ๐ฅโ ), then it is the midpoint of the line segment from (0, ๐) to (1, ๐). Hence the total energy of the springs must be the same as โฐ0 , and thus we must choose ๐ผ = 2. If we continue subdividing dyadically the interval [0, 1], we obtain a partition 1 2 3 ๐๐ = {0, ๐ , ๐ , ๐ , . . . , 1} 2 2 2 for each ๐ โฅ 1. Given a real valued function ๐ข on ๐๐ , we define its energy โฐ๐ (๐ข) as the quadratic form 2๐
(11.3)
โฐ๐ (๐ข) = 2
๐
โ (๐ข( ๐=1
๐โ1 ๐ 2 โ ๐ข( ) )) . 2๐ 2๐
If ๐ข(0) = ๐, ๐ข(1) = ๐, then โฐ0 (๐ข) = (๐ โ ๐)2 , and, as we observed above, min{โฐ1 (๐ข) โถ ๐ข(0) = ๐, ๐ข(1) = ๐} = โฐ0 (๐ข) = (๐ โ ๐)2 , and is attained when ๐ข(1/2) = (๐ + ๐)/2. Inductively, we can verify that min{โฐ๐ (๐ข) โถ ๐ข(0) = ๐, ๐ข(1) = ๐} = โฐ0 (๐ข) for each ๐ โฅ 1, and that is attained at the function ๐ข that satisfies 1 ๐โ1 2๐ โ 1 ๐ (11.4) ๐ข( ๐ ) = (๐ข( ๐โ1 ) + ๐ข( ๐โ1 )) 2 2 2 2
11.1. Discrete energy on the interval
205
for each ๐ โฅ 1 and each 1 โค ๐ โค 2๐โ1 , and thus can be constructed inductively by the algorithm (11.4). In fact, (11.4) describes the dyadic points of the line segment from (0, ๐ข(0)) to (1, ๐ข(1)). By the mean value theorem, if ๐ข is a differentiable function in [0, 1], for each 1 โค ๐ โค 2๐ there is some ๐ก ๐ โ [(๐ โ 1)/2๐ , ๐/2๐ ] such that 1 ๐โ1 ๐ ) โ ๐ข( ๐ ) = ๐ขโฒ (๐ก ๐ ) โ
๐ , 2๐ 2 2 and hence (11.3) can be written as ๐ข(
2๐
2๐
๐ขโฒ (๐ก ) 2 1 โฐ๐ (๐ข) = 2 โ ( ๐๐ ) = โ ๐ขโฒ (๐ก ๐ )2 โ
๐ , 2 2 ๐=1 ๐=1 ๐
which, if ๐ขโฒ is Riemann-integrable on [0, 1], is a Riemann sum of the integral 1
โฐ(๐ข) = โซ ๐ขโฒ (๐ก)2 ๐๐ก,
(11.5)
0
the energy of the function ๐ข on [0, 1]. We note that the minimizers of this energy are the linear functions, which are the harmonic functions in [0, 1], and the continuous limit of algorithm (11.4). Moreover, we can observe that, if we polarize the quadratic form โฐ๐ , we see that the bilinear form 2๐
โฐ๐ (๐ข, ๐ฃ) = 2
๐
โ (๐ข( ๐=1
๐โ1 ๐ ๐โ1 ๐ ) โ ๐ข( ๐ ))(๐ฃ( ๐ ) โ ๐ฃ( ๐ )) 2๐ 2 2 2
converges, for ๐ข, ๐ฃ differentiable in [0, 1] and ๐ขโฒ , ๐ฃโฒ Riemann-integrable, to 1
โฐ(๐ข, ๐ฃ) = โซ ๐ขโฒ (๐ก)๐ฃโฒ (๐ก)๐๐ก, 0
the energy form on the interval studied in Section 1.4. If, say, ๐ข โ ๐ถ 2 ([0, 1]), and ๐ฃ is zero at the boundary points 0 and 1, then we can integrate by parts to obtain 1
โฐ(๐ข, ๐ฃ) = โ โซ ๐ขโณ (๐ก)๐ฃ(๐ก)๐๐ก.
(11.6)
0 โณ
Note that ๐ข is the Laplacian of ๐ข for a one-variable function, and that ๐ข is harmonic (๐ขโณ (๐ก) = 0 for all ๐ก โ [0, 1]) if and only if it is a linear function.
206
11. The Laplacian on the Sierpiลski gasket We can also obtain ๐ขโณ (๐ก) by a discrete limit. Note that we can rewrite 2๐
โฐ๐ (๐ข, ๐ฃ) = โ โ ๐ป๐ ๐ข( ๐=0
๐ ๐ )๐ฃ( ๐ ), 2๐ 2
where ๐ป๐ ๐ข(
๐ 1 ๐โ1 ๐+1 ๐ ) = โ๐ (๐ข( ๐ ) + ๐ข( ๐ ) โ 2๐ข( ๐ )) ๐ 2 2 2 2 2
if 1 โค ๐ โค 2๐ โ 1, 1 1 (๐ข( ๐ ) โ ๐ข(0)), 2โ๐ 2
๐ป๐ ๐ข(0) = and
2๐ โ 1 ) โ ๐ข(1)). 2๐ If, for 1 โค ๐ โค 2๐ โ 1, we define the piecewise linear function ๐ฃ by ๐ป๐ ๐ข(1) =
๐ฃ(
1
2โ๐
(๐ข(
1 if ๐ = ๐ ๐ )={ ๐ 2 0 otherwise,
then
๐ ). 2๐ In fact, one can verify explicitly (Exercise (1)) that โฐ๐ (๐ข, ๐ฃ) = โ๐ป๐ ๐ข( 1
โซ ๐ขโณ (๐ก)๐ฃ(๐ก)๐๐ก = ๐ป๐ ๐ข( 0
๐ ). 2๐
2
As ๐ข โ ๐ถ ([0, 1]), given ๐ > 0 we can choose ๐ large enough so that |๐ขโณ (๐ฅ) โ ๐ขโณ (๐ฆ)| < ๐ for |๐ฅ โ ๐ฆ| โค 2โ๐ . Thus 1
1
1
| โซ ๐ขโณ (๐ก)๐ฃ(๐ก)๐๐ก โ ๐ขโณ ( ๐ ) โซ ๐ฃ(๐ก)๐๐ก| โค โซ |๐ขโณ (๐ก) โ ๐ขโณ ( ๐ )|๐ฃ(๐ก)๐๐ก | | | 2๐ 2๐ | 0
0
0
1
< ๐ โซ ๐ฃ(๐ก)๐๐ก. 0
Since
1
โซ ๐ฃ(๐ก)๐๐ก = 0
1 , 2๐
11.2. Harmonic structure on the Sierpiลski gasket
207
we have that, for any dyadic point ๐ฅ0 in [0, 1], 1 ๐ป๐ ๐ข(๐ฅ0 ) = ๐ขโณ (๐ฅ0 ). lim ๐โโ 2โ๐ The previous limit is also a known result from calculus. We call the ๐ป๐ the sequence of discrete Laplacians on the interval [0, 1].
11.2. Harmonic structure on the Sierpiลski gasket We now proceed to define a Laplacian on the Sierpiลski gasket ๐ introduced in Example 10.20. Recall that ๐ is the self-similar set in the plane that satisfies ๐ = ๐1 (๐) โช ๐2 (๐) โช ๐3 (๐), where the ๐๐ โถ โ2 โ โ2 are the contractions 1 ๐๐ (๐ฅ) = (๐ฅ + ๐ ๐ ), 2 and the points ๐1 = (1/2, โ3/2), ๐2 = (0, 0) and ๐3 = (1, 0) are the vertices of an equilateral triangle (Figure 11.2). As in the case of the interval, we will do this by constructing a sequence of quadratic forms
Figure 11.2. The Sierpiลski gasket.
on approximating points to ๐, whose limit will define a quadratic form on functions on ๐. This quadratic form willl induce a Laplacian on the set ๐, which will also be a limit of discrete difference operators. We start by considering three springs on the sides of an equilateral triangle, with nodes at each of its vertices, at heights ๐, ๐ and ๐, as in
208
11. The Laplacian on the Sierpiลski gasket
Figure 11.3. Three springs with nodes at heights ๐, ๐ and ๐.
Figure 11.3. The energy of this system of springs is now defined to be โฐ0 = (๐ โ ๐)2 + (๐ โ ๐)2 + (๐ โ ๐)2 . Suppose that we want to add nodes at their middle points, with heights ๐ฅ, ๐ฆ, ๐ง, as in Figure 11.4. The energy is now given by โฐ1 = ๐ผ((๐ โ ๐ง)2 + (๐ง โ ๐ฆ)2 + (๐ฆ โ ๐)2 + (๐ง โ ๐)2 + (๐ โ ๐ฅ)2 + (๐ฅ โ ๐ง)2 + (๐ฆ โ ๐ฅ)2 + (๐ฅ โ ๐)2 + (๐ โ ๐ฆ)2 ), where the constant ๐ผ is chosen so that, if ๐ฅ, ๐ฆ and ๐ง minimize โฐ1 , then this minimum is equal to โฐ0 . Thus, we can calculate (Exercise (3)) that the minimizing values are (11.7)
๐ฅโ =
๐ + 2๐ + 2๐ , 5
๐ฆโ =
2๐ + ๐ + 2๐ , 5
๐งโ =
2๐ + 2๐ + ๐ , 5
and that
5 . 3 We observe that the values (11.7) are averages of the values at the vertices, with weights 2 to 1 depending on whether the corresponding vertex is adyacent or opposite from each middle point node. ๐ผ=
Let ๐0 = {๐1 , ๐2 , ๐3 } and define, for each ๐ โฅ 1, the set ๐๐ = ๐1 (๐๐โ1 ) โช ๐2 (๐๐โ1 ) โช ๐3 (๐๐โ1 ). Note that each ๐ฅ โ ๐๐ is of the form ๐๐ค (๐ ๐ ), where, as in Chapter 10, ๐ค = ๐ค 1 ๐ค 2 . . . ๐ค ๐ โ ๐๐ , where ๐๐ is the set of words of length ๐ (with
11.2. Harmonic structure on the Sierpiลski gasket
209
Figure 11.4. We add three more nodes, at the middle points of each previous spring, with heights ๐ฅ, ๐ฆ, ๐ง that minimize the energy โฐ1 .
letters ๐ค๐ = 1, 2, 3), and ๐๐ค = ๐๐ค1 โ ๐๐ค2 โ โฏ โ ๐๐ค๐ .
Figure 11.5. The sets ๐ 0 , ๐ 1 and ๐ 2 . The edges of the graphs join adjacent vertices at each level.
We thus have ๐๐ =
โ
๐๐ค (๐0 ).
๐คโ๐๐
We say that two vertices ๐ฅ, ๐ฆ โ ๐๐ are adjacent, or neighbors, and write ๐ฅ โผ๐ ๐ฆ, if there exists a word ๐ค โ ๐๐ such that ๐ฅ = ๐๐ค (๐ ๐ ) and ๐ฆ = ๐๐ค (๐๐ ), for some ๐, ๐ = 1, 2, 3. Figure 11.5 shows ๐0 , ๐1 and ๐2 , with edges joining neighboring vertices. For a function ๐ข defined on ๐๐ , define the quadratic form 2 5 ๐ (11.8) โฐ๐ (๐ข) = ( ) โ (๐ข(๐ฅ) โ ๐ข(๐ฆ)) . 3 ๐ฅโผ ๐ฆ ๐
210
11. The Laplacian on the Sierpiลski gasket
The quadratic form โฐ๐ defined in (11.8) is positive semidefinite, and from its definition on can verify that โฐ๐ (๐ข) = 0 if and only if ๐ข is constant on ๐๐ . โฐ๐ is called the discrete energy of level ๐ on the Sierpiลski gasket. If ๐ข is a function on ๐๐ , then, for each ๐ = 1, 2, 3, ๐ข โ ๐๐ is a function on ๐๐โ1 . From (11.8) we see that 3
(11.9)
โฐ๐ (๐ข) =
5 โโฐ (๐ข โ ๐๐ ). 3 ๐=1 ๐โ1
Thus, from the previous discussion, if ๐ฃ is a function on ๐๐โ1 , then (11.10) min{โฐ๐ (๐ข) โถ functions ๐ข on ๐๐ such that ๐ข|๐๐โ1 = ๐ฃ} = โฐ๐โ1 (๐ฃ), where the minimizing function ๐ข is calculated using (11.7). The sequence โฐ๐ is called a harmonic structure on ๐. Using (11.10), one can verify by induction that, given a function ๐ on ๐0 , we have that min{โฐ๐ (๐ข) โถ functions ๐ข on ๐๐ such that ๐ข|๐0 = ๐} = โฐ0 (๐). Again, the minimizer function can be calculated recursively using the algorithm (11.7). The points in ๐0 are called the boundary of ๐, and the minimizer function ๐ข is called harmonic with boundary values ๐. Figure 11.6 shows the harmonic function with boundary values given by ๐(๐1 ) = 1 and ๐(๐2 ) = ๐(๐3 ) = 0. If ๐ข is harmonic, we can show that there exists ๐ด > 0 such that, for ๐ โฅ 0, 3 ๐ (11.11) |๐ข(๐ฅ) โ ๐ข(๐ฆ)| โค ๐ด( ) 5 whenever ๐ฅ โผ๐ ๐ฆ. Indeed, let ๐ด = max{|๐ข(๐1 ) โ ๐ข(๐2 )|, |๐ข(๐2 ) โ ๐ข(๐3 )|, |๐ข(๐3 ) โ ๐ข(๐1 )|}. Then clearly (11.11) is satisfied at ๐ = 0, and suppose it is true at ๐ โ 1, for some ๐ โฅ 1. If ๐ฅ โผ๐ ๐ฆ, then either ๐ฅ, ๐ฆ โ ๐๐ โงต ๐๐โ1 or, say, ๐ฅ โ ๐๐โ1 and ๐ฆ โ ๐๐ โงต ๐๐โ1 . Let ๐ค โ ๐๐โ1 , and ๐ฅ๐ = ๐๐ค (๐ ๐ ), for ๐ = 1, 2, 3. If ๐ฆ ๐ is the point in ๐๐ โงต ๐๐โ1 in the cell bounded by the points ๐ฅ1 , ๐ฅ2 , ๐ฅ3 , opposite to ๐ฅ๐ , then, by (11.7), ๐ข(๐ฆ ๐ ) =
๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) , 5
11.2. Harmonic structure on the Sierpiลski gasket 3
i 3
i
2
4
0 1
1 2
0 0 1 2
1
Figure 11.6. Harmonic function with boundary values ๐(๐ 1 ) = 1 and ๐(๐ 2 ) = ๐(๐ 3 ) = 0.
where ๐, ๐, ๐ are the three different numbers 1, 2, 3. Thus |๐ข(๐ฆ ๐ ) โ ๐ข(๐ฆ๐ )| ๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) ๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) | โ | 5 5 |๐ข(๐ฅ๐ ) โ ๐ข(๐ฅ๐ )| 1 3 ๐ 3 ๐โ1 = โค ๐ด( ) , โค โ
๐ด( ) 5 5 5 5
= ||
by the induction hypothesis, and ๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) + 2๐ข(๐ฅ๐ ) | | 5 |๐ข(๐ฅ๐ ) โ ๐ข(๐ฅ๐ )| 2|๐ข(๐ฅ๐ ) โ ๐ข(๐ฅ๐ )| โค + 5 5 ๐โ1 1 3 2 3 ๐โ1 3 ๐ โค โ
๐ด( ) + โ
๐ด( ) = ๐ด( ) . 5 5 5 5 5
|๐ข(๐ฅ๐ ) โ ๐ข(๐ฆ๐ )| = ||๐ข(๐ฅ๐ ) โ
Thus (11.11) holds for every ๐ โฅ 0. If ๐โ =
โ
๐โฅ0
๐๐ ,
211
212
11. The Laplacian on the Sierpiลski gasket
we can use estimate (11.11) to show that ๐ข is a uniformly continuous function on ๐โ (Exercise (4)), which, since ๐โ is dense in ๐ (Exercise (5)), can be extended to a continuous function on ๐ (Exercise (6)).
11.3. The Laplacian on the Sierpiลski gasket Let ๐ข โ ๐ถ(๐), a continuous function on ๐, and for each ๐ โฅ 0, let ๐ข๐ = ๐ข|๐๐ , its restriction to ๐๐ . By (11.10), the sequence โฐ๐ (๐ข๐ ) is increasing. Let โฑ = {๐ข โ ๐ถ(๐) โถ โฐ๐ (๐ข๐ ) is bounded}. Then, for each ๐ข โ โฑ, the limit โฐ(๐ข) = lim โฐ๐ (๐ข๐ ) = sup โฐ๐ (๐ข๐ ) ๐โโ
๐โฅ0
exists. We call โฐ(๐ข) the energy of ๐ข on ๐. Example 11.12. If ๐ข is constant, then โฐ๐ (๐ข๐ ) = 0 for all ๐, and thus ๐ข โ โฑ and โฐ(๐ข) = 0. In fact, since โฐ(๐ข) = sup โฐ๐ (๐ข๐ ), ๐โฅ0
then โฐ(๐ข) = 0 only if ๐ข is a constant function. Example 11.13. If ๐ข is the continuous extension of a harmonic function, then โฐ๐ (๐ข๐ ) = โฐ0 (๐ข0 ) for all ๐, and thus โฐ(๐ข) = โฐ0 (๐ข0 ). Moreover, since โฐ(๐ข) = sup โฐ๐ (๐ข๐ ), ๐โฅ0
then ๐ข minimizes โฐ among all functions in โฑ with boundary values ๐ข0 . We thus also call ๐ข harmonic in ๐. We say that ๐ข โ ๐ถ(๐) is ๐-harmonic if, for each ๐ค โ ๐๐ , the function ๐ข โ ๐๐ค is harmonic. Equivalently, ๐ข is the continuous extension of the function constructed using (11.7) starting from a given function on ๐๐ . Thus, for each ๐ โฅ ๐, โฐ๐ (๐ข๐ ) = โฐ๐ (๐ข๐ ), and thus ๐ข โ โฑ and โฐ(๐ข) = โฐ๐ (๐ข๐ ). Example 11.14. Let ๐ โฅ 0 and ๐ฅ โ ๐๐ . Consider the ๐-harmonic function ๐๐ฅ,๐ (if ๐ = 0 we just say harmonic) constructed from the function on ๐๐ given by 1 ๐ฆ=๐ฅ ๐(๐ฆ) = { 0 ๐ฆ โ ๐ฅ.
11.3. The Laplacian on the Sierpiลski gasket
213
๐๐ฅ,๐ is called the ๐-harmonic spline at ๐ฅ. Note that, for any ๐-harmonic function ๐ข, we have ๐ข = โ ๐ข(๐ฅ)๐๐ฅ,๐ . ๐ฅโ๐๐
In particular, if ๐ = 0, the splines of Example 11.14 correspond to the harmonic functions with boundary values 1, 0, 0, as the one in Figure 11.6. If we denote ๐๐๐ ,0 simply by ๐๐ , we see that ๐ข = ๐ข(๐1 )๐1 + ๐ข(๐2 )๐2 + ๐ข(๐3 )๐3 , and thus ๐1 , ๐2 and ๐3 form a basis for the (3-dimensional) space of harmonic functions on ๐. 2๐ Note that, by the symmetry of ๐ and (11.7), each ๐๐ is a rotation 3 of each other. Also, any other ๐๐ฅ,๐ is either of the form ๐๐๐ค โ
๐๐ โ ๐๐คโ1
or
โ1 ๐๐๐ค โ
๐๐ โ ๐๐คโ1 + ๐๐๐คฬ โ
๐๐ โ ๐๐ค ฬ ,
where ๐ค, ๐คฬ โ ๐๐ , depending on whether ๐ฅ โ ๐0 or ๐ฅ โ ๐๐ โงต๐0 (Exercise (8)). We want to obtain the analogous result of integration by parts in the interval, as in (11.6), for the Sierpiลski gasket. This will give us a way to define the Laplacian on ๐. For this, we first need to define an integral of functions on ๐. The usual Lebesgue integral that we have been using in the previous chapters is not useful for us here, since ๐ is a set of measure zero in the plane. So we have to start by defining a measure on subsets of ๐. Recall that, for any word ๐ค โ ๐๐ , we define the cell ๐๐ค = ๐๐ค (๐). For any given cell, we define 1 , 3๐ where ๐ is the length of the word ๐ค. We now define, for ๐ด โ ๐, (11.15)
(11.16)
๐(๐๐ค ) =
๐(๐ด) = inf { โ ๐(๐๐ ) โถ ๐๐ are cells and ๐ด โ ๐
โ
๐๐ }.
๐
As in the case of the Hausdorff measure that we discussed in Chapter 10, ๐ satisfies the properties of the Lebesgue outer measure. Proposition 11.17. Let ๐ be defined on subsets of ๐ as in (11.16). (1) If ๐ด โ ๐ต, then ๐(๐ด) โค ๐(๐ต).
214
11. The Laplacian on the Sierpiลski gasket (2) If ๐ด = โ๐ ๐ด๐ , then ๐(๐ด) โค โ๐ ๐(๐ด๐ ). (3) If dist(๐ด, ๐ต) > 0, ๐(๐ด โช ๐ต) = ๐(๐ด) + ๐(๐ต).
We leave the proof of Proposition 11.17 as an exercise (Exercise (9)). We can then define a set ๐ด โ ๐ to be measurable if, for any ๐ต โ ๐, ๐(๐ต) = ๐(๐ต โฉ ๐ด) + ๐(๐ต โงต ๐ด). As in the case of the Lebesgue measure, part (3) of Proposition 11.17 implies that every open subset of ๐ is measurable (and hence every closed subset ). Also, every open ๐ โ ๐ can be written as an almost disjoint union โ๐ ๐๐ of cells, so ๐(๐) = โ ๐(๐๐ ). ๐
We, in fact, can prove more. Proposition 11.18. (1) If ๐ด1 , ๐ด2 , . . . are disjoint measurable subsets of ๐, then ๐ด = โ๐ ๐ด๐ is measurable and ๐(๐ด) = โ ๐(๐ด๐ ). ๐
(2) If ๐ด โ ๐ is measurable, its complement ๐ โงต ๐ด is also measurable and ๐(๐ โงต ๐ด) = 1 โ ๐(๐ด). The proofs of these statements are similar to their analogous results on Lebesgue measure and we again leave them as an exercise (Exercise (9)). We can now proceed to define the integral in an analogous way, first for characteristic functions of measurable sets, โซ ๐๐ด ๐๐ = ๐(๐ด), ๐
then for simple functions ๐ = โ๐ ๐๐ ๐(๐ด๐ ), โซ ๐๐๐ = โ ๐๐ ๐(๐ด๐ ), ๐
๐
11.3. The Laplacian on the Sierpiลski gasket
215
and then by approximating any measurable ๐ โฅ 0 with simple functions โซ ๐๐๐ = sup { โซ ๐๐๐ โถ ๐ is simple and 0 โค ๐ โค ๐}. ๐
๐
However, while the Lebesgue measure is traslation invariant and dilations by a positive ๐ฟ induce a factor of ๐ฟ๐ , where ๐ is the dimension of the space, in this case we have the following properties. 11.19. If ๐ is a symmetry of ๐, say, a rotation or a reflection over an axis passing through one of its vertices ๐ ๐ , then โซ ๐ โ ๐๐๐ = โซ ๐๐๐. ๐
๐
To prove this fact, observe that it is enough to prove it for characteristic functions ๐๐ด , and in that case โซ ๐๐ด โ ๐๐๐ = โซ ๐๐โ1 (๐ด) ๐๐ = ๐(๐โ1 (๐ด)), ๐
๐
so we just need to verify that ๐ is invariant under the symmetries of ๐. This follows from the observation that, for any cell ๐๐ค and any symmetry ๐ of ๐, then ๐(๐๐ค ) is also a cell of level ๐, and thus have the same measure. Indeed, if ๐ is the reflection over the axis passing through ๐ ๐ , so it keeps ๐ ๐ fixed and switches ๐๐ and ๐ ๐ , then, for any ๐ค โ ๐๐ , ๐(๐๐ค ) = ๐๐คฬ , where ๐คฬ โ ๐๐ is the word obtained from ๐ค by switching the letters ๐ and ๐. If ๐ is the rotation ๐1 โฆ ๐2 , ๐2 โฆ ๐3 and ๐3 โฆ ๐1 , then ๐(๐๐ค ) = ๐๐คฬ where ๐คฬ is obtained by replacing any letter ๐ of ๐ค by the letter ๐ + 1 (mod 3). 11.20. For any ๐ค โ ๐๐ , โซ ๐๐๐ = ๐๐ค
1 โซ ๐ โ ๐๐ค ๐๐. 3๐ ๐
In order to verify this for characteristic functions, we have to verify that 1 ๐(๐โ1 (๐ด โฉ ๐๐ค )), 3๐ ๐ค which follows from the fact that, if the ๐๐ are cells, ๐(๐ด โฉ ๐๐ค ) =
๐๐คโ1 (๐ด โฉ ๐๐ค ) โ
โ ๐
๐๐
if and only if
๐ด โฉ ๐๐ค โ
โ ๐
๐๐ค (๐๐ ),
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11. The Laplacian on the Sierpiลski gasket
and ๐(๐๐ค (๐๐ )) =
1 ๐(๐๐ ) 3๐
for each cell ๐๐ . Example 11.21. If ๐ is a harmonic function on ๐, then ๐ = ๐(๐1 )๐1 + ๐(๐2 )๐2 + ๐(๐3 )๐3 , where each ๐๐ is the harmonic function with boundary values ๐๐ (๐ ๐ ) = 1 and ๐๐ (๐๐ ) = 0, for ๐ โ ๐. As we have seen above, each ๐๐ is a rotation of the others. Since ๐1 + ๐2 + ๐3 = 1 and ๐(๐) = 1, we have that โซ ๐1 ๐๐ = โซ ๐2 ๐๐ = โซ ๐3 ๐๐ = ๐
๐
Therefore โซ ๐๐๐ = ๐
๐
1 . 3
๐(๐1 ) + ๐(๐2 ) + ๐(๐3 ) . 3
Example 11.22. Let ๐๐ฅ,๐ be the ๐-harmonic spline at ๐ฅ. As we have seen above, if ๐ฅ โ ๐0 , ๐๐ฅ,๐ = ๐๐๐ค โ
๐๐ โ ๐๐คโ1 for some ๐ and ๐ค โ ๐๐ , and thus 1 1 โซ ๐๐ฅ,๐ ๐๐ = โซ ๐๐ โ ๐๐คโ1 ๐๐ = ๐ โซ ๐๐ ๐๐ = ๐+1 . 3 ๐ 3 ๐ ๐ ๐ค
โ1 If ๐ฅ โ ๐๐ โงต ๐0 , then ๐๐ฅ,๐ = ๐๐๐ค โ
๐๐ โ ๐๐คโ1 + ๐๐๐คฬ โ
๐๐ โ ๐๐ค ฬ for some ๐, ๐ and ๐ค, ๐คฬ โ ๐๐ , and thus โ1 โซ ๐๐ฅ,๐ = โซ ๐๐ โ ๐๐คโ1 ๐๐ + โซ ๐๐ โ ๐๐ค ฬ ๐๐ ๐
๐๐ค
๐๐คฬ
1 1 = ๐ โซ ๐๐ ๐๐ + ๐ โซ ๐๐ ๐๐ 3 ๐ 3 ๐ 2 = ๐+1 . 3 We are ready to define the Laplacian. First, consider the polarization โฐ(๐ข, ๐ฃ) of the energy, defined for ๐ข, ๐ฃ โ โฑ. We can see that โฐ(๐ข, ๐ฃ) = lim โฐ๐ (๐ข, ๐ฃ), where in the right-hand side we also denote by ๐ข and ๐ฃ their restrictions to ๐๐ , and, for functions ๐๐ , โฐ๐ (๐ข, ๐ฃ) is the polarization of the quadratic form โฐ๐ , 5 ๐ โฐ๐ (๐ข, ๐ฃ) = ( ) โ (๐ข(๐ฅ) โ ๐ข(๐ฆ))(๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)). 3 ๐ฅโผ ๐ฆ ๐
11.3. The Laplacian on the Sierpiลski gasket
217
In fact, one can show (Exercise (10)) that 5 ๐ (11.23) โฐ๐ (๐ข, ๐ฃ) = โ( ) โ ฮ๐ ๐ข(๐ฅ) ๐ฃ(๐ฅ), 3 ๐ฅโ๐ ๐
where ฮ๐ is the difference operator (11.24)
ฮ๐ ๐ข(๐ฅ) = โ (๐ข(๐ฆ) โ ๐ข(๐ฅ)). ๐ฆโผ๐ ๐ฅ
Note that this sum has two terms if ๐ฅ โ ๐0 , or four terms if ๐ฅ โ ๐๐ โงต ๐0 . ฮ๐ is called the discrete Laplacian at level ๐. We say that ๐ข โ dom ฮ if ๐ข โ โฑ and there exists ๐ โ ๐ถ(๐) such that, for all ๐ฃ โ โฑ with ๐ฃ|๐0 = 0, (11.25)
โฐ(๐ข, ๐ฃ) = โ โซ ๐๐ฃ๐๐. ๐
We write ๐ = ฮ๐ข, and we call it the Laplacian of ๐ข. 11.26. If ๐ข is harmonic, for each ๐ โฅ 1 and all ๐ฅ โ ๐๐ โงต ๐0 , we have that ฮ๐ ๐ข(๐ฅ) = 0 (Exercise (11)). Thus, if ๐ฃ โ โฑ with ๐ฃ|๐0 = 0, โฐ๐ (๐ข, ๐ฃ) = 0 for all ๐, and thus โฐ(๐ข, ๐ฃ) = 0. We thus conclude that, if ๐ข is harmonic, ๐ข โ dom ฮ and ฮ๐ข = 0. 11.27. If ๐ข is ๐-harmonic, ฮ๐ ๐ข(๐ฅ) = 0 for all ๐ > ๐ and ๐ฅ โ ๐๐ โงต ๐๐ . Thus, for any other ๐ฃ โ โฑ, โฐ(๐ข, ๐ฃ) = โฐ๐ (๐ข, ๐ฃ). However, ๐ข โ dom ฮ, because ฮ๐ข(๐ฅ) cannont be defined for ๐ฅ โ ๐๐ . Fix ๐ฅ0 โ ๐โ โงต ๐0 , and let ๐ be so ๐ฅ0 โ ๐๐ . If ๐ฃ = ๐๐ฅ0 ,๐ , then, as we have seen before, ๐ฃ โ โฑ, and of course ๐ฃ|๐0 = 0, because ๐ฅ0 โ ๐0 . Let ๐ข โ dom ฮ. We first note that, since ๐ฃ is ๐-harmonic, 5 ๐ โฐ(๐ข, ๐ฃ) = โฐ๐ (๐ข, ๐ฃ) = โ( ) โ ฮ๐ ๐ข(๐ฅ) ๐ฃ(๐ฅ) 3 ๐ฅโ๐ ๐
5 ๐ = โ( ) ฮ๐ ๐ข(๐ฅ0 ), 3 because ๐ฃ(๐ฅ0 ) = 1 and ๐ฃ(๐ฅ) = 0 for any other ๐ฅ โ ๐๐ . Also โซ ฮ๐ข(๐ฅ)๐ฃ(๐ฅ)๐๐(๐ฅ) โ ฮ๐ข(๐ฅ0 ) โซ ๐ฃ(๐ฅ)๐๐(๐ฅ) ๐
๐
= โซ (ฮ๐ข(๐ฅ) โ ฮ๐ข(๐ฅ0 ))๐ฃ(๐ฅ)๐๐(๐ฅ). ๐
218
11. The Laplacian on the Sierpiลski gasket
Given ๐ > 0, we can choose ๐ large enough so that |ฮ๐ข(๐ฅ) โ ฮ๐ข(๐ฅ0 )| < ๐ for every ๐ฅ โ supp ๐ฃ, because ฮ๐ข is continuous. Thus, for such ๐, and using our result in Example 11.22, | โซ (ฮ๐ข(๐ฅ) โ ฮ๐ข(๐ฅ ))๐ฃ(๐ฅ)๐๐(๐ฅ)| < ๐ โซ ๐ฃ๐๐ = 2 ๐. 0 | | 3๐+1 ๐ ๐ Putting the previous identities and inequalities together we obtain ๐ |( 5 ) ฮ ๐ข(๐ฅ ) โ 2 ฮ๐ข(๐ฅ )| < 2 ๐, ๐ 0 0 | | 3 3๐+1 3๐+1
and thus
(11.28)
| 3 5๐ ฮ ๐ข(๐ฅ ) โ ฮ๐ข(๐ฅ )| < ๐. ๐ 0 0 | |2
We have therefore proven Theorem 11.29, that states that the Laplacian is a limit of normalized difference operators. Theorem 11.29. If ๐ข โ dom ฮ and ๐ฅ0 โ ๐โ โงต ๐0 , then
ฮ๐ข(๐ฅ0 ) = lim
๐โโ
3 ๐ 5 ฮ๐ ๐ข(๐ฅ0 ). 2
We can also prove that the limit in Theorem 11.28 is uniform, in the sense that, given ๐ > 0, we can choose ๐ such that (11.29) is true independently of the particular choice of ๐ฅ0 โ ๐๐ . Moreover, the existence of this uniform limit also implies that ๐ข โ dom ฮ. We have left these facts as exercises (Exercises (12) and (13)).
Exercises
219
Exercises (1) Let ๐ฅ0 โ (0, 1) and โ > 0 so that (๐ฅ0 โ โ, ๐ฅ0 + โ) โ (0, 1). Let ๐ฃ be the piecewise linear function given on [0, 1] by โง0 โช โช ๐ก โ ๐ฅ0 + โ ๐ฃ(๐ก) = โ โจ ๐ฅ0 + โ โ ๐ก โช โ โช โฉ0
๐ก < ๐ฅ0 โ โ ๐ฅ0 โ โ โค ๐ก < ๐ฅ 0 ๐ฅ0 โค ๐ก < ๐ฅ 0 + โ ๐ฅ0 + โ โค ๐ก.
Then, for ๐ข โ ๐ถ 2 ([0, 1]), 1
โซ ๐ขโณ (๐ก)๐ฃ(๐ก)๐๐ก = 0
๐ข(๐ฅ0 โ โ) + ๐ข(๐ฅ0 + โ) โ 2๐ข(๐ฅ0 ) . โ
(2) For functions ๐ข and ๐ฃ as in the previous exercise, 1
1 โซ ๐ขโณ (๐ก)๐ฃ(๐ก)๐๐ก = ๐ขโณ (๐ฅ0 ). โโ0 โ 0 lim
(3) Let ๐(๐ฅ, ๐ฆ, ๐ง) =(๐ โ ๐ง)2 + (๐ง โ ๐ฆ)2 + (๐ฆ โ ๐)2 + (๐ง โ ๐)2 + (๐ โ ๐ฅ)2 + (๐ฅ โ ๐ง)2 + (๐ฆ โ ๐ฅ)2 + (๐ฅ โ ๐)2 + (๐ โ ๐ฆ)2 . Then ๐ takes its minimum at ๐ฅโ =
๐ + 2๐ + 2๐ , 5
๐ฆโ =
2๐ + ๐ + 2๐ , 5
๐(๐ฅโ , ๐ฆโ , ๐งโ ) =
3 ((๐ โ ๐)2 + (๐ โ ๐)2 + (๐ โ ๐)2 ). 5
๐งโ =
2๐ + 2๐ + ๐ , 5
and
(4) If ๐ข is a harmonic function, then it is uniformly continuous on ๐โ , the set of all vertices in ๐. (5) The union ๐โ of all vertices in ๐ is dense in ๐. (6) If ๐ข is a harmonic function, then it can be extended to a continuous function on ๐. (Hint: Use Exercises (4) and (5).)
220
11. The Laplacian on the Sierpiลski gasket
(7) If ๐ข is the continuous extension of a harmonic function in ๐, then ๐ข is a Hรถlder continuous function with exponent ๐ผ=
log(5/3) . log 2
(8) Let ๐ โฅ 1 and ๐ฅ โ ๐๐ . (a) If ๐ฅ โ ๐0 , ๐๐ฅ,๐ = ๐๐๐ค โ
๐๐ โ ๐๐คโ1 for some ๐ = 1, 2, 3, and ๐ค is the word ๐ค = ๐๐ โฏ ๐ โ ๐๐ . (b) If ๐ฅ โ ๐0 , there exist ๐, ๐ = 1, 2, 3 and ๐ค, ๐คฬ โ ๐๐ such that โ1 ๐๐ฅ,๐ = ๐๐๐ค โ
๐๐ โ ๐๐คโ1 + ๐๐๐คฬ โ
๐๐ โ ๐๐ค ฬ .
(9) (a) Prove Proposition 11.17. (b) Prove Propostion 11.18. (10) For functions ๐ข, ๐ฃ on ๐๐ , โ (๐ข(๐ฅ) โ ๐ข(๐ฆ))(๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)) = โ โ ฮ๐ ๐ข(๐ฅ) ๐ฃ(๐ฅ), ๐ฅโผ๐ ๐ฆ
๐ฅโ๐๐
where ฮ๐ is the difference operator ฮ๐ ๐ข(๐ฅ) = โ (๐ข(๐ฆ) โ ๐ข(๐ฅ)). ๐ฆโผ๐ ๐ฅ
(11) If ๐ข is harmonic, then, for each ๐ โฅ 1 and ๐ฅ โ ๐๐ โงต ๐0 , ฮ๐ ๐ข(๐ฅ) = 0. (12) Let ๐ข โ dom ฮ. Then, for any ๐ > 0, there exists ๐ such that, for any ๐ โฅ ๐, | 3 5๐ ฮ ๐ข(๐ฅ) โ ฮ๐ข(๐ฅ)| < ๐ ๐ |2 | for any ๐ฅ โ ๐๐ โงต ๐0 . (13) Let ๐ข โ โฑ and suppose there exists ๐ โ ๐ถ(๐) such that, for any ๐ > 0, we can find ๐ such that, for any ๐ โฅ ๐, | 3 5๐ ฮ ๐ข(๐ฅ) โ ๐(๐ฅ)| < ๐ ๐ | |2 for any ๐ฅ โ ๐๐ โงต ๐0 . Then ๐ข โ dom ฮ and ๐ = ฮ๐ข.
Notes The calculus on the Sierpiลski gasket developed in this chapter was introduced by Jun Kigami in [Kig89], where the motivation from the discretization of the interval is also presented. The idea of using springs to define discrete energy was also used in [YHK97]. The discrete energy can also be obtained from the theory of electrical networks, as in
Notes
221
[Str06] or in [Kig01]. The Laplacian on the Sierpiลski gasket can also be introduced through Brownian motion, as in [Bar98].
Chapter 12
Eigenfunctions of the Laplacian
In Chapter 3, we studied the decomposition of functions in the interval in terms of trigonometric functions, as a means to solve the Dirichlet problem in the disk. The trigonometric functions sine and cosine satisfy the equations ๐2 sin ๐ฅ = โ sin ๐ฅ ๐๐ฅ2
and
๐2 cos ๐ฅ = โ cos ๐ฅ, ๐๐ฅ2
which imply that the trigonometric functions sin ๐๐ฅ and cos ๐๐ฅ are all eigenfunctions of the operator ๐ 2 /๐๐ฅ2 , as they satisfy ๐2 ๐(๐ฅ) = โ๐๐(๐ฅ) ๐๐ฅ2 for some nonnegative ๐ โ โ. In this chapter we study the eigenfunctions and eigenvalues of the Laplacian on the Sierpiลski gasket, defined in Chapter 11. That is, we will study the solutions to the equation ฮ๐ข(๐ฅ) = โ๐๐ข(๐ฅ), for some scalar ๐ and nonzero ๐ข โ dom ฮ, and each ๐ฅ โ ๐ โงต ๐0 . The first natural question is whether such eigenfunctions can be constructed through an interpolation algorithm, as is the case of the harmonic functions, studied above. Though there is no a priori reason for the existence 223
224
12. Eigenfunctions of the Laplacian
of such algorithm, we first note that, in the case of the interval [0, 1], it indeed exists.
12.1. Discrete eigenfunctions on the interval We first observe that if ๐ is an eigenfunction of the Laplacian ฮ = ๐ 2 /๐๐ฅ2 on the interval, then, for a sufficiently large ๐, ๐|๐๐ is an eigenfunction of the difference operator 1 1 ) + ๐ข(๐ฅ + ๐ ) โ 2๐ข(๐ฅ), 2๐ 2 where ๐ข is a function on the dyadic partition ๐๐ of level ๐, and ๐ฅ is of the form ๐/2๐ , 1 โค ๐ โค 2๐ โ 1. Indeed, the eigenfunctions of ฮ are linear combinations of functions of the form ๐(๐ฅ) = ๐๐๐ฅ , since ฮ๐ ๐ข(๐ฅ) = ๐ข(๐ฅ โ
๐ 2 ๐๐ฅ ๐ = ๐2 ๐๐๐ฅ . ๐๐ฅ2 Now 1
1
ฮ๐ ๐(๐ฅ) = ๐๐(๐ฅโ 2๐ ) + ๐๐(๐ฅ+ 2๐ ) โ 2๐๐๐ฅ = โ๐๐ ๐๐๐ฅ , where ๐
๐
๐
โ๐๐ = ๐โ 2๐ + ๐ 2๐ โ 2 = (๐ 2๐+1 โ ๐
โ
๐ 2๐+1
)2 = 4 sinh
2
๐ 2๐+1
.
Thus, the restriction of ๐ to ๐๐ is an eigenfunction of the difference operator ฮ๐ with respect to the eigenvalue โ๐๐ . Moreover, note that, as we have seen in Chapter 11, 4๐ ฮ๐ ๐(๐ฅ) โ ๐โณ (๐ฅ) = ๐2 ๐(๐ฅ) as ๐ โ โ, and ๐ โ ๐2 , 2๐+1 so the discrete eigenvalues, when properly normalized, converge to the true eigenvalues of ฮ. 2
โ4๐ ๐๐ = (2๐+1 )2 sinh
Note that it may happen that ๐|๐๐ โก 0 for some ๐: for instance, if ๐(๐ฅ) = sin(2๐ ๐๐ฅ), then ๐|๐๐ โก 0 for every ๐ โค ๐. Since an eigenfunction ๐ is not identically zero, ๐|๐๐ โข 0 for sufficiently large ๐, so the argument above applies.
12.1. Discrete eigenfunctions on the interval
225
Conversely, we can construct the eigenfunctions of ฮ by extending discrete eigenfunctions of the operators ฮ๐ . Suppose we have a function ๐ข on ๐๐โ1 which is an eigenfunction of ฮ๐โ1 with respect to the eigenvalue โ๐๐โ1 , that is ฮ๐โ1 ๐ข(๐ฅ) = โ๐๐โ1 ๐ข(๐ฅ) for each ๐ฅ โ ๐๐โ1 โงต ๐0 . We want to extend ๐ข to a function on ๐๐ , that we also denote by ๐ข, that satisfies ฮ๐ ๐ข(๐ฅ) = โ๐๐ ๐ข(๐ฅ) for all ๐ฅ โ ๐๐ โงต ๐0 , for some ๐๐ . In particular, for ๐ฅ โ ๐๐ โงต ๐๐โ1 of the form (2๐ + 1)/2๐ , we want ฮ๐ ๐ข(
2๐ + 1 2๐ + 1 ) = โ๐๐ ๐ข( ๐ ), 2๐ 2
and thus 2๐ 2๐ + 2 2๐ + 1 2๐ + 1 ) + ๐ข( ๐ ) โ 2๐ข( ๐ ) = โ๐๐ ๐ข( ๐ ), 2๐ 2 2 2 so ๐ข satisfies, for each ๐, 2๐ + 1 ๐ ๐+1 (12.1) (2 โ ๐๐ )๐ข( ๐ ) = ๐ข( ๐โ1 ) + ๐ข( ๐โ1 ). 2 2 2 Note that ๐/2๐โ1 , (๐ + 1)/2๐โ1 โ ๐๐โ1 so, provided ๐๐ โ 2, we have the extension algorithm ๐ข(
(12.2)
๐ ๐+1 ๐ข( ๐โ1 ) + ๐ข( ๐โ1 ) 2๐ + 1 2 2 . ๐ข( ๐ ) = 2 2 โ ๐๐
The equation (12.2) tells us how to extend ๐ข from ๐๐โ1 to ๐๐ , if we want ๐ข to be an eigenfunction of ฮ๐ with respect to โ๐๐ . Now, the eigenfunction equation must also be satisfied at points in ๐๐โ1 โงต ๐0 , so we also want ๐ ๐ ฮ๐ ๐ข( ๐โ1 ) = โ๐๐ ๐ข( ๐โ1 ) 2 2 for 1 โค ๐ โค 2๐โ1 โ 1, which gives the equation (12.3)
(2 โ ๐๐ )๐ข(
๐ 2๐ โ 1 2๐ + 1 ) = ๐ข( ๐ ) + ๐ข( ๐ ). 2 2 2๐โ1
By (12.2), ๐โ1 ๐ ๐ข( ๐โ1 ) + ๐ข( ๐โ1 ) 2๐ โ 1 2 2 ๐ข( ๐ ) = , 2 2 โ ๐๐
226
12. Eigenfunctions of the Laplacian
and substituting in (12.3) we obtain ๐โ1 ๐+1 ๐ ) + ๐ข( ๐โ1 ) = (2 โ 4๐๐ + ๐2๐ )๐ข( ๐โ1 ). 2๐โ1 2 2 Since ๐ข is assumed to be an eigenfunction of ฮ๐โ1 with respect to the eigenvalue โ๐๐โ1 , we also have ๐ข(
๐โ1 ๐+1 ๐ ) + ๐ข( ๐โ1 ) = (2 โ ๐๐โ1 )๐ข( ๐โ1 ), 2๐โ1 2 2 so we obtain the equation ๐ข(
2 โ ๐๐โ1 = 2 โ 4๐๐ + ๐2๐ , which can be written as (12.4)
๐๐โ1 = ๐๐ (4 โ ๐๐ ).
We have proven that we can always extend an eigenfunction of ฮ๐โ1 with respect to the eigenvalue โ๐๐โ1 to an eigenfunction of ฮ๐ with respect to โ๐๐ , provided ๐๐ โ 2 and the condition (12.4) holds. Together with (12.2), this provides an algorithm to construct all eigenfunctions of ฮ. Indeed, since the roots of (12.4) are ๐๐ = 2 ยฑ โ4 โ ๐๐โ1 , we can start from an eigenfunction of ฮ๐0 with respect to ๐๐0 , for some ๐0 , and extend to each ๐ > ๐0 choosing ๐๐ as any of these roots. Note that, if ๐ค = 2โโ4 โ ๐ง, using the principal branch of โ4 โ ๐ง on โโงต[4, โ), then ๐ง ๐ค = + ๐(|๐ง|2 ), 4 and we can conclude that 4๐ ๐๐ converges if we choose the minus sign for all but finitely many ๐ (Exercise (1)). Note that, if ๐๐0 โ 2, the starting eigenfunction on ๐๐0 will be an extension of its restriction to ๐๐0 โ1 , with respect to the eigenvalue โ๐๐0 โ1 given by (12.4). If ๐๐0 = 2, then, by (12.1) and (12.3), ๐ข( and
๐ ๐+1 ) + ๐ข( ๐ โ1 ) = 0, 2๐0 โ1 2 0
0 โค ๐ โค 2๐0 โ1 โ 1,
2๐ โ 1 2๐ + 1 1 โค ๐ โค 2๐0 โ1 โ 1, ) + ๐ข( ๐0 ) = 0, 2 ๐0 2 so we either have ๐ข|๐๐ โ1 โก 0 and ๐ข|๐๐ โงต๐๐ โ1 equal to an alternating 0 0 0 sequence of ยฑ1, or the other way around, as is shown in Figure 12.1. ๐ข(
12.2. Discrete eigenfunctions on the Sierpiลski gasket
227
Their limit corresponds to the eigenfunctions of the form sin(๐๐๐ฅ) and
Figure 12.1. Eigenfunctions of ฮ3 with respect to the eigenvalue ๐3 = 2. Note that they correspond to the restrictions to ๐ 3 of the functions sin(4๐๐ฅ) (top) and cos(4๐๐ฅ) (bottom).
cos(๐๐๐ฅ), respectively.
12.2. Discrete eigenfunctions on the Sierpiลski gasket We now study the eigenfunctions and eigenvalues of the Laplacian on the Sierpiลski gasket ๐ and, in particular, the possiblity of constructing them through an interpolation process as in the case of the interval. This time we donโt have any explicit formulae nor identities for the eigenfunctions, so we cannot prove directly that their restrictions to each ๐๐ are discrete eigenfunctions of the difference operators ฮ๐ given by (11.24). However, in this section we show how to extend an eigenfunction of ฮ๐โ1 on ๐๐โ1 to the next level, provided certain conditions are satisfied, and we discuss whether this process generates all eigenfunctions of the Laplacian ฮ on ๐. Assume ๐ข is an eigenfunction of ฮ๐โ1 on ๐๐โ1 with respect to the eigenvalue โ๐๐โ1 , so it satisfies the equation (12.5)
ฮ๐โ1 ๐ข(๐ฅ) = โ๐๐โ1 ๐ข(๐ฅ)
for every ๐ฅ โ ๐๐โ1 โงต๐0 . We want to extend to a function on ๐๐ , which we also denote by ๐ข, which will be an eigenfunction of ฮ๐ with eigenvalue โ๐๐ . Let ๐ฅ1 , ๐ฅ2 , ๐ฅ3 โ ๐๐โ1 be the boundary points of a cell ๐๐ค , for some
228
12. Eigenfunctions of the Laplacian
๐ค โ ๐๐โ1 . Let ๐ฆ1 , ๐ฆ2 , ๐ฆ3 โ ๐๐ โฉ ๐๐ค be the points in ๐๐ โงต ๐๐โ1 inside the cell, where each ๐ฆ ๐ is opposite to each ๐ฅ๐ , as in Figure 12.2.
Figure 12.2. The vertices in the cell ๐ ๐ค , for ๐ค โ ๐๐ . ๐ฅ1 , ๐ฅ2 , ๐ฅ3 โ ๐๐โ1 , while ๐ฆ 1 , ๐ฆ 2 , ๐ฆ 3 โ ๐๐ โงต ๐๐โ1 .
We want to construct ๐ข so that, for each ๐ฆ ๐ , ฮ๐ ๐ข(๐ฆ ๐ ) = โ๐๐ ๐ข(๐ฆ ๐ ). By (11.24), this gives us the three equations (12.6a)
๐ข(๐ฅ2 ) + ๐ข(๐ฅ3 ) + ๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 ) = (4 โ ๐๐ )๐ข(๐ฆ1 ),
(12.6b)
๐ข(๐ฅ1 ) + ๐ข(๐ฅ3 ) + ๐ข(๐ฆ1 ) + ๐ข(๐ฆ3 ) = (4 โ ๐๐ )๐ข(๐ฆ2 ),
(12.6c)
๐ข(๐ฅ1 ) + ๐ข(๐ฅ2 ) + ๐ข(๐ฆ1 ) + ๐ข(๐ฆ2 ) = (4 โ ๐๐ )๐ข(๐ฆ3 ),
which can be added to obtain (12.7)
(2 โ ๐๐ )(๐ข(๐ฆ1 ) + ๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 )) = 2(๐ข(๐ฅ1 ) + ๐ข(๐ฅ2 ) + ๐ข(๐ฅ3 )).
If ๐๐ โ 2, we obtain the relation ๐ข(๐ฆ1 ) + ๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 ) =
2(๐ข(๐ฅ1 ) + ๐ข(๐ฅ2 ) + ๐ข(๐ฅ3 )) , 2 โ ๐๐
which can be used, by adding ๐ข(๐ฆ ๐ ) to its corresponding equation in (12.6a)โ(12.6c), to obtain (12.8)
2๐ข(๐ฅ๐ ) + (4 โ ๐๐ )(๐ข(๐ฅ๐ ) + ๐ข(๐ฅ๐ )) = (5 โ ๐๐ )๐ข(๐ฆ ๐ ), 2 โ ๐๐
12.2. Discrete eigenfunctions on the Sierpiลski gasket
229
where ๐, ๐, ๐ are the three distinct numbers 1, 2, 3. Thus, if ๐๐ โ 5, we have the extension algorithm (12.9)
๐ข(๐ฆ ๐ ) =
2๐ข(๐ฅ๐ ) + (4 โ ๐๐ )(๐ข(๐ฅ๐ ) + ๐ข(๐ฅ๐ )) . (2 โ ๐๐ )(5 โ ๐๐ )
Note that (12.9) reduces to the extension algorithm (11.7) for harmonic functions if ๐๐ = 0. The resulting function ๐ข on ๐๐ will be an eigenfunction of ฮ๐ if it also satifies ฮ๐ ๐ข(๐ฅ) = โ๐๐ ๐ข(๐ฅ) for each ๐๐โ1 โงต ๐0 . Suppose that ๐ฅ1 above is not in ๐0 . Then ๐ฅ1 belongs to two cells in level ๐ โ 1, as in
Figure 12.3. If ๐ฅ1 โ ๐๐โ1 โงต ๐ 0 , then it belongs to two cells of level ๐ โ 1, with vertices ๐ฅ2 , ๐ฅ3 , ๐ฅ4 , ๐ฅ5 โ ๐๐โ1 and ๐ฆ 1 , ๐ฆ 2 , ๐ฆ 3 , ๐ฆ 4 , ๐ฆ5 , ๐ฆ 6 โ ๐๐ โงต ๐๐โ1 .
Figure 12.3. Thus, ฮ๐ ๐ข(๐ฅ1 ) = โ๐๐ ๐ข(๐ฅ1 ) implies the equation ๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 ) + ๐ข(๐ฆ4 ) + ๐ข(๐ฆ5 ) = (4 โ ๐๐ )๐ข(๐ฅ1 ).
230
12. Eigenfunctions of the Laplacian
As each ๐ข(๐ฆ ๐ ), ๐ = 2, 3, 4, 5, satisfies (12.9), we see that (12.10)
๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 ) + ๐ข(๐ฆ4 ) + ๐ข(๐ฆ5 ) =
4(4 โ ๐๐ )๐ข(๐ฅ1 ) + (6 โ ๐๐ )(๐ข(๐ฅ2 ) + ๐ข(๐ฅ3 ) + ๐ข(๐ฅ4 ) + ๐ข(๐ฅ5 )) . (2 โ ๐๐ )(5 โ ๐๐ )
Now, ๐ข(๐ฅ1 ) also satisfies ฮ๐โ1 ๐ข(๐ฅ1 ) = โ๐๐โ1 ๐ข(๐ฅ1 ), so we have ๐ข(๐ฅ2 ) + ๐ข(๐ฅ3 ) + ๐ข(๐ฅ4 ) + ๐ข(๐ฅ5 ) = (4 โ ๐๐โ1 )๐ข(๐ฅ1 ), which substituting in (12.10) gives us ๐ข(๐ฆ2 ) + ๐ข(๐ฆ3 ) + ๐ข(๐ฆ4 ) + ๐ข(๐ฆ5 ) =
4(4 โ ๐๐ )๐ข(๐ฅ1 ) + (6 โ ๐๐ )(4 โ ๐๐โ1 )๐ข(๐ฅ1 ) , (2 โ ๐๐ )(5 โ ๐๐ )
so 4(4 โ ๐๐ )๐ข(๐ฅ1 ) + (6 โ ๐๐ )(4 โ ๐๐โ1 )๐ข(๐ฅ1 ) = (4 โ ๐๐ )๐ข(๐ฅ1 ). (2 โ ๐๐ )(5 โ ๐๐ ) It is not hard to see that this must hold for every ๐ฅ โ ๐๐โ1 โงต ๐0 , and, since ๐ข is an eigenfunction of ฮ๐โ1 , it cannot be zero at all points, we obtain the relation 4(4 โ ๐๐ ) + (6 โ ๐๐ )(4 โ ๐๐โ1 ) = (4 โ ๐๐ )(2 โ ๐๐ )(5 โ ๐๐ ), which we can rewrite as (6 โ ๐๐ )๐๐โ1 = (6 โ ๐๐ )๐๐ (5 โ ๐๐ ). If ๐๐ โ 6, we obtain the condition (12.11)
๐๐โ1 = ๐๐ (5 โ ๐๐ ).
We summarize the previous analysis in Proposition 12.12. Proposition 12.12. Suppose that ๐๐ โ 2, 5, 6, and that we have the condition (12.11). (1) If ๐ข is an eigenfunction of ฮ๐โ1 on ๐๐โ1 with respect to the eigenvalue โ๐๐โ1 and is extended to ๐๐ by (12.9), then we obtain an eigenfunction of ฮ๐ with respect to the eigenvalue โ๐๐ . (2) If ๐ข is an eigenfunction of ฮ๐ on ๐๐ with respect to the eigenvalue โ๐๐ , then ๐ข|๐๐โ1 is an eigenfunction of ฮ๐โ1 with respect to โ๐๐โ1 .
12.2. Discrete eigenfunctions on the Sierpiลski gasket
231
The discrete process described in Proposition 12.12 can be used to construct eigenfunctions of the Laplacian ฮ on ๐. Indeed, note that the roots of (12.11) are given by ๐๐ =
5 ยฑ โ25 โ 4๐๐โ1 . 2
Using the principal branch of โ5 โ ๐ง on โ โงต [5, โ), we observe that, if ๐ค=
5 โ โ25 โ 4๐ง , 2
then ๐ค=
๐ง + ๐(|๐ง|2 ) 5
as ๐ง โ 0, and thus 3 ๐ 5 ๐๐ 2 converges if we start from some |๐๐0 | < 5 and choose the minus sign for all but infinitely many ๐ (Exercise (3)). Moreover, the functions ๐ข defined by Proposition 12.12 converge to a uniformly continuous function on ๐โ (Exercise (4)), which satisfies (12.13)
3 ๐ 3 5 ฮ๐ ๐ข(๐ฅ) = โ 5๐ ๐๐ ๐ข(๐ฅ) 2 2 3 for all ๐ฅ โ ๐โ โงต ๐0 . Thus, if 5๐ ๐๐ โ ๐, we can continuously extend ๐ข 2 to a function on ๐ in dom ฮ that satisfies ฮ๐ข = โ๐๐ข, and thus is an eigenfunction with respect to the eigenvalue โ๐. We described how to construct an eigenfunction of ฮ through this interpolation process, so we now dicuss whether we can obtain all eigenfunctions of ฮ in this way. First, we deal with the question of whether we can obtain any eigenvalue โ๐ as a limit (12.13) starting from a proper ๐ ๐0 . An eigenfunction ๐ข that satisfies ๐ข|๐0 = 0 is called a Dirichlet eigenfunction, and we say that an eigenvalue is a Dirichlet eigenvalue if it has a corresponding Dirichlet eigenfunction. Note that, if โ๐ is an eigenvalue that is not a Dirichlet eigenvalue, its corresponding eigenspace is at most 3-dimensional, because #๐0 = 3
232
12. Eigenfunctions of the Laplacian
and otherwise we could obtain a linear combination of 4 or more of its eigenfunctions to obtain 0 on ๐0 . Moreover, for any ๐1 โ 2, (12.9) defines an eigenfunction of ฮ1 for any choice of ๐ข(๐1 ), ๐ข(๐2 ) and ๐ข(๐3 ), so it defines a 3-dimensional space of eigenfunctions and thus, by taking the minus sign for every ๐ in the roots of (12.11), defines a 3-dimensional space of eigenfunctions with respect to each eigenvalue โ๐, if ๐ is a limit (12.13). To see which points in โ are such limits, define 5 โ โ25 โ 4๐ง 2 and, for |๐ง| < 2 (so to avoid ๐ง = 2), ๐(๐ง) =
ฮจ(๐ง) = lim 5๐ ๐๐ (๐ง), ๐โโ
๐
where ๐ is the composition of ๐ with itself ๐ times. ฮจ is well defined in this way (Exercise (3)), and by the above discussion, for any |๐1 | < 2, if 3 ๐ = 5ฮจ(๐1 ), 2 then โ๐ is an eigenvalue of ฮ. The convergence of the limit that defines ฮจ is uniform in compact subsets of ๐ต2 (0), so ฮจ is holomorphic in ๐ต2 (0). Moreover, since ๐โฒ (0) = 1/5, we see that each 5๐ ๐๐ (๐ง) has derivative 1 at 0, so ฮจโฒ (0) = 1. By the open mapping theorem of complex analysis (see, for instance, [Ull08, Theorem 5.7]), we see that ฮจ(๐ต2 (0)) contains a ball ๐ต๐ (0) for some ๐ > 0. Therefore, given any non-Dirichlet eigenvalue in the disc of radius 15๐/2, this eigenvalue and its corresponding eigenfunctions are the result of the discrete algorithm starting from some |๐1 | < 2. Now, for a larger non-Dirichlet eigenvalue โ๐, we choose ๐0 large enough so that |๐| < 3 โ
5๐0 ๐/2. Thus 3 ๐ = 5๐0 ฮจ(๐๐0 ) 2 for some ๐๐0 โ ๐ต2 (0), so we need to make sure that we can construct a 3dimensional eigenspace of ฮ๐0 with respect to โ๐๐0 . However, observe that the system of #(๐๐0 โงต ๐0 ) linear equations ฮ๐0 ๐ข(๐ฅ) = โ๐๐0 ๐ข(๐ฅ), can be written as ๐๐ข(๐ฅ) = ๐๐ฅ ,
12.3. Dirichlet eigenfunctions
233
where ๐ is the linear operator given, for each ๐ฅ โ ๐๐0 โงต ๐0 , by ๐๐ข(๐ฅ) = โ ๐ข(๐ฆ) โ (4 โ ๐๐0 )๐ข(๐ฅ), ๐ฆโผ๐ ๐ฅ ๐ฆโ๐0
and โ๐ข(๐ ๐ ) ๐๐ฅ = { 0
๐ฅ โผ๐ ๐ ๐ ๐ฅ is not a neighbor of a point ๐ ๐ โ ๐0 .
Thus, either we have a unique solution for each choice of ๐ข(๐1 ), ๐ข(๐2 ), ๐ข(๐3 ), so we have a 3-dimensional space of eigenfunctions with respect to โ๐๐0 , or โ๐๐0 is a Dirichlet eigenvalue, because in that case we would have solutions when all ๐๐ฅ = 0, and thus โ๐ would be a Dirichlet eigenvalue. We have proven that any eigenfunction with respect to a non-Dirichlet eigenvalue can be constructed through the discrete process. The Dirichlet eigenfunctions will be discussed in Section 12.3.
12.3. Dirichlet eigenfunctions In this section we discuss the construction of the Dirichlet eigenfunctions of ฮ on the Sierpiลski gasket ๐. For this, we will analyze more carefully the discrete process, keeping track on the number of linearly independent eigenfunctions constructed, in order to conclude that we have constructed all of them. We first construct all discrete Dirichlet eigenfunctions of the operators ฮ๐ , ๐ โฅ 1. We start with the Dirichlet eigenfunctions of ฮ1 on ๐1 , so we need to solve the three equations ฮ1 ๐ข(๐ฅ) = โ๐1 ๐ข(๐ฅ), one for each of the three points ๐ฅ โ ๐1 โงต ๐0 , with the Dirichlet condition ๐ข(๐1 ) = ๐ข(๐2 ) = ๐ข(๐3 ) = 0. By Exercise (5), we have three linearly independent solutions shown in Figure 12.4, with corresponding eigenvalues โ2 and โ5, the latter with multiplicity 2. Note that these are two of the forbidden eigenvalues of Proposition 12.12, which makes sense as their restrictions to ๐0 are not eigenfunctions of ฮ0 . Also, observe that one of the Dirichlet eigenfunctions with ๐1 = 5 is the rotation by 2๐/3 of the other, and that we have
234
12. Eigenfunctions of the Laplacian
Figure 12.4. The Dirichlet eigenfunctions of ฮ1 on ๐ 1 , corresponding to ๐1 = 2, 5 and 5, respectively. For simplicity, we only show the values of the eigenfunction at the vertices, and we donโt show the ones that have value zero.
a third eigenfunction corresponding to the next rotation. However, the latter is not linearly independent of the other two. If we remove the Dirichlet condition, we have two more linearly independent solutions for ๐1 = 2 and one more for ๐1 = 5, shown in Figure 12.5. Note that, for the case ๐1 = 2,
Figure 12.5. Eigenfunctions of ฮ1 on ๐ 1 corresponding to ๐1 = 2, 2 and 5, respectively. The third rotation of the first two of them is also an eigenfunction corresponding to ๐1 = 2, but is not linearly independent.
๐ข(๐1 ) + ๐ข(๐2 ) + ๐ข(๐3 ) = 0, which is a necessary condition given by (12.7). For ๐1 = 5, we have ๐ข(๐1 ) = ๐ข(๐2 ) = ๐ข(๐3 ), which follows from (12.8) (Exercise (6)). We now move on to the eigenvalues of ฮ2 on ๐2 . As #(๐2 โงต ๐0 ) = 12, we need to construct twelve linearly independent Dirichlet eigenfunctions. Although we could just solve explicitly the system of equations (12.14)
ฮ2 ๐ข(๐ฅ) = โ๐2 ๐ข(๐ฅ)
12.3. Dirichlet eigenfunctions
235
for ๐ฅ โ ๐2 โงต ๐0 , with ๐ข(๐ ๐ ) = 0, we will construct them in such a way that we can generalize for any ๐ โฅ 2. First, six of the Dirichlet eigenfunctions correspond to the extensions of the eigenfunctions of ฮ1 shown in Figure 12.4, with eigenvalues โ๐2 where 5 ยฑ โ25 โ 4๐1 ๐2 = . 2 Thus, we have six eigenfunctions with ๐2 given by 5 + โ17 5 โ โ17 5 + โ5 5 โ โ5 , , , , 2 2 2 2 the last two with multiplicity 2. The other six Dirichlet eigenfunctions are not extensions from any eigenfunction on ๐1 , so they must correspond to any of the forbidden values ๐2 = 2, 5 or 6; otherwise, their restrictions would be Dirichlet eigenfunctions on ๐1 , by Proposition 12.12. For ๐2 = 2, note that the restriction of such eigenfunction to each cell ๐ ๐ โฉ ๐2 must correspond to one of eigenfunctions with ๐1 = 2 shown in Figures 12.4 or 12.5, since the same difference equation must be satisfied. However, since (12.14) must also be satisfied in the points ๐1 โงต ๐0 , this impose an extra restriction of how the ๐1 -eigenfunctions are pasted into ๐2 . In fact, one sees that it is impossible to do this because, when one starts at one of the corners, the next one is either determined by this condition if one chooses to start with the eigenfunction of Figure 12.4, and then it is impossible to past the third one, or it is already impossible to paste the second one if we choose one of those shown in Figure 12.5. See Figure 12.6. Therefore, there are no Dirichlet eigenfunctions of ฮ2 with respect to the eigenvalue โ2. For ๐2 = 5, we now see that it is possible to paste two consecutive eigenfunctions shown in Figure 12.4, and their rotations, so we obtain the three linearly independent Dirichlet eigenfunctions shown in Figure 12.7. For ๐2 = 6, we first consider the eigenfunctions of ฮ1 with ๐1 = 6. These are shown in Figure 12.8. These are not Dirichlet eigenfunctions, of course, but they can be pasted consecutively (as in the case of ๐2 = 5) to form Dirichlet eigenvalues of ฮ2 , as is shown in Figure 12.9. Note that not only each is a rotation of the other, but each one is โcenteredโ at a point in ๐1 , the point where two of the eigenfunctions from Figure 12.8
236
12. Eigenfunctions of the Laplacian
Figure 12.6. Failed attempts to construct a Dirichlet eigenfunction on ๐ 2 with ๐2 = 2.
Figure 12.7. Three linearly independent Dirichlet eigenfunctions with ๐2 = 5, constructed by pasting two consecutive eigenfunctions on ๐ 1 .
Figure 12.8. The three linearly independent eigenfunctions ฮ1 with ๐1 = 6. Note that each is a rotation of the other.
where pasted. These form the last three linearly independent Dirichlet eigenfunctions of ฮ2 on ๐2 . We can now generalize these constructions for all ๐ โฅ 3. Since #(๐๐ โงต ๐0 ) =
3๐+1 โ 3 2
12.3. Dirichlet eigenfunctions
237
Figure 12.9. Three linearly independent Dirichlet eigenfunctions of ฮ2 on ๐ 2 with ๐2 = 6. Note that each is centered at a point in ๐ 1 .
(Exercise (8)), there must be (3๐+1 โ 3)/2 linearly independent Dirichlet eigenfunctions of ฮ๐ . First, note that, as in the case for ๐ = 2, there are no Dirichlet eigenfunctions with ๐๐ = 2, for ๐ โฅ 3. The argument is the same as above, and we leave it as an exercise (Exercise (7)). Also, note that if ๐๐โ1 = 6, then 5 ยฑ โ25 โ 4๐๐โ1 5ยฑ1 = = 2, 3, 2 2 so we cannot extend such Dirichlet eigenfunctions of ฮ๐โ1 using the negative sign above. Thus, in order to count the eigenfunctions of ฮ๐ that are extensions from ๐๐โ1 , we first need to count the discrete Dirichlet eigenfunctions with respect to the eigenvalue โ6. These are constructed as in the case ๐ = 2 above. Indeed, for each ๐๐ , we can paste two of the eigenfunctions in Figure 12.8 centered at each point ๐ฅ โ ๐๐โ1 , as in Figure 12.9. Hence, we have (3๐ โ 3)/2 such eigenfunctions. On ๐๐โ1 , we have then (3๐โ1 โ 3)/2 Dirichlet eigenfunctions with ๐๐โ1 = 6, so they extend to Dirichlet eigenfunctions of ฮ๐ with ๐๐ = 3. The number of Dirichlet eigenfunctions with ๐๐โ1 โ 6 is then 3๐ โ 3 3๐โ1 โ 3 โ = 3๐โ1 , 2 2 and they extend to 2 โ
3๐โ1 eigenfunctions of ฮ๐ with ๐๐ =
5 ยฑ โ25 โ 4๐๐โ1 . 2
Thus, the number of Dirichlet eigenfunctions of ฮ๐ extended from ๐๐โ1 is (3๐โ1 โ 3)/2 + 2 โ
3๐โ1 = (5 โ
3๐โ1 โ 3)/2 which, together with the
238
12. Eigenfunctions of the Laplacian
ones with ๐๐ = 6 that we already constructed, make a total of 8 โ
3๐โ1 โ 6 5 โ
3๐โ1 โ 3 3๐ โ 3 + = . = 4 โ
3๐โ1 โ 3. 2 2 2 Thus, we need to construct 3๐+1 โ 3 3๐โ1 + 3 โ (4 โ
3๐โ1 โ 3) = 2 2 more eigenfunctions. They will correspond to ๐๐ = 5. To construct these, first note that we can paste together a chain of eigenfunctions with ๐1 = 5 (as the last two in Figure 12.4 and their rotations) in each of the cells ๐๐ค , for ๐ค โ ๐๐โ1 , surrounding each downward triangular cycle formed with points in ๐ ๐ โงต ๐ ๐โ1 , for ๐ = 1, 2, . . . , ๐ โ 1. We show this for the largest and one of the next largest downward triangular cycles for ๐3 in Figure 12.10. As #๐ ๐ โงต ๐ ๐โ1 = 3๐ ,
Figure 12.10. Two of the Dirichlet eigenfunctions of ฮ3 with ๐3 = 5, chained around triangles with vertices in ๐ 1 โงต ๐ 0 and ๐ 2 โงต ๐ 1 , respectively.
we have that the number of downward triangular cycles with vertices in ๐ ๐ โงต ๐ ๐โ1 is 3๐โ1 , and thus the total number of such cycles is 3๐โ1 โ 1 . 2 We obtain two more linearly independent eigenfunctions by pasting a chain if eigenfunctions along the edge from vertex ๐1 to ๐2 , and another one from ๐2 to ๐3 . We thus have a total of (3๐โ1 + 3)/2 eigenfunctions with ๐๐ = 5, as required. 1 + 3 + 32 + 3๐โ2 =
We have thus constructed all Dirichlet eigenfunctions of the discrete Laplacian ฮ๐ on ๐๐ . As in Section 12.2, starting from each eigenfunction on ๐๐0 with ๐๐0 , the sequence obtained from discrete converges to
12.3. Dirichlet eigenfunctions
239
a uniformly continuous function on ๐โ , which extends continuously to a Dirichlet eigenfunction ๐ข of ฮ on ๐ with respect to the eigenvalue โ๐, where 3 ๐ = lim 5๐ ๐๐ . 2 ๐โโ However, recall that, for each ๐ > ๐0 , 5 + ๐๐ โ25 โ 4๐๐โ1 , 2 where all but infinitely many ๐๐ = โ1. This means that, for each finite sequence ๐๐0 +1 , ๐๐0 +2 , . . . , ๐๐ of ยฑ1, we obtain a distinct eigenvalue โ๐ with a distinct eigenfunction ๐ข when choosing ๐๐ = โ1 for all ๐ > ๐. Therefore, we obtain a sequence of Dirichlet eigenvalues and eigenfunctions from each starting ๐๐0 . ๐๐ =
In Figure 12.11 we show the Dirichlet eigenfunctions that we ob-
Figure 12.11. Dirichlet eigenfunctions of ฮ on ๐ with respect to the eigenvalues โ๐, with ๐ โ 16.816, ๐ โ 240.1686 and ๐ โ 920.6197, respectively, obtained from ๐1 = 2 when choosing ๐๐ = โ1 for all ๐ โฅ 2 in the first case, ๐2 = 1 and ๐๐ = โ1 for all ๐ โฅ 3 in the second, and ๐2 = ๐3 = 1 and ๐๐ = โ1 for all ๐ โฅ 4 in the third case.
tain when we start from the Dirichlet eigenfunction with respect to the eigenvalue โ2 shown in Figure 12.4, where ๐ข(๐ฅ) = 1 for ๐ฅ โ ๐1 โงต ๐0 . We obtain the first eigenfunction shown when we choose ๐๐ = โ1 for all ๐ โฅ 2. The first terms of the sequence ๐๐ in this case are 5 โ โ17 5 โ โ15 + 2โ17 5 โ โ15 + 2โ15 + 2โ17 , , , ... , 2 2 2 and one can approximate its normalized limit by 2,
3 lim 5๐ ๐๐ โ 16.816. 2 ๐โโ
240
12. Eigenfunctions of the Laplacian
We obtain the second eigenfunction when choosing ๐2 = 1 and ๐๐ = โ1 for all ๐ โฅ 3, so the first terms of the sequence ๐๐ are 5 + โ17 5 โ โ15 โ 2โ17 5 โ โ15 + 2โ15 โ 2โ17 , , , ... , 2 2 2 and its limit 3 lim 5๐ ๐๐ โ 240.1686. 2 ๐โโ We obtain the third eigenfunction in Figure 12.11 when choosing ๐2 = ๐3 = 1 and ๐๐ = โ1 for all ๐ โฅ 4, and in this case 3 lim 5๐ ๐๐ โ 920.6197. 2 ๐โโ In Figure 12.12 we show the eigenfunctions obtained starting with ๐1 = 2,
Figure 12.12. Dirichlet eigenfunctions of ฮ on ๐ with respect to the eigenvalues โ๐, with ๐ โ 55.8858, ๐ โ 172.3645 and ๐ โ 1032.0357.
5, corresponding to the second eigenfunction in Figure 12.4, and the same choice of starting sequences ๐2 , ๐3 as above. In this case we obtain the limits ๐ โ 55.8858, ๐ โ 172.3645, ๐ โ 1032.0357, respectively. 12.15. The argument in Section 12.2 can also be applied here to conclude that, if โ๐ is a Dirichlet eigenvalue of ฮ on ๐, then ๐ is the limit of (3/2)5๐ ๐๐ , starting from ๐1 = 2, ๐๐0 = 5 or ๐๐0 = 6, from an appropiately chosen ๐0 and a particular choice of ๐๐ , ๐ > ๐0 (Exercise (5)). Our construction above produces all Dirichlet eigenfunctions with respect to a Dirichlet eigenvalue โ๐. 12.16. If โ๐ is a Dirichlet eigenvalue, so is โ5๐ ๐, for each ๐ โฅ 1. Indeed, if ๐ is the discrete limit starting from ๐๐0 , then ๐ ฬ = 5๐ ๐ is the limit ฬ +๐ = ๐๐ . starting from ๐๐ 0 0
Exercises
241
It is also possible to prove that we can construct all eigenfunctions ๐ข with respect to a Dirichlet eigenvalue through this discrete algorithm, even those where ๐ข|๐0 โ 0. The argument involves Neumann derivatives at the vertices, which we wonโt discuss here. The reader can find the argument in [Str06].
Exercises (1) Let ๐(๐ง) = 2 โ โ4 โ ๐ง, for |๐ง| < 4, using the principal branch of โ4 โ ๐ง on โ โงต [4, โ). 1 (a) ๐(๐ง) = ๐ง + ๐(|๐ง|2 ) as ๐ง โ 0. 4 (b) Given |๐1 | < 4, the sequence defined by ๐๐ = ๐(๐๐โ1 ) and ๐ง๐ = 4๐ ๐๐ for ๐ โฅ 1 satisfies ๐ง๐ โ ๐ง๐โ1 = ๐(2โ๐ ). (c) ๐ง๐ is Cauchy and hence converges. (2) Let ๐๐ be the sequence defined in Exercise (1) with ๐1 = 2. Then 4๐ ๐๐ โ ๐2 . 5 โ โ25 โ 4๐ง 25 , for |๐ง| < , with the principal branch of 2 4 โ5 โ ๐ง on โ โงต [5, โ). 1 (a) ๐(๐ง) = ๐ง + ๐(|๐ง|2 ) as ๐ง โ 0. 5 25 (b) Given |๐1 | < , the sequence defined by ๐๐ = ๐(๐๐โ1 ) and 4 ๐ง๐ = 5๐ ๐๐ for ๐ โฅ 1 satisfies
(3) Let ๐(๐ง) =
๐ง๐ โ ๐ง๐โ1 = ๐(5โ๐ ). (c) ๐ง๐ is Cauchy and hence converges. (4) The sequence functions resulting from the discrete process on the Sierpiลski gasket constructs a uniformly continuous function on ๐โ . (5) Calculate the eigenvalues and eigenvectors of the matrix 4 โ1 โ1 (โ1 4 โ1) . โ1 โ1 4
242
12. Eigenfunctions of the Laplacian
(6) If ๐๐ = 5, the equation (12.8) implies that ๐ข(๐ฅ1 ) = ๐ข(๐ฅ2 ) = ๐ข(๐ฅ3 ). (7) For ๐ โฅ 2, ฮ๐ does not have any Dirichlet eigenfunctions on ๐2 with respect to the eigenvalue โ2. (8) (a) For any ๐ โฅ 0, #๐๐ =
3๐+1 + 3 . 2
(b) For any ๐ โฅ 1, 3๐+1 โ 3 . 2 (9) If โ๐ is a Dirichlet eigenvalue of ฮ on ๐, then there exists ๐0 and a sequence ๐๐ such that 3 ๐ = lim 5๐ ๐๐ , 2 ๐โโ with ๐๐0 = 2, 5 or 6. #(๐๐ โงต ๐0 ) =
(10) Let ๐ข be the Dirichlet eigenfunction obtained by discrete with ๐1 = 2, and choosing ๐๐ = โ1 for all ๐ โฅ 2. This is the first eigenfunction shown in Figure 12.11. Then ๐ข(๐ฅ) > 0 for all ๐ฅ โ ๐ โงต ๐0 .
Notes The algorithm described in this chapter for the construction of the eigenfunctions, as well as the relation for the discrete eigenvalues between levels, is known as the spectral decimation method and is due to Tadashi Shima [Shi91], who based his work from the results in [Ram84]. Our discussion follows the analysis presented in [Str06].
Chapter 13
Harmonic functions on post-critically finite sets
In this chapter we discuss how to construct the Laplacian on other fractal sets, analogous to the construction of the Laplacian on the Sierpiลski gasket.
13.1. Post-critically finite sets Let ๐พ โ โ๐ be a self-similar set with respect to the contractions ๐1 , ๐2 , . . . , ๐๐ . Recall that ๐พ is nonempty, compact, and satisfies ๐พ = ๐1 (๐พ) โช ๐2 (๐พ) โช . . . โช ๐๐ (๐พ). We assume that the contractions ๐๐ are restricted to ๐พ, so we see them as functions ๐๐ โถ ๐พ โ ๐พ. We use the notation of Chapter 10, so ๐๐ , for ๐ โฅ 1, is the set of words ๐ค = ๐ค 1 ๐ค 2 . . . ๐ค ๐ of length ๐ with each ๐ค ๐ โ {1, 2, . . . , ๐}. We define ๐๐ค = ๐๐ค1 โ ๐๐ค2 โ โฏ โ ๐๐ค๐ and ๐พ๐ค = ๐๐ค (๐พ). For ๐ = 0 we define ๐0 = {โ
}, where โ
is the empty word, ๐โ
is the identity function, and ๐พโ
= ๐พ. Each ๐พ๐ค , with ๐ค โ ๐๐ , is called a cell of level ๐. We have that ๐พ = โ๐คโ๐ ๐๐ค (๐พ) where, if ๐ผ๐ ๐ is the contraction constant of ๐๐ , diam(๐พ๐ค ) โค ๐ผ๐ค diam(๐พ), 243
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13. Harmonic functions on post-critically finite sets
with ๐ผ๐ค = ๐ผ๐ค1 ๐ผ๐ค2 โฏ ๐ผ๐ค๐ . Since each ๐ผ๐ < 1, then diam(๐พ๐ค ) โ 0 if ๐ โ โ. Thus, for each sequence ๐ = ๐ค 1 , ๐ค 2 , . . ., if we denote ๐๐ = ๐ค 1 ๐ค 2 . . . ๐ค ๐ , then โ๐โฅ1 ๐พ๐๐ contains exactly one point of ๐พ. Conversely, if ๐ฅ โ ๐พ, there exists a sequence ๐ such that (13.1)
โ
๐พ๐๐ = {๐ฅ}.
๐โฅ1
For a self-similar set ๐พ, define the set (13.2)
๐=
โ
๐พ๐ โฉ ๐พ๐ .
๐โ ๐
Then ๐ is the set of all overlaps between the images ๐พ๐ of ๐พ under the contractions. ๐ is called the critical set. We assume ๐ โ โ
, and define the post-critical set of ๐พ by (13.3)
๐0 =
โ โ
๐๐คโ1 (๐),
๐โฅ1 ๐คโ๐๐
Observe that ๐0 is the union of all possible pre-images, under any number of iterations of the ๐๐ , of the critical set. We say that ๐พ is a post-critically finite set, denoted as PCF set, if ๐0 is finite. Example 13.4. Consider the Sierpiลski gasket ๐ discussed in the previous chapters. Its critical set ๐ consists of the three middle points of its sides, which clearly are the images of the triangle vertices ๐1 , ๐2 , ๐3 (see Figure 11.2) under the contractions ๐1 , ๐2 , ๐3 . Thus ๐1 , ๐2 , ๐3 โ ๐0 . Now, since ๐1 โ ๐ 2 nor ๐1 โ ๐ 3 , and ๐1 is the fixed point of ๐1 , the further preimages of ๐1 contain only the point ๐1 . Similarly for ๐2 and ๐3 , which are the fixed points of ๐2 and ๐3 , respectively, we have that ๐0 = {๐1 , ๐2 , ๐3 }. Therefore ๐ is a PCF set. Example 13.5. The interval ๐ผ = [0, 1], seen as a self-similar set with contractions ๐1 (๐ฅ) = ๐ฅ/2 and ๐2 (๐ฅ) = ๐ฅ/2+1/2, is also a PCF set. Indeed, ๐ = {1/2} and ๐0 = {0, 1}. Note that, in both of the previous examples, ๐0 is known as the boundary of ๐ and ๐ผ, respectively. In general, ๐0 is called the boundary of the PCF set ๐พ. Example 13.6. The Hata tree set ๐พ of Example 10.22, with contractions 2 1 ๐1 (๐ง) = ๐๐ง,ฬ ๐2 (๐ง) = ๐ง ฬ + , 3 3
13.2. Harmonic structures and discrete energy
245
โ3 1 where ๐ = + ๐. Its critical set consists only of the point 1/3 (see 2 6 Figure 10.6), which is given by 1 = ๐1 (๐) = ๐2 (0). 3 Thus ๐, 0 โ ๐0 . Now ๐ = ๐1 (1), so 1 โ ๐0 . Since 0, ๐ โ ๐พ1 โงต ๐พ2 and 1 โ ๐พ2 โงต ๐พ1 , and 0 and 1 are the fixed points of ๐1 and ๐2 , respectively, we conclude that ๐0 = {๐, 0, 1}. For each ๐ โฅ 1, we define ๐๐ = ๐1 (๐๐โ1 ) โช ๐2 (๐๐โ1 ) โช . . . โช ๐๐ (๐๐โ1 ). ๐๐ is the set of vertices of level ๐. We see that ๐๐ =
โ
๐๐ค (๐0 )
๐คโ๐๐
and, by (13.3), each ๐๐โ1 โ ๐๐ . The set of all vertices is denoted by ๐โ , so ๐โ =
โ
๐๐ .
๐โฅ0
By (13.1), ๐โ is dense in ๐พ (Exercise (1)). If we denote ๐0 = {๐1 , ๐2 , . . . , ๐๐ }, where ๐ = #๐0 , then each ๐ฅ โ ๐๐ is of the form ๐ฅ = ๐๐ค (๐ ๐ ) for some ๐ค โ ๐๐ and ๐ = 1, 2, . . . , ๐. We say that two vertices ๐ฅ, ๐ฆ โ ๐๐ are adjacent, or neighbors, and write ๐ฅ โผ๐ ๐ฆ, if there exists ๐ค โ ๐๐ such that ๐ฅ = ๐๐ค (๐ ๐ ) and ๐ฆ = ๐๐ค (๐๐ ), for some ๐, ๐. That is, ๐ฅ, ๐ฆ โ ๐๐ค (๐0 ), so they belong to the same cell ๐พ๐ค .
13.2. Harmonic structures and discrete energy For a nonempty finite set ๐, let ๐(๐) be the vector space of real valued functions on ๐. Thus, if #๐ = ๐, then ๐(๐) โ
โ๐ . ๐(๐) has the inner product โจ๐ข, ๐ฃโฉ = โ ๐ข(๐ฅ)๐ฃ(๐ฅ), ๐ฅโ๐
so โจโ
, โ
โฉ is essentially the dot product in โ๐ .
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13. Harmonic functions on post-critically finite sets
The standard basis of ๐(๐) is given by the functions ๐{๐ฅ} , ๐ฅ โ ๐. A linear operator ๐ on ๐(๐) can be identified with the ๐ ร๐ matrix which, with respect to the standard basis, has entries ๐๐ฅ๐ฆ = โจ๐๐{๐ฅ} , ๐{๐ฆ} โฉ. Any bilinear form ๐ต on ๐(๐) is induced by an operator ๐ via ๐ต(๐ข, ๐ฃ) = โโจ๐๐ข, ๐ฃโฉ. We use the negative sign for technical reasons that will be clear below. If ๐ต is symmetric, that is ๐ต(๐ข, ๐ฃ) = ๐ต(๐ฃ, ๐ข) for any ๐ข, ๐ฃ โ ๐(๐), then ๐ is symmetric. If ๐ต is symmetric, we denote ๐ต(๐ข, ๐ข) simply by ๐ต(๐ข), so ๐ข โฆ ๐ต(๐ข) is a quadratic form on ๐(๐). The symmetric bilinear form โฐ is a Dirichet form if it satisfies (1) โฐ(๐ข) โฅ 0 for all ๐ข โ ๐(๐); (2) โฐ(๐ข) = 0 if and only if ๐ข is constant; and (3) โฐ(๐ข)ฬ โค โฐ(๐ข) for every ๐ข โ ๐(๐), where ๐ข(๐ฅ) < 0 โง0 ๐ข(๐ฅ) ฬ = ๐ข(๐ฅ) 0 โค ๐ข(๐ฅ) โค 1 โจ ๐ข(๐ฅ) > 1 โฉ1 is the cut of ๐ข by [0, 1]. 13.7. Any bilinear form โฐ given by โฐ(๐ข, ๐ฃ) = โ ๐๐ฅ๐ฆ (๐ข(๐ฅ) โ ๐ข(๐ฆ))(๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)), ๐ฅ,๐ฆโ๐
with ๐๐ฅ๐ฆ โฅ 0, is a Dirichlet form if and only if, for each ๐ฅ0 , ๐ฆ0 โ ๐, we can find ๐ฅ1 , ๐ฅ2 , . . . , ๐ฅ๐ = ๐ฆ0 such that ๐๐ฅ๐โ1 ๐ฅ๐ > 0. We just need to observe that โฐ(๐ข) = โ ๐๐ฅ๐ฆ (๐ข(๐ฅ) โ ๐ข(๐ฆ))2 ๐ฅ,๐ฆโ๐
is a nonnegative linear combination of the squares (๐ข(๐ฅ) โ ๐ข(๐ฆ))2 , so (1) and (3) are clearly satisfied, and that โฐ(๐ข) = 0 when, for each pair ๐ฅ, ๐ฆ โ ๐, ๐๐ฅ๐ฆ = 0 or ๐ข(๐ฅ) = ๐ข(๐ฆ), so the condition implies that this can only happen if ๐ข is constant. Note that โฐ(๐ข, ๐ฃ) is the polarization of โฐ(๐ข).
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247
Let ๐พ be a PCF set and consider the sequence ๐๐ of vertices of level ๐, defined for ๐ โฅ 0 as above. For each ๐ โฅ 0, let โฐ๐ be a Dirichlet form on ๐(๐๐ ). We say that the sequence โฐ๐ is a harmonic structure on ๐พ if is satisfies: โข (self-similarity) there exist numbers 0 < ๐1 , ๐2 , . . . , ๐๐ < 1 such that ๐
(13.8)
1 โฐ๐โ1 (๐ข โ ๐๐ , ๐ฃ โ ๐๐ ) ๐ ๐=1 ๐
โฐ๐ (๐ข, ๐ฃ) = โ
for all ๐ข, ๐ฃ โ (๐๐ ) and ๐ โฅ 1; and โข (compatibility) for each ๐ โฅ 1 and ๐ฃ โ ๐(๐๐โ1 ), (13.9)
โฐ๐โ1 (๐ฃ) = min{โฐ๐ (๐ข) โถ ๐ข โ ๐(๐๐ ), ๐ข|๐๐โ1 = ๐ฃ}.
We call each Dirichlet form โฐ๐ on ๐(๐๐ ) the energy on ๐๐ . For each ๐ข โ ๐(๐๐ ) and ๐ = 1, 2, . . . , ๐, ๐ข โ ๐๐ defines a function on ๐๐โ1 , so selfsimilarity of the harmonic structure โฐ๐ means that the energy on each level can be distributed over each cell ๐พ๐ , weighted by the 1/๐๐ (although they are not really weights). We canโinductivelyโverify that the selfsimilarity property (13.8) implies that (13.10)
โฐ๐ (๐ข, ๐ฃ) = โ ๐คโ๐๐
1 โฐ (๐ข โ ๐๐ค , ๐ฃ โ ๐๐ค ), ๐๐ค 0
for any ๐ข, ๐ฃ โ ๐(๐๐ ), where ๐๐ค = ๐๐ค1 ๐๐ค2 โฏ ๐๐ค๐ . Hence, the harmonic structure is determined by the initial energy โฐ0 on ๐0 and the numbers ๐1 , ๐2 , . . . , ๐๐ . The compatibility property (13.9) means that the energy is preserved between levels. Inductively, using and (13.10) we can see that, given ๐ฃ โ ๐(๐0 ), (13.11)
โฐ0 (๐ฃ) = min{โฐ๐ (๐ข) โถ ๐ข โ ๐(๐๐ ), ๐ข|๐0 = ๐ฃ}
for every ๐ โฅ 1. We can then prove by induction that, given โฐ0 and ๐1 , ๐2 , . . . , ๐๐ , if โฐ๐ is given by (13.10) and (13.11) is satisfied with ๐ = 1, then (13.11) is satisfied for every ๐ โฅ 1 (Exercise (4)). If ๐ข โ ๐(๐๐ ) satifies (13.11), we say that ๐ข is harmonic on ๐๐ . A harmonic function is a function ๐ข on ๐โ such that each ๐ข|๐๐ is harmonic on ๐๐ , for every ๐.
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13. Harmonic functions on post-critically finite sets
Example 13.12. Consider the interval [0, 1] with contractions as above. The vertices ๐๐ of level ๐ correspond to the dyadic partition 1 2 2๐ โ 1 , , . . . , , 1}. 2๐ 2๐ 2๐ The sequence of energies defined in (11.3), ๐๐ = {0,
2๐
โฐ๐ (๐ข) = 2
๐
โ (๐ข( ๐=1
๐โ1 ๐ 2 ) โ ๐ข( ๐ )) , 2๐ 2
is a harmonic structure on ๐ผ, as discussed in Section 11.1. Harmonic functions are restrictions to ๐โ of linear functions. Example 13.13. Similarly, the sequence of energies on the vertices of the Sierpiลski gasket ๐ discussed in Section 11.2, 2 5 ๐ โ (๐ข(๐ฅ) โ ๐ข(๐ฆ)) , โฐ๐ (๐ข) = ( ) 3 ๐ฅ,๐ฆโ๐ ๐ฅโผ๐ฆ
๐
is a harmonic structure on ๐. Harmonic functions can be constructed by interpolation, as discussed in that section. Example 13.14. Consider now the Hata tree set ๐พ described in Example 13.6. We have that ๐0 = {๐, 0, 1}. For โ > 0, we consider the Dirichlet form on ๐(๐0 ) given by (13.15)
โฐ0 (๐ข) = (๐ข(0) โ ๐ข(1))2 + โ(๐ข(0) โ ๐ข(๐))2 .
โฐ0 is a Dirichlet form, as discussed in 13.7. For ๐1 , ๐2 > 0, we have, for ๐ข โ ๐(๐1 ), โฐ1 (๐ข) =
1 ((๐ข(0) โ ๐ข(๐))2 + โ(๐ข(0) โ ๐ข(1/3))2 ) ๐1 1 + ((๐ข(1/3) โ ๐ข(1))2 + โ(๐ข(1/3) โ ๐ข(๐))2 ), ๐2
where ๐ = ๐2 (๐) (see Figure 13.1). Given ๐ข(๐) = ๐ผ, ๐ข(0) = ๐ฝ and ๐ข(1) = ๐พ, we want to find ๐ข(1/3) = ๐ฅฬ and ๐ข(๐) = ๐ฆ ฬ such that the quadratic function 1 1 ๐(๐ฅ, ๐ฆ) = ((๐ผ โ ๐ฝ)2 + โ(๐ฝ โ ๐ฅ)2 ) + ((๐ฅ โ ๐พ)2 + โ(๐ฅ โ ๐ฆ)2 ) ๐1 ๐2 ฬ and ๐1 , ๐2 such that 0 < ๐1 , ๐2 < 1 and takes its minimum at (๐ฅ,ฬ ๐ฆ), ๐(๐ฅ,ฬ ๐ฆ)ฬ = โ(๐ผ โ ๐ฝ)2 + (๐ฝ โ ๐พ)2 .
13.2. Harmonic structures and discrete energy
249
c
1/ 3
0
1 d
Figure 13.1. The set ๐ 1 of vertices of level 1 of the Hata tree set.
By elementary calculus ๐ takes its minimum at ๐ฅฬ = ๐ฆ ฬ =
(13.16)
โ๐2 ๐ฝ + ๐1 ๐พ โ๐2 + ๐1
and ๐(๐ฅ,ฬ ๐ฆ)ฬ =
1 โ (๐ผ โ ๐ฝ)2 + (๐ฝ โ ๐พ)2 . ๐1 โ๐2 + ๐1
Thus 1 =โ ๐1
and
โ = 1, โ๐2 + ๐1
so 1 1 and ๐2 = 1 โ 2 = 1 โ ๐12 . โ โ Note that 0 < ๐1 , ๐2 < 1 if and only if โ > 1. Thus, we have a family of harmonic structures on the Hata set, one for each number โ > 1. (13.17)
๐1 =
By (13.16), a harmonic function ๐ข on ๐1 satisfies 1 1 )๐ข(0) + 2 ๐ข(1). 2 โ โ Note that (13.18) is a convex combination of ๐ข(0) and ๐ข(1), and that it does not depend on the value of ๐ข(๐). By the observations above, (13.18) provides an algorithm to construct a harmonic function on ๐โ . Figure 13.2 shows harmonic functions with โ = 2 and boundary values ๐ข = ๐{1} and ๐ข = ๐{๐} . (13.18)
๐ข(1/3) = ๐ข(๐) = (1 โ
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13. Harmonic functions on post-critically finite sets
Figure 13.2. Harmonic functions on the Hata tree set with โ = 2 and boundary values ๐ข(๐) = ๐ข(0) = 0 and ๐ข(1) = 1 on the left, and ๐ข(๐) = 1 and ๐ข(0) = ๐ข(1) = 0 on the right.
13.3. Discrete Laplacians Let ๐ be a finte set and ๐ป a symmetric operator on ๐(๐). ๐ป is a Laplacian on ๐ if (a) ๐ป is nonpositive definite; (b) ๐ป๐ข = 0 if and only if ๐ข is a constant; and (c) ๐ป๐ฅ๐ฆ โฅ 0 for all ๐ฅ, ๐ฆ โ ๐, ๐ฅ โ ๐ฆ. Let โฐ be the bilinear form induced by ๐ป, so โฐ(๐ข, ๐ฃ) = โโจ๐ป๐ข, ๐ฃโฉ. It is clear that โฐ(๐ข) โฅ 0 for all ๐ข if and only if ๐ป is nonpositive definite, by definition. By the spectral theorem, โฐ(๐ข) = 0 if and only if ๐ข is a constant is equivalent to saying that ๐ป๐ข = 0 if and only if ๐ข is a constant (Exercise (6)). We have Proposition 13.19. Proposition 13.19. The bilinear form โฐ induced by ๐ป is a Dirichlet form if and only if ๐ป is a Laplacian on ๐.
13.3. Discrete Laplacians
251
Proof. We have seen that (1) and (2) of the definition of a Dirichlet form are equivalent to (a) and (b) of the definition of a Laplacian. Now, โฐ(๐ข, ๐ฃ) = โโจ๐ป๐ข, ๐ฃโฉ = โ โ ๐ป๐ข(๐ฅ)๐ฃ(๐ฅ) ๐ฅโ๐
= โ โ ๐ป๐ฅ๐ฆ ๐ข(๐ฆ)๐ฃ(๐ฅ) ๐ฅ,๐ฆโ๐
=
1 โ ๐ป (๐ข(๐ฅ) โ ๐ข(๐ฆ))(๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)), 2 ๐ฅ,๐ฆโ๐ ๐ฅ๐ฆ
using the fact that โ ๐ป๐ฅ๐ฆ = 0 ๐ฆโ๐
for each ๐ฅ, because ๐ป is zero on a constant. Hence, if ๐ป๐ฅ๐ฆ โฅ 0 for all pairs ๐ฅ โ ๐ฆ, โฐ is a Dirichlet form by 13.7. Suppose some ๐ป๐ฅ๐ฆ < 0. By multiplying by a positive constant, we can assume ๐ป๐ฅ๐ฆ = โ1. Let ๐ก < 0 and define ๐ข on ๐ by setting ๐ข(๐ฅ) = 1, ๐ข(๐ฆ) = ๐ก and ๐ข(๐ง) = 0 for all ๐ง โ ๐, ๐ง โ ๐ฅ, ๐ฆ. Then โฐ(๐ข) =
1 โ ๐ป (๐ข(๐ง) โ ๐ข(๐ค))2 = โ(1 โ ๐ก)2 + ๐ด๐ก2 + ๐ต, 2 ๐ง,๐คโ๐ ๐ง๐ค
for some ๐ด, ๐ต โฅ 0. Since ๐ก < 0, the cut ๐ขฬ of ๐ข by [0, 1] is ๐ข(๐ฅ) = 1 and ๐ข(๐ง) = 0 for all ๐ง โ ๐ฅ, so โฐ(๐ข)ฬ = โ1 + ๐ต. Thus โฐ(๐ข) โ โฐ(๐ข)ฬ = 2๐ก + (๐ด โ 1)๐ก2 < 0 if ๐ด โค 1 or 0>๐ก>โ
2 , ๐ดโ1
so โฐ doesnโt satisfy (3) of the definition of Dirichlet form.
โก
Hence, if โฐ๐ is a harmonic structure on the PCF set ๐พ, then we have a sequence of Laplacians ๐ป๐ on ๐๐ , where each ๐ป๐ induces โฐ๐ . The self-similarity condition (13.8) gives a relation between each ๐ป๐โ1 and ๐ป๐ , which then defines an extension of ๐ป๐โ1 on ๐๐โ1 to ๐๐ . We leave this as an exercise (Exercise (9)). To write the compatiblity condition (13.9) in terms of Laplacians, we make the following observations. Let ๐ a finite set and ๐ โ ๐ a proper
252
13. Harmonic functions on post-critically finite sets
subset. If ๐ป is a Laplacian on ๐, we write the matrix of ๐ป, which we also denote by ๐ป, in blocks ๐ ๐ป=( ๐ฝ
(13.20)
๐ฝ๐ก ), ๐
where ๐ โถ ๐(๐) โ ๐(๐), ๐ฝ โถ ๐(๐) โ ๐(๐ โงต๐) and ๐ โถ ๐(๐ โงต๐) โ ๐(๐ โงต๐). Hence, if we write ๐ข โ ๐(๐) as ๐ข ๐ข = ( 0) , ๐ข1 where ๐ข0 = ๐ข|๐ and ๐ข1 = ๐ข|๐ โงต๐ , then (13.21)
๐ป๐ข = (
๐๐ข0 + ๐ฝ ๐ก ๐ข1 ). ๐ฝ๐ข0 + ๐๐ข1
๐ is invertible. Indeed, for ๐ฃ โ ๐(๐ โงต ๐), consider its extension ๐ข โ ๐(๐) with ๐ข|๐ = 0 and ๐ข|๐ โงต๐ = ๐ฃ. Then, if โฐ is the Dirichlet form induced by ๐ป, by (13.21) we have โฐ(๐ข) = โโจ๐ป๐ข, ๐ขโฉ = โโจ (
๐ฝ๐ก๐ฃ 0 ) , ( ) โฉ = โโจ๐๐ฃ, ๐ฃโฉ. ๐๐ฃ ๐ฃ
Thus ๐๐ฃ = 0 implies โฐ(๐ข) = 0, so ๐ข must be constant and therefore ๐ฃ = 0. Note that this also implies that ๐ is negative definite. We can thus write โฐ(๐ข) = โโจ๐๐ข0 + ๐ฝ ๐ก ๐ข1 , ๐ข0 โฉ โ โจ๐ฝ๐ข0 + ๐๐ข1 , ๐ข1 โฉ = โจ(๐ โ ๐ฝ ๐ก ๐ โ1 ๐ฝ)๐ข0 , ๐ข0 โฉ โ โจ๐(๐ข1 + ๐ โ1 ๐ฝ๐ข0 ), ๐ข1 + ๐ โ1 ๐ฝ๐ข0 โฉ Given ๐ข0 on ๐, we see that โฐ(๐ข) is minimal when โจ๐(๐ข1 + ๐ โ1 ๐ฝ๐ข0 ), ๐ข1 + ๐ โ1 ๐ฝ๐ข0 โฉ = 0, because ๐ is negative definite. Thus we require (13.22)
๐ข1 + ๐ โ1 ๐ฝ๐ข0 = 0
and โฐ(๐ข) = โฐโฒ (๐ข0 ), where โฐโฒ is the quadratic form induced by ๐ โ ๐ฝ ๐ก ๐ โ1 ๐ฝ. Equations (13.21) and (13.22) imply that a minimizer function ๐ข satifies (13.23)
๐ป๐ข|๐ โงต๐ = 0.
13.3. Discrete Laplacians
253
13.24. If ๐ข โ ๐(๐) satisfies (13.23), then, for all ๐ฅ โ ๐, min ๐ข(๐ฆ) โค ๐ข(๐ฅ) โค max ๐ข(๐ฆ). ๐ฆโ๐
๐ฆโ๐
If we have any of the above equalities for some ๐ฅ โ ๐ โงต ๐, then ๐ข is constant. Thus, a minimizer function satisfies the maximum principle. Indeed, for any ๐ฅ โ ๐, ๐ป๐ข(๐ฅ) = โ ๐ป๐ฅ๐ฆ ๐ข(๐ฆ) = โ ๐ป๐ฅ๐ฆ (๐ข(๐ฆ) โ ๐ข(๐ฅ)), ๐ฆโ๐
๐ฆโ๐
because โ๐ฆโ๐ ๐ป๐ฅ๐ฆ = 0 for all ๐ฅ. Thus, if ๐ฅ โ ๐ โงต ๐, ๐ป๐ข(๐ฅ) = 0 and thus (13.25)
โ ๐ป๐ฅ๐ฆ (๐ข(๐ฆ) โ ๐ข(๐ฅ)) = 0. ๐ฆโ๐
If ๐ข takes its maximum at ๐ฅ โ ๐ โงต ๐, (13.25) implies that ๐ข(๐ฆ) = ๐ข(๐ฅ) whenever ๐ป๐ฅ๐ฆ > 0, because each ๐ป๐ฅ๐ฆ โฅ 0 and ๐ข(๐ฆ) โ ๐ข(๐ฅ) โค 0, so ๐ข = ๐ข(๐ฅ) on the set of points ๐ฆ such that ๐ป๐ฅ๐ฆ > 0. By 13.7, for any ๐ฆ โ ๐ there exists a sequence ๐ฆ0 = ๐ฅ, ๐ฆ1 , ๐ฆ2 , . . ., ๐ฆ๐ = ๐ฆ such that ๐ป๐ฆ๐โ1 ๐ฆ๐ > 0, so recursively the above argument shows that ๐ข(๐ฆ) = ๐ข(๐ฅ), and ๐ข is a constant. If we apply the previous analysis with ๐ = ๐๐ and ๐ = ๐๐โ1 , we see that the compatibility condition (13.9) is equivalent to (13.26)
๐ป๐โ1 = ๐ โ ๐ฝ ๐ก ๐ โ1 ๐ฝ,
where ๐, ๐ฝ and ๐ are the blocks of ๐ป๐ as in (13.20). Note that the minimizer function ๐ข โ ๐(๐๐ ), for a given ๐ฃ โ ๐(๐๐โ1 ), is given by extending (13.27)
๐ข|๐๐ โงต๐๐โ1 = โ๐ โ1 ๐ฝ๐ฃ.
By (13.21), ๐ข is a minimizer for a given ๐ฃ โ ๐(๐๐โ1 ) if (13.28)
๐ป๐ ๐ข|๐๐ โงต๐๐โ1 = 0.
By (13.11) and (13.28), ๐ข is harmonic if (13.29)
๐ป๐ (๐ข|๐๐ )(๐ฅ) = 0
for all ๐ฅ โ ๐๐ โงต ๐0 . Since (13.29) is a linear equation, the set of harmonic functions is a vector space. If #๐0 = ๐, this vector space is ๐-dimensional. Moreover, (13.27) provides an algorithm to construct a harmonic function by interpolation to each level ๐ from level ๐ โ 1.
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13. Harmonic functions on post-critically finite sets
13.30. As a consequence of (13.28) and 13.24, we have that harmonic functions satisfy the maximum principle: if ๐ข is harmonic, then min ๐ข โค ๐ข(๐ฅ) โค max ๐ข ๐0
๐0
for all ๐ฅ โ ๐โ , and we have any of the equalities for some ๐ฅ โ ๐โ โงต ๐0 only if ๐ข is a constant. 13.31. If ๐ข is harmonic, then it is uniformly continuous in ๐โ , so it has a unique continuous extension to ๐พ. To see this, choose first ๐ large enough so that, if ๐ฅ โผ๐ ๐ฆ, at most one of them belong to ๐0 . If ๐ข is harmonic and not a constant, then (13.32) max{|๐ข(๐ฅ) โ ๐ข(๐ฆ)| โถ ๐ฅ โผ๐ ๐ฆ} < max{|๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐0 }, where ๐ฃ = ๐ข|๐0 is the boundary value of ๐ข. By iterating (13.27), for each ๐ค โ ๐๐ there exists an operator ๐๐ค โถ ๐(๐0 ) โ ๐(๐๐ค (๐0 )) such that ๐ข|๐๐ค (๐0 ) = ๐๐ค ๐ฃ. Note that, if ๐ฃ(๐ฅ) ฬ = ๐ฃ(๐ฅ) โ
1 โ ๐ฃ(๐ฆ), ๐ ๐ฆโ๐ 0
then max{|๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐0 } = max{|๐ฃ(๐ฅ) ฬ โ ๐ฃ(๐ฆ)| ฬ โถ ๐ฅ, ๐ฆ โ ๐0 }, and โ๐ฅโ๐ ๐ฃ(๐ฅ) ฬ = 0. By Exercise (12) and (13.32), there exists ๐พ๐ค < 1 0 such that max{|๐๐ค ๐ฃ(๐ฅ) โ ๐๐ค ๐ฃ(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐0 } โค ๐พ๐ค max{|๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐0 }, and thus max{|๐ข(๐ฅ) โ ๐ข(๐ฆ)| โถ ๐ฅ โผ๐ ๐ฆ} โค ๐พ max{|๐ฃ(๐ฅ) โ ๐ฃ(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐0 }, where ๐พ = max{๐พ๐ค โถ ๐ค โ ๐๐ } < 1. Thus, for any ๐ โฅ 1, if ๐ฅ โผ๐๐ ๐ฆ then |๐ข(๐ฅ) โ ๐ข(๐ฆ)| โค ๐ด๐พ๐ , which implies that ๐ข is uniformly continuous because ๐ด๐พ๐ โ 0. We say that ๐ข โ ๐ถ(๐พ) is harmonic if ๐ข|๐โ is a harmonic function.
13.4. The Laplacian on a PCF set
255
13.4. The Laplacian on a PCF set A Laplacian on a PCF set can be constructed in the same way that we constructed a Laplacian on the Sierpiลski gasket in Section 11.3. For a given ๐ข โ ๐ถ(๐พ), the sequence โฐ๐ (๐ข), where we are denoting the restriction of ๐ข to ๐๐ simply by ๐ข, is increasing, so we define โฑ = {๐ข โ ๐ถ(๐พ) โถ โฐ๐ (๐ข) is bounded} and define the bilinear form โฐ on โฑ by โฐ(๐ข, ๐ฃ) = lim โฐ๐ (๐ข, ๐ฃ). We also call โฐ(๐ข) the energy of ๐ข on ๐พ. As in the case of the Sierpiลski gasket, โฑ contains constant and harmonic functions, as well as ๐-harmonic functions, where a function ๐ข โ ๐ถ(๐พ) is called ๐-harmonic if ๐ข โ ๐๐ค is harmonic for each ๐ค โ ๐๐ . Let ๐ > 0 be the unique number such that ๐ ๐1๐ + ๐2๐ + . . . + ๐๐ = 1.
(13.33)
For each cell ๐พ๐ค , we define ๐(๐พ๐ค ) = ๐๐ค๐ . By (13.33) we have that โ ๐(๐พ๐ค ) = 1 ๐คโ๐๐
and, as in the case of the Sierpiลski gasket, if we define for each ๐ด โ ๐พ ๐(๐ด) = inf { โ ๐(๐๐ ) โถ ๐๐ are cells and ๐ด โ ๐
โ
๐๐ },
๐
the ๐ satisfies the analogous to Propositions 11.17 and 11.18 in Chapter 11. We thus obtain a self-similar measure on ๐พ, and hence an integral on ๐พ with respect to the measure ๐. As above, we say that ๐ข โ dom ฮ if ๐ข โ โฑ and there exists ๐ โ ๐ถ(๐พ) such that, for all ๐ฃ โ โฑ with ๐ฃ|๐0 = 0, (13.34)
โฐ(๐ข, ๐ฃ) = โ โซ ๐๐ฃ๐๐. ๐พ
We write ๐ = ฮ๐ข, and we call it the Laplacian of ๐ข. If ๐ข is harmonic, for each ๐ โฅ 1 we have ๐ป๐ ๐ข(๐ฅ) = 0 for all ๐ฅ โ ๐๐ โงต ๐0 , and thus โฐ(๐ข, ๐ฃ) = โฐ0 (๐ข, ๐ฃ)
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13. Harmonic functions on post-critically finite sets
for any ๐ฃ โ โฑ. Hence, if further ๐ฃ|๐0 = 0, we have โฐ(๐ข, ๐ฃ) = 0 and therefore ฮ๐ข(๐ฅ) = 0 for all ๐ฅ โ ๐พ โงต ๐0 . For ๐ฅ0 โ ๐โ โงต ๐0 , let ๐ large enough so that ๐ฅ0 โ ๐๐ and let ๐๐ฅ0 ,๐ be the ๐-harmonic function such that ๐๐ฅ0 ,๐ |๐๐ = ๐{๐ฅ0 } . As in the case of the Sierpiลski gasket, we call ๐๐ฅ0 ,๐ the ๐-harmonic spline on ๐ฅ0 . Write ๐๐ฅ0 ,๐ = โซ ๐๐ฅ0 ,๐ ๐๐. ๐พ
Thus, if ๐ข โ dom ฮ, for large ๐ we have โซ ฮ๐ข ๐๐ฅ0 ,๐ ๐๐ โ ฮ๐ข(๐ฅ0 )๐๐ฅ0 ,๐ , ๐พ
because ฮ๐ข is continuous. Also โฐ(๐ข, ๐๐ฅ0 ,๐ ) = โฐ๐ (๐ข, ๐๐ฅ0 ,๐ ) = โ โ ๐ป๐ ๐ข(๐ฅ) ๐๐ฅ0 ,๐ (๐ฅ) = โ๐ป๐ ๐ข(๐ฅ0 ), ๐ฅโ๐๐
so we have that ฮ๐ข(๐ฅ0 ) โ
1 ๐ป (๐ฅ ). ๐๐ฅ0 ,๐ ๐ 0
We can make this argument precise to prove Theorem 13.35, the generalization of Theorem 11.29 to PCF sets. Theorem 13.35. Let ๐ข โ โฑ. ๐ข โ dom ฮ if and only if there exists ๐ โ ๐ถ(๐พ) such that 1 lim max {|| ๐ป๐ (๐ฅ) โ ๐(๐ฅ)|| โถ ๐ฅ โ ๐๐ โงต ๐0 } = 0. ๐
๐โโ
๐ฅ,๐
In such case, ๐ = ฮ๐ข. We leave the details as an exercise (Exercise (13)).
Exercises
257
Exercises (1) ๐โ is dense in ๐พ. (2) Let ๐ be the Sierpiลski gasket, ๐ โฅ 1, and ๐ฅ โ ๐๐ โงต ๐๐โ1 . Then there exists a unique ๐ค โ ๐๐โ1 and ๐, ๐ = 1, 2, 3 such that ๐ฅ = ๐๐ค๐ (๐๐ ) = ๐๐ค๐ (๐ ๐ ). (3) Let ๐ be the Sierpiลski gasket, ๐ โฅ 1, 0 โค ๐ < ๐, and ๐ฅ โ ๐๐ such that ๐ฅ โ ๐๐+1 โงต ๐๐ . Then there exists a unique ๐ค โ ๐๐ and ๐, ๐ = 1, 2, 3 such that ๐ฅ = ๐๐ค๐๐. . .๐ (๐๐ ) = ๐๐ค๐๐. . .๐ (๐ ๐ ), where the words ๐ค๐๐ . . . ๐ and ๐ค๐๐ . . . ๐ of length ๐ have ๐ โ 1 โ ๐ repeated ๐โs and ๐โs, respectively. (4) Let โฐ0 be a Dirichlet form on ๐(๐0 ) and 0 < ๐1 , ๐2 , . . . , ๐๐ < 1. If โฐ๐ is given by (13.10) and (13.11) is satisfied with ๐ = 1, then (13.11) is satisfied for every ๐ โฅ 1. (5) Calculate all the harmonic structures of the interval, self-similar with the contractions ๐1 (๐ฅ) = ๐ฅ/2 and ๐2 (๐ฅ) = ๐ฅ/2 + 1/2 on the real line. (6) Let ๐ be a finite set, ๐ป a symmetric nonpositive definite operator on ๐(๐), and โฐ(๐ข, ๐ฃ) โ โจ๐ป๐ข, ๐ฃโฉ the bilinear form induced by ๐ป. The following are equivalent. (a) โฐ(๐ข) = 0 if and only if ๐ข is a constant. (b) ๐ป๐ข = 0 if and only if ๐ข is a constant. (Hint: Use the spectral theorem.) (7) Calculate the initial Laplacian ๐ป0 of the harmonic structure of the Sierpiลski gasket. (8) Calculate the Laplacian ๐ป0 of the harmonic structure of the Hata tree set of Example 13.14. (9) For each ๐ = 1, 2, . . . , ๐, let ๐
๐ โถ ๐(๐๐ ) โ ๐(๐๐โ1 ) be given by ๐
๐ ๐ข = ๐ข โ ๐๐ . Then the self-similarity condition (13.8) is equivalent to ๐
1 ๐ก ๐
๐ ๐ป๐โ1 ๐
๐ . ๐ ๐=1 ๐
๐ป๐ = โ
258
13. Harmonic functions on post-critically finite sets
(10) For each ๐ค โ ๐๐ , let ๐
๐ค โถ ๐(๐๐ ) โ ๐(๐0 ) be given by ๐
๐ค ๐ข = ๐ข โ ๐๐ค . Then (13.10) is equivalent to 1 ๐ก ๐ป๐ = โ ๐
๐ค ๐ป0 ๐
๐ค . ๐ ๐ค ๐คโ๐ ๐
(11) Calculate the Laplacian ๐ป1 of the harmonic structure of the Hata tree set and verify (13.26) explicitly. (12) (a) Let ๐ a finite set and โ be the set of equivalence classes of ๐(๐) modulo constants, that is, under the equivalence relation ๐ข โผ ๐ฃ if and only if ๐ข โ ๐ฃ is a constant. Then โ is finite dimensional, โ๐ขโ = max{|๐ข(๐ฅ) โ ๐ข(๐ฆ)| โถ ๐ฅ, ๐ฆ โ ๐} and is a norm on โ. (b) The set {๐ข โ ๐(๐) โถ โ๐ฅโ๐ ๐ข(๐ฅ) = 0, โ๐ขโ โค 1} is compact. (13) Prove Theorem 13.35.
Notes The development of the Laplacian on post-critically finite sets was done by Kigami in [Kig93], where he proves that harmonic functions satisfy the linear equations involving discrete Laplacians, and hence form a finite dimensional space. The number ๐ defined by equation (13.33) is equal to the Hausdorff dimension of ๐พ with respect to the effective resistence metric, which was developed in [Kig94]. If ๐ต๐ (๐ฅ) is a ball in ๐พ with respect to this metric, then we can prove that ๐(๐ต๐ (๐ฅ)) โผ ๐๐ for small ๐, where ๐ is the measure defined in this chapter, and thus ๐ satisfies the analogous result to the HardyโLittlewood Theorem 7.4 on a PCF set [Sรกe12]. Further results on the analysis on PCF sets can be found in the text [Kig01].
Appendix A
Some results from real analysis
In this appendix we review some basic results from real analysis. With a few exceptions that may be useful for the discussion in the text, we wonโt include the proofs of the results listed here. All can be found in standard introductory texts as [Gau09] or [BS92] for the case of the real line, [Spi65] or [Fle77] for Euclidean space, and [Fol99] or [MH93] for metric and Banach spaces.
A.1. The real line The set of real numbers โ is a complete ordered field. It is unique up to isomorphism. With complete we mean that it satisfies the least upper bound axiom: Axiom (Completeness). If ๐ด โ โ is nonempty and bounded, then it has a least upper bound. A sequence ๐ฅ๐ in โ converges to ๐ฅ, and we write ๐ฅ๐ โ ๐ฅ, if, given any ๐ > 0, there exists ๐ such that ๐ โฅ ๐ implies |๐ฅ๐ โ ๐ฅ| < ๐. Any convergent sequence is a Cauchy sequence: given ๐ > 0, there exists ๐ such that ๐, ๐ โฅ ๐ implies |๐ฅ๐ โ ๐ฅ๐ | < ๐. The converse is also true. Theorem A.1. A sequence in โ converges if and only if it is a Cauchy sequence. 259
260
A. Some results from real analysis
In fact, Theorem A.1 is equivalent to the completeness axiom. It is clear that convergent sequences are bounded. The converse is not true, but we have the following fact. Theorem A.2 (BolzanoโWeierstrass). If ๐ฅ๐ is bounded, then it has a convergent subsequence. It turns out that the BolzanoโWeierstrass theorem is also equivalent to the completeness axiom. A function ๐ โถ ๐ด โ โ, where ๐ด โ โ, is continuous at ๐ฅ0 โ ๐ด if given ๐ > 0, there exists ๐ฟ > 0 such that |๐ฅ โ ๐ฅ0 | < ๐ฟ implies |๐(๐ฅ) โ ๐(๐ฅ0 )| < ๐, whenever ๐ฅ โ ๐ด. ๐ is continuous at ๐ฅ0 if, and only if, for every sequence ๐ฅ๐ in ๐ด that converges to ๐ฅ0 , then we have ๐(๐ฅ๐ ) โ ๐(๐ฅ0 ). Sums and products of continuous functions are continuous. Quotients are also continuous, provided they are well defined. ๐ is continuous on ๐ด if it is continuous at each point of ๐ด. ๐ is uniformly continuous on ๐ด if, for each ๐ > 0, there exist ๐ฟ > 0 such that |๐ฅ โ ๐ฆ| < ๐ฟ, for any ๐ฅ, ๐ฆ โ ๐ด, implies |๐(๐ฅ) โ ๐(๐ฆ)| < ๐. If ๐ is uniformly continuous and ๐ฅ๐ is a Cauchy sequence in ๐ด, then ๐(๐ฅ๐ ) is also a Cauchy sequence. This has the following consequence. A limit point of ๐ด is a point ๐ฅ โ โ such that there exists a sequence ๐ฅ๐ in ๐ด such that ๐ฅ๐ โ ๐ฅ. ๐ฅ may or may not be in ๐ด. Theorem A.3. If ๐ is uniformly continuous on ๐ด and ๐ฅ is a limit point of ๐ด, then ๐ can extended to a continuous function on ๐ด โช {๐ฅ}. Given a sequence ๐ฅ๐ in ๐ด that converges to ๐ฅ, the extension of ๐ to ๐ฅ is well-defined by the limit of ๐(๐ฅ๐ ), which converges because it is a Cauchy sequence. The union ๐ดฬ of ๐ด and its limit points is called the closure of ๐ด. Thus, by Theorem A.3, if ๐ is uniformly continuous on ๐ด, then it can be extended continuously to its closure ๐ด.ฬ Let ๐๐ โถ ๐ด โ โ be a sequence of functions on ๐ด. We say that ๐๐ converges pointwise to ๐ โถ ๐ด โ โ if, for each ๐ฅ โ ๐ด, ๐๐ (๐ฅ) โ ๐(๐ฅ). We usually just write to ๐๐ โ ๐ to denote that ๐๐ converges pointwise to ๐. We say that ๐๐ converges uniformly to ๐ if, given ๐ > 0, there exists ๐ such that ๐ โฅ ๐ implies |๐๐ (๐ฅ) โ ๐(๐ฅ)| < ๐ for all ๐ฅ โ ๐ด. In this case we write ๐๐ โ ๐. Theorem A.4. If each ๐๐ is continuous and ๐๐ โ ๐, then ๐ is continuous.
A.2. Topology
261
A series โ ๐๐ converges to ๐ if the sequence ๐ ๐ of partial sums, ๐ ๐ = ๐1 + ๐2 + . . . + ๐๐ , converges to ๐ . By Theorem A.1, if โ |๐๐ | converges, then โ ๐๐ converges. If โ |๐๐ | converges, we write โ |๐๐ | < โ and say that โ ๐๐ converges absolutely. Thus, any absolutely convergent series converges. Not every convergent series is absolutely convergent; we say that such a series converges conditionally. A test for conditional convergence is the following: Theorem A.5 (Dirichletโs test). Let ๐๐ , ๐๐ be sequences in โ satisfying the following. (1) Each ๐๐ > 0, ๐๐+1 โค ๐๐ and ๐๐ โ 0 (we write ๐๐ โ 0). (2) The sequence ๐ ๐ = ๐1 + ๐2 + . . . ๐๐ is bounded. Then the series โ ๐๐ ๐๐ converges. A series โ ๐๐ of functions converges pointwise or uniformly if the corresponding sequence of partial sums ๐ ๐ (๐ฅ) = ๐1 (๐ฅ) + . . . + ๐๐ (๐ฅ) converges pointwise or uniformly, respectively. Thus, if the series โ ๐๐ converges uniformly to ๐ and each ๐๐ is continuous, then ๐ is also continuous. We have the following test for uniform convergence of a series. Theorem A.6 (Weierstrass ๐-test). Let ๐๐ be a sequence of functions on ๐ด satisfying the following: (1) Each ๐๐ is bounded, with |๐๐ (๐ฅ)| โค ๐๐ for all ๐ฅ โ ๐ด. (2) โ ๐๐ < โ. Then the series โ ๐๐ converges uniformly.
A.2. Topology A set ๐ด โ โ is closed if it contains its limit points. Equivalently, ๐ด is closed if ๐ดฬ = ๐ด. ๐ด is open if, for each ๐ฅ โ ๐ด, there exists ๐ > 0 such that (๐ฅ โ ๐, ๐ฅ + ๐) โ ๐ด. A set is open if, and only if, itโs complement is closed. Any union of open sets in โ is open, and any intersection of closed sets is closed. Finite intersections of open sets are open, while finite unions of closed sets are closed. A countable intersection of open sets is called a ๐บ ๐ฟ set, and a countable union of closed set is called an ๐น๐ set.
262
A. Some results from real analysis
A set ๐ต โ ๐ด is said to be open in ๐ด is there exists an open set ๐ in โ such that ๐ต = ๐ด โฉ ๐. Similarly, ๐ต is closed in ๐ด if there exists a closed set ๐ธ in โ such that ๐ต = ๐ด โฉ ๐ธ. ๐ต is open in ๐ด if, and only if, ๐ด โงต ๐ต is closed in ๐ด. If ๐ โถ ๐ด โ โ is a function, then ๐ is continuous on ๐ด if, and only if, for any open set ๐ in โ its preimage ๐โ1 (๐) is open in ๐ด, and equivalently, if for any closed set ๐ธ in โ its preimage ๐โ1 (๐ธ) is closed in ๐ด. By the BolzanoโWeierstrass theorem, if ๐ด is closed and bounded, then any sequence in ๐ด has a convergent subsequence, and its limit is also in ๐ด (we say it converges in ๐ด). In fact, if any sequence in ๐ด has a subsequence that converges in ๐ด, then ๐ด must be closed and bounded. As a consequence, if ๐ is continuous on ๐ด and ๐ด is closed and bounded, then its image ๐ด is also closed and bounded. An open cover for a set ๐ด โ โ is a collection {๐๐ผ } of open sets such that ๐ด โ โ๐ผ ๐๐ผ . We say that the open cover {๐๐ผ } for ๐ด has a finite subcover if we can choose a finite subcollection ๐๐ผ1 , ๐๐ผ2 , . . . , ๐๐ผ๐ that also covers ๐ด. We have the following result. Theorem A.7 (HeineโBorel). Let ๐ด โ โ. The following are equivalent. (1) ๐ด is closed and bounded. (2) Every open cover for ๐ด has a finite subcover. A set that satisfies the statements of the HeineโBorel theorem is called compact. We also have the following results for continuous functions on compact sets. Theorem A.8. Let ๐ด โ โ be compact and ๐ โถ ๐ด โ โ continuous on ๐ด. Then: (1) ๐ is uniformly continuous. (2) ๐(๐ด) is compact. In particular, if ๐ด is compact and ๐ is continuous on ๐ด, then ๐ takes its maximum and its minimum value in ๐ด.
A.3. Riemann integration Let ๐ โถ [๐, ๐] โ โ be bounded and ๐ซ = {๐ฅ0 = ๐ < ๐ฅ1 < . . . < ๐ฅ๐ = ๐} a partition of [๐, ๐]. The lower and upper sums of ๐ with respect to ๐ซ are
A.3. Riemann integration
263
given by ๐
๐
๐ฟ(๐, ๐ซ) = โ ๐๐ (๐ฅ๐ โ ๐ฅ๐โ1 ) and
๐(๐, ๐ซ) = โ ๐๐ (๐ฅ๐ โ ๐ฅ๐โ1 ),
๐=1
๐=1
respectively, where ๐๐ = inf{๐(๐ฅ) โถ ๐ฅ โ [๐ฅ๐โ1 , ๐ฅ๐ ]} and ๐๐ = sup{๐(๐ฅ) โถ ๐ฅ โ [๐ฅ๐โ1 , ๐ฅ๐ ]}. We say that ๐ is Riemann-integrable on [๐, ๐] if, for any ๐ > 0, there exists a partition ๐ซ such that ๐(๐, ๐ซ) โ ๐ฟ(๐, ๐ซ) < ๐. In such case, the unique number ๐ผ that satisfies ๐ฟ(๐, ๐ซ) < ๐ผ < ๐(๐๐ซ) for all partitions is called the integral of ๐, and is denoted by ๐
๐
โซ ๐(๐ฅ)๐๐ฅ, ๐
โซ ๐, or simply โซ ๐. ๐
Linear combinations and products of Riemann-integrable functions are Riemann-integrable, and we have ๐
๐
โซ โ ๐๐ = โ โซ ๐๐ . ๐=1
๐=1
Also, if ๐ โค ๐ on [๐, ๐], then โซ ๐ โค โซ ๐. If ๐ is Riemann-integrable, then so is |๐| and | โซ ๐| โค โซ |๐|. | | We also have that, for any ๐ โ (๐, ๐), ๐
๐
๐
โซ ๐ = โซ ๐ + โซ ๐. ๐
๐
๐
Continuous and monotone functions are Riemann-integrable, as well as piecewise continuous or piecewise monotone functions. A set ๐ด โ โ is of measure zero if for each ๐ > 0 there exists intervals {๐ผ๐ } such that ๐ด โ โ๐ ๐ผ๐ and โ |๐ผ๐ | < ๐, where |๐ผ๐ | denotes the length ot ๐ผ๐ . Finite and countable infinite sets are of measure zero. There exist also uncountable sets of measure zero, as the Cantor set (see Chapter 5). We have the following criterion for Riemann integrability. Theorem A.9. A bounded function ๐ โถ [๐, ๐] โ โ is Riemann-integrable if, and only if, the set where ๐ is not continuous is of measure zero.
264
A. Some results from real analysis
A Riemann-integrable function can be approximated by continuous functions, in the following sense. Theorem A.10. Let ๐ โถ [๐, ๐] โ โ be Riemann-integrable, and let ๐ such that |๐(๐ฅ)| โค ๐ for all ๐ฅ โ [๐, ๐]. For any ๐ > 0, there exists a continuous function ๐ on [๐, ๐] such that |๐(๐ฅ)| โค ๐
and
โซ |๐ โ ๐| < ๐.
We include the proof of this theorem here, as its ideas are useful for the discussion in the text (see Chapter 3). Proof. Given ๐ > 0, let ๐ซ = {๐ฅ0 = ๐ < ๐ฅ1 < . . . < ๐ฅ๐ = ๐} be a partition so that ๐(๐, ๐ซ) โ ๐ฟ(๐, ๐ซ) < ๐/2. In particular, |๐(๐, ๐ซ) โ โซ ๐| < ๐ . | | 2 Note that ๐(๐, ๐ซ) is the integral of the function โ on [๐, ๐] given by โ(๐ฅ) = ๐๐ , if ๐ฅ โ [๐ฅ๐โ1 , ๐ฅ๐ ), which satisfies โ โฅ ๐. Hence |๐(๐, ๐ซ) โ โซ ๐| = โซ |โ โ ๐|. | | Let ๐ฟ > 0 so that
๐ . 4๐๐ Let ๐ be the continuous function on [๐, ๐] such that it is equal to ๐๐ on each interval [๐ฅ๐โ1 + ๐ฟ, ๐ฅ๐ โ ๐ฟ], linear from ๐๐โ1 to ๐๐ on each [๐ฅ๐โ1 โ ๐ฟ, ๐ฅ๐โ1 +๐ฟ], ๐ = 2, . . . , ๐, linear from ๐(๐) to ๐1 on [๐ฅ0 , ๐ฅ0 +๐ฟ], and linear from ๐๐ to ๐(๐) on [๐ฅ๐ โ๐ฟ, ๐ฅ๐ ]. As each |๐๐ | โค ๐, we have that |๐| โค ๐. Also ๐ฟ
0, there exists a partition ๐ซ such that ๐(๐, ๐ซ) โ ๐ฟ(๐, ๐ซ) < ๐. If ๐ is Riemann integrable, its integral is denoted by โซ ๐(๐ฅ)๐๐ฅ, ๐
โซ ๐, or simply โซ ๐. ๐
As in the case of the Riemann integral of single variable functions on an interval, linear combinations and products of Riemann-integrable func๐ ๐ tions are Riemann-integrable, we have โซ โ๐=1 ๐๐ = โ๐=1 โซ ๐๐ , and, if ๐ โค ๐ on ๐
, then โซ ๐ โค โซ ๐. If ๐ is Riemann-integrable on ๐
, then so is |๐| and | โซ ๐| โค โซ |๐|. We also have that, for any partition ๐ซ of ๐
with subrectangles, each ๐|๐ is Riemann-integrable for each ๐ โ ๐ซ and โซ ๐ = โ โซ ๐. ๐
๐โ๐ซ ๐
As above, a set ๐ด โ โ๐ is of measure zero if for each ๐ > 0 there exist rectangles {๐
๐ } such that ๐ด โ โ๐ ๐
๐ and โ vol(๐
๐ ) < ๐. The analog
A.5. Complete metric spaces
267
of Theorem A.9 is true for multivariable functions: a bounded function ๐ โถ ๐
โ โ is Riemann-integrable if and only if the set where ๐ is not continuous is of measure zero. If we write ๐
= ๐ ร ๐, where ๐ and ๐ are rectangles in โ๐ and โ๐ , respectively, with ๐ = ๐ + ๐, we have the identity, for a continuous ๐ โถ ๐
โ โ, (A.12)
โซ ๐ = โซ ( โซ ๐๐ฅ (๐ฆ)๐๐ฆ)๐๐ฅ, ๐
๐
๐
where, for each ๐ฅ โ ๐ and ๐ฆ โ ๐, ๐๐ฅ (๐ฆ) = ๐(๐ฅ, ๐ฆ). The assumption that ๐ is continuous is needed because ๐๐ฅ might not be a Riemann-integrable function on ๐ for all ๐ฅ โ ๐, given a Riemann-integrable function ๐ on ๐
. See [Spi65, Theorem 3.10] for a precise version of (A.12) for general Riemann-integrable functions. If ๐
โ โ๐ is a closed rectangle, ๐ด โ ๐
and ๐ โถ ๐
โ โ is Riemannintegrable, we define โซ ๐ = โซ ๐ โ
๐๐ด , ๐ด
๐
where ๐๐ด is the characteristic function of the set ๐ด, provided ๐๐ด is Riemann integrable, which occurs when the set of boundary points1 of ๐ด is of measure zero.
A.5. Complete metric spaces The Euclidean space โ๐ with the Euclidean distance induced by the norm ๐ฅ โฆ |๐ฅ| is an example of a metric space. A metric space is a set provided with a metric ๐ โถ ๐ ร ๐ โ [0, โ) that satisfies, for all ๐ฅ, ๐ฆ, ๐ง โ ๐, (1) ๐(๐ฅ, ๐ฆ) = 0 if and only if ๐ฅ = ๐ฆ, (2) ๐(๐ฅ, ๐ฆ) = ๐(๐ฆ, ๐ฅ), and (3) ๐(๐ฅ, ๐ฆ) โค ๐(๐ฅ, ๐ง) + ๐(๐ง, ๐ฆ). A sequence ๐ฅ๐ in a metric space ๐ converges to ๐ฅ if, for any ๐ > 0, there exists ๐ such that ๐(๐ฅ๐ , ๐ฅ) < ๐. A convergent sequence ๐ฅ๐ is clearly a Cauchy sequence: given ๐ > 0, there exists ๐ such that ๐(๐ฅ๐ , ๐ฅ๐ ) < ๐ for any ๐, ๐ โฅ ๐. 1
A point ๐ฅ is in the boundary of ๐ด if, for all ๐ > 0, ๐ด โฉ ๐ต๐ (๐ฅ) โ โ
and ๐ต๐ (๐ฅ) โงต ๐ด โ โ
.
268
A. Some results from real analysis
In general, though, not every Cauchy sequence converges in a metric space. A complete metric space is a metric space in which every Cauchy sequence converges. By Theorem A.1, the Euclidean space is complete. If ๐ is a real or complex vector space, โ โ
โ โถ ๐ โ [0, โ) is a norm if it satisfies, for any vectors ๐ฅ, ๐ฆ โ ๐, (1) โ๐ฅโ = 0 if and only if ๐ฅ = 0, (2) โ๐๐ฅโ = |๐|โ๐ฅโ for any scalar ๐, and (3) โ๐ฅ + ๐ฆโ โค โ๐ฅโ + โ๐ฆโ. The properties above imply that, if โ โ
โ is a norm on ๐, then ๐(๐ฅ, ๐ฆ) = โ๐ฅ โ ๐ฆโ is a metric, and thus ๐ is metric space. If ๐ is a complete normed vector space, we call it a Banach space. A series โ ๐ฅ๐ in the normed space ๐ converges to ๐ฅ if the sequence ๐ of partial sums ๐ ๐ = โ๐=1 ๐ฅ๐ converges to ๐ฅ. Equivalently, ๐
|| โ ๐ฅ โ ๐ฅ|| โ 0. ๐ || || ๐=1
A series converges absolutely if the series โ โ๐ฅ๐ โ converges. If ๐ is a Banach space and the series โ ๐ฅ๐ converges absolutely, then it converges. It turns out that this fact is equivalent to the completeness of ๐, as stated in Theorem A.13 due to Banach. Theorem A.13. The normed vector space ๐ is a Banach space if and only if every absolutely convergent series converges in ๐. An inner product space in the vector space ๐ is a scalar form โจโ
, โ
โฉ on ๐ ร ๐ that satisfies (1) โจ๐ฅ, ๐ฅโฉ โฅ 0, and โจ๐ฅ, ๐ฅโฉ = 0 if and only if ๐ฅ = 0, (2) โจ๐ฅ, ๐ฆโฉ = โจ๐ฆ, ๐ฅโฉ for any ๐ฅ, ๐ฆ โ ๐, and (3) โจ๐๐ฅ + ๐๐ฆ, ๐งโฉ = ๐โจ๐ฅ, ๐งโฉ + ๐โจ๐ฆ, ๐งโฉ for any scalars ๐, ๐ and ๐ฅ, ๐ฆ, ๐ง โ ๐. Two vectors ๐ฅ, ๐ฆ โ ๐ are called orthogonal, and we write ๐ฅ โ ๐ฆ, if โจ๐ฅ, ๐ฆโฉ = 0. If ๐ โ ๐, ๐ฅ is orthogonal to ๐, and we write ๐ฅ โ ๐, if
A.5. Complete metric spaces
269
๐ฅ is orthogonal to every vector in ๐. The orthogonal complement of ๐, denoted as ๐โ , is given by all the orthogonal vectors to ๐. ๐ โ is always a vector subspace of ๐. If ๐ is a real vector space, then an inner product is a positive symmetric bilinear form. An inner product induces the norm โ๐ฅโ = โโจ๐ฅ, ๐ฅโฉ. The norm satisfies the parallelogram identity โ๐ฅ + ๐ฆโ2 + โ๐ฅ โ ๐ฆโ2 = 2โ๐ฅโ2 + 2โ๐ฆโ2 , and the polarization identity, given by 1 โจ๐ฅ, ๐ฆโฉ = (โ๐ฅ + ๐ฆโ2 โ โ๐ฅ โ ๐ฆโ2 ) 4 in the real case, and by 1 โจ๐ฅ, ๐ฆโฉ = (โ๐ฅ + ๐ฆโ2 โ โ๐ฅ โ ๐ฆโ2 + ๐โ๐ฅ + ๐๐ฆโ2 โ ๐โ๐ฅ โ ๐๐ฆโ2 ) 4 in the complex case. If ๐ is a complete inner product space, then it is called a Hilbert space. Note that a Hilbert space is a Banach space whose norm is induce by an inner product. If ๐ is a Hilbert space and ๐ is a closed vector subspace, then, for each ๐ฅ โ ๐, there is ๐ฆ0 โ ๐ closest to ๐ฅ, that is, โ๐ฆ0 โ ๐ฅโ โค โ๐ฆ โ ๐ฅโ for all ๐ฆ โ ๐ . ๐ฆ0 is called the orthogonal projection of ๐ฅ in ๐ , and satisfies ๐ฅ โ๐ฆ0 โ ๐ . As a consequence, if ๐ is a proper closed subspace of ๐, then ๐ โ is nontrivial.
Acknowledgments
I would like to thank all the students who have taken this course, both in IAS/PCMI and Colima, who provided the motivation to work on this text. Their questions and commentary to the manuscript were very valuable during its writing. I would also like to thank the reviewers of the first version of the manuscript, as their observations lead to several improvements to this book. I finally want to thank Eriko Hironaka, Ina Mette, Marcia C. Almeida, John F. Brady Jr., and Abigail Lawson at the AMS, who took care of all the logistics in the publication of this book.
271
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Index
๐น๐ set, 261 ๐บ ๐ฟ set, 261 Abel means, 43 summable, 43, 55 theorem, 43 adyacent vertex, 209, 245 almost everywhere, 98 average, 54, 137, 143 Banach space, 126, 268 theorem, 268 Bernsteinโs theorem, 69 Besselโs inequality, 51 better kernels, 148, 158, 165, 173 BolzanoโWeierstrass theorem, 260 boundary, 210, 244 Cantor set, 188 Cauchy sequence, 259, 267 cell, 192, 243 Cesร ro summable, 55 summation, 54 sums, 54 Chebyshevโs inequality, 140 closed set, 261 closure, 260 compact, 262
complete, 259, 268 connected set, 266 continuous, 260 uniformly, 260 contraction, 191 contraction constant, 191 convergence absolute, 46, 48, 261 conditional, 261 in measure, 179 mean-square, 59, 164, 168 nontangential, 146 pointwise, 260 radial, 29 uniform, 260 convolution, 128 cover ๐ฟ-cover, 185 critical set, 244 cube, 85 dyadic, 85 strictly thinner, 88 diameter, 185 Dirichlet eigenfunction, 231, 233 eigenvalue, 231 form, 246 principle, 11 problem, 10, 27
277
278 test, 261 discrete energy, 204, 210 discrete Laplacian, 207, 217 domain, 6 dominated convergence theorem, 118 dyadic decomposition, 51 eigenfunction, 223 eigenvalue, 223 energy form, 9, 212, 216, 247, 255 minimizers, 9 extended real line, 96 Fatouโs lemma, 111 Fejรฉr kernel, 58 theorem, 55 flounces function, 42 Fourier coefficients, 39 inversion formula, 157 series, 40 transform, 151 Fubiniโs theorem, 123
Index Hilbert space, 269 transform, 171, 173 Hopf lemma, 33 integrable function, 113, 115 locally integrable, 138 Riemann-integrable, 263, 266 square integrable, 130 integral complex valued function, 115 extended real valued function, 113 improper, 74 nonnegative function, 108 nonnegative simple function, 105 on the Sierpiลski gasket, 214 Riemann, 263, 266 self-similarity, 215 symmetry, 215 isometry, 160, 172
Gaussian kernel, 154, 158 golden fractal, 199 good kernels, 27, 58, 79, 81
Laplace equation, 3 Laplacian, 3, 217, 250, 255 polar coordinates, 11, 35 Lebesgue differentiation theorem, 143 measure, 94 space, 124, 130 Lipschitz function, 53
Hรถlder continuous, 53, 69 Hardy-Littlewood maximal function, 138 theorem, 139 harmonic ๐-harmonic function, 212, 217 conjugate, 4, 167 function, 3, 13, 210, 212, 217, 247, 254 harmonic structure, 210, 247 compatibility, 247 self-similarity, 247 Harnack inequality, 33 Hata tree set, 197, 244, 248 Hausdorff dimension, 185, 188 measure, 186 heat kernel, 81, 154 HeineโBorel theorem, 262
maximal function, 138 uncentered, 142 maximum princple, 20, 253, 254 measurable ๐น๐ set, 94 ๐บ ๐ฟ set, 94 closed cube, 92 closed set, 93, 214 complement, 89, 214 countable intersection, 92 countable union, 90, 214 function, 96 open set, 93, 214 pointwise limit, 97 set, 89, 214 union, 90 measure countable additivity, 94, 214 countable subadditivity, 87
Index exterior, 86 invariance, 96 Lebesgue, 94 monotone continuity, 94 monotonicity, 87 on the Sierpiลski gasket, 213 outer, 85, 186 regularity, 95 zero, 86, 263, 266 metric space, 267 Minkowskiโs inequality, 124, 132 monotone convergence theorem, 109 negative part, 102 neighbor, 209, 245 nontangential limit, 146 norm, 268 open cover, 262 open set, 261 open set condition, 193 orthogonal transformation, 31 orthogonality, 49, 59, 268 Parsevalโs identity, 60 PCF set, 244 Plancherelโs theorem, 162 Poisson integral, 27, 75, 164, 167 kernel, 24, 71, 154, 158, 167 polarization identity, 160 positive part, 102 post-critical set, 244 post-critically finite set, 244 punctured ball, 30 RiemannโLebesgue lemma, 53, 152 sawtooth function, 40, 46 section of a function, 123 of a set, 120 self-similar, 191, 243 sharkteeth function, 41, 46 Sierpiลski gasket, 196, 207, 244 similitude, 193 simple function, 99 approximation, 100 reduced form, 100
279 singularity isolated, 29 removable, 30 snowflake set, 197 spherical coordinates, 7 spline, 213, 256 integral, 216 step function, 100 subordination, 81 symmetry lemma, 25, 32 Tonelliโs theorem, 124 translation, 31 trigonometric polynomial, 37, 58 unitary operator, 162 upper half-space, 5, 71 Vitalliโs covering lemma, 140 volume, 85 weak type, 139, 174, 182 Weierstrass ๐-test, 261 word, 191, 208, 243
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