Morrey Spaces: Introduction and Applications to Integral Operators and Pde's, Volume I 9781498765510, 9780367459154, 9780429085925, 9781003029076, 1498765513

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Morrey Spaces

Monographs and Research Notes in Mathemacs Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky

About the Series This series is designed to capture new developments and summarize what is known over the en!re field of mathema!cs, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students, and prac!!oners. Interdisciplinary books appealing not only to the mathema!cal community, but also to engineers, physicists, and computer scien!sts are encouraged. This series will maintain the highest editorial standards, publishing well-developed monographs as well as research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publica!on for current material where the style of exposi!on reflects a developing topic. Spectral Methods Using Mulvariate Polynomials On The Unit Ball Kendall Atkinson, David Chien, and Olaf Hansen Glider Representaons Frederik Caenepeel, Fred Van Oystaeyen La!ce Point Idenes and Shannon-Type Sampling Willi Freeden, M. Zuhair Nashed Summable Spaces and Their Duals, Matrix Transformaons and Geometric Properes Feyzi Basar, Hemen Du!a Spectral Geometry of Paral Differenal Operators (Open Access) Michael Ruzhansky, Makhmud Sadybekov, Durvudkhan Suragan Linear Groups: The Accent on Infinite Dimensionality Martyn Russel Dixon, Leonard A. Kurdachenko, Igor Yakov Subbo"n Morrey Spaces: Introducon and Applicaons to Integral Operators and PDE’s, Volume I Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim Morrey Spaces: Introducon and Applicaons to Integral Operators and PDE’s, Volume II Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim For more informa!on about this series please visit: h$ps://www.crcpress.com/Chapman--HallCRC-Monographs-and-Research-Notes-in-Mathema!cs/bookseries/CRCMONRESNOT

Morrey Spaces Introduction and Applications to Integral Operators and PDE’s, Volume II

Yoshihiro Sawano Chuo University

Giuseppe Di Fazio University of Catania

Denny Ivanal Hakim Bandung Institute of Technology

First edition published 2020 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN c 2020 Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Sawano, Yoshihiro, author. | Di Fazio, Giuseppe, 1963- author. | Hakim, Denny Ivanal, author. Title: Morrey spaces : introduction and applications to integral operators and PDE’s / Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim. Description: First edition. | Boca Raton : C&H/CRC Press, 2020. | Series: Chapman & Hall/CRC monographs and research notes in mathematics | Includes bibliographical references and index. | Contents: Banach function lattices -- Fundamental facts in functional analysis -Polynomials and harmonic functions -- Various operators in Lebesgue spaces -- BMO spaces and Morrey-Campanato spaces -- BMO spaces and Morrey-Campanato spaces -- General metric measure spaces -- Weighted Lebesgue spaces -- Approximations in Morrey spaces -- Predual of Morrey spaces -- Linear and sublinear operators in Morrey spaces. Identifiers: LCCN 2019060107 | ISBN 9781498765510 (v. 1 ; hardback) | ISBN 9780367459154 (v. 2 ; hardback) | ISBN 9780429085925 (v. 1 ; ebook) | ISBN 9781003029076 (v. 2 ; ebook) Subjects: LCSH: Banach spaces. | Harmonic analysis. | Differential equations, Partial--Numerical solutions. | Differential equations, Elliptic--Numerical solutions. | Integral operators. Classification: LCC QA322.2 .S29 2020 | DDC 515/.732--dc23 LC record available at https://lccn.loc.gov/2019060107 ISBN: 9780367459154 (hbk) ISBN: 9781003029076 (ebk) Typeset in CMR by Nova Techset Private Limited, Bengaluru & Chennai, India

Contents

Preface

xi

Acknowledgement Notation in this book 11 Multilinear operators and Morrey spaces 11.1

11.2

11.3

Multi-Morrey spaces . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Multi-Morrey spaces . . . . . . . . . . . . . . . . . . 11.1.2 Multilinear Hardy–Littlewood maximal operator . . . 11.1.3 Multilinear singular integral operators . . . . . . . . 11.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Multilinear fractional integral operators . . . . . . . . . . . 11.2.1 Multilinear fractional integral operators of Kenig– Stein-type . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Multilinear fractional integral operators of Grafakos type . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Generalized Morrey/Morrey–Campanato spaces 12.1

12.2

Generalized Morrey spaces . . . . . . . . . . . . . . . . . . . 12.1.1 Definition of generalized Morrey spaces . . . . . . . . 12.1.2 The class Gq . . . . . . . . . . . . . . . . . . . . . . . 12.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Boundedness properties of operators in generalized Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Hardy–Littlewood maximal operator in generalized Morrey spaces and the class Z0 . . . . . . . . . . . . 12.2.2 Singular integral operators on generalized Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Generalized fractional integral operators in generalized Morrey spaces . . . . . . . . . . . . . . . . . . . . . .

xiii xv 1 1 2 3 5 9 10 10 12 14 15 17 17 18 23 33 33 34 41 47

v

vi

Contents

12.3

12.4

12.2.4 Generalized fractional maximal operators in generalized Morrey spaces . . . . . . . . . . . . 12.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . Generalized Morrey–Campanato spaces . . . . . . . . 12.3.1 Generalized Morrey–Campanato spaces over the line (0, ∞) . . . . . . . . . . . . . . . . . . . . . 12.3.2 Generalized Morrey Campanato spaces over Rn 12.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . half . . . . . . . . . . . .

13 Generalized Orlicz–Morrey spaces 13.1

13.2

13.3

13.4

Generalized Orlicz–Morrey spaces of the first kind . . . . . 13.1.1 (ϕ, Φ)-average and generalized Orlicz–Morrey spaces of the first kind . . . . . . . . . . . . . . . . . . . . . 13.1.2 Hardy–Littlewood maximal operator in generalized Orlicz–Morrey spaces of the first kind . . . . . . . . . 13.1.3 Singular integral operators in generalized Orlicz–Morrey spaces of the first kind . . . . . . . . . 13.1.4 Generalized fractional integral operators in generalized Orlicz–Morrey spaces of the first kind . . . . . . . . . 13.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Orlicz–Morrey spaces of the second kind . . . . 13.2.1 Generalized Orlicz–Morrey spaces of the second kind 13.2.2 Hardy–Littlewood maximal operator in generalized Orlicz–Morrey spaces of the second kind . . . . . . . 13.2.3 Singular integral operators in generalized Orlicz– Morrey spaces of the second kind . . . . . . . . . . . 13.2.4 Generalized fractional integral operators in generalized Orlicz–Morrey spaces of the second kind . . . . . . . 13.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Difference of two generalized Orlicz–Morrey spaces . . . . . 13.3.1 The space L2 (Rn ) ∩ L3 (Rn ) . . . . . . . . . . . . . . 13.3.2 The space MpL log L (Rn ) . . . . . . . . . . . . . . . . . 13.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14.2

Morrey spaces on domains . . . . . . . . . . . . . . . . . 14.1.1 Morrey spaces over the half space Rn+ . . . . . . . 14.1.2 Morrey spaces over domains . . . . . . . . . . . . Sobolev–Morrey spaces on domains . . . . . . . . . . . . 14.2.1 Interpolation property of Sobolev–Morrey spaces

61 63 66 67 73

14 Morrey spaces over metric measure spaces 14.1

57 60 61

73 74 79 84 85 88 89 89 91 92 93 95 95 96 97 99 99 103

. . . . .

. . . . .

103 103 107 109 109

Contents

14.3

14.4

14.5

14.6

14.2.2 Extension property of Sobolev–Morrey spaces over Lipschitz domains . . . . . . . . . . . . . . . . . . . . Morrey–Campanato spaces on domains . . . . . . . . . . . . 14.3.1 Morrey–Campanato spaces on domains . . . . . . . . 14.3.2 Morrey–Campanato spaces and H¨older–Zygmund spaces on domains . . . . . . . . . . . . . . . . . . . 14.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Morrey spaces for non-doubling measures on Rn . . . . . . 14.4.1 Morrey spaces for non-doubling measures on Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Equivalent norm of doubling type . . . . . . . . . . . 14.4.3 Maximal inequalities . . . . . . . . . . . . . . . . . . 14.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Morrey spaces on the metric measure space setting . . . . . 14.5.1 Fundamental properties of Morrey spaces on locally doubling metric measure spaces . . . . . . . . . . . . 14.5.2 Gauss Morrey space . . . . . . . . . . . . . . . . . . . 14.5.3 An example of Morrey spaces for non-doubling measures on metric measure spaces . . . . . . . . . . 14.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 Weighted Morrey spaces 15.1

15.2

Weighted Morrey spaces of Komori–Shirai-type . . . . . . . 15.1.1 The structure of weighted Morrey spaces of Komori–Shirai-type . . . . . . . . . . . . . . . . . . . 15.1.2 Maximal operators in weighted Morrey spaces of Komori–Shirai-type . . . . . . . . . . . . . . . . . . . 15.1.3 Singular integral operators acting on weighted Morrey spaces of Komori–Shirai-type . . . . . . . . . . . . . 15.1.4 Fractional maximal operators and fractional integral operators in weighted Morrey spaces of Komori– Shirai-type . . . . . . . . . . . . . . . . . . . . . . . . 15.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Weighted Morrey spaces of Samko-type . . . . . . . . . . . 15.2.1 The structure of weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 The class Bα . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Maximal operator in weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Singular integral operators in weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . . . . . . 15.2.5 Fractional maximal operators in weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . .

vii

110 112 113 116 118 119 119 120 122 126 127 127 130 131 137 138 143 145 145 147 150

152 156 156 157 161 163 168 170

viii

Contents

15.3

15.4

15.2.6 Fractional integral operators in weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . . 15.2.7 Vector-valued estimates in weighted Morrey spaces of Samko-type . . . . . . . . . . . . . . . . . . . . . . . 15.2.8 Dual weighted estimates of Stein-type . . . . . . . . . 15.2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Weighted local Morrey spaces . . . . . . . . . . . . . . . . . 15.3.1 Maximal operators in weighted local Morrey spaces . 15.3.2 Singular integral operators in weighted local Morrey spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Fractional maximal operators and fractional integral operators in weighted local Morrey spaces . . . . . . 15.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 Morrey-type spaces 16.1

16.2

16.3

Morrey-type spaces . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Local and global Morrey-type spaces . . . . . . . . . 16.1.2 Maximal and sharp maximal inequalities for local Morrey-type spaces . . . . . . . . . . . . . . . . . . . 16.1.3 Singular integral operators in Morrey-type spaces . . 16.1.4 Fractional integral operators in Morrey-type spaces . 16.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . General Morrey-type spaces . . . . . . . . . . . . . . . . . . 16.2.1 Elementary properties of general Morrey-type spaces 16.2.2 Maximal operator in general Morrey-type spaces . . 16.2.3 Singular integral operators in general Morrey-type spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Fractional maximal operators in general Morrey-type spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17.2

Calder´ on product . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Calder´ on product . . . . . . . . . . . . . . . . . . . 17.1.2 Calder´ on product of Morrey spaces . . . . . . . . . 17.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Pointwise multipliers . . . . . . . . . . . . . . . . . . . . . 17.2.1 Pointwise multipliers on general Banach lattices . . 17.2.2 Pointwise multipliers in Morrey spaces . . . . . . . 17.2.3 Pointwise multipliers in local Morrey spaces . . . . 17.2.4 Pointwise multipliers in generalized Morrey spaces .

181 188 189 190 191 197 201 205 206 213

17 Pointwise product 17.1

176

213 214 216 217 218 218 219 219 224 229 233 235 236 239

. . . . . . . . .

239 239 243 247 248 248 250 255 256

Contents

17.3

17.4

17.5

17.2.5 Pointwise multipliers for BMOϕ (Rn ) . . . . . . . . 17.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Olsen’s inequalities . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Olsen’s inequality for fractional maximal operators 17.3.2 Olsen’s inequality for fractional integral operators . 17.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Multipliers from homogeneous Besov spaces to Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n s . . . . . 17.4.1 The Besov space B˙ p1 (Rn ) with 0 < s ≤ p s 17.4.2 The pointwise multipliers from B˙ p1 (Rn ) to Lp (Rn ) n with 0 < s ≤ . . . . . . . . . . . . . . . . . . . . p 17.4.3 Ho’s vector-valued Morrey spaces and pointwise multiplier spaces . . . . . . . . . . . . . . . . . . . 17.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix . . . . . .

261 268 270 270 272 274

. .

274 275

.

277

. . .

278 281 282

18 Real interpolation of Morrey spaces 18.1

18.2

18.3

18.4

18.5

Real interpolation . . . . . . . . . . . . . . . . . . . . . . . 18.1.1 Real interpolation functors . . . . . . . . . . . . . . . 18.1.2 Theorem on powers . . . . . . . . . . . . . . . . . . . 18.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Description of interpolation spaces . . . . . . . . . . . . . . 18.2.1 Weighted Lebesgue space of monotonic functions . . 18.2.2 Spaces defined via M↑ (0, ∞) . . . . . . . . . . . . . . 18.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Real interpolation of Lebesgue spaces . . . . . . . . . . . . 18.3.1 Description of real interpolation spaces for a pair of Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . 18.3.2 Application of the A-B majorization to real interpolation of Lebesgue spaces . . . . . . . . . . . . 18.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Real interpolation of Morrey spaces . . . . . . . . . . . . . 18.4.1 Real interpolation of Morrey(-type) spaces . . . . . . 18.4.2 Interpolation of generalized local Morrey spaces with the family G . . . . . . . . . . . . . . . . . . . . . . . 18.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 Complex interpolation of Morrey spaces 19.1

Complex interpolation . . . . . . . . . . . . . . . . . . . . . 19.1.1 Taylor expansion of exp . . . . . . . . . . . . . . . . 19.1.2 Doestch’s three-line lemma . . . . . . . . . . . . . . .

287 287 287 293 297 297 297 303 307 308 308 310 312 312 312 314 319 319 323 323 324 327

x

Contents

19.2

19.3

19.1.3 Complex interpolation functors . . . . . . . . . . . 19.1.4 A dense subspace of the first complex interpolation space . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1.5 Relations between the first and the second complex interpolation functors . . . . . . . . . . . . . . . . . 19.1.6 Complex interpolation of Banach function spaces . 19.1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Complex interpolation of Morrey spaces . . . . . . . . . . 19.2.1 General description of complex interpolation of Morrey spaces . . . . . . . . . . . . . . . . . . . . . 19.2.2 Complex interpolation of Morrey spaces . . . . . . 19.2.3 Interpolation of closed subspaces of generalized Morrey spaces . . . . . . . . . . . . . . . . . . . . . 19.2.4 Interpolation of different closed subspaces of generalized Morrey spaces . . . . . . . . . . . . . . 19.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

331

.

336

. . . .

337 340 343 344

. .

345 346

.

348

. . .

356 359 359

Bibliography

365

Index

407

Preface

There are many problems related to Morrey spaces, which make us feel that Morrey spaces themselves are insufficient. So we embark on generalizations. Chapters 11, 12 and 13 are oriented to generalization of the function spaces above. Chapter 11 handles multilinear operators and Morrey spaces. We propose multilinear Morrey spaces to handle multilinear operators. We will need them because multilinear operators map a sequence of functions to a function. Morrey spaces are equipped with two parameters. We seek generalizations of these two parameters in Chapters 12 and 13. As we see in Chapter 1, Morrey spaces have a parameter which describes the local integrability and one which describes the global integrability. Chapter 12 generalizes the global integrability. Extending this global parameter allows us to include many function spaces such as intersection of Lebesgue spaces, uniformly local Lebesgue spaces and so on. Chapter 13 generalizes the local integrability. There are a couple of ways to proceed in this direction. We will show that each generalization via the methods above is independent. Generalizing the global integrability parameters allows us to compensate for the failure of the boundedness of operators in the endpoint cases. Chapters 14 and 15 distort the symmetry of Euclidean space. Chapter 14 proposes a general framework, while Chapter 15 explains some development in the theory of weighted Morrey spaces. Unlike weighted Lebesgue spaces, the theory of weighted Morrey spaces is not mature. Nevertheless, for the power weights we can completely characterize the condition for which operators are bounded. The remaining chapters, Chapters 16, 17, 18 and 19, concern interpolation of Morrey spaces. Especially Chapters 18 and 19 also address some open problems of the description of the interpolation of Morrey spaces. Chapter 16 can be located also as the application of Chapter 7. Chapter 16 is connected with real interpolation of function spaces as we will see in Chapter 18. Chapter 17 discusses the pointwise product of functions. Chapter 17 is independent of interpolation. However, as we will see in Chapter 19, Chapter 17 will be a key observation for the complex interpolation. This will be a preparatory observation to Chapter 19. One of the difficult problems in the study of Morrey spaces is to describe interpolation spaces. Chapters 18 and 19 consider real and complex interpolations, respectively. Chapter 19 will give a description of complex interpolations of Morrey spaces. However, to the best knowledge of the authors, there is no

xi

xii

Preface

complete description of real interpolation of Morrey spaces. Some special cases can be covered in this book. However, in general it is an open problem. Schwartz distributions are important in many fields of mathematics. However, doing this makes the book even thicker. So we discarded this topic. We could not include many other related function spaces such as Besov Morrey spaces, Triebel-Lizorkin Morrey spaces, Besov-type spaces and TriebelLizorkin type spaces. These spaces are nowadays called smoothness Morrey spaces. These spaces are based on Besov spaces and Triebel-Lizorkin spaces, which can be found in Triebel’s textbooks [481, 482, 483]. We remark that these spaces cover Morrey spaces as a special case as is shown in [306, Proposition 4.1]. We refer to [263, Theorem 9.6] and [354, Corollary 6.2] for the counterpart of generalized Morrey spaces. We refer to [224, 520, 521, 529, 531] for the results of this direction. Since we do not deal with smoothness Morrey spaces in this book, we content ourselves with listing the papers containing the definition of the function spaces [358, 244], which are succeeded by [306, 307, 201, 420, 471] and are further expanded by [199, 518, 519, 258, 265] to various directions, as well as the textbooks [136, 405, 484, 532, 510, 511]. We remark that there are works [80, 332, 333] behind these works. See [446, 531] for more details.

Acknowledgement

We are thankful to the following individuals: Ali Akbulut, Ryutaro Arai, Tserendorj Batbold, Neal Bez, David Cruz-Uribe SFO, Victor I. Burenkov, Javier Duoandikoetxea, Fatih Deringoz, Eridani, Maria Stella Fanciullo, Bruno Franchi, Michele Frasca, Sadek Gala, Arash M. Ghorbanalizadeh, Loukas Georgios Grafakos, Vagif Guliyev, Hendra Gunawan, Cristian Gutierrez, Sabir G. Hasanov, Naoya Hatano, Kwok-Pun Ho, Takeshi Iida, Mitsuo Izuki, Takashi Izumi, Eder Kikianty, Yasuo Komori-Furuya, Pier Domenico Lamberti, Ermanno Lanconelli, Sang Hyuk Lee, Yiyu Liang, Liguang Liu, Fumi-Yuki Maeda, Mieczyslaw Mastylo, Katsuo Matsuoka, Alexander Meskhi, Akihiko Miyachi, Yoshihiro Mizuta, Eiichi Nakai, Shohei Nakamura, Van Hanh Nguyen, Toru Nogayama, Takahiro Noi, Mehriban Omarova, Takahiro Ono, Maria Alessandra Ragusa, Hiroki Saito, Daniel Salim, Enji Sato, Christopher Schwanke, Raul Serapioni, Saad R. El-Shabrawy, Minglei Shi, Tetsu Shimomura, Satoru Shirai, Idha Sihwaningrum, Takuya Sobukawa, Sakoto Sugano, Hitoshi Tanaka, Ryotaro Tanaka, Naohito Tomita, Lorenzo Tuccari Yohei Tsutsui, Tino Ullrich, Hidemitsu Wadade, Wen Yuan, Kozo Yabuta, Da Chun Yang, Dong Yong Yang, Sibei Yang, Kaoru Yoneda, Hiroko Yoshida, Pietro Zamboni, Ciqiang Zhuo. We wish to thank the following institutions: Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan, University of Catania, Bandung Institute of Technology, Tokyo Metropolitan University and Instituto Nazionale di alta Matematica (Indam) for financial support. We thank Professor Pidatella for the permission to use a hand-writing of Professor Chiarenza as cover of the book. Special thanks to Professor Filippo Chiarenza who has been a friend and a mentor.

xiii

Notation in this book

Sets and set functions (1) The metric open ball defined by `2 is usually called a ball. We denote by B(x, r) the ball centered at x of radius r. Namely, we write B(x, r) ≡ {y ∈ Rn : kx − yk < r} when x ∈ Rn and r > 0. Given a ball B, we denote by c(B) its center and by r(B) its radius. We write B(r) instead of B(o, r), where o ≡ (0, 0, . . . , 0). (2) By a “cube” we mean a compact cube whose edges are parallel to the coordinate axes. The metric closed ball defined by `∞ is called a cube. If a cube has center x and radius r, we denote it by Q(x, r). Namely, we write   Q(x, r) ≡ y = (y1 , y2 , . . . , yn ) ∈ Rn : max |xj − yj | ≤ r j=1,2,...,n

when x = (x1 , x2 , . . . , xn ) ∈ Rn and r > 0. For a cube Q or a right-open cube Q, define Q(Q) to be the set of all cubes of the same type contained in Q and Q] (Q) to be the set of all cubes of the same type containing Q. From the definition of Q(x, r), its volume is (2r)n . We write Q(r) instead of Q(o, r). Given a cube Q, we denote by c(Q) the center of Q and by `(Q) the sidelength of Q: `(Q) = |Q|1/n , where |Q| denotes the volume of the cube Q. The symbol Q denotes the set of all cubes. (3) Given Q ∈ Q and k > 0, k Q means the cube concentric to Q with sidelength k `(Q). Given a ball B and k > 0, we denote by k B the ball concentric to B with radius k r(B). (4) For ν ∈ Z and m = (m1 , m2 , . . . , mn ) ∈ Zn , we define Qνm ≡  n  Y mj mj + 1 , . Denote by D = D(Rn ) the set of such cubes. ν ν 2 2 j=1 The elements in D(Rn ) are called dyadic cubes. (5) Let E be a measurable set. Then we denote its indicator function by χE . If E has positive measure and E is integrable over f , then denote by mE (f ) the average of f over E. |E| denotes the volume of E. xv

xvi

Notation in this book

(6) If we are working on Rn , then B denotes the set of all balls in Rn , while Q denotes the set of all cubes in Rn . Be careful because B can be used for a different purpose: When we are working on a measure space (X, B, µ), then B stands for the set of all Borel sets. (7) Let X be a topological space. Then OX is the set of all open subsets of X. (8) A tacit understanding is that by a “cube” we mean a closed cube whose edges are parallel to the coordinate axes. However, we say that dyadic cubes are also cubes. (9) We define the upper half space Rn+ and the lower half space Rn− by Rn± ≡ {(x0 , xn ) ∈ Rn : ±xn > 0}.

(10.1)

Numbers (1) The constants C and c denote positive constants that may change from one occurrence to another. Because the two constants c can be different, the inequality 0 < 2c < c is by no means a contradiction. When we add a subscript, for example, this means that the constant c depends upon the parameter. It can happen that the constants with subscript differ according to the above rule. In particular, we prefer to use cn for various constants that depend on n, when we do not want to specify its precise value. (2) Let A, B ≥ 0. Then A . B and B & A mean that there exists a constant C > 0 such that A ≤ CB, where C depends only on the parameters of importance. The symbol A ∼ B means that A . B and B . A happen simultaneously, while A ' B means that there exists a constant C > 0 such that A = CB. For example, for positive sequences {Ak }∞ k=1 ∞ and {Bk }∞ k=1 “Ak ∼ Bk as k → ∞” means that {log(Ak /Bk )}k=1 is a bounded sequence. (3) When we need to emphasize or keep in mind that the constant C depends on the parameters α, β, γ etc: (1) (2) (3) (4)

Instead Instead Instead Instead

of of of of

A . B, A & B, A ∼ B, A ' B,

we we we we

write write write write

A .α,β,γ,... A &α,β,γ,... A ∼α,β,γ,... A 'α,β,γ,...

B. B. B. B.

(a) We define N ≡ {1, 2, . . .}, Z ≡ {0, ±1, ±2, . . .} and N0 ≡ {0, 1, . . .}. (b) We denote by K either R or C, the coefficient field under consideration. p (c) For a ∈ Rn , we write hai ≡ 1 + |a|2 . (d) We occasionally identify Rm+n with Rm × Rn .

Notation in this book

xvii

Function spaces (1) We use · for functions; f = f (·). (2) The function spaces are tacitly on Rn unless otherwise stated. (3) Let X be a Banach space. We denote its norm by k · kX . (4) Let Ω be an open set in Rn . Then Cc∞ (Ω) denotes the set of smooth functions with compact support in Ω. (5) Let 1 ≤ j ≤ n. The symbol xj denotes not only the j-th coordinate but also the function x = (x1 , x2 , . . . , xn ) 7→ xj . (6) The space L2 (Rn ) is the Hilbert space of square integrable functions on Rn whose inner product is given by Z hf, gi = f (x)g(x)dx. (10.2) Rn

(7) Let E be a measurable set and f be a measurable function with respect Z 1 f. to the Lebesgue measure. Then write mE (f ) ≡ |E| E (8) Let 0 < η < ∞, E be a measurable set, and f be a positive measurable (η) function with respect to the Lebesgue measure. Then write mE (f ) ≡ 1 mE (f η ) η . (9) Let 0 < η < ∞. We define the powered Hardy–Littlewood maximal operator M (η) by ! η1 Z 1 η (η) |f (y)| dy M f (x) ≡ sup . |B(x, R)| B(x,R) R>0 (10) The space C(Rn ) denotes the set of all continuous functions on Rn . (11) The space BC(Rn ) denotes the set of all bounded continuous functions on Rn . (12) Occasionally we identify the value of functions with functions. For example sin x denotes the function on R defined by x 7→ sin x. (13) Given a Banach space X , we denote by X ∗ its dual space. (14) Let µ be a measure on a measure space (X, B, µ). Given a µ-measurable set A with positive µ-measure and a function f , we write Z 1 f (x)dµ(x). mQ (f ) ≡ µ(A) A (η)

1

Let 0 < η < ∞. Then define mQ (f ) ≡ mQ (f η ) η whenever f is positive.

xviii

Notation in this book

(15) If notational confusion seems likely, for the Hardy–Littlewood maximal operator M , we use [ ] to denote M [f + g + h]. (16) For j = 1, 2, . . . , n, ∂xj =

∂ ∂xj

stands for the partial derivative. The n P ∂2 symbol ∆ stands for the Laplacian ∂xj 2 . j=1

(17) We denote the Lp (Rn )-norm by k·kLp . For other function spaces such as H¨ older’s continuous function space C γ (Rn ) of order γ > 0, we use k · kC γ to stress the function spaces. (18) When we consider function spaces on a domain Ω, we denote by C γ (Ω) the H¨ older continuous function space of order γ. (19) A quasi-norm over a linear space X enjoys positivity, homogeneity, and quasi-triangle inequality: for some α ≥ 1, kf + gkX ≤ α( kf kX + kgkX ) (f, g ∈ X ). However, to simplify, we frequently omit the word “quasi”. Likewise, we abbreviate the word “quasi-Banach space” to Banach space. (20) The ( Kronecker delta function defined on a set X is given by δjk ≡ 1 (j = k), for j, k ∈ X. 0 (j 6= k). (21) When two normed spaces X, Y are isomorphic with equivalence of norms, we write X ≈ Y . (22) For subsets A, B of a linear space V and v ∈ V , define the Minkovski sum by v + A ≡ {v + a : a ∈ A},

A + B ≡ { a + b : a ∈ A, b ∈ B}.

(23) When two topological spaces X , Y are homeomorphic, we write X ≈ Y. (24) When A and B are sets, A ⊂ B stands for the inclusion of sets. If, in addition, both A and B are topological spaces, and if the natural embedding mapping A → B is continuous, we write A ,→ B in the sense of continuous embedding. If A and B are quasi-normed spaces with the k

embedding constant k > 0, then A ,→ B.

Chapter 11 Multilinear operators and Morrey spaces

Does the Fourier series of a function converge back to the original periodic function? This is one of the most fundamental questions in harmonic analysis. Since the Fourier transform, which is the most elementary operator in this branch, and the Fourier series have a lot to do with each other, one of the goals in harmonic analysis is investigating the boundedness of the Fourier transform. In T, the one-dimensional torus, Carleson gave a complete answer; any Fourier series of L2 (T) does converge back to the original function almost everywhere. Later, Hunt [211] and other people [16, 69, 137, 248, 250] extended the results and simplified the proof. The Hilbert transform is a crucial tool for this purpose. However, in higher dimensions, there has yet to be a complete answer. Unfortunately, the ball Fourier multiplier fails to be bounded on Lp (Rn ) for any p ∈ (1, 2) ∪ (2, ∞) if n ≥ 2; see the textbook [136]. As is symbolized by this fact, investigating the Fourier transform in higher dimensions is difficult. One applicable tool to tackle the problem of Fourier analysis in higher dimensions is to consider multilinear operators. Chapter 11 scratches the surface of this branch and discusses the contribution of Morrey spaces. Section 11.1 proposes the notion of multi-Morrey spaces; we extend the definition of Morrey spaces to the multilinear setting. We investigate the multilinear Hardy–Littlewood maximal operator and multilinear singular integral operators in Section 11.1. We take up multilinear fractional integral operators in Section 11.2. Before we go further, let us review how to use the symbol ⊗. For functions f1 , f2 , . . . , fm defined on Rn , we define ⊗m j=1 fj by ⊗m j=1 fj (y1 , y2 , . . . , ym ) =

m Y

fj (yj )

(y1 , y2 , . . . , ym ∈ Rn ).

j=1

11.1

Multi-Morrey spaces

One standard method to describe the property of multilinear operators is considering the multilinear boundedness taking the form kT (f1 , f2 , . . . , fm )kY . kf1 kX1 kf2 kX2 · · · kfm kXm 1

2

Morrey Spaces

for (f1 , f2 , . . . , fm ) ∈ X1 × X2 × · · · × Xm . Here, each Xj , j = 1, 2, . . . , m is a quasi-Banach space, Y is also a quasi-Banach space and T is a multilinear p operator. We can consider the case Xj = Mqjj (Rn ) for j = 1, 2, . . . , m. However, it seems that this choice of the Xj ’s loses the potential of Morrey spaces. To take advantage of the operator T , we propose the extension of Morrey spaces to the (multilinear) function spaces adapted to our multilinear setting of T . Section 11.1.1 defines multi-Morrey spaces. The boundedness property of the multilinear maximal operator is investigated in Section 11.1.2. Multilinear singular integral operators are considered in Section 11.1.3.

11.1.1

Multi-Morrey spaces

We solidify our idea above. We content ourselves with the definition together with some simple observations and some examples. To be able to handle multilinear fractional integral operators, we define the quantity called the multi-Morrey norm as follows: Definition 1. Let 0 < q1 , q2 , . . . , qm , p ≤ ∞. Also let f1 , f2 , . . . , fm ∈ L0 (Rn ). Define the multi-Morrey norm of (f1 , f2 , . . . , fm ) by 1

1

k(f1 , f2 , . . . , fm )kMp(q

1 ,q2 ,...,qm )

≡ sup |Q| p Q∈Q

q Z m  Y j 1 . |fj (yj )|qj dyj |Q| Q j=1

Example 1. Let 0 < q1 , q2 , . . . , qm , p ≤ ∞. In order that there exist f1 , f2 , . . . , fm ∈ L0 (Rn ) such that 0 < k(f1 , f2 , . . . , fm )kMp(q we must have

1 p

≤

m P j=1

1 qj .

1 ,q2 ,...,qm )

0 m m Q P 1 p kfj kMpq j for satisfies p1 = pj . Then k(f1 , f2 , . . . , fm )kM(q ,q ,...,q ) ≤ 1

j=1

0

m

2

j

j=1

n

all f1 , f2 , . . . , fm ∈ L (R ). Hence, we are interested in the opposite inequality to Lemma 1. However, as the following example shows, this can actually fail. Example 2. Let δ denote the Dirac measure on the real line which concentrates unit point mass at the origin. An example of the case where m = 2, n = 1, f1 = δ and f2 = δ(· − 1) shows that in general, the multi-Morrey norm is strictly smaller than the m-fold product of the Morrey norms. In fact, for the exponents 1 < p1 , p2 < ∞, kf1 kMp11 kf2 kMp12 = ∞ × ∞ = ∞. Meanwhile sup(b − a) a 0, according to Chapter 3, the operator (f1 , f2 , . . . , fm ) ∈ L2 (Rn ) × L∞ (Rn )m−1 m X Y 7→ hf1 , ϕjν iL2 hfk , τjν iL2 ϕjν j∈Z,ν∈Zn

k=2

is a multilinear Calder´ on–Zygmund operator. We will consider another example of T . Such an example of T is usually R generated by the Fourier transform, given by fb(ξ) = Rn f (x)e−2πix·ξ dx for f ∈ L1 (Rn ), which is beyond the reach of this book.

6

Morrey Spaces

Example 4. We use the Fourier transform f 7→ fˆ in this example. Modeling after the classical Mihlin linear multiplier condition, Coifman and Meyer considered |∂ α σ(ξ1 , ξ2 , . . . , ξm )| . (|ξ1 |+|ξ2 |+· · ·+|ξm |)−|α| for (ξ1 , ξ2 , . . . , ξm ) ∈ (Rn )m \ {0} and α ∈ (N0 n )m satisfying |α| ≤ M for some large M . Let f1 , f2 , . . . , fm ∈ Cc∞ (Rn ). We consider the multilinear operator T [f1 , f2 , . . . , fm ](x) Z 2πix·(ξ1 +···+ξm ) = σ(ξ1 , ξ2 , . . . , ξm )fb1 (ξ1 ) · · · fc dξ1 · · · dξm m (ξm )e (Rn )m

for x ∈ Rn . Then T satisfies all these requirements. We prove a fundamental inequality for the above operators. Theorem 4. Let 1 < q1 , q2 , . . . , qm < ∞ and 0 < q ≤ p < ∞ satisfy m P 1 1 n ∞ qj = q . For all functions f1 , f2 , . . . , fm ∈ Lc (R ),

1 p

≤

j=1

kT [f1 , f2 , . . . , fm ]kMpq . k(f1 , f2 , . . . , fm )kMp(q

1 ,q2 ,...,qm )

.

Proof Similar to Theorem 2; see Exercise 3. Corollary 5. Let 1 < p1 , p2 , . . . , pm , p < ∞, 1 < q1 , q2 , . . . , qm , q < ∞ satisfy m m P P 1 1 1 n ∞ and = qj ≤ pj , p1 = pj q qj . Then for all f1 , f2 , . . . , fm ∈ Lc (R ), j=1

j=1

kT [f1 , f2 , . . . , fm ]kMpq .

m Y j=1

kfj kMpq j .

(11.5)

j

Proof Combine Lemma 1 and Theorem 4. We boil down Corollary 5 to Morrey spaces. Theorem 6. Let 1 < p1 , p2 , . . . , pm , p < ∞, 1 < q1 , q2 , . . . , qm , q < ∞ satisfy m m P P 1 1 1 qj ≤ pj , p1 = pj and q = qj . The operator T can be extended to j=1

j=1

m a bounded linear operator from Mpq11 (Rn ) × Mpq22 (Rn ) × · · · × Mpqm (Rn ) to p n Mq (R ). Use the same symbol T to denote this extension. Then (11.5) holds m for all (f1 , f2 , . . . , fm ) ∈ Mpq11 (Rn ) × Mpq22 (Rn ) × · · · × Mpqm (Rn ).

Proof We follow the strategy taken in Definition 95 in the first book. This is possible because we assume 1 < q ≤ p < ∞ and 1 < qj ≤ pj < ∞ for all j = 1, 2, . . . , m. Then similar to Theorem 4, we can extend T . In the first fpq 2 (Rn ) × · · · × M fpq m (Rn ) step, we obtain an extension T from Mpq11 (Rn ) × M 2 m p n to Mq (R ). In the next step, we will obtain an extension T from Mpq11 (Rn ) × fpq 3 (Rn ) × · · · × M fpq m (Rn ) to Mpq (Rn ). In m steps, we obtain the Mpq22 (Rn ) × M 3 m desired extension.

Multilinear operators and Morrey spaces

7

Theorem 7. Suppose that we have an m-linear singular integral operator T associated with a Calder´ on–Zygmund kernel K satisfying (11.2), (11.3) and (11.4). Suppose that T maps Lq1 (Rn )×Lq2 (Rn )×· · ·×Lqm (Rn ) boundedly into Lq (Rn ) for some 1 < q1 , q2 , . . . , qm ≤ ∞ satisfying min(q1 , q2 , . . . , qm ) < ∞. Then T can be extended to a mapping that maps L1 (Rn )×L1 (Rn )×· · ·×L1 (Rn ) boundedly into WL1/m (Rn ). Proof Suppose that we have fj ∈ L1 (Rn )∩Lqj (Rn )\{0} for each j. Write m Q kfl kL1 . By a scaling argument, it suffices to show that P ≡ l=1

kχ{T [f1 ,f2 ,...,fm ]|>2m } kL1/m . P assuming that kf1 kL1 = kf2 kL1 = · · · = kfm kL1 > 0. Fix j = 1, 2, . . . , m for a moment. Form the Calder´ on–Zygmund decomposition of fj at level 1. We have a decomposition fj = gj + bj for each j with the following properties together with collections {Qkj }k∈Kj of cubes: We write bkj ≡ χQkj (fj − mQkj (fj )) for each k ∈ Kj here and below. •

1

|Qkj | . kfj kL1 = P m .

P k∈Kj

• mQkj (|fj |) ∼ 1 for each k ∈ Kj . • kgj kL∞ ≤ 2n . P k • bj = bj . k∈Kj

Along with this decomposition, we will prove the following two model inequalites: kχ{|T [g1 ,g2 ,...,gm ]|>1} kL1/m . P (11.6) and kχ{|T [g1 ,g2 ,...,gs ,bs ,bs+1 ,...,bm ]|>1} kL1/m . P.

(11.7)

Estimate (11.6) is easy to prove:  kχ{|T [g1 ,g2 ,...,gm ]|>1} kL1/m ≤ (kT [g1 , g2 , . . . , gm ]kLq )qm . 

m Y

qm kgj kLqj 

j=1 1

1

1

Since kgj kLqj . (kgj kL1 ) qj . (kfj kL1 ) qj = P mqj , we obtain (11.6). We move on to the proof of (11.7). By (11.3), we have |T [g1 , g2 , . . . , gs , bs , bs+1 , . . . , bm ](x)| m X X X Y L1 k . ··· kbj j kL1 n(m−s)+1 (L + L ) 1 2 j=s+1 k ∈K k ∈K k ∈K s+1

s+1

s+2

s+2

m

m

.

8

Morrey Spaces m S

for x ∈ Rn \

S

j=1 k∈Kj

√ 2 nQkj , where L1 ≡

m P

m P

k

j=s+1

`(Qj j ) and L2 ≡

|x −

j=s+1

k

c(Qj j )|. Consequently, |T [g1 , g2 , . . . , gs , bs , bs+1 , . . . , bm ](x)| .

X

X

···

km ∈Km

ks+1 ∈Ks+1 ks+2 ∈Ks+2

=



m Y

k



k

+ |x −

k

1 k c(Qj j )|)n+ m−s

k

1

k

We integrate this estimate over Rn \

m S

kbj j kL1  .

S

j=1 k∈Kj

√ 2 nQkj and then use the

H¨ older inequality to obtain 

m−s

kT [g1 , g2 , . . . , gs , bs , bs+1 , . . . , bm ]k

m S

1

L m−s Rn \

S

j=1 k∈Kj

Z .



m Y

.

m Y

k



k

kj ∈Kj

 X 

j=s+1


 √ 2 nQk j 1  m−s

1

`(Qj j ) m−s

X

Rn j=s+1

kbj j kL1



1 m−s

(`(Qj j ) + |x − c(Qj j )|)n+ m−s

kj ∈Kj

j=s+1

`(Qj j ) m−s

kj j=s+1 (`(Qj )

`(Qj j )

X

1

k

m Y

X

k

1

k

(`(Qj j ) + |x − c(Qj j )|)n+ m−s 1  m−s

kbj j kL1 

dx

k

kbj j kL1 

kj ∈Kj 1

. kf1 kL1 = P m , proving (11.7). As an application of the endpoint boundedness of multilinear singular integral operators, we obtain an estimate of the oscillation. Proposition 8. Let 0 < λ < 1, f1 , f2 , . . . , fm ∈ L1 (Rn ) and let Q ∈ D. If we set m Y α ≡ Cλ m2Q (|fl |), l=1 n

Gα (Q) ≡ {x ∈ R : |T [χ2Q f1 , χ2Q f2 , . . . , χ2Q fm ](x)| > α}, where Cλ  1, then |Gα (Q)| ≤ λ|Q| and for all x, x0 ∈ Q \ Gα (Q) |T [f1 , f2 , . . . , fm ](x) − T [f1 , f2 , . . . , fm ](x0 )| .λ

∞ X k=1

In particular, ωλ (T [f1 , f2 , . . . , fm ]; Q) .λ

∞ P k=1

2−k

Qm

l=1

2−k

m Y

m2k Q (|fl |).

l=1

m2k Q (|fl |).

Multilinear operators and Morrey spaces

9

Proof We write F (~y ) = f1 (y1 )f2 (y2 ) · · · fm (ym ) = ⊗m y) j=1 fj (~ for y1 , y2 , . . . , ym ∈ Rn . Then |T [f1 , f2 , . . . , fm ](x) − T [f1 , f2 , . . . , fm ](x0 )| . |T [χ2Q f1 , χ2Q f2 , . . . , χ2Q fm ](x)| + |T [χ2Q f1 χ2Q , f2 , . . . , χ2Q fm ](x0 )|  −n−1 ∞ Z m X X + `(Q)  |x − yj | |F (~y )|d~y . k=1

(2k+1 Q)m \(2k Q)m

j=1

Thus, by the mean-value theorem, we have |T [f1 , f2 , . . . , fm ](x) − T [f1 , f2 , . . . , fm ](x0 )| . |T [χ2Q f1 , χ2Q f2 , . . . , χ2Q fm ](x)| + |T [χ2Q f1 , χ2Q f2 , . . . , χ2Q fm ](x0 )| ∞ m X Y + 2−k m2k Q (|fl |). k=1

l=1

Since there exists D ≥ 1 such that |Gα (Q)| ≤

m DY kfl kL1 (2Q) α

! m1

l=1

thanks to the weak boundedness of T , we have the desired control of |Gα (Q)|. Corollary 9. Let T be an m-linear singular integral operator associated with a Calder´ on–Zygmund kernel K. Then if 1 ≤ p1 , p2 , . . . , pm < ∞ and 0 < p < ∞ satisfy m 1 X 1 = , p j=1 pj then T extends to a bounded operator from Lp1 (Rn )×Lp2 (Rn )×· · ·×Lpm (Rn ) to WLp (Rn ). If in addition pj > 1 for each j, then T extends to a bounded operator from Lp1 (Rn ) × Lp2 (Rn ) × · · · × Lpm (Rn ) to Lp (Rn ). Proof Since we can use the Lerner–Hyt¨onen decomposition, this is a consequence of the boundedness of the multilinear maximal operator.

11.1.4

Exercises

Exercise 1. Let 0 < q1 , q2 , . . . , qm < ∞, 0 < r1 , r2 , . . . , rm < ∞ and 0 < q ≤ m P 1 p < ∞ satisfy rj ≤ qj for j = 1, 2, . . . , m and 1q = qj . Then show that j=1

Mp(q1 ,q2 ,...,qm ) (Rn ) ⊂ Mp(r1 ,r2 ,...,rm ) (Rn ).

10

Morrey Spaces

Exercise 2. Let K0 : S mn−1 → C be a C 1 -function. If we set p r ≡ |x1 |2 + |x2 |2 + · · · + |xn |2 , x x xm  1 2 K(x1 , x2 , . . . , xn ) ≡ K0 , ,..., r r r mn for (x1 , x2 , . . . , xm ) ∈ R , then show that K satisfies (11.2) and (11.3). Exercise 3. Let 1 < p1 , p2 , . . . , pm , p < ∞, 1 < q1 , q2 , . . . , qm , q < ∞ satisfy m m P P 1 1 1 qj ≤ pj , p1 = pj and q = qj . Let T be a singular integral operator. j=1

j=1

n Also let f1 , f2 , . . . , fm ∈ L∞ c (R ).

(1) Let Q be a given cube. Show that 1

|T [f1 , f2 , . . . , fm ](x)| . |Q|− p

m Y j=1

kfj kMpq j j

if x ∈ 3Q and 3Q does not intersect supp(fj ) for some j = 1, 2, . . . , m. (2) Complete the proof of Theorem 4.

11.2

Multilinear fractional integral operators

As we saw, the boundedness of the fractional integral operator Iα can be viewed more precisely using Morrey spaces than Lebesgue spaces. Hence, it is natural to expect the same thing in the multilinear setting. Here, we present a definition of multilinear fractional integral operators. Roughly there are two definitions, the one of Kenig–Stein-type and the one of Grafakos-type. The former appears in Section 11.2.1 and the latter appears in Section 11.2.2. In Section 11.2.2 we will use what we obtained in Section 11.2.1.

11.2.1

Multilinear fractional integral operators of Kenig–Stein-type

We consider the following multilinear fractional integral operators. Definition 3. The m-fold multilinear fractional integral operator of Kenig– Stein-type Iα , 0 < α < mn is given by Z f1 (y1 )f2 (y2 ) · · · fm (ym )d~y Iα [f1 , f2 , . . . , fm ](x) ≡ (x ∈ Rn ), mn−α (|x − y | + · · · + |x − y |) n m 1 m (R ) where m ∈ N and f1 , f2 , . . . , fm ∈ M+ (Rn ).

Multilinear operators and Morrey spaces Let F ∈ L0 (Rmn ). If we write Z Iα F (x1 , x2 , . . . , xm ) ≡ (Rn )m

11

F (y1 , y2 , . . . , ym )d~y (|x1 − y1 | + · · · + |xm − ym |)mn−α

(11.1)

  for x1 , x2 , . . . , xm ∈ Rn , then Iα ⊗m j=1 fj (x, x, . . . , x) = Iα [f1 , f2 , . . . , fm ](x). As we did for the classical fractional integral operator, we decompose fractional integral operators: Lemma 10. Let f1 , f2 , . . . , fm ∈ M+ (Rn ). Then X
Iα [f1 , f2 , . . . , fm ] . `(Q)α m(3Q)m ⊗m j=1 fj χQ . Q∈D

Proof The proof is omitted since it is similar to the classical case. In terms of multi-Morrey spaces, we describe the boundedness property of Iα . Theorem 11. Let m ∈ N and 0 < α < mn,

1 < q1 , q2 , . . . , qm ≤ ∞,

0 < q ≤ p < ∞,

0 1. From Lemma 13 and H¨older’s inequality, we have Jα [f1 , f2 ](x) ∞ X X . l=−∞ Q∈Dl ∞ X

.

l

2



1 u1

 + u1 −α 2

kf1 kLu1 (Q(x,2−l )) kf2 kLu2 (Q(x,2−l )) χQ (x)

(Rn )

X

2l(2n−α) m3Q (M (u1 ) f1 )m3Q (M (u2 ) f2 )χQ (x).

l=−∞ Q∈Dl (Rn )

Thus, we are in the position of using Theorem 12 to have the desired result. Theorem 15. Let m ∈ N and 0 < α < mn, 1 < q1 ≤ p1 < ∞, 1 < q2 ≤ p2 < ∞, 0 < t ≤ s < ∞. Define p, q, s and t by 1 1 1 = + , p p1 p2

1 1 1 = + , q q1 q2

1 1 α = − , s p n

q t = . p s

Assume s < min(q1 , q2 ). Then for all f1 ∈ Mpq11 (Rn ) and f2 ∈ Mpq22 (Rn ), kJα [f1 , f2 ]kMst . kf1 kMpq 1 kf2 kMpq 2 . 1

2

Proof Let v ∈ (s, min(q1 , q2 )). Then using the averaging technique of Morrey spaces, we have kJα [f1 , f2 ]kMst

Z

X

X

∞

l(n−α) χQ

. 2 f1 (· + y)f2 (· − y)dy 1

v |Q| v Q(2−l )

l=−∞ Q∈Dl (Rn ) L (Q) Ms t



X ∞ X

2l(2n−α) m3Q (M (v) f1 )m3Q (M (v) f2 )χQ . .

s

l=−∞ Q∈Dl (Rn ) Mt

Thus, we are again in the position of using Theorem 12 to have the desired result.

11.2.3

Exercises

Exercise 4. Show that Iα χ[0,1]mn (x) ∼ (1 + |x|)α−mn for x ∈ Rmn . Hint: Two cases must be considered; x ∈ [−2, 2]n or otherwise. Exercise 5. [326, Theorem 7.2] Suppose that n ≥ 2 and 1 < r, s < ∞ with 1 1 1 1 1 1 1 q p = r + s and 2 < p < n. If we let q = p − n , then show that kf · gkL . k∇f kLr kgkLs + kf kLr k∇gkLs for all f, g ∈ Cc∞ (Rn ) using |f | . I1 [|∇f |] and |g| . I1 [|∇g|] and the operator Iα given by (11.1).

Multilinear operators and Morrey spaces

11.3

15

Notes

Section 11.1 General remarks and textbooks in Section 11.1 Multilinear operators acting on Lebesgue spaces can be found in [136], for example. See the textbook [288, §4.4] for multilinear oscillatory singular integral operators. Section 11.1.1 See [221, 222] for multi-Morrey spaces. A passage to the case where Rn is equipped with a Radon measure is done in [195]. Section 11.1.2 The endpoint boundedness of multilinear singular integral operators, obtained in Theorem 7, is due to Grafakos and Torres [133]. We considered the oscillation of the image of multilinear singular integral operators in Proposition 8. We followed the paper [256]. We refer to [257] and [287] for the multilinear Hardy–Littlewood maximal operator acting on Lebesgue spaces and for multilinear Hardy operators acting on CBMO spaces, respectively. Section 11.1.3 See [133] for the fundamental facts on the boundedness in Lebesgue spaces. See [197, 210, 259, 437] for the boundedness property of multilinear singular integral operators in Morrey spaces. See [474] for the boundedness property of multilinear singular integral operators in Morrey spaces with variable exponent. See [475] for the boundedness property of multilinear singular integral operators in Morrey spaces with general Radon measures. See [285, 433, 439, 440, 503, 507, 508] for the boundedness property of multilinear singular integral operators in Herz–Morrey spaces. See [443, 445, 468, 492] for commutators. We can find a different type of multilinear singular integral operators acting on Morrey spaces [274, 473].

Section 11.2 General remarks and textbooks in Section 11.2 The definition of multilinear fractional integral operators of Kenig–Steintype goes back to [231].

16

Morrey Spaces

Section 11.2.1 Fan and Gao dealt with the boundedness of the bilinear fractional integral operator of Kenig–Stein-type equipped with the rough kernel in Morrey spaces [110, Theorem 2.2]. See [110, §4] for the boundedness property of the bilinear fractional integral operator on local Morrey spaces. Iida, Sato, Sawano and Tanaka considered the Morrey boundedness in [221, 222, 223, 217]. A passage to the non-doubling case is done in [438]. The operator Iα [f1 , f2 ] acting on Morrey spaces is investigated by many authors in many settings such as the generalized Morrey spaces [33], the weighted setting [170, 216], the case equipped with the rough kernel [215, 437] and the non-doubling setting [475]. See also [86, 512] for the case of commutators generated by Iα and other functions. We can also consider the following variant: Suppose that we have ρj ∈ R and nj ∈ N satisfying ρj < nj for each j = 1, 2, . . . , k, We can consider k Q multi-parameter Riesz potentials in Rn = Rnk , which is given by j=1

Z Rρ µ(x) =

k Y

|xk − yk |ρk −nk dµ(y1 , y2 , . . . , yk ).

Rn j=1

See [451] for more about this operator. Section 11.2.2 The operator Jα has a lot to do with the bilinear Hilbert transform. The definition of multilinear fractional integral operators of Kenig–Stein-type goes back to [134]. One of the important problems in harmonic analysis is to investigate the boundedness property of the bilinear Hilbert transform. A conjecture of Calder´ on in 1964 concerned possible extensions of H to a bounded bilinear operator on products of Lebesgue spaces. A remarkable fact is that H maps 2 Lp1 (R) × Lp2 (R) to Lp (R) boundedly if 1 < p1 ≤ ∞, 1 < p2 ≤ ∞, < p < ∞ 3 1 1 1 and = + [247, 249]. p p1 p2 See [469] for the endpoint case and [327] for the weighted case. Recently He and Yan investigated fractional integral operators of Grafakos type acting on Morrey spaces [196]. See also [193, 194] for more. See [528] for the action of multilinear operators acting on Morrey spaces.

Chapter 12 Generalized Morrey/Morrey–Campanato spaces

Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces cannot handle. The Morrey spaces, which have been handled in the previous chapters, are called classical Morrey spaces. However, classical Morrey spaces are insufficient to describe the boundedness properties. To this end, parameters p and q need to be generalized. Section 12.1 defines generalized Morrey–Campanato spaces, which are investigated in Section 12.2. In addition, Section 12.1 considers a reasonable generalization of the parameter p. Section 12.3 discusses examples of generalize Morrey–Campanato spaces.

12.1

Generalized Morrey spaces

We go back to Euclidean space with the Lebesgue measure. Let 0 < q ≤ p < ∞. We defined the classical Morrey space Mpq (Rn ) to be the set of all f ∈ L0 (Rn ) for which the quantity kf kMpq ≡

sup (x,r)∈Rn+1 +

|Q(x, r)|

1 1 p−q

Z

! q1 q

|f (y)| dy Q(x,r)

is finite, where Q(x, r) = {y ∈ Rn : |x1 − y1 |, |x2 − y2 |, . . . , |xn − yn | ≤ r}. To describe the endpoint case or to describe the intersection space, it is occasionally useful ton generalize the parameter p: let us suppose that p comes n from the function t p . Consequently, we envisage the situation where t p is replaced by a general function ϕ(t). Likewise, we will replace q by a function Φ later in Section 13.1. As is seen from this discussion, there are a couple of generalizations. Following [376], when we replace p by a function, we call it the ϕ-generalization. When we replace q by a function, which we will do in Section 13.1, we call it the Φ-generalization. Actually, we mix both of them in Section 13.1. n We define the generalized Morrey space Mϕ q (R ) in Section 12.1.1. We will show that many function spaces fall under the scope of generalized Morrey spaces. We especially consider what condition we postulate on ϕ in Section 17

18

Morrey Spaces

12.1.2. We investigate the boundedness properties of the Hardy–Littlewood maximal operator, singular integral operators, (generalized) fractional integral operators and (generalized) fractional maximal operators in Sections 12.2.1, 12.2.2 12.2.3 and 12.2.4. It is notable that the postulate on ϕ is different in each occasion. In particular, unlike Section 12.2.1 we need to consider the integral condition in Section 12.2.2. A similar thing applies to Sections 12.2.3 and 12.2.4.

12.1.1

Definition of generalized Morrey spaces

From the observation above, we present the following definition following the notation by Sawano, Sugano and Tanaka [417]: Definition 5. Let 0 < q ≤ ∞ and ϕ : (0, ∞) → [0, ∞) be a function which does not satisfy ϕ ≡ 0. n (1) The generalized Morrey space Mϕ q (R ) is defined as the set of all f ∈ 0 n L (R ) such that

kf kMϕq ≡

sup

ϕ(r)

(x,r)∈Rn+1 +

1 |Q(x, r)|

! q1

Z

q

|f (y)| dy

< ∞. (12.1)

Q(x,r)

n (2) The weak generalized Morrey space WMϕ q (R ) is defined as the set of all f ∈ L0 (Rn ) such that kf kWMϕq ≡ sup kλχ(λ,∞] (|f |)kMϕq < ∞. λ>0

Formally, we consider the case where q = ∞. Example 6. Let ϕ : (0, ∞) → [0, ∞) be a function which satisfies ϕ(t) ≥ 1 n ∞ n for all t ∈ (0, 1). Then Mϕ ∞ (R ) equals as a set to L (R ), if ϕ is bounded. n ϕ Otherwise M∞ (R ) is trivial. Since we did not consider this case for Morrey spaces, we usually omit the case where q = ∞. Although we tolerate the case where ϕ(t) = 0 for some t > 0, it turns out that there is no need to consider such a possibility. Remark that the terminology weight, used in [127, 128] and so on, stands for the generalization of p into ϕ. Before we go into more detail, clarifying remarks may be in order. Remark 1. n (1) Nakai [336] defined the generalized Morrey space Mϕ q (R ) to be the set 0 n of all f ∈ L (R ) such that

kf kMϕq ≡

sup (x,r)∈Rn+1 +

ϕ(r)−1

1 |Q(x, r)|

Z

! q1 |f (y)|q dy

< ∞.

Q(x,r)

(12.2)

Generalized Morrey/Morrey–Campanato spaces

19

Some people prefer to use kf kMϕq ≡

1 ϕ(r)

sup (x,r)∈Rn+1 +

! q1

Z

|f (y)|q dy

< ∞.

Q(x,r)

(2) Some people prefer to include the case where ϕ depends on x as well: For f ∈ L0 (Rn ), 0 < q < ∞ and a function ϕ : Rn × (0, ∞) → [0, ∞), we can consider the norm given by ! q1 Z 1 sup ϕ(x, r) |f (y)|q dy < ∞. kf kMϕq ≡ |Q(x, r)| Q(x,r) (x,r)∈Rn+1 +

See [261, 312, 336] for example. n

We can recover the Lebesgue space Lq (Rn ) by letting ϕ(t) ≡ t q for t > 0 as we have mentioned. To compare Morrey spaces with generalized Morrey spaces, we occasionally call Morrey spaces classical Morrey spaces. In addition n to the function ϕ(t) = t p , we consider the following typical functions: Example 7. (1) The function ϕ ≡ 1 generates L∞ (Rn ) thanks to the Lebesgue convergence theorem. n

(2) Let 0 < q < ∞ and a ∈ R. Define ϕ(t) ≡ t q (log(e + t))a for t > 0. We n remark that Mϕ q (R ) 6= {0} if and only if a ≤ 0. In fact, we have kf kMϕq =

(log(e + r))a kf kLq (Q(x,r)) .

sup (x,r)∈Rn+1 +

Thus, if f ∈ L0 (Rn ) satisfies kf kMϕq < ∞ and if a > 0, then we have kf kLq (Q(x,r)) = 0 for any cube Q(x, r). Thus, f = 0 a.e. Conversely if n a ≤ 0, then Lq (Rn ) ⊂ Mϕ q (R ). n

n

(3) Let 0 < q ≤ p1 < p2 < ∞. Then ϕ(t) = t p1 + t p2 , t > 0 can be used to express the intersection of Mpq 1 (Rn ) ∩ Mpq 2 (Rn ). In general, for 0 < q < ∞ and ϕ1 , ϕ2 : (0, ∞) → [0, ∞) satisfying ϕ1 (t1 ) 6= 0 and n ϕ1 +ϕ2 n ϕ2 1 (Rn ) with ϕ(t2 ) 6= 0 for some t1 , t2 > 0 Mϕ q (R ) ∩ Mq (R ) ≈ Mq equivalence of norms. n

n

(4) Let 0 < q ≤ p1 < p2 < ∞. Let ϕ(t) = χQ∩(0,∞) (t)t p1 + χ(0,∞)\Q (t)t p2 n p1 n p2 n 0 n for t > 0. Then Mϕ q (R ) = Mq (R )∩Mq (R ) and for any f ∈ L (R ) kf kMϕq = max{kf kMpq 1 , kf kMpq 2 }. (5) We can consider the norm sup (x,r)∈Rn ×(0,1)

|Q(x, r)|

1 p

1 |Q(x, r)|

! q1

Z

|f (y)|q dy

Q(x,r) n

for 0 < q < p < ∞. In fact, we take ϕ(t) = t p χ(0,1] (t).

20

Morrey Spaces

(6) Likewise, we can consider the norm kf kMqp ≡

|Q(x, r)|

sup

1 |Q(x, r)|

1 p

x∈Rn ,r∈[1,∞)

! q1

Z

q

|f (y)| dy Q(x,r)

n

for 0 < q < p < ∞. In fact, we take ϕ(t) = t p χ[1,∞) (t). (7) Let 0 < q < ∞. The uniformly locally Lq -integrable space Lquloc (Rn ) is the set of all f ∈ L0 (Rn ) for which ! q1

Z

q

|f (y)| dy

sup (x,r)∈Rn ×(0,1)

! q1

Z

q

|f (y)| dy

= sup x∈Rn

Q(x,r)

Q(x,1)

n

is finite. As before, if we let ϕ(t) ≡ t q χ(0,1] (t) for t > 0, then we obtain n q Lquloc (Rn ) = Mϕ q (R ). We can define the weak uniformly locally L q n 0 n integrable space WLuloc (R ) in a similar manner. For f ∈ L (R ), the norm is given by kf kWLquloc ≡ sup λkχ(λ,∞] (|f |)kLquloc . λ>0

(8) We can consider the norm kf kMpq ≡

1

sup (x,r)∈Rn ×(0,1)

(q)

|Q(x, min(1, r))| p mQ(x,r) (f ) n

n

n

for 0 < q ≤ p < ∞. In fact, we take ϕ(t) = min(1, t p − q )t q . Sometimes the space Mpq (Rn ) generated by this norm is called the modified Morrey space. The fifth example deserves a name. We define the small Morrey space as follows: Definition 6. Let 0 < q < p < ∞. The small Morrey space mpq (Rn ) is the set of all f ∈ L0 (Rn ) for which the quantity kf kmpq ≡

sup (x,r)∈Rn ×(0,1)

|Q(x, r)|

1 p

1 |Q(x, r)|

! q1

Z

|f (y)|q dy

Q(x,r)

is finite. The weak small Morrey space Wmpq (Rn ) is defined in a similar manner. For f ∈ L0 (Rn ), the norm is given by kf kWmpq ≡ sup λkχ(λ,∞] (|f |)kmpq . λ>0

Remark that small Morrey spaces are sometimes called local Morrey spaces; see [244, Definition 0.1], for example. The following estimate of the norm of the indicator functions can be found in many places; see [103, Proposition A] for example.

Generalized Morrey/Morrey–Campanato spaces

21

Example 8. Let x ∈ Rn and r > 0. Then n

n

kχQ(x,r) kMϕq = sup ϕ(t) min(t− q , r− q ). t>0

In fact, simply observe that  1 n n |Q(x, min(r, R))| q kχQ(x,r) kMϕq ≡ sup ϕ(R) = sup ϕ(t) min(t− q , r− q ). |Q(x, R)| t>0 R>0 The following min(1, q)-triangle inequality holds: Lemma 16. Let 0 < q < ∞ and ϕ : (0, ∞) → (0, ∞) be a function. Then kf + gkMϕq min(1,q) ≤ kf kMϕq min(1,q) + kgkMϕq min(1,q) n for all f, g ∈ Mϕ q (R ).

Proof This is similar to classical Morrey spaces: Use the min(1, q)-triangle inequality for the Lebesgue space Lq (Rn ). See Exercise 7. n We show that the (quasi-)norm of Mϕ q (R ) is complete.

Proposition 17. Let 0 < q < ∞, and let ϕ : (0, ∞) → [0, ∞) be a function n which does not satisfy ϕ ≡ 0. Then Mϕ q (R ) is a quasi-Banach space and if n ϕ q ≥ 1, then Mq (R ) is a Banach space. Proof The norm inequality follows from Lemma 16. The proof of the completeness is a routine, which we omit. n Proposition 17 guarantees that the (quasi-)norm of Mϕ q (R ) is complete. n ϕ However, it may happen that Mq (R ) = {0} as is seen from Example 9. We check that this extraordinary thing never happens if ϕ satisfies a mild condition.

Proposition 18. Let 0 < q < ∞, and let ϕ : (0, ∞) → [0, ∞) be a function which fails ϕ ≡ 0. Then the following are equivalent: n n ϕ (a) L∞ c (R ) ⊂ Mq (R ). n (b) Mϕ q (R ) 6= {0}. n

(c) sup ϕ(t) min(t− q , 1) < ∞. t>0

Proof It is clear that (a) implies (b). n Assume (b). Then there exists f ∈ Mϕ q (R ) \ {0}. We may assume that f (0) 6= 0 and that x = 0 is the Lebesgue point of |f |q . Since f ∈ Lqloc (Rn ), by (q) the Lebesgue differentiation theorem, mQ(r) (f ) ∼ 1 for all 0 < r < 1. Here, the implicit constants depend on f . Thus, ! q1 Z 1 sup ϕ(t) ∼ sup ϕ(t) |f (y)|q dy ≤ kf kMϕq < ∞. |Q(t)| Q(t) 0 0. Then L∞ c (R ) ⊂ ϕ n Mq (R ).

(3) We let ϕ(t) ≡ `−B (t), t > 0. For a ∈ R, we set f (x) ≡ (1+|x|)−a , x ∈ Rn . n ∞ n Let us see that Mϕ q (R ) is close to L (R ) if β1 ≥ 0. (1) By the Lebesgue differentiation theorem, kf kL∞ ≤ 0 if β1 < 0, so n that f = 0 a.e. Consequently, if β1 < 0, then Mϕ q (R ) = {0}. n (2) Let β1 ≥ 0 > β2 . Then f ∈ Mϕ q (R ) if and only if a < 0. n (3) Let β1 , β2 ≥ 0. Then f ∈ Mϕ q (R ) if and only if a ≤ 0.

The following scaling law can be found in many places; see [354, Lemma 2.5] for example. Lemma 19. Let 0 < η, q < ∞ and ϕ : (0, ∞) → (0, ∞) be a function. ϕη n η n Then f ∈ Mϕ q (R ) if and only if |f | ∈ Mq/η (R ). Furthemore in this case k|f |η kMϕη = kf kMϕq η . q/η

Generalized Morrey/Morrey–Campanato spaces

23

Proof Let f ∈ L0 (Rn ). We content ourselves with showing the equality k|f |η kMϕη = kf kMϕq η . We calculate q/η

k|f |η kMϕη = q/η

sup Q=Q(a,r)

ϕ(r)η kf kLq (Q) η η

|Q| q

= kf kMϕq η

by using k|f |η k ηq = kf kq η . The nesting property holds like classical Morrey spaces. Lemma 20. Let 0 < q1 ≤ q2 < ∞ and ϕ : (0, ∞) → (0, ∞) be a function. n n ϕ Then Mϕ q2 (R ) ,→ 1,→Mq1 (R ). Proof This is similar to classical Morrey spaces. See Exercise 8.

12.1.2

The class Gq

It will be demanding to consider all possible functions ϕ. We will single out good functions. We answer the question of what functions are good. Definition 7. Let n0 < q ≤ ∞. A function ϕ ∈ M↑ (0, ∞) belongs to the class n Gq if t− q ϕ(t) ≥ s− q ϕ(s) for all 0 < t ≤ s < ∞. It should be noted that Gq1 ⊂ Gq2 if 0 < q2 < q1 < ∞. Here, we list a series of functions in Gq . Example 10. Let 0 < q < ∞. (1) Let u ∈ R, and let ϕ(t) = tu for t > 0. Then ϕ belongs to Gq if and only if 0 ≤ u ≤ nq . (2) Let 0 < u ≤

n q,

L  1, and let ϕ(t) =

tu for t > 0. Then ϕ log(L + t)

belongs to Gq . (3) If ϕ1 , ϕ2 ∈ Gq , then ϕ1 + ϕ2 , max(ϕ1 , ϕ2 ), min(ϕ1 , ϕ2 ) ∈ Gq . (4) Let 0 ≤ u  1, and let ϕ(t) =

tu for t ≥ 0. Then ϕ ∈ / Gq because log(e + t)

ϕ is not increasing. (5) Note that G∞ is the set of all constant functions. So we often ignore the case where q = ∞. We start with a simple observation that any function in Gq enjoys the doubling property: n

Proposition 21. If ϕ ∈ Gq with 0 < q < ∞, then ϕ(r) ≤ ϕ(2r) ≤ 2 q ϕ(r) for all r > 0.

24

Morrey Spaces

Proof The left inequality is a consequence of the fact that ϕ ∈ M↑ (0, ∞), −n q while the right inequality follows from the fact that t 7→ t ϕ(t) is decreasing. The next proposition justifies that we can naturally use the class Gq . Proposition 22. Let 0 < q 0

∗

n ϕ n that Mϕ q (R ) ≈ Mq (R ) with equivalence of norms.

Proof Let f ∈ L0 (Rn ). By decomposing Q into [`(Q)/t0 + 1]n cubes of the same size, we have (q) (q) mQ0 (f ). If we let mQ (f ) . sup Q0 ∈Q(Q),`(Q0 )=t0

ϕ1 (t0 ) ≡ inf0 ϕ(t) t≥t

then it is easy to see that ϕ1 (t) ≤ ϕ(t) for all t > 0. Hence kf kMϕq 1 ≤ kf kMϕq . Furthermore, (q)

(q)

sup

ϕ(`(Q))mQ0 (f )

sup ϕ(`(Q))mQ (f ) . sup

inf

. sup

inf

sup

ϕ(`(Q))mQ0 (f )

= sup

inf

sup

ϕ1 (t0 )mQ0 (f )

= sup

inf

sup

ϕ1 (`(Q0 ))mQ0 (f )

Q∈Q t0 ∈(0,`(Q)] Q0 ∈Q(Q),`(Q0 )=t0

Q∈Q

(q)

Q∈Q t0 ∈(0,`(Q)] Q0 ∈Q: `(Q0 )=t0

(q)

Q∈Q t0 ∈(0,`(Q)] Q0 ∈Q: `(Q0 )=t0

(q)

Q∈Q t0 ∈(0,`(Q)] Q0 ∈Q: `(Q0 )=t0

≤ kf kMϕq 1 . Thus, it follows that kf kMϕq 1 ≤ kf kMϕq . kf kMϕq 1 . Next, if we let n

n

n

ϕ∗ (t) ≡ t q sup ϕ1 (t0 )t0− q = sup ϕ1 (st)s− q t0 ≥t

(t > 0),

s≥1

then kf kMϕq 1 = kf kMϕq ∗ . In fact, since ϕ1 is increasing, ϕ∗ is increasing. From the definition of ϕ∗ , ϕ∗ (t) ≥ ϕ1 (t) for all t > 0 and thus kf kMϕq 1 ≤ kf kMϕq ∗ . On the other hand, for any r > 0 and ε > 0, we can find r0 ≥ r such that n

n

ϕ∗ (r) ≤ (1 + ε)r q ϕ1 (r0 )r0− q . Thus for any cube Q(x, r), ∗

ϕ

(q) (r)mQ(x,r) (f )

0

≤ (1 + ε)ϕ1 (r )

1 |Q(x, r0 )| (q)

≤ (1 + ε)ϕ1 (r0 )mQ(x,r0 ) (f ) ≤ (1 + ε)kf kMϕq 1 .

Z

! q1 q

|f (y)| dy Q(x,r)

Generalized Morrey/Morrey–Campanato spaces

25

Taking the supremum over x and r, we have kf kMϕq ∗ ≤ (1 + ε)kf kMϕq 1 . Since ε > 0 is arbitrary, we conclude kf kMϕq ∗ ≤ kf kMϕq 1 . In view of Proposition 22, it follows that we can always suppose that ϕ ∈ Gq . We may further suppose that there exists δ > 0 such that ϕ(r) ≤ δ −1 and that

n

r− q ϕ(r) > δ

(0 < r ≤ 1)

(12.3)

(1 < r < ∞).

(12.4)

As the next proposition shows, Gq is a good class in addition to the nice property that it naturally arises in generalized Morrey spaces. Proposition 23. Let ϕ ∈ Gq with 0 < q < ∞. Then we can find a continuous function ϕ∗ ∈ Gq such that ϕ∗ is strictly decreasing and that ϕ ∼ ϕ∗ . Proof We look for ϕ∗ in a couple of steps. (1) Consider ϕ0 (t) ≡ t

n q

Z

2t

ϕ(s) t

ds s

n q +1

2

Z =

ϕ(ts) 1

ds s

n q +1

(t > 0).

Since ϕ is increasing, the function ϕ0 is increasing. Also, by the fact that ϕ is doubling, we see that ϕ0 and ϕ are equivalent. Thus, we may assume that ϕ is continuous. (2) We define  ψ(t) ≡ ϕ(t)χ(0,1] (t) +

1 − e−t 1 − e−1

 nq ϕ(1)χ(1,∞) (t)

(t > 0). n

n

Since we can check that t ∈ (0, 1] 7→ t−n q ψ(t) and t ∈ [1, ∞) 7→ t− q ψ(t) are both decreasing, t ∈ (0, ∞) 7→ t− q ψ(t) is decreasing. Likewise, we can check that ψ is increasing. Thus, ψ ∈ Gq . Since ψ + ϕ ' ϕ, we can assume that ϕ is strictly increasing in (1, ∞) and that ϕ is continuous. (3) Finally, we consider ϕ∗ (t) ≡

∞ X ϕ(2k t) k=0

2kN

(t > 0),

n + 1. Then it is easy to check that ϕ∗ is equivalent to ϕ q since each term is dominated by 2−k ϕ. Likewise, since ϕ ∈ Gq , ϕ∗ ∈ Gq . Finally since ϕ is strictly increasing in (1, ∞), the function ϕ(2k t) is strictly increasing on (2−k , ∞) for any k ∈ N. Since k ∈ N is arbitrary, ϕ∗ is strictly increasing. where N ≡

26

Morrey Spaces Thus, we are led to the following definition:

Definition 8. The class W stands for the set of all positive continuous functions on (0, ∞). That is, W = C((0, ∞), (0, ∞)). We apply what we have obtained to small Morrey spaces. n Example 11. Let 0 < q < ∞. Let us see how we modify ψ in Mψ q (R ) to ϕ n obtain the equivalent space Mq (R ) with ϕ ∈ Gq . n

n

(1) Let ϕ(t) ≡ max(t p , 1) and ψ(t) ≡ t p χ(0,1] (t) for t > 0. Then mpq (Rn ) = n ϕ n Mψ q (R ) = Mq (R ). n

(2) Let ϕ(t) ≡ max(ta , 1) with a ≥ nq and ψ(t) ≡ t q χ(0,1] (t) for t > 0. Then n n ϕ Lquloc (Rn ) = Mψ q (R ) = Mq (R ). Note that ϕ ∈ Gq ∩ W and that ψ ∈ Gq \ W in both cases. So far we have shown that we may assume that ϕ ∈ Gq . As a result, we may assume that ϕ is doubling. This observation makes the definition of the norm k · kMϕq more flexible. Remark 2. In (12.1), cubes can be replaced with balls; equivalent norms will be obtained. Precisely use the norm given by kf kMϕq ≡

sup (x,r)∈Rn+1 +

ϕ(r)

1 |B(x, r)|

Z

! q1 |f (y)|q dy

B(x,r)

to go through the same argument given for the norm defined by means of cubes. Related to this definition, we give some definitions related to the class Gq . Although we showed that it is sufficient to limit ourselves to Gq , we still feel that this class is too narrow as the function of ϕ(t) = t log(e + t−1 ) shows. Consequently, it is convenient to relax the condition on ϕ. The following definition will serve this purpose. Definition 9 (Almost decreasing, Almost increasing). Let ` > 0. (1) A function ϕ : (0, `] → (0, ∞) is almost decreasing if ϕ(s) . ϕ(t) for all 0 ≤ t < s ≤ `. (2) A function ϕ : (0, `] → (0, ∞) is almost increasing if ϕ(s) & ϕ(t) for all 0 ≤ t < s ≤ `. (3) A function ϕ : (0, ∞) → (0, ∞) is almost decreasing if ϕ(s) . ϕ(t) for all 0 ≤ t < s < ∞. (4) A function ϕ : (0, ∞) → (0, ∞) is almost increasing if ϕ(s) & ϕ(t) for all 0 ≤ t < s < ∞.

Generalized Morrey/Morrey–Campanato spaces

27

The implicit constants in these inequalities are called the almost decreasing/increasing constants. Example 12. (1) Let a, b be real parameters with b 6= 0. Let ϕa,b (t) = ta (log(e + t))b for t > 0. Then ϕa,b is almost increasing for any a > 0. If a < 0, then ϕa,b is almost decreasing. (2) The function ϕ(t) = t + sin 2t, t > 0 is almost increasing but not increasing. (3) The function ϕ(t) = 1 + et (1 − cos t), t > 0 is not almost increasing because ϕ(2πm) = 1 and ϕ(2πm + π) = 1 + 2e2πm+π for all m ∈ N. (4) Let 0 < p, q < ∞ and β1 , β2 ∈ R. We write ( (1 + | log r|)β1 (0 < r ≤ 1), B ` (r) ≡ (1 + | log r|)β2 (1 < r < ∞). Set B ≡ (β1 , β2 ) and n

ϕ(t) ≡ t p `−B (t)

(t > 0)

as we did in Example 9. We observe that ϕ is almost increasing for n all p > 0 and β1 , β2 ∈ R. We also note that t 7→ ϕ(t)t− q is almost decreasing if and only if p = q and β1 ≤ 0 ≤ β2 or p > q. Note that ϕ is an equivalent to a function ψ ∈ Gq if and only if either p = q, β1 ≤ 0 ≤ β2 or p > q since we can neglect the effect of ”log”. (5) Let E ⊂ (0, ∞) be a non-Lebegsgue measurable set. Then ϕ = 1 + χE is almost increasing and almost decreasing. Note that ϕ is not measurable. (6) A simple but still standard example is as follows: Let m ∈ N and define n

tq ϕ(t) ≡ lm (t)

(t > 0),

where lm (t) is given inductively by l0 (t) ≡ t,

lm (t) ≡ log(3 + lm−1 (t))

(m = 1, 2, . . .)

for t > 0. The for all 0 < q < ∞ and m ∈ N, ϕ ∈ Gq . As the following lemma shows, we can always replace an almost increasing function with an increasing function. Lemma 24. If a function ϕ : (0, ∞) → (0, ∞) is almost increasing with the almost increasing constant C0 > 0, then there exists an increasing function ψ ∈ M+ (0, ∞) such that ϕ(t) ≤ ψ(t) ≤ C0 ϕ(t) for all t > 0.

28

Morrey Spaces Proof Simply set ψ(t) ≡ sup ϕ(s), t > 0. 0 0. Proof We may reexamine Example 8. Here, we give a direct proof. It is easy to see that kχQ(x,R) kMϕq ≥ ϕ(R). To prove the opposite inequality we consider 1 q ϕ(r) 1 kχQ(y,r)∩Q(x,R) kL |Q(y, r)| q for any cube Q = Q(y, r). When R ≤ r, then ϕ(r)

kχQ(y,r)∩Q(x,R) kLq |Q(y, r)|

1 q

≤ ϕ(r)

kχQ(x,R) kLq |Q(y, r)|

1 q

≤ ϕ(R)

kχQ(x,R) kLq 1

|Q(y, R)| q

= ϕ(R).

since ϕ ∈ Gq . When R > r, then ϕ(r)

kχQ(y,r)∩Q(x,R) kLq |Q(y, r)|

1 q

≤ ϕ(r)

kχQ(y,r) kLq 1

|Q(y, r)| q

= ϕ(r) ≤ ϕ(R)

again by virtue of the fact that ϕ ∈ Gq . Following [354, Corollary 2.3], we obtain a direct consequence of the above quantitative information. Corollary 26. Let 0 < q < ∞ and ϕ ∈ Gq . If N0 > n Mϕ q (R ).

n q,

then (1 + | · |)−N0 ∈

n

Proof Since ϕ ∈ Gq , we have ϕ(t)t− q ≤ ϕ(1) for all t ≥ 1. we have k(1 + | · |)−N0 kMϕq .

∞ X j=1

∞ X kχQ(j) kMϕq ϕ(j) ≤ < ∞. max(1, j − 1)N0 max(1, j − 1)N0 j=1

Here, for the second inequality we invoked Lemma 25. We now consider the role of ϕ. We did not tolerate the case where p = ∞ n p n when we define Mpq (Rn ). If we define M∞ q (R ) similar to Mq (R ), then ∞ n ∞ n Mq (R ) = L (R ) by the Lebesgue differentiation theorem. The next theorem concerns a situation close to this. n ∞ n Theorem 27. Let 0 < q < ∞ and ϕ ∈ Gq . Then, Mϕ q (R ) ⊂ L (R ) if and ϕ− inf ϕ(t)

n ∞ n only if inf ϕ(t) > 0. In this case, Mϕ q (R ) ≈ L (R ) ∩ Mq t>0

equivalence of norms.

t>0

(Rn ) with

Generalized Morrey/Morrey–Campanato spaces

29

Proof If inf ϕ(t) > 0, then by the Lebesgue differentiation theorem we t>0

obtain 1 |f (x)| = lim r↓0 |B(x, r)| q

Z

|f (y)|q dy

B(x,r) q

ϕ(r) 1 lim ≤ q r↓0 inf ϕ(t) |B(x, r)|

Z

t>0

≤

|f (y)|q dy

B(x,r)

1 kf kMϕq q inf ϕ(t)q

t>0

n q ∞ n for all f ∈ Mϕ q (R ) and all Lebesgue points x of |f | . Therefore, f ∈ L (R ). n ∞ n ϕ Assume that Mq (R ) ⊂ L (R ). Then by the closed graph theorem, n kf kL∞ . kf kMϕq for all f ∈ Mϕ q (R ). If we choose f = χB(a,r) , then 1 . ϕ(r). This shows that inf ϕ(t) > 0. t>0

n Finally, by taking ϕ1 ≡ inf ϕ(t) and ϕ2 ≡ ϕ − ϕ1 , we obtain Mϕ q (R ) ≈ t>0

ϕ− inf ϕ(t)

L∞ (Rn )∩Mq (Rn ) with equivalence of norms from the general formula n n ϕ1 ϕ1 +ϕ2 2 (R ) ≈ Mq (Rn ) ∩ Mϕ Mq q (R ) with equivalence of norms. t>0

We also investigate the reverse inclusion to Theorem 27. n ∞ n Theorem 28. Let 0 < q < ∞ and ϕ ∈ Gq . Then, Mϕ q (R ) ⊃ L (R ) if and only if sup ϕ(t) < ∞. t>0

Proof If sup ϕ(t) < ∞, then by letting ψ(t) ≡ sup ϕ(s), t > 0 we have t>0

s>0

n L∞ (Rn ) ≈ Mψ q (R ) with equivalence of norms. Since ϕ ≤ ψ, we have n n ∞ n ϕ n ϕ Mψ q (R ) ⊂ Mq (R ). Thus L (R ) ⊂ Mq (R ). n ∞ n ϕ Conversely, if L (R ) ⊂ Mq (R ), then by the closed graph theorem the embedding norm is finite. Hence ϕ(r) = kχQ(r) kMϕq . kχQ(r) kL∞ = 1 for all r > 0. Then sup ϕ(t) < ∞. t>0

By combining these two theorems, we obtain the following result. Corollary 29. Let 0 < q < ∞ and ϕ ∈ Gq . Then, the following are equivalent: (1) log ϕ ∈ L∞ (0, ∞), i.e., 0 < inf ϕ(t) ≤ sup ϕ(t) < ∞, t>0

t>0

n (2) L∞ (Rn ) = Mϕ q (R ). n Consequently, if log ϕ grows or decays slowly we can say that Mϕ q (R ) is close to L∞ (Rn ). The next example shows that when the support of functions is torn apart, the norm does not increase even in the case of generalized Morrey spaces. We will say that a family {Qλ }λ∈Λ of cubes is non-overlapping if {Int(Qλ )}λ∈Λ is disjoint.

30

Morrey Spaces

Example 13. Let 0 < q < ∞ and ϕ ∈ Gq . Suppose that we have a collection ∞ ∞ ∞ of cubes {Qj }∞ j=1 = {Q(aj , rj )}j=1 such that {3Qj }j=1 = {Q(aj , 3rj )}j=1 is ∞ non-overlapping and is contained in Q = Q(a0 , r0 ). Let {fj }j=1 be a collection n of functions in Mϕ q (R ) satisfying n

kfj kLq ≤ ϕ(r0 )−1 rj q ,

supp(fj ) ⊂ Qj ,

(12.5)

kfj kMϕq ≤ 1 for each j ∈ N. Then we claim f ≡

∞ X

(12.6)

n fj ∈ Mϕ q (R ). Since f is supported

j=1

in Q, we have only to consider cubes Q(x, r) contained in 3Q; if a cube Q0 = Q(x, r) intersects both Q and Rn \ 3Q, then its triple 3Q0 engulfs Q and the radius of Q is smaller than that of Q0 . Thus, we have kf kMϕq .

ϕ(r)

sup , Q(x,r)⊂3Q (x,r)∈Rn+1 +

1 |Q(x, r)|

! q1

Z

q

|f (y)| dy

.

Q(x,r)

With this in mind, we suppose that Q(x, r) is contained in 3Q = Q(x0 , 3r0 ). (q) We fix a cube Q(x, r) and estimate ϕ(r)mQ(x,r) (f ). Let J1 ≡ {j ∈ N : Q(x, r) ∩ Q(aj , rj ) 6= ∅, rj ≤ r}, J2 ≡ {j ∈ N : Q(x, r) ∩ Q(aj , rj ) 6= ∅, rj > r}. Accordingly we set q  q1 X 1  Ii ≡ ϕ(r) fj (y) dy  |Q(x, r)| Q(x,r) j∈Ji 

Z

for i = 1, 2. As for I1 , we use (12.5) and the fact that {Q(aj , rj )}∞ j=1 is non-overlapping:  q1

 I1 ≤ ϕ(r) 

1 |Q(x, r)|

X j∈J1

 q1



ϕ(r0 )−q rj n  . 

1 |Q(x, r)|

X

rj n  . 1.

j∈J1

As for I2 , we have to consider only one summand, since Q(aj , 3rj ) engulfs Q(x, r) if j ∈ J2 : we can deduce I2 ≤ 1 using (12.6). As an application of Example 13, we present another example. Denote by [t] the integer part of t ∈ R. Example 14. Let ϕ ∈ Gq with 0 < q < ∞. We fix a ∈ Rn . Let {sk }∞ k=1 ⊂ (0, 1] be a sequence which decreases to 0.

Generalized Morrey/Morrey–Campanato spaces

31

Keeping in mind inf ϕ(t)q/n t−1 = ϕ(1)q/n ,

0 0. (1) Let q ≥ 1. Find the necessary and sufficient condition for ϕ ∈ Gq . S ϕ∗ n n (2) Find ϕ∗ ∈ Gq such that Mϕ 2 (R ) = M2 (R ) with norm equivalence. q≥1 ∗ n 5 n (3) Describe Mϕ 2 (R ) ∩ L (R ) in terms of Orlicz spaces.

Exercise 7. [406, Lemma 3.7] Prove Lemma 16 using Minkowski’s inequality and the p-convexity. Exercise 8. [406, Lemma 3.12] Prove Lemma 20 reexamining the proof of Theorem 19 in the first book. Exercise 9. Let 0 < q < p < ∞ and R > 0. Also let f ∈ L0 (Rn ). Consider 1 (q) the norm kf kMqp ;R ≡ sup |Q(x, r)| p mQ(x,r) (f ) motivated by Example x∈Rn ,r∈[R,∞)

7. Then show that kf kMqp ∼R,p,q kf kMqp ;R . Exercise 10. Show that in the definition of generalized Morrey spaces, we can use both cubes and balls; both result in the equivalent norms. Hint: Mimic the idea used in Exercise 13 in the first book.

12.2

Boundedness properties of operators in generalized Morrey spaces

Having set down the properties of generalized Morrey spaces, we are now interested in the boundedness properties of operators.

34

12.2.1

Morrey Spaces

Hardy–Littlewood maximal operator in generalized Morrey spaces and the class Z0

After we generalize the parameter p in the space Mpq (Rn ), we realize that the boundedness of the maximal operator is obtained due to the condition on q ∈ (1, ∞). Theorem 34. Let 1 < q < ∞ and ϕ ∈ Gq . Then kM f kMϕq . kf kMϕq for all f ∈ L0 (Rn ). We observe that we did not require anything other than ϕ ∈ Gq as evidence of the fact that the parameter q plays a central role for the boundedness of the Hardy–Littlewood maximal operator. Proof Once we assume ϕ ∈ Gq , the proof of this theorem will be an adaptation of the classical case. Fix a cube Q(x, r). We need to prove (q) ϕ(r)mQ(x,r) (M f ) . kf kMϕq . Let f1 ≡ χQ(x,5r) f and f2 ≡ f − f1 . We need to prove (q) ϕ(r)mQ(x,r) (M f1 ) . kf kMϕq (12.1) and (q)

ϕ(r)mQ(x,r) (M f2 ) . kf kMϕq .

(12.2)

The proof of (12.1) follows from the boundedness of the Hardy–Littlewood maximal operator and the fact that ϕ(5r) ' ϕ(r) for any r > 0. As for (12.2), we use M [χRn \5Q f ](y) . sup mR (|f |) (y ∈ Q) R∈Q] (Q)

with Q = Q(x, r). Then by virtue of the fact that ϕ is increasing, we obtain (q)

ϕ(r)mQ(x,r) (M f2 ) . ϕ(r) × ≤

sup

mR (|f |)

R∈Q] (Q)

sup

ϕ(`(R))mR (|f |)

R∈Q] (Q)

. kf kMϕ1 ≤ kf kMϕq , n where for the last inequality, we used the nesting property of Mϕ q (R ), 1 ≤ q < ∞.

Any function ϕ : (0, ∞) → (0, ∞) will do. Hence, we have the following boundedness for small Morrey spaces. Example 15. Let 1 < q ≤ p < ∞. Then M is bounded on mpq (Rn ). Hence, n p n M is bounded on Lquloc (Rn ). In fact, M is bounded on Mϕ q (R ) ∼ mq (R ) n thanks to Theorem 34, where ϕ(t) = max(t p , 1), t > 0. Similar to Theorem 34, we can prove the boundedness of the Hardy– Littlewood maximal operator.

Generalized Morrey/Morrey–Campanato spaces

35

Theorem 35. Let 1 ≤ q < ∞ and ϕ ∈ Gq . Then kM f kWMϕq . kf kMϕq for all f ∈ L0 (Rn ). Proof Simply resort to the weak-(1, 1) boundedness of M and modify the proof of Theorem 34. We prove the vector-valued inequality. Unlike the usual maximal inequality, we need integral condition (12.3). Theorem 36. Let 1 < q < ∞, 1 < u ≤ ∞ and ϕ ∈ Gq . Assume in addition that Z ∞ ds 1 . (r > 0). (12.3) ϕ(s)s ϕ(r) r n ∞ ∞ ϕ ϕ ϕ Then k{M fj }∞ j=1 kMq (`u ) . k{fj }j=1 kMq (`u ) for all {fj }j=1 ⊂ Mq (R ).

Proof When u = ∞, the result is clear from Theorem 34. The proof is essentially the same as the classical case except that we truly use (12.3). The proof of the estimate of the inner term remains unchanged except in that we need to generalize the parameter p to the function ϕ. Let fj,1 ≡ χ5Q fj and fj,2 ≡ fj − fj,1 . We can handle fj,1 ’s in a standard manner as before. Going through a similar argument to the classical case, we will have (q) ϕ(`(Q))mQ (k{M fj,2 }∞ j=1 k`u )

.

∞ X

ϕ(`(Q))m2k Q (k{fj }∞ j=1 k`u ).

(12.4)

k=1

See Exercise 12. If we use the definition of the generalized Morrey norm, we obtain (q)

ϕ(`(Q))mQ (k{M fj,2 }∞ j=1 k`u ) .

∞ X ϕ(`(Q)) ϕ k{fj }∞ j=1 kMq (`u ) . ϕ(2k `(Q))

(12.5)

k=1

Since ϕ ∈ Gq , we obtain

Z ∞ ∞ X ϕ(`(Q)) ϕ(`(Q)) . dt. If we use (12.3) k ϕ(2 `(Q)) `(Q) ϕ(t)t

k=1

∞ X ϕ(`(Q)) and ϕ ∈ Gq , then . 1. Inserting this estimate into (12.5), we ϕ(2k `(Q)) k=1 obtain the counterpart to the classical case.

Since (12.3) is an important condition, we are interested in its characterization. In fact, we have the following useful one. Theorem 37. Assume that ϕ ∈ M+ (0, ∞) is an almost increasing function. Then the following are equivalent: (1) ϕ satisfies (12.3). (2) There exists m0 ∈ N such that ϕ(2m0 r) > 2ϕ(r) for all r > 0.

(12.6)

36

Morrey Spaces

Proof Assume that (12.3) holds. Let m00 be a positive integer such that 0 ϕ(2m0 −1 r) ≤ 2ϕ(r) for all r > 0. Thus, since ϕ is almost increasing, log m00 ≤ 2ϕ(r)

0

Z r

2m0 −1 r

ds 1 . ϕ(s)s ϕ(r)

Since ϕ(r) > 0, we have an upper bound M ∈ N for m00 . Thus, if we set m0 ≡ M + 1, we obtain the desired number m0 . If (12.6) holds, then Z r

∞

∞

X ds = ϕ(s)s j=1

Z

∞

2m0 j r

2m0 (j−1) r

X ds . ϕ(s)s j=1

Z

2m0 j r

ds

2m0 (j−1) r

2j ϕ(r)s

.

1 . ϕ(r)

As we have mentioned, we need (12.3) for the vector-valued maximal inequality. We give an example showing that (12.3) is absolutely necessary: By no means is (12.3) artificial as the following proposition shows: Proposition 38. Let 1 < q < ∞, 1 < u < ∞ and ϕ ∈ Gq . Assume in addition that ∞ ϕ ϕ k{M fj }∞ (12.7) j=1 kWMq (`u ) . k{fj }j=1 kMq (`u ) . 0 n holds for all sequences of {fj }∞ j=1 ⊂ L (R ). Then (12.3) holds.

We exclude the case where u = ∞, since (12.7) still holds without (12.3). Proof Assume to the contrary; for all m ∈ N ∩ [2, ∞), there would exist rm > 0 such that ϕ(2m rm ) ≤ 2ϕ(rm ) for r = rm . Let us consider fj = χ[1,j] (m)χB(2j rm )\B(2j−1 rm ) for j ∈ N. Observe

  u1

m

X

 ϕ χB(2j rm )\B(2j−1 rm )  k{fj }∞ . j=1 kMq (`u ) =

j=1

ϕ Mq

m ϕ ϕ As a result, k{fj }∞ j=1 kMq (`u ) ≤ kχB(2m rm ) kMq ≤ ϕ(2 rm ) . ϕ(rm ). Let x ∈ B(rm ). For j > m, we have M fj (x) = 0. Meanwhile for 1 ≤ j ≤ m, we have Z 1 fj (y)dy M fj (x) ≥ |B(x, 2j+1 rm )| B(x,2j+1 rm ) Z 1 j−1 χ j (y)dy ≥ |B(x, 2j+1 rm )| B(2j rm ) B(2 rm )\B(2 rm ) 2n − 1 1 = ≥ n. 4n 4

Consequently, m X

k{M fj }∞

u k ≥ ϕ(r ) ϕ m j=1 ` WM 1

j=1



1 4n

u

1

∼ ϕ(rm ) m u .

Generalized Morrey/Morrey–Campanato spaces

37

By our assumption, we have

1 ∞

ϕ ϕ(rm )m u . k{M fj }∞ j=1 k`u WMϕ . k{fj }j=1 kMq (`u ) . ϕ(rm ), 1

or equivalently m≤D where D is independent of m. This contradicts the fact that m ∈ N ∩ [2, ∞) is arbitrary. Example 16. Let 1 ≤ q ≤ p < ∞. Then



∞



k{M fj }∞ j=1 k`u Wmp . k{fj }j=1 k`u mp

p n ({fj }∞ j=1 ⊂ mq (R ))

fails. Hence



∞



k{M fj }∞ j=1 k`u Wmp . k{fj }j=1 k`u mp

p n ({fj }∞ j=1 ⊂ mq (R ))

q

1

q

q

fails. In particular,

k{M fj }∞

j=1 k`u WLq

uloc



. k{fj }∞ j=1 k`u Lq

uloc

q n ({fj }∞ j=1 ⊂ Luloc (R ))

n

fails. In Zfact, let ϕ(t) ≡ max(t p , 1) for t > 0 as before. Then ϕ fails (12.3) ∞ dr 1 because ∼ log , 0 < u < 1. ϕ(r) u u Proposition 38 led us to the conclusion that (12.3) is fundamental. The following proposition will be fundamental in the study of the boundedness of operators in generalized Morrey spaces. Theorem 39. If ψ ∈ M+ (0, ∞) ∩ L1loc (0, ∞) and D > 0 satisfy Z ∞ dt ψ(t) ≤ Dψ(r) (r > 0), t r then Z ∞ r

ψ(t)tε

dt rε ≤ · t 1 − Dε

Z

∞

ψ(t)t−1 dt ≤

r

D · ψ(r)rε 1 − Dε

(r > 0) (12.8)

for all 0 < ε < D−1 . Proof Let

Z Ψ(r) ≡

∞

ψ(t)t−1 dt (r > 0).

r

For 0 < r < R Z R Z R Z R dt dt ε dt ε ψ(t)tε = [−Ψ(t)tε ]R + Ψ(t)εt ≤ Ψ(r)r + εD ψ(t)tε . r t t t r r r

38

Morrey Spaces

Therefore

R

Z

ψ(t)tε

r

dt 1 D ≤ Ψ(r)rε ≤ ψ(r)rε . t 1 − εD 1 − εD

It remains to let R → ∞. We change variables in Theorem 39. Theorem 40. Let ψ ∈ M+ (0, ∞) satisfy Z r dt ψ(t) ≤ Dψ(r) t 0

(r > 0)

for some D > 0 independent of r > 0. If 0 < ε < D−1 , then Z r Z r dt 1 D dt −ε ψ(t) 1+ε ≤ r r−ε ψ(r). ψ(t) ≤ t 1 − Dε t 1 − Dε 0 0 Proof Set η(t) ≡ ψ

  1 t

(t > 0).

Then our assumption reads as: Z ∞ dt η(t) ≤ Dη(r) t r

(r > 0).

Thus, Z

∞

η(t) r

dt t1−ε

≤

1 rε 1 − Dε

Z

r

ψ(t) 0

dt D ≤ rε η(r) t 1 − Dε

(r > 0)

according to Theorem 39. If we express this inequality in terms of ψ, we obtain the desired result. In the next proposition, based on [354, Proposition 2.7], we further characterize and apply our key assumption (12.3). Proposition 41. Let ϕ ∈ M+ (Rn ) be a doubling function, that is, ϕ(s) ∼ ϕ(r) as long as 21 ≤ 2r ≤ 2. There exists a constant ε > 0 such that tε rε . ϕ(t) ϕ(r)

(t ≥ r)

(12.9)

holds if and only if (12.3) holds, or equivalently, ϕ satisfies (12.8) for some ε > 0. If one of these conditions is satisfied, then Z ∞ ds 1 . (r > 0) (12.10) us u ϕ(s) ϕ(r) r for all 0 < u < ∞, where the implicit constant depends only on u.

Generalized Morrey/Morrey–Campanato spaces

39

Proof The implication (12.3) =⇒ (12.8) follows from Theorem 39. Assume (12.8). Then Z 2t tε dv rε . . ϕ(t) v 1−ε ϕ(v) ϕ(r) t thanks to the doubling property of ϕ, proving (12.9). If we assume (12.9), then Z ∞ ε Z ∞ ε Z ∞ r s 1 ds ds ds . = = , 1+ε 1+ε ϕ(s)s ϕ(s) s ϕ(r) s εϕ(r) r r r which implies (12.3). Note that (12.9) also implies (12.10) because ϕu satisfies (12.9) as well. Let 0 < u < ∞ be fixed. Inequality (12.10) is necessary for (12.3); simply apply Proposition 41 to ϕu . Example 17. Let ϕ ∈ Gq satisfy (12.3). Then Z ∞ rε 1 ds . (r > 0). ϕ(s)s1−ε ϕ(r) r Hence

sε rε . ϕ(s) ϕ(r) whenever 0 < r ≤ s < ∞. As a result, Z 1 dr < ∞. ϕ(r) r 0 n

Let ϕ(t) ≡ max(tZ p , 1) for t > 0 as before. Then ϕ clearly fails to satisfy this ∞ dr condition, since = ∞. r 1 We generalize condition (12.3) as follows: Definition 10 (Zγ , Zγ ). Let γ ∈ R. (1) The (upper) Zygmund class Zγ is defined as the set of all ϕ ∈ M+ (0, ∞) for which lim ϕ(r) = 0 and r↓0

r

Z

ϕ(t)t−γ−1 dt . ϕ(r)r−γ

(r > 0),

0

(2) The (lower) Zygmund class Zγ is defined as the set of all ϕ ∈ M+ (0, ∞) for which lim ϕ(r) = 0 and r↓0

Z

∞

ϕ(t)t−γ−1 dt . ϕ(r)r−γ

r

Note that (12.3) reads as

1 ϕ

∈ Z0 .

(r > 0).

40

Morrey Spaces

Example 18. Let ϕ(t) ≡ tp , t > 0 with 1 < p < ∞, and let γ ∈ R. (1) ϕ ∈ Zγ if and only if p > γ. (2) ϕ ∈ Zγ if and only if p < γ. (3) 1 ∈ Zγ if and only if γ < 0. n We present an example of functions in Mϕ q (R ).

Example 19. [354, Proposition 2.11] Let 0 < q < ∞ and ϕ ∈ Gq . Define f≡

∞ X χ[2−j−1 ,2−j ]n , ϕ(2−j ) j=−∞

g ≡ sup j∈Z

χ[0,2−j ]n . ϕ(2−j )

We claim that the following are equivalent; n (a) f ∈ Mϕ q (R ), n (b) g ∈ Mϕ q (R ),

(c)

1 ϕ

n

∈ Zq .

Let 0 < u < q. Note that f ≤ g ≤ 2n M (u) f , where M (u) denotes the powered Hardy–Littlewood maximal operator. Observe that M (u) is bounded n on Mϕ q (R ). Thus, (a) and (b) are equivalent, since f is expressed as f = f0 (k · k`∞ ), that is, there exists a function f0 : [0, ∞) → R such that f (x) = f0 (kxk`∞ ) for all x ∈ Rn , where k · k`∞ denotes the `∞ -norm of Rn ; it follows that (a) and (c) are equivalent. Example 20. [354, Proposition 2.11] Let 0 < q < ∞ and ϕ ∈ Gq . Define ϕ† ∈ M↓ (0, ∞) by n ϕ† (t)≡ϕ(t)t− q (12.11) ∞ ∞ X X χ[0,2−j ]n 1 1 for t > 0. Define h ≡ . Then . and that −j ) −j ) −l ) ϕ(2 ϕ(2 ϕ(2 j=−∞ j=l

l X

1 1 n . † −l for all l ∈ Z if and only if h ∈ Mϕ q (R ). † −j ϕ (2 ) ϕ (2 ) j=∞ n To verify this, we let f, g be as in Example 19. Suppose first h ∈ Mϕ q (R ). Then   q  q1 Z ∞ X 1 1   ϕ(2−l )  dx ≤ khkMϕq . |[0, 2−l ]n | [0,2−l ]n ϕ(2−j ) j=l

Thus

∞ X j=l

khkMϕq 1 ≤ for all l ∈ Z. This implies that f ≤ g ≤ −j ϕ(2 ) ϕ(2−l )

h . f , where f and g are defined in Example 19. Thus, from Example 19, l X 1 1 . † −l holds as well. † −j ϕ (2 ) ϕ (2 ) j=∞

Generalized Morrey/Morrey–Campanato spaces

41

Conversely, assume that l X

1 1 . † −l † (2−j ) ϕ ϕ (2 ) j=∞ and

∞ X

(12.12)

1 1 . ϕ(2−j ) ϕ(2−l )

j=l

(12.13)

n hold for all l ∈ Z. Then h ∼ f from (12.13). Thus, f ∈ Mϕ q (R ) by (12.12), ϕ n from which it follows that h ∈ Mq (R ). n We further present some examples of functions in Mϕ q (R ) using [101]. n

Lemma 42. Let 0 < q < ∞ and ϕ ∈ Gq ∩ Z− q . Then the function ψ ≡ ϕ(| · |) n belongs to Mϕ q (R ). n

Proof First note that ϕ ∈ Z− q is equivalent to Z r 1 ϕ(t)q tn−1 dt . ϕ(r)q (r > 0). rn 0

(12.14)

Note that ϕ(| · |) is radial decreasing, so that for all a ∈ Rn and r > 0, 1 |B(a, r)|

! q1

Z

q

≤

ϕ(|x|) dx B(a,r)

1 |B(r)|

! q1

Z

q

ϕ(|x|) dx

.

(12.15)

B(r)

Combining (12.14) and (12.15) and using the spherical coordinate, we obtain the desired result.

12.2.2

Singular integral operators on generalized Morrey spaces

Let T be a singular integral operator which we considered in Section 4.5 n in the first book. To define the function T f for f ∈ Mϕ q (R ) we follow the same strategy as that for f ∈ Mpq (Rn ). To this end, we need to establish the following estimate: Lemma 43. Let 1 < q < ∞ and ϕ ∈ Gq satisfy n kf kMϕq for all f ∈ L∞ c (R ). We note that

1 ϕ

1 ϕ

∈ Z0 . Then kT f kMϕq .

∈ Z0 appeared once again.

Proof Let Q be a fixed cube. Then we need to prove  ϕ(`(Q))

1 |Q|

Z

q

|T f (y)| dy Q

 q1 . kf kMϕq .

42

Morrey Spaces

To this end, we split f according to 2Q: f1 ≡ χ2Q f , f2 ≡ f − f1 . As for f1 , we have    q1  q1 Z Z 1 1 q q ≤ ϕ(`(Q)) ϕ(`(Q)) |T f1 (y)| dy |T f1 (y)| dy |Q| Q |Q| Rn   q1 Z 1 q . ϕ(`(Q)) |f1 (y)| dy |Q| Rn   q1 Z 1 q . ϕ(`(Q)) |f (y)| dy |Q| 2Q . kf kMϕq . As for f2 , we use the size condition of K, the integral kernel of T , to have the local estimate: For y ∈ Q ! Z ∞ Z Z 1 |f (y)|dy . |f (y)|dy d`. |T f2 (y)| . n `n+1 B(c(Q),`) `(Q) Rn \2Q |y − c(Q)| By the definition of the norm, Lemma 20 and (12.3), we obtain Z ∞ 1 1 |T f2 (y)| . dr · kf kMϕq . kf kMϕq . ϕ(`(Q)) `(Q) rϕ(r) It remains to integrate this pointwise estimate. To carry out our program of proving the boundedness of singular integral operators, we need to investigate the predual and the predual. Definition 11 (Generalized block space). Let 1 < q < ∞ and ϕ ∈ Gq . 0

(1) A (ϕ, q)-block is a function A ∈ Lq0 (Rn ) supported on a cube Q satisfying ϕ(`(Q)) kAkLq0 ≤ . In this case A is a (ϕ, q)-block supported on Q. 1 |Q| q (2) The generalized block space Hqϕ0 (Rn ) is the set of all f ∈ L0 (Rn ) for ∞ P which it can be written f = λj Aj for some sequence {Aj }∞ j=1 of j=1

1 n ϕ (ϕ, q)-blocks and {λj }∞ j=1 ∈ ` (R ). The norm kf kHq0 is the infimum of ∞ ∞ k{λj }∞ j=1 k`1 where {Aj }j=1 and {λj }j=1 run over all expressions above. 0

Example 21. Let A be a non-zero Lq (Rn )-function supported on a cube Q. ϕ(`(Q)) Then B ≡ A is a (ϕ, q)-block supported on Q. 1 |Q| q kAkLq0 n

We have an equivalent expression if ϕ(t) & t q for all t > 0.

Generalized Morrey/Morrey–Campanato spaces

43 n

Proposition 44. Let 1 < q < ∞ and ϕ ∈ Gq be such that ϕ(t) & t q for all t > 0. Then f ∈ L0 (Rn ) belongs to Hqϕ0 (Rn ) if and only if f admits a ∞ P λj Aj , for some sequence {Aj }∞ decomposition: f = λ0 B+ j=1 of (ϕ, q)-blocks j=1

1 n supported on cubes of volume less than or equal to 1 and {λj }∞ j=1 ∈ ` (R ) q0 n and B ∈ L (R ) with unit norm. Furthermore, the norm kf kHϕ0 is equivalent q ∞ ∞ to the infimum of k{λj }∞ j=0 k`1 where {Aj }j=1 , B and {λj }j=1 run over all expressions above.

Proof Let Q be a cube with `(Q)  1. Then we can say that A is a (ϕ, q)0 block suppported on Q if 2A has the Lq (Rn )-norm 1 and A is supported on Q. Consequently, in the decomposition in Definition 11 any block Aj with the cube Qj satisfying |Qj |  1 can be combined into a block supported on “Rn ”. The following lemma justifies the definition above: Lemma 45. Let 1 < q < ∞ and ϕ ∈ Gq . If A is a (ϕ, q)-block and f ∈ n ϕ Mϕ q (R ), then kA · f kL1 ≤ kf kMq . Proof Simply mimic the case of classical Morrey spaces. It is easy to see that Hqϕ0 (Rn ) is a normed space. We extend Lemma 343 in the first book to generalized Morrey spaces. Lemma 46. Let 1 < q < ∞, ϕ ∈ Gq , and let f ∈ Hqϕ0 (Rn ). LetPD be a large number depending only on ϕ. Then f can be decomposed as f = λ(Q)b(Q), Q∈D P where λ(Q) ≥ 0 for each Q, λ(Q) ≤ Dkf kHϕ0 and each b(Q) is a (ϕ, q)q

Q∈D

block supported in Q. Proof Similar to the case of the predual of classical Morrey spaces. Example 22. Let 1 < q < ∞, ϕ ∈ Gq , and let f ∈ Hqϕ0 (Rn ). Assume n that inf t− q ϕ(t) > 0, so that Proposition 44 is applicable. Then f can be t>0 P decomposed as f = B + λ(Q)b(Q), where λ(Q) ≥ 0 for each Q and Q∈D,|Q|≤1 P kBkLq0 + λ(Q) . kf kHϕ0 and b(Q) is a (ϕ, q)-block supported in Q. Q∈D

q

About the definition above, the following proposition is fundamental: n Proposition 47. Let 1 < q < ∞ and ϕ ∈ Gq . Let f ∈ Mϕ q (R ) and g ∈ ϕ n ϕ ϕ Hq0 (R ). Then kf · gkL1 ≤ kf kMq kgkH 0 . q

Proof Similar to the classical case. Use Lemma 45. Corollary 48. Let 1 < q < ∞ and ϕ ∈ Gq . Then every function in Hqϕ0 (Rn ) is locally integrable.

44

Morrey Spaces

Proof Simply combine Lemma 25, Proposition 47 and the fact that χQ ∈ n Mϕ q (R ) for any cube Q. We can prove the Fatou property of Hqϕ0 (Rn ). Proposition 49. Let 1 < q < ∞ and ϕ ∈ Gq . Assume in addition that ϕ satisfies (12.3). Suppose that f and fk , that each fk ∈ Hqϕ0 (Rn ) ∩ M+ (Rn ), (k = 1, 2, . . .), satisfies kfk kHϕ0 ≤ 1 and that fk ↑ f a.e. as k → ∞. Then f ∈ Hqϕ0 (Rn ) and kf kHϕ0 ≤ 1.

q

q

P

Proof By Lemma 46 fk can be decomposed as fk =

λk (Q)bk (Q),

Q∈D

where λk (Q) ≥ 0 for each Q and X

λk (Q) ≤ D

(12.16)

Q∈D

and bk (Q) is a (ϕ, q)-block supported in Q and kbk (Q)kLq0 ≤

ϕ(`(Q)) 1

|Q| q

.

(12.17) 0

Using (12.16), (12.17) and the weak-compactness of the Lebesgue space Lq (Q) we now apply a diagonalization argument and, hence, we can select an increasing sequence {kj }∞ j=1 of integers that satisfies the following: lim λkj (Q) = λ(Q),

(12.18)

j→∞

0

lim bkj (Q) = b(Q) in the weak-topology of Lq (Q),

j→∞

(12.19)

where b(Q) is a (ϕ, q)-block supported in Q. We set X f0 ≡ λ(Q)b(Q). Q∈D

Then, by Fatou’s lemma and (12.16), X X λ(Q) ≤ lim inf λkj (Q) ≤ D, j→∞

Q∈D

which implies f0 ∈ Hqϕ0 (Rn ). We will verify that Z lim j→∞

Q0

(12.20)

Q∈D

Z fkj (x)dx =

f0 (x)dx

(12.21)

Q0

for all Q0 ∈ D(Rn ). Once (12.21) is established, we will see that f = f0 . Hence f ∈ Hqϕ0 (Rn ) by virtue of the Lebesgue differentiation theorem because

Generalized Morrey/Morrey–Campanato spaces

45

0

at least we know that f0 locally in Lq (Rn ). By the definition of fkj , we have ∞ X

Z fkj (x)dx = Q0

Z

X

λkj (Q)

bkj (Q)(x)dx. Q0

l=−∞ Q∈D(Q0 )∪D ] (Q0 )

Note that 1

1

kbkj (Q)kL1 ≤ |Q0 ∩Q| q kbkj (Q)kLq0 ≤

ϕ(`(Q))|Q ∩ Q0 | q |Q|

1 q

≤

ϕ(`(Q)) 1

|Q| q

(12.22)

for any cube Q containing Q0 . We distinguish two cases. If n

lim ϕ(t)t− q = 0,

t→∞

then for all ε > 0 there exists l ∈ N such that Z ∞ X X λk (Q) j

Q0

l=N Q∈D( Rn )∩(D(Q0 )∪D ] (Q0 ))

bkj (Q)(x)dx < ε.

Thus, we are in the position of using the Lebesgue convergence theorem based on Example 17. Thus we obtain (12.21). If n

lim ϕ(t)t− q > 0,

t→∞

then we go through a similar argument See Exercise 13. Z to obtain (12.21). Z Since fk ↑ f a.e., we must have f (x)dx = f0 (x)dx for all Q0 ∈ Q0

Q0

D(Rn ) by (12.21). This yields f = f0 a.e., by the Lebesgue differentiation theorem, and, hence, f ∈ Hqϕ0 (Rn ). Since we have verified f ∈ Hqϕ0 (Rn ), it follows that   Z kf kHϕ0 = sup fk (x)g(x)dx : k = 1, 2, . . . , kgkMϕq ≤ 1 ≤ 1. q

Rn

This completes the proof of the theorem. n We can establish the duality Hqϕ0 (Rn ) - Mϕ q (R ) in the completely same manner as the classical case once Proposition 49 is proven.

Theorem 50. Let 1 < q < ∞, and let ϕ ∈ Gq satisfy

1 ϕ

∈ Z0 .

n (1) The dual of Hqϕ0 (Rn ) is Mϕ q (R ).

fϕ (Rn ) is Hϕ0 (Rn ). (2) The dual of M q q Proof The same as the classical case. As we discussed for classical Morrey spaces, we can establish that singular integral operators are bounded on generalized Morrey spaces.

46

Morrey Spaces

Theorem 51. Let 1 < q < ∞, and let ϕ ∈ Gq satisfy ϕ1 ∈ Z0 . Then any n singular integral operator, which is initially defined for L∞ c (R )-functions, ϕ can be naturally extended to a bounded linear operator on Mq (Rn ). Proof Similar to the classical case. See Exercise 14. Recall that we did not use (12.3) for the proof of the boundedness of the Hardy–Littlewood maximal operator. However for the proof of the boundedness of singular integral operators, (12.3) is absolutely necessary as the following proposition shows: Proposition 52. Let 1 < q < ∞ and ϕ ∈ Gq . If kR1 f kWMϕ1 . kf kMϕq for 1 n all f ∈ L∞ c (R ), then ϕ ∈ Z0 , where R1 denotes the first Riesz transform. Proof Let V ≡ {x = (x1 , x2 , . . . , xn ) ∈ Rn : 2x1 > |x|}. Assume that ϕ1 ∈ / m Z0 , so that for any m ∈ N ∩ [3, ∞) there exists rm > 0 such that ϕ(2 rm ) ≤ 2ϕ(rm ). Then, consider fm = χV ∩B(2m−1 rm )\B(2rm ) . Let x ∈ V ∩ B(rm ). If y ∈ V ∩ B(2m−1 rm ) \ B(2rm ), then x − y ∈ V and rm ≤ |x − y| ≤ 2m rm . Thus, Z x1 − y1 fm (y)dy R1 fm (x) = |x − y|n+1 n R Z y1 = dy n+1 V ∩B(2m−1 rm )\B(2rm ) |y| Z 1 & dy n V ∩B(2m−1 rm )\B(2rm ) |y| ∼ log m.

(12.23)

χB(rm ) . M χV ∩B(rm ) .

(12.24)

Since V is a cone, we have

n We use this estimate and the boundedness of M on Mϕ q (R ) to obtain

ϕ(rm ) . kχB(rm ) kMϕq . kM χV ∩B(rm ) kMϕq . kχV ∩B(rm ) kMϕq . By using the inequality log m . |R1 fm (x)| for x ∈ V ∩ B(rm ) and the boundu edness of R1 on Mϕ q (` ), we have ϕ(rm ) log m . kR1 fm kWMϕ1 . kfm kMϕq ≤ kχB(2m rm ) kMϕq . ϕ(2m rm ) ∼ ϕ(rm ).

Generalized Morrey/Morrey–Campanato spaces

47

This implies log m ≤ D where D is independent of m, contradictory to the fact that m ≥ 3 is arbitrary. Hence, there exists some m0 ∈ N such that ϕ (2m0 r) > 2ϕ(r). Thus, the integral condition (12.3) holds. We disprove that T cannot be extended to a bounded linear operator on mpq (Rn ). Example 23. Let 1 ≤ q ≤ p < ∞. Then both kR1 f kWmpq . kf kmpq

n (f ∈ L∞ c (R ))

and kR1 f kWLquloc . kf kLquloc

n (f ∈ L∞ c (R ))

n

p fail. In fact, Z ∞let ϕ(t) = min(t , 1), t > 0 as in Example 7. Then ϕ fails (12.3) dr because = ∞. ϕ(r) 1

We supplement a characterization. n fϕ n Lemma 53. A function f ∈ Mϕ q (R ) belongs to Mq (R ) if and only if

lim χB(R) χ[0,R] (|f |)f = 0

(12.25)

R→∞ n in Mϕ q (R ).

We end Section 12.2.2 with extension to the vector-valued inequality. We u n ϕ use the standard notation Mϕ q (` , R ) and k · kMq (`u ) to define the vectorvalued norm. We write



∞

{fj }∞

j=1 Mϕ (`u ) ≡ k{fj }j=1 k`u Mϕ q

q

n ∞ n ϕ u ϕ for {fj }∞ j=1 ⊂ Mq (R

). The space Mq (` , R ) stands for the set of all {fj }j=1 ∞

for which {fj }j=1 Mϕ (`u ) is finite. q

Theorem 54. Let 1 < q < ∞, 1 < u < ∞ and ϕ ∈ Gq . Also let T be a singular integral operator. Assume in addition that (12.3) holds. Then ∞ ∞ ϕ u n ϕ ϕ k{T fj }∞ j=1 kMq (`u ) . k{fj }j=1 kMq (`u ) for all {fj }j=1 ∈ Mq (` , R ). Proof The proof is similar to Theorem 36. We will also note that, for all q u n ∞ ∞ {gj }∞ j=1 ∈ L (` , R ), k{T gj }j=1 kLq (`u ) . k{gj }j=1 kLq (`u ) . See the proof of the weighted vector-valued boundedness of singular integral operators in the first book.

12.2.3

Generalized fractional integral operators in generalized Morrey spaces

To consider an operator like (1 − ∆)−1 , we are oriented to considering Z ρ(|x − y|) Iρ f (x) = f (y) dy |x − y|n n R

48

Morrey Spaces

for any suitable function f on Rn , where ρ ∈ M+ (0, ∞) is a suitable function. Generalized Morrey spaces allow us to consider more general fractional integral operators. Needless to say, (1 − ∆)−1 and Iα fall under the scope of this framework. Let us discuss what condition we need in order to guarantee that Iρ enjoys some boundedness property. We always assume that ρ satisfies the Dini condition for Iρ . Z 1 ρ(s) ds < ∞, (12.26) s 0 so that Iρ χQ (x) is finite for any cube Q and x ∈ Rn . In addition, we also assume that ρ satisfies the “growth condition”: there exist constants C > 0 and 0 < 2k1 < k2 < ∞ such that Z k2 r ρ(s) sup ρ(s) . ds, r > 0. (12.27) r s k1 r 2 0 such that ρ(r) 1 ≤ ≤D (12.28) D ρ(s) whenever r > 0 and s > 0 satisfy r ≤ 2s ≤ 4r. We quantify the above observation. Proposition 55. If ρ Z∈ M+ (0, ∞) satisfies the doubling condition (12.28), r ds then sup ρ(s) ≤ 2D2 ρ(s) . r r s 2 ≤s≤r 2 Z r ds Proof Keeping in mind = log 2 < 1, we calculate 1 s 2r D sup ρ(s) ≤ Dρ(r) = r log 2 2 ≤s≤r

Z

r r 2

ds D2 ρ(r) ≤ s log 2

Z

r r 2

ds ρ(s) ≤ 2D2 s

Z

r

ρ(s) r 2

ds . s

Example 24. Let 0 < α < ∞. (1) ρ(t) ≡ tα , t > 0, which generates Iα , satisfies the doubling condition. (2) ρ(t) ≡

tα , t > 0 satisfies the doubling condition. log(e + t)

(3) ρ(t) ≡ tα e−t , t > 0 satisfies the growth condition but fails the doubling condition. (4) Let 0 ≤ γ < ∞ and β1 , β2 ∈ R. We set ( (1 + | log t|)β1 `B (t) ≡ (1 + | log t|)β2

(0 < t ≤ 1), (1 < t < ∞).

Generalized Morrey/Morrey–Campanato spaces

49

as before. Then ρ(t) ≡ tγ `B (t), t > 0 satisfies (12.26) if and only if γ = 0 > −1 > β1 or γ > 0. Meanwhile (12.28) is always satisfied. It is noteworthy that we can tolerate the case γ = 0 if β1 < −1. To check that our example is not so artificial we introduce the Bessel kernel. s

Definition 12 ((1 − ∆)− 2 f ). Let s > 0. The function Gs , given by 1 Gs (x) = lim ε↓0 (2π)n

Z Rn

exp(−|εξ|2 )eix·ξ dξ s (1 + |ξ|2 ) 2

(x ∈ Rn ),

is called the Bessel kernel of order s. One defines the Bessel potential of order s s s (1 − ∆)− 2 f by (1 − ∆)− 2 f = Gs ∗ f . The above examples are natural in some sense but somewhat artificial because the second example and the third one do not appear naturally in the context of other areas of mathematics. Here, we present some other examples related to partial differential equations. Example 25. Note that the solution to (1 − ∆)f = g, where f is an unknown function and g is a given function is given by g = (1 − ∆)−1 f. Although it is impossible to find Iρ χB(R) (y), y ∈ Rn and R > 0, we still have a partial but important estimate.
Lemma 56. Let ρ ∈ M+ (0, ∞). Then inequality ρ˜ R2 . Iρ χB(R) (x) holds
whenever x ∈ B R2 and R > 0.
Proof Take x ∈ B R2 . We write the integral in full: Z Iρ χB(R) (x) = Rn

ρ(|x − y|) χB(R) (y)dy = |x − y|n

Z B(R)

ρ(|x − y|) dy. |x − y|n

A geometric observation shows that B(x, R2 ) ⊆ B(R). Hence, we have Z Iρ χB(R) (x) ≥

B(x, R 2 )

ρ(|x − y|) dy ∼ |x − y|n

Z 0

R 2

ρ(s) R
ds = ρ˜ . s 2

Note that we only use the spherical coordinates to obtain the last integral. In the case of the radially symmetric functions θ, we can calculate Iρ [θ(| · |)χB(R)c ](x) for x small. Lemma 57. For every R > 0 and θ ∈ M+ (0, ∞) satisfying the doubling condition, the inequality    Z ∞ Z ∞ R θ(t)ρ(t) θ(t)ρ(t) dt . Iρ [θ(| · |)χB(R)c ](x) . dt x∈B . t t 3 2R 2R/3

50

Morrey Spaces

Proof We prove  the  right-hand inequality, the left-hand inequality being R similar. Let x ∈ B . A geometric observation shows that |x − y| ∼ |y| for  3 2R all y ∈ Rn \ B . Since θ satisfies the doubling condition, then 3 Z θ(|y|)ρ(|x − y|) Iρ [θ(| · |)χB(R)c ](x) = dy |x − y|n n R \B(R) Z ρ(|x − y|) ≤ θ(|y|) dy |x − y|n n R \B(x,2R/3) Z  θ(|x − y|) ρ(|y|) dy  = 2R |y|n Rn \ B 3 Z ∞ θ(t)ρ(t) dt. . t 2R/3 It remains to write the most right-hand side in terms of the spherical coordinates. Z r ρ(t) For convenience, write ρ˜(r) ≡ dt for r > 0. Sometimes, we are t 0 interested in the case where matters are reduced to classical fractional integral operators. Proposition 58. Let ρ ∈ M+ (0, ∞) satisfy (12.27). Then the following are equivalent for α > 0: (a) ρ(r) . rα for all r > 0. (b) ρ˜(r) . rα for all r > 0. Z Proof Clearly (a) implies (b), since 0

r

sα−1 ds =

1 α r . Let us see that α

(b) implies (a). Combining (12.27) with (b), we obtain Z k2 r Z k2 r ρ(s) ρ(s) sup ρ(s) . ds . ds . (k2 r)α ∼ rα . r s s k1 r 0 2 0). (12.29) ϕ(r) 0 t tϕ(t) ψ(r) r

Generalized Morrey/Morrey–Campanato spaces

51

Note that the left-hand side of (12.29) equals Z r Z ∞ Z ∞ ρ(t) ρ(t) ρ(t) −1 ϕ(r) dt + dt = dt. t tϕ(t) tϕ(max(r, t)) 0 r 0 Proof We start with the necessity: Assume that Iρ is bounded from ψ n n Mϕ 1 (R ) to M1 (R ). Let r > 0. By Lemma 56 and the doubling property of ψ, we obtain Z Z kIρ χB(r) kMψ 1 1 1 ρ˜(r) . n Iρ χB(r) (x)dx ≤ n Iρ χB(r) (x)dx ≤ . r B(r/2) r B(r/2) ψ(r) ψ n n Since ψ ∈ G1 and Iρ is assumed bounded from Mϕ p (R ) to M1 (R ), it follows −1 that ρ˜(r) . ψ(r) kχB(r) kMϕp . Since kχB(r) kMϕp ∼ ϕ(r), we conclude ρ˜(r) . ϕ(r) . Let gr (x) ≡ ϕ(|x|)−1 χB(r)c (x) for x ∈ Rn . By Lemma 57 with θ = ϕ−1 , ψ(r) we have Z ∞  r −1 ρ(t) dt . ψ kIρ gr kMψ . ψ(r)−1 kgr kMϕ1 . ψ(r)−1 . 1 tϕ(t) 6 r

Thus, the necessity part of Theorem 59 is proven. n We move on to the sufficiency. For a ball B(z, r) and f ∈ Mϕ p (R ), we let f1 ≡ f χB(z,2r) and f2 ≡ f − f1 . Then a geometric observation shows B(z, r) ⊂ B(y, 3r) for all y ∈ B(z, 2r). Hence, by Fubini’s theorem and the normalization, ! Z Z Z ρ(|x − y|) |Iρ f1 (x)|dx ≤ |f (y)| dy dx |x − y|n B(z,r) B(z,r) B(z,2r) ! Z Z ρ(|x − y|) dx dy ≤ |f (y)| |x − y|n B(z,2r) B(y,3r) Z Z ρ(|x|) = |f (y)|dy × dx. n B(z,2r) B(3r) |x| By the use of the definition of the generalized Morrey norm k · kMϕq , (12.29) and the doubling condition of ψ, we obtain Z |Iρ f1 (x)|dx . ρ˜(3r)ϕ(2r)−1 rn kf kMϕp B(z,r)

. ρ˜(3r)ϕ(3r)−1 rn kf kMϕp . ψ(3r)−1 rn kf kMϕp . ψ(r)−1 rn kf kMϕp . Thus, the estimate for f1 is valid. As for f2 , we let x ∈ B(z, r). Then Z Z ρ(|x − y|) ρ(|x − y|) |Iρ f2 (x)| ≤ |f (y)| dy ≤ |f (y)| dy n |x − y| |x − y|n B(z,2r)c B(x,r)c

52

Morrey Spaces

and decomposing the right-hand side dyadically, we obtain |Iρ f2 (x)| ≤

∞ Z X j=1

|f (y)|

B(x,2j r)\B(x,2j−1 r)

ρ(|x − y|) dy . |x − y|n

Z

∞

2k1 r

ρ(t)dt · kf kMϕp . tϕ(t)

If we use (12.29) once again and the doubling condition on ψ, then we obtain |Iρ f2 (x)| . ψ(r)−1 . Thus, the estimate for f2 is valid as well. The following example explains why generalized Morrey spaces are natural function spaces. (1 + r)s for max(1, log r−1 ) 0 n r > 0. Let ρ(r) ≡ rs exp(−κr), ϕ(r) ≡ rs for r > 0. Z rThen ρ ∈ ZL ∞(R ) and it ρ(t) ρ(t) satisfies (12.27). Furthermore, if 0 < r < 1, ϕ(r)−1 dt+ dt ∼ t tϕ(t) 0 r Z r Z ∞ ρ(t) ρ(t) ψ(r)−1 and if r ≥ 1, ϕ(r)−1 dt+ dt . ψ(r)−1 . Thus it follows t tϕ(t) 0 r n s that k(1 − ∆)− 2 f kMψ .s kIρ f kMψ . kf k ns for all f ∈ M1s (Rn ). This

Example 26. Let s ∈ (0, n) and κ > 0. Define ψ(r) ≡

1

M1

1

s

calculation shows we cannot delete max(1, log r−1 ) and that (1 − ∆)− 2 does n not map Mpq (Rn ) to L∞ (Rn ) when = s and 1 < q ≤ p < ∞. p Example 26 convinces us that generalized Morrey spaces occur naturally. 1 1 α = − . s p n Then Iα does not map mp1Z(Rn ) to ms1 (Rn ), since ρ(t) Z≡ tα , t ≥ 0 and ϕ(t) ≡ ∞ ∞ s n ρ(t) ρ(t) dt = ∞ instead of dt . ϕ(t)− p . max(t p , 1), t > 0 satisfies tϕ(t) tϕ(t) r r If we consider the truncated fractional maximal operator iα given by Z f (x − y) iα f (x) ≡ dy (x ∈ Rn ), n−α |y| B(1)

Example 27. Let 1 < p < s < ∞ and 0 < α < n satisfy

then iα maps mp1 (Rn ) to ms1 (Rn ), since ϕ(r)−1

Z 0

r

ρ(t)χ(0,1) (t) dt + t

Z r

∞

ρ(t)χ(0,1) (t) s dt . ϕ(t)− p . tϕ(t)

Example 28. Let 0 < s < n. Define ϕ(r) ≡ rs and ψ(r) ≡ (1 + r)−s `−1,0 (r) for r > 0. Let ρ(r) ≡ rn Gs (r), where Gs denotes the Bessel kernel, the kernel ρ(r) ˜ of (1 − ∆)s/2 . Observe that ρ˜(r) ∼ min(rs , 1). Hence ϕ(r) ∼ min(1, r−s ). Note also that ( Z ∞ log(e/r) (r < 1), ρ(t) dt ∼ tϕ(t) rn−s Gs (r) (r ≥ 1). r

Generalized Morrey/Morrey–Campanato spaces 53 Z ∞ ρ˜(r) ρ(t) Then + dt ∼ ψ(r)−1 for r > 0. Hence, it follows from Theorem ϕ(r) tϕ(t) r 59 that kIρ f kMψ . kf kMϕ1 , extending Proposition 26. This triple (ρ, ϕ, ψ) 1

ρ ϕ

fulfills the assumption (12.29). However, and (12.30) fails.

∈ / Z0 since

ρ(r) ϕ(r)

= o(1) as r ↓ 0

We will give a result, which improves Example 26. We prove the Adams type estimate of fractional integral operators in generalized Morrey spaces. Theorem 60. Let 1 < p < q < ∞ and ϕ ∈ Gp . Assume that ρ ∈ M+ (0, ∞) satisfies (12.27). n (1) The generalized fractional integral operator Iρ is bounded from Mϕ p (R ) p/q

to Mϕ q

(Rn ) if −1

r

Z

ϕ(r)

0

ρ(t) dt + t

Z r

∞

p ρ(t) dt . ϕ(r)− q tϕ(t)

(12.30)

n

for all r > 0. If ϕ ∈ Z− p , then (12.30) is necessary for the boundedness n ϕp/q (Rn ). of Iρ from Mϕ p (R ) to Mq p/q

n ϕ (2) Assume ϕρ ∈ Z0 . Then Iρ is bounded from Mϕ p (R ) to Mq and only if ρ˜(r) . ϕ(r)1−p/q (r > 0).

Consequently, if

ρ ϕ

(Rn ) if (12.31)

∈ Z0 , condition (12.30) simplies to (12.31).

Before the proof a helpful remark may be in order. Remark 3. (1) The first half of the “only if” part (12.31) is clear from Theorem 59 with ψ = ϕp/q . ρ ϕ

∈ Z0 , it is easy to check that (12.31) implies (12.30). Z ∞ ρ(s) ρ(r) Indeed, if we use ϕρ ∈ Z0 and ϕ ∈ Gp , then ds . . Since ρ sϕ(s) ϕ(r) Z ∞r ρ(s) ρ˜(k2 r) satisfies the growth condition, we have ds . . If we use sϕ(s) ϕ(r) r Z ∞ ρ(t) (12.31) and the doubling condition on ϕ, then we obtain dt . tϕ(t) r −p ϕ(r) q .

(2) Once we assume

(3) For the “if” part we only need the following estimate of Hedberg-type, see Lemma 61 below. We first show the necessity.

54

Morrey Spaces

Proof of ZTheorem 60, necessity According to Theorem 59, we have ∞ ρ(t) only to show dt . ϕ(2R)−p/q . By virtue of Lemma 57, we obtain 2R tϕ(t) Z

∞

2R

ρ(t) dt ∼ tϕ(t)

1 Rn

! q1

Z

q

Iρ g(x) dx B(

R 3

)

.

1 p

ϕ(R) q

kIρ [θ(| · |)χB(R)c ]k

Since Iρ is bounded, we obtain Z ∞ 1 1 ρ(t) . dt . p kθ(| · |)χB(R)c kMϕ p p ϕ(R) q ϕ(R) q 2R tϕ(t)

p/q

Mϕ q



1

ϕ(| · |)

.

.

Mϕ p

n

Recall that we are assuming ϕ ∈ Z− p . We now invoke Lemma 42 to conclude Z ∞ ρ(t) 1 1 dt . p . p . q tϕ(t) ϕ(R) ϕ(2R) q 2R Thus, necessity is proven. As we have mentioned, we want an estimate of Hedberg-type. We may p ask ourselves whether inf ϕ(r)− q can be removed. That is, we want to justify r>0

assuming sup ϕ(t) = ∞. However, it can happen that sup ϕ(t) < ∞ as an t>0

t>0

example below shows. Lemma 61. Let 1 ≤ p < q < ∞, and let ϕ ∈ Gp ∩ W satisfy (12.30). n Assume that ρ ∈ M+ (0, ∞) satisfies (12.27). If f ∈ Mϕ p (R ) has norm 1, q p −p then |Iρ f (x)| . mB (|Iρ f | ) . mB (M f ) + inf ϕ(u) . u>0

Once Lemma 61 is established, we can conclude the proof of (the sufficiency part of) Theorem 60 as follows: We choose an arbitrary ball B ⊂ B(z; r). If we take the average of Lemma 61 over B and if we multiply both sides by ϕ(r)p , then we obtain Z Z ϕ(r)p ϕ(r)p q |Iρ f (x)| dx . M f (x)p dx + 1 . 1 |B| B |B| B n by virtue of the boundedness of the maximal operator M on Mϕ p (R ); see Theorem 34. The ball B being arbitrary, we obtain the desired result.

Proof of Lemma 61 Recall that k1 and k2 appeared in condition (12.27) Z k2 r ρ(s) on ρ. Write ρ∗ (r) ≡ ds for r > 0. Then, for given x ∈ Rn and s k1 r Z −1 ∞ X X ρ∗ (2j r) r > 0 we have |Iρ f (x)| . + |f (y)|dy. Let ΣI and (2j r)n B(x,2j r) j=−∞ j=0

Generalized Morrey/Morrey–Campanato spaces

55

ΣII be the first and second summations above. We now invoke the overlapping property: −1 X

∞ X

χ[2j k1 r,2j k2 r] . χ(−∞,2−1 k2 r] ,

j=−∞

χ[2j k1 r,2j k2 r] . χ[k1 r,∞) .

(12.32)

j=0

As a result, we have −1 X

ρ∗ (2j r) ≤

j=−∞

−1 Z X j=−∞

2j k2 r

2j k1 r

ρ(s) ds . s

k2 r

Z 0

ρ(s) ds = ρ˜(k2 r) s

and   Z ∞ ∞ X ρ(s) ρ(s)   . χ[2j k1 r,2j k2 r] (s) ds . ds. j ϕ(2 r) sϕ(s) k1 r k1 r sϕ(s) j=0

∞ X ρ∗ (2j r) j=0

Thus, ΣI .

Z

−1 X

∞

ρ∗ (2j r)M f (x) . ρ˜(k2 r)M f (x) . ϕ(r)1−p/q M f (x) thanks

j=−∞

to (12.31). Meanwhile ΣII .

∞ X ρ∗ (2j r) j=0

ϕ(2j r)

Z

∞

kf kMϕp . k1 r

Z

ρ(s) ds. We use sϕ(s)

ρ ϕ

∈ Z0 or

∞

p ρ(t) dt . ϕ(r)− q . By the doubling tϕ(t) rp property of ϕ, we obtain ΣII . ϕ(r)− q . Hence,
|Iρ f (x)| . ϕ(r)1−p/q M f (x) + ϕ(r)−1 (12.33)

(12.30) now. If we use (12.30), then

for all r > 0. First assume M f (x) ≤ inf ϕ(r)−1 . Then, the conclusion is immediate from r>0

(12.33). Next, we assume that M f (x) > inf ϕ(r)−1 holds. Since kf kMϕp = 1, we r>0

(p)

have 1 ≥ ϕ(r)mQ(x,r) (f ) ≥ ϕ(r)mQ(x,r) (|f |), or equivalently, mQ(x,r) (|f |) ≤ ϕ(r)−1 for all r > 0. This implies mQ(x,r) (|f |) ≤ sup ϕ(R)−1 for all r > 0. R>0

Since r > 0 and x ∈ Rn are arbitrary, it follows that M f (x) ≤ sup ϕ(r)−1 . r>0

We can thus find R > 0 such that M f (x) = 2ϕ(R) and, with this R, we can obtain the desired estimate. p/q

n ϕ In order that Iρ be bounded from Mϕ (Rn ), we must have p (R ) to Mq p ρ˜(r) . ϕ(r)− q according to Theorem 59 with ψ = ϕp/q . ϕ(r) We note that if ρ(r) = rα , with 0 < α < n, then Iρ = Iα is the classical fractional integral operator, also known as the Riesz potential, which is

56

Morrey Spaces

bounded from Lp (Rn ) to Lq (Rn ) if and only if p1 − 1q = α n , where 1 < p, q < ∞. The necessary part is usually proven by using the scaling arguments. Theorem 60 characterizes the kernel function ρ for which Iρ is bounded from Lp (Rn ) to Lq (Rn ) for 1 < p < q < ∞. We have the following result: Corollary 62. Let 1 < p < q < ∞. Then the operator Iρ is bounded from n n Lp (Rn ) to Lq (Rn ) if and only if ρ(r) . r p − q for all r > 0. For ρ(r) = rα , Corollary 62 further reads that the operator Iρ is bounded from Lp (Rn ) to Lq (Rn ) if and only if α = np − nq , where 1 < p < q < ∞. With Theorems 59 and 60 we can characterize the function ρ for which Iρ is bounded from one Morrey space to another. The next corollary generalizes the previous characterization in Corollary 62. Corollary 63. Assume that the parameters p, q, s, t and α satisfy 1 < q ≤ p < ∞, 1 < t ≤ s < ∞, 0 < α < n and 1 α 1 = − , s p n

t q = . s p

Let ρ : (0, ∞) → (0, ∞) be a function satisfying the growth condition. Then the generalized fractional integral operator Iρ is bounded from Mpq (Rn ) to Mst (Rn ) precisely when ρ(r) . rα . We show by examples that two statements in Theorem 60 are of independent interest. As before we write ( (1 + | log r|)β1 (0 < r ≤ 1), B ` (r) ≡ (1 + | log r|)β2 (1 < r < ∞). This function is used to describe the “log”-growth and “log”-decay properties. Also, we fix p and q so that 1 < p < q < ∞. The key properties we are interested in are summarized in the following table: ρ ϕ

Example Example Example Example

29 30 31 32

∈ Z0 ϕ ∈ Z + + + − − − − −

−n p

(12.30) + + + +

(12.31) + + + +

In the above, + means “true” and − means “false”.   Example 29. Let λ < 0 satisfy 0 < pq − 1 λ < n and − np < λ. Take p

µ1 , µ2 ∈ R arbitrarily. Define ϕ(r) ≡ r−λ `(−µ1 −,µ2 ) (r) and ρ(r) ≡ ϕ(r)1− q for r > 0. Then this pair (ρ, ϕ) fulfills the assumptions ϕρ ∈ Z0 and ϕ ∈ n

p

−p 1− q Z and Z ∞ in Theorem 60. Indeed, for r > 0 we have ρ˜(r) ∼ ρ(r) = ϕ(r) ρ(t) ρ(r) dt ∼ . tϕ(t) ϕ(r) r

Example 30 is an endpoint case of the above example.

Generalized Morrey/Morrey–Campanato spaces 57   Example 30. Let µ1 , µ2 ≥ 0. Set α ≡ np − nq and βi ≡ pq − 1 µi for i = 1, 2. n

p

Define ϕ(r) ≡ r p `(−µ1 ,−µ2 ) (r) for r > 0 and ρ ≡ ϕ1− q . We note that ρ˜ ∼ ρ. p Then this pair (ρ, ϕ) fulfills the assumptions ϕρ = ϕ− q ∈ Z0 and (12.30) but n

ϕ∈ / Z− p since `(−µ1 ,−µ2 ) ∈ / Z0 . The next example concerns the case where the spaces are close to L∞ (Rn ) and the smoothing order of Iρ is “almost 0”.   Example 31. Let µ1 , µ2 < 0. Set β1 ≡ pq − 1 µ1 + 1 ∈ (1, ∞) and β2 ≡   p − 1 µ2 − 1 ∈ (−1, ∞). Write B ≡ (β1 , β2 ). Define ρ ≡ `B as we did in q n

Example 9 and let ϕ ≡ `(µ1 ,µ2 ) . Then this pair (ρ, ϕ) fulfills ϕ ∈ / Z− p and ρ (β1 −µ1 ,β2 −µ2 ) assumption (12.30) but ϕ = ` ∈ / Z0 . More precisely, we have R ∞ ρ(t) (β1 −1,β2 +1) ρ˜ ∼ ` since β1 > 1, and r tϕ(t) dt ∼ `(β1 −µ1 −1,β2 −µ2 +1) (r) for r > 0. We consider a case where the target space is close to L∞ (Rn ). Example 32. Let 1 < p, q < ∞.  Also let  α, β1 , µ1 , µ2 satisfy 0 < α < p n , µ + β < 1, µ < 0. Set β ≡ − 1 µ2 − 1 ∈ (−1, ∞). Write B ≡ 1 1 2 2 p q (β1 , β2 ). Define ρ(r) ≡ min(1, rα )`B (r) as we did in Example 9, and let ϕ(r) ≡ n max(1, r−α )`(µ1 ,µ2 ) (r) for r > 0. Then this pair (ρ, ϕ) fulfills ϕ ∈ / Z− p and ρ˜(r) / Z0 More precisely, assumption (12.30) but ϕρ ∈ ∼ `(β1 −µ1 ,β2 −µ2 +1) (r) ϕ(r) Z ∞ ρ(t) and dt ∼ `(β1 −µ1 −1,β2 −µ2 +1) (r) for r > 0. tϕ(t) r Based upon these preliminary results and Lemma 57, we will prove Theorems 60–59. We remark that (12.30) includes (12.31). We prove an estimate. Once we n prove Lemma 61 below, we can obtain the boundedness of Iρ from Mϕ p (R ) p/q

to Mϕ (Rn ) as we will see below. Here, we use the fact that the Hardy– q n Littlewood maximal operator M is bounded on Mϕ p (R ), if p > 1 and ϕ is almost decreasing; see Theorem 34.

12.2.4

Generalized fractional maximal operators in generalized Morrey spaces

We discuss the boundedness property of the generalized fractional maximal operator Mρ , which is defined by Z ρ(r) Mρ f (x) ≡ sup |f (y)|dy (x ∈ Rn ), r>0 |B(x, r)| B(x,r) where f ∈ L1loc (Rn ) and ρ is a suitable function from (0, ∞) to [0, ∞).

58

Morrey Spaces

Example 33. Let 0 ≤ α < n. (1) If we let ρ(t) ≡ tα , t > 0, then we obtain the usual fractional maximal operator Mα ; Mρ = Mα . (2) If we let ρ(t) ≡ min(tα , 1)χ(0,1] (t), t > 0, then we obtain the operator mα called the local fractional maximal operator, where for f ∈ L0 (Rn ) Z rα mα f (x) ≡ sup |f (y)|dy (x ∈ Rn ) 0 0 be fixed. We utilize the pointwise estimate ρ(R)χB(R) ≤ Mρ χB(2R) , and the doubling condition of ϕ to obtain ρ(R) .

kρ(R)χB(R) kWMψ 1

ψ(R)

.

kMρ χB(2R) kMψ 1

ψ(R)

.

kχB(2R) kMϕq ϕ(R) ∼ , ψ(R) ψ(R)

as desired. Our result completely characterizes the boundedness of Mρ on generalized Morrey spaces. Theorem 65. Let 0 < a < 1 < q < ∞. Let ϕ ∈ Gq . Then, Mρ is bounded ϕa n n from Mϕ q (R ) to Ma−1 q (R ) if and only if ρ and ϕ satisfy ρ(R) . ϕ(R)1−a

(12.34)

for all R > 0. The proof hinges on the following inequality of Hedberg type: Lemma 66. Let 0 < a < 1 < q < ∞ and ϕ ∈ Gq . Also let ρ : (0, ∞) → (0, ∞) n satisfy (12.34). Then Mρ f (x) . M f (x)a for x ∈ Rn for any f ∈ Mϕ q (R ) ϕ with kf kMq ≤ 1. Once Lemma 66 is proven, we have only to resort to the scaling law (Lemma n 19) and the boundedness of M on Mϕ q (R ). Proof of Lemma 66 It should be noted that ϕ is bijective. Let R > 0. By using the definition of M , we obtain Z ρ(R) |f (y)|dy ≤ ρ(R)M f (x) . ϕ(R)1−a M f (x) |B(x, R)| B(x,R)

Generalized Morrey/Morrey–Campanato spaces

59

and ρ(R) |B(x, R)|

Z |f (y)|dy . ρ(R)

kf kMϕ1

B(x,R)

ϕ(R)

. ϕ(R)−a .

Thus, it follows that Z  ρ(R) |f (y)|dy . min ϕ(R)1−a M f (x), ϕ(R)−a |B(x, R)| B(x,R)  ≤ sup min t1−a M f (x), t−a t>0

= M f (x)a . Since R > 0 being arbitrary, we obtain the desired result. The weak boundedness of Mρ can be characterized in a similar way. Corollary 67. Let 0 < a < 1 ≤ q < ∞. Let ϕ ∈ Gq . Then, Mρ is bounded ϕa n n from Mϕ q (R ) to WMa−1 q (R ) if and only if ρ and ϕ satisfy (12.34) for all R > 0. We obtain the vector-valued inequality for Mρ on generalized Orlicz– Morrey spaces and weak generalized Orlicz–Morrey spaces. Theorem 68. Let 0 < a < 1 < q < ∞ and 1 ≤ u < ∞. Also let ϕ ∈ Gq . ϕ n (1) If ρ and ϕ satisfy (12.3) and (12.34), then for {fj }∞ j=1 ⊂ Mq (R ),



∞

{Mρ fj }∞

(12.35) j=1 Mϕ (`u ) . {fj }j=1 Mϕ (`u ) . q

q

n ϕ (2) Conversely, if (12.35) holds for {fj }∞ j=1 ⊂ Mq (R ), then ρ, ϕ and ψ satisfy (12.34). Moreover, under the assumption that ρ ∼ ϕ/ψ, inequality (12.35) holds if and only if ϕ satisfies (12.3).

Proof (1) Using (12.34), we may assume that ρ = ϕ1−a . Since ϕ is a doubling function and 0 < a < 1, we have Mϕ1−a fj . Iϕ1−a |fj |. Thus,   ∞ ∞ k{Mρ fj }∞ j=1 k`u . k{Iϕ1−a fj }j=1 k`u . Iϕ1−a k{fj }j=1 k`u . Since we can verify (12.30) by the use of (12.3), it remains to resort to ϕa n n the boundedness of Iϕ1−a from Mϕ q (R ) to Ma−1 q (R ). n ∞ (2) Fix f ∈ Mϕ q (R ). Let fj ≡ χ{1} (j)f . Then applying (12.35) to {fj }j=1 , ϕ n we see that Mρ is bounded on M1 (R ). Hence, by Theorem 65, we conclude that inequality (12.34) holds.

Under the assumption that ρ ∼ ϕ/ψ, we will prove that inequality (12.35) holds if and only if ϕ satisfies (12.3). To do this, it is

60

Morrey Spaces enough to show that (12.3) follows from (12.35). Now, assume that the integral condition (12.3) fails. Then, for any m ∈ N, there exists rm > 0 such that ϕ(2m rm ) ≤ 2ϕ(rm ). For each j ∈ N, we set fj ≡ χ[1,m] (j)χB(2j rm )\B(2j−1 rm ) . Then we have kfj kMϕq (`u ) ≤ kχB(2m rm ) kMϕq ∼ ϕ(2m rm ) ≤ 2ϕ(rm ).

(12.36)

Since θ ∈ Gn and ρ ∼ ϕ/ψ = ϕ/θ(ϕ), ρ(r) . ρ(s) for all r ≤ s. Due to this fact and the inequality Mρ fj & ρ(2j rm )χB(rm ) , we have kMρ fj kMψ (`u ) & k{ρ(2j rm )}m j=1 k`u χB(rm ) kMψ 1

1

1

& ρ(2rm )ψ(2rm )m u 1

& ϕ(rm )m u .

(12.37)

We combine inequalities (12.36) and (12.37) with the boundedness of Mρ ψ u u from Mϕ q (` ) to M1 (` ) to obtain m ≤ D, where D is independent of m, contradictory to the fact that m ∈ N is arbitrary. Thus, the integral condition (12.3) holds.

12.2.5

Exercises

Exercise 11. [344] Let 0 < q < ∞. Suppose that ϕ ∈ G1 and that f is supported in B(a, r). Assume in addition   ! q1 Z   1 M ≡ sup ϕ(s) |f (x)|q dx : B(b, s) ⊂ B(a, 3r) < ∞.   |B(b, s)| B(b,s) n ϕ Then show that f ∈ Mϕ q (R ) and that M = kf kMq .

Exercise 12. Prove (12.4) by reexamining the proof of Theorem 383 in the first book. Exercise 13. Prove (12.21) using Example 22. Exercise 14. [406, Theorem 4.32] Prove Theorem 51 by mimicking Definition 96 in the first book. Exercise 15. [314, Remark 7.1] Let 0 < α < n, β ∈ R and γ ∈ (0, n−α n ). n Write p ≡ n−α . Set ρ(r) ≡ rα (log(2 + r−1 ))−β . Define f (y) ≡ |y|−n (log(1 + |y|−1 ))−1 (log(1 + log(1 + |y|−1 )))−γ−1 χB(1) (y) for y ∈ Rn . (1) Show that f ∈ L1 (B(1)). (2) Show that Iρ f (x) ≥ Iρ [f χB(|x|/2) ](x) & |x|α−n (log(1 + |x|−1 ))−1 (log(1 + log(1 + |x|−1 )))−γ . Z (3) Prove that Iρ f (x)p (log(1 + Iρ f (x)))pβ−1 dx = ∞. B(1)

Generalized Morrey/Morrey–Campanato spaces

61

Exercise 16. [314, Remark 7.2] Let 0 < α < n, β ∈ R and δ ∈ (0, ∞). Write n p ≡ n−α . Suppose further that we have parameters γ ∈ (p(β + δ) − 1, ∞) α −1 −β )) for r > 0. Define and ε ∈ (0, 1+γ p − β − δ). Set ρ(r) ≡ r (log(2 + r −n −1 −δ−ε−1 n f (y) ≡ |y| (log(1 + |y| )) χB(1) (y) for y ∈ R . (1) Show that f ∈ L logδ L(Rn ). (2) Show that Iρ f (x) & |x|α−n (log(1 + |x|−1 ))−δ−ε . (3) Disprove that f ∈ Lp logγ L(Rn ).

12.3

Generalized Morrey–Campanato spaces

Here, we generalize BMO spaces. For f ∈ L1loc (Rn ), recall that its BMOnorm is given by Z Z 1 1 f (x) − kf k∗ ≡ sup mQ (|f − mQ (f )|) = sup f (y)dy dx. |Q| Q Q∈Q Q∈Q |Q| Q We seek its generalization. We will include Morrey–Campanato spaces. We work in the half line (0, ∞) in Section 12.3.1, while we work in the whole space in Section 12.3.2.

12.3.1

Generalized Morrey–Campanato spaces over the half line (0, ∞)

We work on the half line (0, ∞) first. On (0, ∞), it is reasonable to define BMO(0, ∞), the BMO space over (0, ∞), to be the set of all locally integrable functions f integrable over (0, 1) for which kf kBMO(0,∞)

1 = sup a≥0,r>0 r

Z

a+r

|f (x) − m(a,a+r) (f )|dx < ∞. a

We will consider a generalization to relate this with Morrey–Campanato spaces and to consider the multiplier spaces later. Definition 13. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . We define BMOϕ (0, ∞), the generalized Morrey–Campanato space generated by ϕ, to be the set of all f ∈ L1loc (0, ∞) ∩ L1 (0, 1) for which Z Z ϕ(r) a+r 1 a+r sup f (t) − f (s)ds dt < ∞. r a a≥0,r>0 r a

62

Morrey Spaces It should be noted that kf kBMOϕ (0,∞)

ϕ(r) ≡ sup inf c∈C r a≥0,r>0

Z

a+r

|f (t) − c| dt.

(12.1)

a

We will work on the whole space Rn later. Example 34. If we define ϕ(t) ≡ log t, t > 0, then ϕ ∈ BMO(0, ∞), since ψ(x) ≡ log |x|, x ∈ Rn belongs to BMO(Rn ). Let ϕ ∈ M↓ (0, ∞) satisfy Φ∗ (r) ≡

Z 1

max(r,2)

1 ϕ

∈ Gn . Set

dt , ϕ(t)t

Φ∗ (r) ≡

Z

2

min(r,1)

dt ϕ(t)t

(r > 0).

(12.2)

The function Φ∗ is increasing and the function Φ∗ is decreasing. The function Φ∗ is different from the Fenchel–Legendre transform. It is easy to show that ϕ, Φ∗ , Φ∗ are all doubling and hence r Z r . Φ∗ (t)tn−1 dt (r > 0). (12.3) rn Φ∗ (r) ∼ rn Φ∗ 2 0 We check how fast Φ∗ grows. Z Lemma 69. For all r ≥ 2, 0

r

r dt . . ∗ Φ (t) ϕ(r)Φ∗ (r)

Proof Since ϕ ∈ M↑ (0, ∞), Φ∗ (r) ≥

log max(r, 2) . Note that ϕ(1)

d max(2, t) 1 1 = − dt log max(2, t) log max(2, t) (log max(2, t))2

(t > 2).

Thus, we conclude  Z r Z r Z r dt d max(2, t) r dt . . 1 + dt ∼ ∗ (t) Φ log max(t, 2) dt log max(2, t) log r 0 0 0 Z r Z r dt dt log r for r ≥ 2. Meanwhile, Φ∗ (r) = ≤ = . Consequently, ϕ(t)t ϕ(r)t ϕ(r) 1 1 we obtain the desired result. We will show that Φ∗ is an element in BMOϕ (0, ∞). Lemma 70. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Z 1 r+h ∗ 1 (1) For all r ≥ 0 and h ∈ (0, r], |Φ (s) − Φ∗ (r + h)|ds . . h r ϕ(h) Z 1 r+h ∗ 1 (2) For all r ≥ 0 and h ∈ (r, ∞), |Φ (s) − Φ∗ (r)|ds . . h r ϕ(h) In particular, Φ∗ is an element in BMOϕ (0, ∞).

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63

Proof (1) This is clear because ϕ does not differ so much from ϕ(r) on [r, r + h]. (2) We observe for all s ∈ [r, r + h], 0 ≤ Φ∗ (s) − Φ∗ (r) =

Z

max(s,2)

max(r,2)

Since log

dt 1 max(s, 2) ≤ log . ϕ(t)t ϕ(max(s, 2)) max(r, 2)

max(s, 2) max(r + h, 2) ≤ log ≤ 1, it follows that max(r, 2) max(r, 2) 0 ≤ Φ∗ (s) − Φ∗ (r) ≤

1 1 ≤ . ϕ(max(s, 2)) ϕ(h)

Thus, it remains to take the average over [0, h]. In the same manner we can prove the following: Corollary 71. Let ϕ ∈ M↓ (0, ∞) satisfy belongs to BMOϕ (0, ∞).

12.3.2

1 ϕ

∈ Gn . Then the function Φ∗

Generalized Morrey Campanato spaces over Rn

We now embark on the space BMOϕ (Rn ). Keeping in mind the definition of generalized Morrey spaces, we define this space as follows: Definition 14. Let ϕ ∈ M↓ (0, ∞) satisfy

1 ϕ

∈ Gn . For f ∈ L1loc (Rn ), set

kf kBMOϕ ≡ sup ϕ(`(Q))mQ (|f − mQ (f )|). Q∈Q

The generalized Morrey–Campanato space BMOϕ (Rn ) is the set of all locally integrable functions f for which the norm kf kBMOϕ is finite. We give some examples of functions in BMOϕ (Rn ). Example 35. (1) If ϕ ≡ 1, then BMOϕ (Rn ) = BMO(Rn ). (2) If ϕ(r) ≡ r−θ , r > 0 with 0 < θ < 1, then BMOϕ (Rn ) = Lipθ (Rn ). Example 36. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Hence, we are working in the situation where BMOϕ (Rn ) is close to Lipschitz spaces. Fix a ∈ Rn and 0 < r < 1. Let Z 2 dt Φ∗ (R) ≡ (R > 0) ϕ(t)t min(R,1)

64

Morrey Spaces

as before. If we define h(x) ≡ χB(a,r) (x)(exp(iΦ∗ (|x − a|)) − exp(iΦ∗ (r))) = exp(iΦ∗ (min(r, |x − a|))) − exp(iΦ∗ (r))

(x ∈ Rn ),

then we obtain khkBMOϕ . 1, since exp(i·) is a Lipschitz function. Observe also that |h| ≤ 2χB(a,r) . Example 34 shows how to create a local function ϕ ∈ BMO(0, ∞) starting from a global function ψ ∈ BMO(Rn ). We will consider the opposite direction. Theorem 72. Let ϕ ∈ M↓ (0, ∞) be such that t ∈ (0, ∞) 7→ ϕ(t)t ∈ (0, ∞) is increasing. Also let Φ ∈ BMOϕ (0, ∞). If we set Ψ ≡ Φ(| · |), then Ψ ∈ BMOϕ (Rn ). Proof We need to show that ϕ(r)mB(x,r) (|Ψ−mB(x,r) (Ψ)|) . 1, where the implicit constant is independent of x ∈ Rn and r > 0. This can be rephrased as the problem of finding a constant cB(x,r) satisfying I ≡ ϕ(r)mB(x,r) (|Ψ − cB(x,r) |) . 1, where once again the implicit constant is independent of x ∈ Rn and r > 0. Since ϕ is a doubling function, we may assume either x = 0 or |x| > 2r. Suppose first that x = 0. Then with cB(x,r) = cB(0) = m(0,r) (Φ), we have Z ϕ(r) I= |Ψ(y) − m(0,r) (Φ)|dy |B(r)| B(r) Z r ϕ(r) = |Φ(r) − m(0,r) (Φ)|rn−1 dr |B(r)| 0 Z ϕ(r) r |Φ(r) − m(0,r) (Φ)|dr. ≤ r 0   y Suppose instead that |x| > 2r. Then define Ω ≡ : y ∈ B(x, r) . We set |y| D ≡ {rω : r ∈ (|x| − r, |x| + r), ω ∈ Ω}. Then |D| ∼ |B(x, r)|. Consequently, we have only to find a constant cD such that ϕ(r)mD (|Ψ − cD |) . 1. By the change of the polar coordinate, we can reduce matters to Z ϕ(r) |x|+r |Φ(r) − cD |dy . 1. 2r |x|−r Once we reduce matters to this inequality, we see that cD ≡ m(|x|−r,|x|+r) (Φ) suffices. About the functions Φ∗ , Φ∗ , we need the following estimate: Lemma 73. Let ϕ ∈ M↓ (0, ∞) satisfy

1 ϕ

∈ Gn .

(1) For any f ∈ BMOϕ (Rn ) and (a, r) ∈ Rn × (0, ∞) with 1 ≤ max(r, |a|), |mQ(a,r) (|f |)| . mQ(1) (|f |) + kf kBMOϕ (Φ∗ (r) + Φ∗ (|a|)).

Generalized Morrey/Morrey–Campanato spaces

65

(2) For any f ∈ BMOϕ (Rn ) and (a, r) ∈ Rn × (0, ∞) with 1 > max(r, |a|), |mQ(a,r) (|f |)| . mQ(1) (|f |) + kf kBMOϕ Φ∗ (r). Proof (1) Notice that Q(1), Q(a, r) ⊂ Q(r + 2|a|). Thus, Z

2r+4|a|

mQ(a,r) (|f |) . mQ(1) (|f |) +

sup mQ(b,t) (|f − mQ(b,t) (f )|)

r ∗

b∈Rn

dt t

. mQ(1) (|f |) + Φ (6|a|)kf kBMO+ ϕ . . mQ(1) (|f |) + Φ∗ (|a|)kf kBMO+ ϕ (2) We use Q(a, r), Q(1) ⊂ Q(3) and argue as before. n 1 n We define BMO+ ϕ (R ) to be the set of all f ∈ Lloc (R ) for which

kf kBMO+ ≡ kf kL1 (Q(1)) + kf kBMOϕ ϕ is finite. We will show that the operator norm of n f ∈ BMO+ ϕ (R ) 7→ mQ(a,r) (|f |)

is comparable to Φ∗ (|a|) + Φ∗ (r) + Φ∗ (r). In view of Lemma 73, we have the estimate from above. We will prove the following proposition to prove the estimate from below. Proposition 74. For all a ∈ Rn and r > 0, sup{mQ(a,r) (|f |) : kf kBMO+ = 1} & max(Φ∗ (|a|), Φ∗ (r), Φ∗ (r)). ϕ Proof Recall that the functions Φ∗ and Φ∗ are doubling and decreasing. We distinguish two cases. (1) If |a| ≥ 4nr, then we use Q(a, r)\B(r) = ∅ to underestimate mQ(a,r) (|f |). We need to show sup{mQ(a,r) (|f |) : kf kBMO+ = 1} & max(Φ∗ (|a|), Φ∗ (r)). ϕ

(12.4)

If we integrate these pointwise estimates, then we obtain (12.4). If we let f ≡ Φ∗ (| · −a|), then f (x) & Φ∗ (r) for all x ∈ Q(a, r). Meanwhile, if we let f ≡ Φ∗ (| · |), then f (x) & Φ∗ (|a|) for all x ∈ Q(a, r) since Φ∗ is doubling. If we integrate these pointwise estimates, then we obtain (12.4).

66

Morrey Spaces

(2) If |a| ≤ 4nr, we need to show sup{mQ(a,r) (|f |) : kf kBMO+ = 1} & max(Φ∗ (r), Φ∗ (r)). ϕ

(12.5)

We can find a small cube Q(z, r/4) such that Q(z, r/4)∩Q(r/4) = ∅ and that Q(z, r/4) ⊂ Q(a, r). Use this cube to underestimate mQ(a,r) (|f |). If we let f ≡ Φ∗ (| · −a|) as before, then f (x) & Φ∗ (r) for all x ∈ Q(a, r). Meanwhile, if we let f ≡ Φ∗ (|·−z|), then f (x) & Φ∗ (r) for all x ∈ Q(r/8) since Φ∗ is doubling. If we integrate these pointwise estimates, then we obtain (12.5). We supplement Lemma 73. Recall that L1uloc (Rn ) is the set of all f ∈ for which the norm kf kL1uloc ≡ sup kχB(a,1) f kL1 is finite. See Exam-

L1loc (Rn )

a∈Rn

ple 7. Once we can use the L1uloc (Rn )-norm, there is no need to use the function Φ∗ . Lemma 75. For all f ∈ BMOϕ (Rn ) ∩ L1uloc (Rn ), a ∈ Rn and r > 0 mQ(a,r) (|f − mQ(a,r) (f )|) . (kf kBMOϕ + kf kL1uloc )Φ∗ (r). Proof When r ≥ 1, we use H¨older’s inequality to obtain mQ(a,r) (|f − mQ(a,r) (f )|) . kf kL1uloc . (kf kBMOϕ + kf kL1uloc )Φ∗ (r). When r ≤ 1, we use mQ(a,r) (|f − mQ(a,r) (f )|) ≤ |mQ(a,r) (|f − mQ(a,r) (f )|) − mQ(a,1) (|f − mQ(a,1) (f )|)| + mQ(a,1) (|f − mQ(a,1) (f )|) . kf kBMOϕ Φ∗ (r) + mQ(a,1) (|f − mQ(a,1) (f )|) and argue as before.

12.3.3

Exercises

Exercise 17. Let 1 ≤ p < ∞. Show that Mp1 (Rn ) is continuously embedded into L1uloc (Rn ). Show also that this embedding is strict. Hint: Consider the constant function 1 to show that this embedding is strict. Exercise 18. [351, Lemma 2.4] Let ϕ ∈ M↓ (0, ∞) satisfy

1 ϕ

∈ Gn .

(1) Show that 2−n ϕ(2s) ≤ ϕ(t) ≤ ϕ(s) whenever 0 < t < s < 2t. (2) If we define Φ∗ and Φ∗ by (12.2), then show that Φ∗ and Φ∗ are doubling. (3) Verify (12.3).

Generalized Morrey/Morrey–Campanato spaces

12.4

67

Notes

Section 12.1 General remarks and textbooks in Section 12.1 It seems that Dzhumakaeva and Nauryzbaev initially considered generalized Morrey spaces in [92]. Actually, they replaced B(x, r) by a general set E. However, generalized Morrey spaces presented in this book can be traced back to the work by Zorko [544] as well as the works by Mizuhara [312], Nakai [336] and Guliyev [139]. See [78] as well. We refer to [388, Chapter 10] for generalized Morrey spaces including generalized fractional integral operators and the Hardy–Littlewood maximal operators acting on generalized Morrey spaces. Section 12.1.1 fpq (Rn ) See [160, Definition 1] for modified Morrey spaces. We also use M n n p p fq (R ) to denote for Mq (R ). However, to avoid confusion, we employed M n n p (R ) to denote the modified (R ) and M the closed subspace spanned by L∞ q c Morrey space. See also [176]. Transirico, Troisi and Vitolo considered small Morrey spaces over open sets [480]. See also [67]. As an example, we can consider small Morrey spaces; see [160, Theorem 1] for Example 15. See [161] for fractional integral operators. We calculated the norm of the indication function of cubes in Example 8; see [339, Lemma 3.3] and [241, Lemma 4.1] as well as [103, Proposition A] for Example 8. See [406, Example 3.10] for Example 9. We investigated the Morrey space Mpq (µ) defined in [418] exhaustively. See [401, 414] for fractional integral operators, [402, 421] for sharp maximal operators, [403] for generalized Morrey spaces considered in Section 12.1, [413] for commutators, [419] for equivalent norms, [422] for Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces and [423] for predual spaces. See [472] for the generalization to geometrically doubling quasi-metric measure spaces. See the survey [404]. Property (12.32) is in [417]. Proposition 22 is due to Nakai, see [339, p.446]. We refer also to [336, 344] and [417, Section 1]. Lemma 25, which concerns the norm of the indicator function of cubes in generalized Morrey spaces, is proven in [103, Proposition A] and [339, Lemma 3.3]. See [241, Lemma 4.1] and [105] as well. There are many equivalent and different definitions for generalized Morrey spaces. For more details on generalized Morrey spaces we refer to [312, 336, 343, 344, 415]. Applications to Schr¨odinger operators can be found in [245] See [241, Definition 2.5] for the definition of the predual of generalized Morrey spaces.

68

Morrey Spaces

The class of functions Zγ ((0, `]) and so on is from [392, Section 2]. The generalized fractional integral operator Iρ was initially investigated in [340]. Nowadays many authors have been culminating important observations about Iρ especially in connection with Morrey spaces. These spaces cover Lebesgue spaces as special cases and seem to describe the behavior of Iρ well. In order to highlight what we will prove in this book, we take up the works [50, 100, 105, 178, 314, 336, 341], where we formulated sufficient conditions on ρ for n Iρ to be bounded in Morrey spaces Mϕ p (R ) with 1 ≤ p < ∞ and ϕ a function from (0, ∞) to itself. See [314] for further examples of ρ. We characterize the boundedness by estimating the norm of the characteristic functions of balls and the function ϕ(| · |), as well as the value of the corresponding fractional integrals. We followed [406] for various examples of ϕ; see [406, Example 3.4]. Here, we will recall some works related to generalized fractional integral operators and generalized Morrey spaces. Condition (12.29) is known to be sufficient in [341, Theorem 3.2]. The authors generalized Morrey spaces to various directions in [177, 336, 341, 347, 403]. Nakai defined generalized Morrey spaces in [336]. Weak-type Morrey spaces are defined in [341]. The work [403] is a passage from [336] to metric measure spaces whose underlying measure fails the doubling condition. A further generalization is done in [177]. As another generalization, in [347], Morrey spaces are generalized to martingale Morrey spaces. The authors applied generalized Morrey spaces to grasp the limiting case in [313]. Generalized fractional integral operators, which were initiated by Gadiyev [118], are investigated further in [33, 166, 167, 323, 413, 448, 462]. The works [166, 167] are oriented to the boundedness of Iρ and Guliyev and Mustafayev used ϕ1 ∈ Z0 and (12.30) as a sufficient condition. The author worked in the setting of non-doubling measure spaces in [448], and the integral condition showed up in [448, Theorem 2.1]. The author extended the GagliardoNirenberg inequality for Iα , the case where ρ(t) ≡ tα , to that for Iρ in [323]. An intersection of two classical Morrey spaces with the same parameter p can be regarded as a generalized Morrey space. Sugano investigated the boundedness of fractional integral operators on intersections of Morrey spaces [462], where she postulated conditions stronger than the integral condition and (12.30). Other operators such as commutators are taken up in [162, 413]. Guliyev, Karaman, Seymur and Shukurov essentially considered (12.30) in order to show the boundedness of the fractional maximal operator of order α in [162, (5.1)]. In order to show the boundedness of commutators generated by BMO functions and Iα , in [413], the authors postulated ϕ on ϕ1 ∈ Z0 and (12.30). The case where ϕ depends on x as well can be covered especially by the group of Guliyev. See [173] for the operator Tα whose property is similar to the one in Iα . See [11] for the maximal operator and singular integral operators. We followed [406, Definition 3.5] for the name of small Morrey spaces; see Definition 6. Proposition 17, showing the completeness of generalized Morrey

Generalized Morrey/Morrey–Campanato spaces

69

spaces, can be found in [406, Proposition 3.8]. Proposition 18, which considers the non-triviality of generalized Morrey spaces, is [354, Lemma 2.2(2)]. Section 12.1.2 Nakai investigated the relation between L∞ (Rn ) and generalized Morrey spaces; see [344, Proposition 3.3] for Theorems 27 and 28. Usually using the Fourier transform, we discuss the Bessel potential; see [460, Theorems 1.13 and 1.14]. We followed [406, Example 3.15] for examples of functions in Gq ; see 10. As we saw in Proposition 21, functions in Gq enjoy the doubling property; see [406, Proposition 3.16]. We followed [406, Example 3.21] for a modification of ϕ as in Example 11. We compared two different generalized Morrey spaces in Corollary 30; see [344, Corollary 4.11]. We followed [406, Example 3.24] to have almost increasing/decreasing functions; see Example 12. See [344, Proposition 3.3] for Corollary 29. The closed n subspaces of Mϕ q (R ) were considered in [185]; see also [406, Lemma 3.36] and [406, Proposition 3.37] for Lemma 32 and Proposition 33, respectively.

Section 12.2 General remarks and textbooks in Section 12.2 The fundamental papers are [139, 312, 336] together with the Russian book [140]. The integral condition is necessary even for the weak-type boundedness of the Hardy–Littlewood maximal operator; see [188] for Proposition 38. Natasha Samko considered the boundedness property of operators in vanishing generalized Morrey spaces in [393, 394]. Section 12.2.1 The boundedness of the maximal operator acting on generalized Morrey spaces is investigated by many researchers. See [336, Theorem 1], [344, Corollary 5.6] and [403, Theorem 2.3] for Theorem 34. Nakai obtained the weak boundedness of the maximal operator acting on generalized Morrey spaces initially in [336, Theorem 1] and then [344, Corollary 6.2] after removing the unnecessary assumption posed on ϕ in [336, Theorem 1]; see Theorem 35. Nakai considered Theorems 39 and 40 in [336, Lemma 2] and Burenkov and Guliyev refined them [50]. See the works by Eridani and Utoyo [104] and by Guliyev [141] for more about the boundedness of the maximal operator. Liu considered for the sharp maximal inequality on generalized Mor1 ∈ Z0 Liu used rey spaces [272], where instead of the integral condition ϕ n ϕ(x, 2r) ≤ C0 ϕ(x, r) for some C0 ∈ (1, 2 ) independent of x ∈ Rn and r > 0. See also [277, Lemma 5] for the sharp maximal inequality. We considered the vector-valued extension of the boundedness of M from small Morrey spaces to weak small Morrey spaces in Example 16; see [406, Example 4.8].

70

Morrey Spaces

Section 12.2.2 See Nakai and Yabuta [352] for singular integral operators on Orlicz– Morrey spaces. Guliyev, Mizuhara and Nakai investigated the boundedness of singular integral operators; see [139], [312] and [336, Theorem 2] for Lemma 43. Komori-Furuya and Mizuhara investigated the predual of generalized Morrey spaces; see [241, Lemma 4.2] for Proposition 47. Zorko considered predual spaces of generalized Morrey spaces in [544, p. 589]. Here we followed the fashion in [241, 442]. See [241, Definition 2.3] and [442, Definition 4] for Definition 11(1) and [241, Definition 2.5] and [442, Definition 5] for Definition 11(2). Functions of predual spaces are locally integrable; see [412, Lemma 2.2 (A), (B)] for Corollary 48. Example 22, an equivalent norm, can be [406, Example 4.28]. As we have seen in Theorem 50, the Fatou property allows us to characterize a predual space of predual spaces. In a different setting of the definition of atoms Zorko proved Theorem 50(1) including Proposition 47 in [544, Proposition 5]; see also [299, Theorem 2.3] and [406, Theorem 4.31]. Proposition 49, the Fatou property of predual spaces, can be found in [299, Theorem 2.2] and [406, Proposition 4.30]. We remark that the key idea is the observation of Izumi, Sato and Yabuta [226]. Nakai investigated the action of singular integral operators on generalized Morrey spaces; see [336, Theorem 2] for Theorem 51. As is seen in Proposition 52, the condition considered by Nakai is also necessary; see [188, Theorem 14]. Guliyev also investigated the boundedness of singular integral operators [141] including the case of rough kernels [144]. Liu and Chen proved the boundendness of the Bochner–Riesz operator [282, Theorem 1.3]. Eroglu obtained the boundedness of oscillatory integral operators [93]. Liu handled Toeplitz type operators related to general singular integral operators [277, 209]. See [147, 174] for commutators . The predual of generalized Morrey spaces can be found in [406]; see [406, Example 4.23] and [406, Proposition 4.24] for Example 21 and Proposition 44, respectively. We recorded [406, Example 4.12] and [406, Example 4.34] as counterexamples of the boundedness of singular integral operators in Examples 17 and 23. Section 12.2.3 It seems that Pustylnik initially considered generalized fractional integral operators in [372]. See also [350] for more. Eridani considered generalized fractional integral operators in [98, Theorem 1] and modified generalized fractional integral operators in [98, Theorem 2]. See the work by Eridani and Utoyo [104] for more. The functions ρZand ρ˜ are subject to the power function rα at the same r ρ(t) dt for r > 0; see [406, Proposition 4.52] for Propotime, where ρ˜(r) ≡ t 0 ψ n n sition 58. The boundedness of Iρ from Mϕ p (R ) to M1 (R ) can be found in [101, Theorem 1.3]; see for Theorem 59. Motivated by the work of KomoriFuruya and Mizuhara [241], Shirai investigated the relation between blocks

Generalized Morrey/Morrey–Campanato spaces

71

and generalized Morrey functions; see [442, Lemma 2] for Lemma 45. Motivated by the technique obtained in [425], Sawano obtained Lemma 46 [406, Theorem 4.27]. As we saw in Example 26, generalized Morrey spaces can compensate for the failure of the boundedness of operators in Morrey spaces; see [426, Theorem 5.1] and [101, Example 5.1]. We can find similar investigations in these works; see [426, Theorem 5.1], [101, Example 5.1], [426, Theorem 5.1] and [101, Example 5.1], [101, Theorems 1.1 and 1.2], [101, Corollary 1.5] and [101, Corollary 1.6] for Examples 27 and 28, Theorem 60 and Corollaries 62 and Corollary 63, respectively. See [97, 143] for the boundedness of fractional integral operators with rough kernels. See [346] for more, where ϕ depends on x as well. Fan considered the boundedness of sublinear operators and their commutators on generalized local Morrey spaces [109]. Section 12.2.4 The boundedness of Mρ acting on generalized Morrey spaces is investigated by Hakim, Nakai and Sawano; see [188, Theorem 1], [188, (15)], [188, Theorem 1] and [188, Theorem 8] for Proposition 64, Lemma 66 which is the auxiliary estimate of Hedberg type, Theorems 65 and 68, respectively. Under certain conditions on ρ and ϕ, Mρ is bounded on generalized Morrey spaces; see [188, Corollary 1] for Corollary 67. See [101, Lemma 3.1] for Lemma 61 which is also the auxiliary estimate of Hedberg type. See [212] for the Bessel potential acting on generalized Morrey spaces. We compared Iρ χB , where B is a ball, in Lemmas 56 and 57; see [101] for this. We used [406, Example 4.73] for examples of generalized fractional integral operators; see Example 33. Gogatishvili and Mustafayev compared kIα f kMϕq and kMα f kMϕq ; see [127, Theorem 1] and [128, Theorem 1.4], which extend the result [8, Theorem 4.2] of Adams and Xiao to generalized Morrey spaces. We followed [406, Proposition 4.46] for Proposition 55.

Section 12.3 General remarks and textbooks in Section 12.3 It seems that there is no textbook on this topic. Section 12.3.1 Spanne considered generalized Morrey spaces on domains. See [430] and [351, p. 210] for Definition 13. See [430, Lemma 1] for (12.1) which concerns an equivalent expression of the norm k · kBMOϕ (0,∞) . Spanne gave an example of functions in BMOϕ (0, ∞); see [430, Theorem 2] for Theorem 72. We refer to [348] for an exhaustive detail of generalized Morrey–Campanato spaces.

72

Morrey Spaces

Section 12.3.2 The definition of generalized Morrey space goes back to [336]. The notation used in this book can be found in [415]. We calculated the norm of the indicator function of balls in Lemma 25. See [103, Proposition A] for example. Theorem 27, which compares L∞ (Rn ) and generalized Morrey spaces, is due to Hakim and Sawano [185, Theorem 8]. Nakai gave an example of functions in generalized Morrey spaces; see [339, Lemma 3.4] Example 13. See [339, p. 452] for Example 14. ∗

n The star subspace Mϕ q (R ) has canonical approximation; see [532, Remark 2. 36] for Lemma 31. The boundedness of the Hardy–Littlewood maximal operator on generalized Morrey spaces goes back to Nakai [336, Theorem 1], who assumes ϕ1 ∈ Z0 . This assumption was removed in [403, Theorem 2.3] and [344]. Theorem 39 goes back to [336, Lemma 2]. See [417] for some examples and more explanation about these two conditions (a) and (b) in Proposition 58. Definition 10 goes back to Zygmund. See [390, Definition 2.13]. The definition of W0 goes back to [390, Definition 2.7]. Persson and Samko investigated the boundedness property of weighted Hardy operators [367]. Theorem 59 goes back to [101, Theorem 1.3]. Example 26 goes back to [426, Theorem 5.1]. See also [464]. Example 28 is from [101, Example 5]. The two statements in Theorem 60 go back to [101, Theorem 1.2] and [101, Theorem 1.1], respectively. We refer to [101, Example 2], [101, Example 1], [101, Example 3] and [101, Example 4] for Examples 29, 30, 31 and 32, respectively. See [100, 178, 180, 463] for more about Iρ . For the proof of Theorem 68, we used the idea from [418] where the authors established the vector-valued boundedness for fractional maximal operators in Morrey spaces with non-doubling measures. Theorem 68 is [188, Theorem 1]. The proof of the sufficient part in Theorems 59–60 is similar to, but not the same as, that in [178, 341]. In this book, we do not assume that ρ satisfies the doubling condition nor that ϕ is onto, as we did in [178].

Section 12.3.1 Nakai and Yabuta investigated the relation between ϕ and Φ∗ ; see [351, Lemma 2.5] for Lemma 69. Estimates for BMOϕ (Rn ) are due to Nakai and Yabuta; see [351, Lemma 3.1] for Lemma 73 and [351, Lemma 3.2] for Lemma 75. See [21] for generalized local Morrey spaces.

Chapter 13 Generalized Orlicz–Morrey spaces

We will do the ϕ-generalization and the Φ-generalization at the same time. So, we are interested in (ϕ, Φ)-generalizations. There are a couple of (ϕ, Φ)generalizations and we will discuss their difference. Chapter 13 aims to distinguish between two different classes of function spaces, which are both called Orlicz–Morrey spaces. More precisely, two function spaces are mainly considered. Morrey spaces and Orlicz spaces were originally considered to extend and supplement Lebesgue spaces. Generalized Morrey spaces are mainly used to describe the property of the generalized Riesz potential, or differential operators such as ∆α (1 − ∆)β , α, β ∈ R. Orlicz spaces can describe the endpoint cases of fundamental operators such as the Hardy–Littlewood maximal operators, fractional integral operators, or singular integral operators. Recently, the situation where they are mixed has arisen. There are at least two types of generalized Orlicz–Morrey spaces: Orlicz–Morrey spaces of the first kind and Orlicz–Morrey spaces of the second kind. The former is investigated in Section 13.1, while the latter is investigated in Section 13.2. Section 13.3 shows that these two scales are actually different. Deringoz, Guliyev and Samko also considered the one of the third kind. But we will consider it in a different context.

13.1

Generalized Orlicz–Morrey spaces of the first kind

Generalized Morrey spaces of the first kind are defined by the use of the average which we do not take up so far. Section 13.1 defines this new average and then discusses the boundedness property of operators: Section 13.1.1 is the setup to define generalized Orlicz–Morrey spaces of the first kind. We investigate the boundedness properties of the Hardy–Littlewood maximal operator, singular integral operators and fractional integral operators in Sections 13.1.2, 13.1.3 and 13.1.4, respectively.

73

74

Morrey Spaces

13.1.1

(ϕ, Φ)-average and generalized Orlicz–Morrey spaces of the first kind

Defining Orlicz–Morrey spaces of the first kind needs the new notion of averages unlike that of the second kind. We start with the definition of the (ϕ, Φ)-average over a cube Q. Definition 15 ((ϕ, Φ)-average). Let ϕ : (0, ∞) → (0, ∞) be a function and Φ : [0, ∞) → (0, ∞) a Young function. For a cube Q and f ∈ L0 (Q), define the (ϕ, Φ)-average over Q by     Z |f (x)| ϕ(`(Q)) Φ kf k(ϕ,Φ);Q ≡ inf λ > 0 : dx ≤ 1 . |Q| λ Q Let Q be a cube and f ∈ L0 (Q). Write µQ (E) ≡ ϕ(`(Q))|Q|−1 |E ∩ Q| for a measurable set E. It should be noted that kf k(ϕ,Φ);Q = kf kLΦ (µQ ) .

(13.1)

Here and below in Section 13.1.1 we suppose that Φ is a Young function ˜ is its conjugate given by Φ(s) ˜ and that Φ ≡ sup(st − Φ(s)). t>0

We present an example of calculation. Example 37. Let ϕ : (0, ∞) → (0, ∞) be an arbitrary function, and let Φ : [0, ∞) → [0, ∞) be a bijective Young function. By an elementary calculus, 1 for all cubes Q(x, r). k1k(ϕ,Φ);Q(x,r) = −1 Φ (ϕ(r)−1 ) We define generalized Orlicz–Morrey spaces of the first kind via the (ϕ, Φ)average. Definition 16 (Generalized Orlicz–Morrey spaces of the first kind). Let ϕ : (0, ∞) → (0, ∞) be a function, and let Φ : [0, ∞) → [0, ∞) be a Young function. (1) For f ∈ L0 (Rn ), define kf kLϕΦ ≡ sup kf k(ϕ,Φ);Q . The function space Q∈Q

n Lϕ Φ (R ), the generalized Orlicz–Morrey space of the first kind, is defined as the set of all f ∈ L0 (Rn ) for which the norm kf kLϕΦ is finite. n (2) The function space WLϕ Φ (R ), the weak generalized Orlicz–Morrey space of the first kind, is defined as the set of all f ∈ L0 (Rn ) for which the norm kf kWLϕΦ ≡ sup λkχ(λ,∞] (|f |)kLϕΦ is finite. λ>0

n p

p n n (3) When ϕ(t) = t , t > 0, write LpΦ (Rn ) ≡ Lϕ Φ (R ) and WLΦ (R ) ≡ ϕ n WLΦ (R ).

Generalized Orlicz–Morrey spaces

75

Note that (

)

 

|f |

λ>0 :

Φ λ ϕ ≤ 1 M

kf kLϕΦ = inf

(13.2)

1

for all f ∈ L0 (Rn ). Example 38. Let Φ(t) = Lq logr L(t) ≡ tq log max(e, tr ) for t  1, where ϕ n n q ≥ 1 and r ≥ 0. In this case we write Lϕ Lq logr L (R ) instead of LΦ (R ). If ϕ q = 1, we omit q to write LL logr L (Rn ) and if r = 1, we omit r to write n Lϕ Lq log L (R ). Here, we give examples of the function spaces the generalized Orlicz– Morrey spaces can cover. n Example 39. For ϕ(t) ≡ tn , t > 0 and Φ(t) ≡ t2 + t3 , t ≥ 0, Lϕ Φ (R ) is 2 n 3 n isomorphic to the Orlicz space L (R ) ∩ L (R ). n ∞ n Example 40. As a special case where ϕ ≡ 1, we have Lϕ Φ (R ) ≈ L (R ) with ϕ n ∞ equivalence of norms. Indeed, it is not so hard to see that LΦ (R ) ←- L (Rn ). n Then kf kΦ;Q ≤ kf kLϕΦ To see the converse, we take f ∈ Lϕ Φ (R ) arbitrarily.! Z 1 |f (x)| Φ dx ≤ 1. If a sequence for all cubes Q ∈ Q, which implies |Q| Q kf kLϕΦ {Qj }∞ j=1 of cubes shrinks to a Lebesgue point x of Φ ◦ f , then ! ! Z 1 |f (y)| |f (x)| = lim Φ dy ≤ 1. Φ j→∞ |Qj | Q kf kLϕΦ kf kLϕΦ j

The set of all Lebesgue points of Φ ◦ f being almost equal to Rn , we see that n ∞ n Lϕ Φ (R ) ,→ L (R ). Recall that G1 is the class of all increasing functions ϕ such that r ∈ (0, ∞) 7→ ϕ(r)r−n ∈ (0, ∞) is decreasing. As the counterpart to Proposition 22, we prove the following proposition: Proposition 76. Let Φ : [0, ∞) → [0, ∞) be a Young function. Then for n any function ϕ : (0, ∞) → (0, ∞) there exists ϕ∗ ∈ G1 such that Lϕ Φ (R ) ≈ ϕ∗ n LΦ (R ) with equivalence of norms. Proof Let f ∈ L0 (Rn ). By Proposition 22, there exist ϕ∗ ∈ G1 and D > 1 such that D−1 kf kMϕ1 ≤ kf kMϕ∗ ≤ Dkf kMϕ1 . Thus, since Φ is a Young 1 function, ( )

 

|f |

kf kLϕΦ ≤ inf λ > 0 : Φ ≤D λ Mϕ∗ 1 ( )

 

|f |

≤ inf λ > 0 :

Φ Dλ ϕ∗ ≤ 1 M 1

= Dkf k

∗ Lϕ Φ

.

Likewise, we can show that kf kLϕ∗ ≤ Dkf kLϕΦ . Φ

76

Morrey Spaces

Lemma 77. Let Φ : [0, ∞) → [0, ∞) be a bijective Young function, and let ϕ ∈ G1 . Then 1 Φ−1 (ϕ(r)−1 )

≤ kχQ(x,r) kLϕ˜ ≤ Φ

D Φ−1 (ϕ(r)−1 )

((x, r) ∈ Rn+1 + ).

Proof The left inequality follows from the definition, while the right inequality follows from a geometric observation and the fact that ϕ and Φ−1 are doubling. Lemma 78. Let Φ : [0, ∞) → [0, ∞) be a bijective Young function such that ˜ is also bijective, and let ϕ ∈ G1 . Then its conjugate Φ ˜ −1 (ϕ(r)−1 ) kχQ(x,r) kLϕΦ ∼ ϕ(r)Φ

((x, r) ∈ Rn+1 + ),

˜ where Φ(s) = sup(st − Φ(t)), s ≥ 0 denotes the conjugate function. t>0

˜ −1 (r) ≤ 2r. Proof Simply combine Lemma 77 with r ≤ Φ−1 (r)Φ We now prove the embedding relation from generalized Morrey spaces of the first kind to Morrey spaces. Proposition 79. Let ϕ ∈ G1 and Φ : [0, ∞) → [0, ∞) be a Young function e are bijective. For r > 0 set ψ(r) ≡ kχQ(r) kLϕ . Then ψ ∈ G1 such that Φ and Φ Φ ψ n n and Lϕ Φ (R ) ,→ M1 (R ).

Proof From the definition of ψ, ψ is increasing. Let us check that r ∈ (0, ∞) 7→ r−n ψ(r) ∈ [0, ∞) is decreasing, or equivalently, for any s > r, we prove that s−n ψ(s) ≤ r−n ψ(r). To this end, we have only to show that   rn ϕ(`(Q)) |Q ∩ Q(s)|Φ n ≤1 |Q| s ψ(r) for any cube Q. Once we fix `(Q), the left-hand side attains the maximum when Q is centered at 0. Consequently, we have only to show   ϕ(R) rn n n ≤1 min(R , s )Φ Rn sn ψ(r) for any fixed R > 0. Once we fix s and r, the left-hand side attains its maximum at R = s, since ϕ ∈ G1 . Thus, we have only to show   rn ϕ(s)Φ n ≤ 1. s ψ(r)  n  Since s−n ϕ(s) and sn Φ snrψ(r) are both decreasing in s, we have only to
show ϕ(r)Φ ψ(r)−1 ≤ 1. However, this is trivial by Lemma 77. Consequently ψ ∈ G1 .

Generalized Orlicz–Morrey spaces

77

We prove inclusion. Fix a cube Q once again. Then we need to show that ψ(`(Q))mQ (|f |) . kf kLϕΦ . We may assume that the right-hand side equals 1; ϕ(`(Q))mQ (Φ(|f |)) ≤ 1 for any Q. We observe that   ψ(`(Q)) ˜ ψ(`(Q)) χQ . |f |χQ ≤ Φ(|f |) + Φ ϕ(`(Q)) ϕ(`(Q)) for all r > 0. If we integrate this inequality over Q, we obtain   Z ϕ(`(Q)) ˜ ψ(`(Q)) . ψ(`(Q))mQ (|f |) ≤ Φ(|f (x)|)dx + ϕ(`(Q))Φ |Q| ϕ(`(Q)) Q If we use Lemma 78, then we obtain ψ(`(Q))mQ (|f |) . 1. n ∞ n We consider the condition under which Lϕ Φ (R ) is embedded into L (R ).

Corollary 80. Let Φ : [0, ∞) → [0, ∞) be a Young function such that Φ and e are bijective and ϕ ∈ G1 . If inf ϕ(t) > 0, then Lϕ (Rn ) ,→ L∞ (Rn ). Φ Φ t>0

Proof In Proposition 79, simply observe that inf ψ(t) > 0 and that t>0

n Mψ 1 (R )

,→ L∞ (Rn ).

n We provide some examples of functions in Lϕ Φ (R ).

Example 41. Let Φ : [0, ∞) → [0, ∞) be a bijective Young function, and let n 0 ϕ ∈ G1 . Then for 0 < t < r, there exists a function f ∈ Lϕ Φ (R ) ∩ L (4r) with the following properties: kf kLϕΦ . 1,

|supp(f )| ∼

rn ϕ(t) , ϕ(r)

f = Φ−1 (ϕ(t)−1 )χsupp(f ) .

Since 0 < t < Lr and ϕ ∈ G1 , we can choose an integer k, κ ∈ N so that k≤

rn ϕ(t) < k + 1, tn ϕ(r)

κn ≤ k < (κ + 1)n .

r rn ϕ(t) ≤ rn . Thus t ≤ . ϕ(r) κ We set Q0 ≡ Q(4r). We divide equally Q0 into (κ + 1)n cubes Qj , j = 2 1, 2, . . . , (κ + 1)n . Since t ≤ r, we can choose a cube Rj = Q(c(Qj ), t) κ+1 concentric to Qj , so that {Rj } is disjoint. Define Note that k ∼ κn . Since ϕ ∈ G1 , κn tn ≤ ktn ≤

(κ+1)n

f≡

X j=1

Φ−1 (ϕ(t)−1 )χRj .

78

Morrey Spaces

Arithmetic shows |supp(f )| = (κ + 1)n |R1 | = (κ + 1)n tn ∼ ktn ∼

rn ϕ(t) . ϕ(r)

n ϕ It remains to check that f ∈ Lϕ Φ (R ) and that kf kLΦ . 1. We choose a n cube Q with volume R . We distinguish three cases. (κ+1)n

(1) Let 0 < R ≤ t. Then Lemma 77 together with Φ(f ) =

X χRj shows ϕ(t) j=1

that kf k(ϕ,Φ);Q ≤ 1. (2) Assume t < R < r. Then a geometical observation shows that the k number of Rj which intersects Q is less than or comparable to n |Q|. r Thus Z ϕ(R) |Q|ktn ϕ(R) ϕ(R) Φ(|f (x)|)dx .n .n . 1. n |Q| Q r ϕ(t) ϕ(r) (2) Assume R ≥ r. Since ϕ ∈ G1 , we have Z Z ϕ(R) ϕ(4R) Φ(|f (x)|)dx ≤ Φ(|f (x)|)dx . 1. |Q| Q |Q0 | Q0 We end Section 13.1.1 with a theorem showing that Φ reflects local inten ∞ n grability. In order to avoid the case where Lϕ Φ (R ) is embedded into L (R ), we suppose inf ϕ(t) = 0. t>0

Theorem 81. Let Φ, Ψ : [0, ∞) → [0, ∞) be bijective Young functions, and let ϕ ∈ G1 . Assume that inf ϕ(t) = 0 and t>0

Ψ−1 (r) = 0. r→∞ Φ−1 (r) S ψ n n 0 n Then (Lϕ LΨ (R ) 6= ∅. Φ (R ) ∩ Lc (R )) \ lim

(13.3)

ψ∈G1

ψ n p0 n n p1 n In the case where Lϕ Φ (R ) = Mq0 (R ) and LΨ (R ) = Mq1 (R ), condition 1

1

(13.3) reads as q0 < q1 , since Φ−1 (r) = r q0 and Ψ−1 (r) = r q1 for r ≥ 0. Proof Let k ∈ N be fixed for the time being. Keeping in mind that inf ϕ = 0 and that we have (13.3), we take tk ∈ (0, 2−k ) such that

Φ−1 ϕ(tk )−1 ≥ 4k Ψ−1 ϕ(tk )−1 ,
! Φ−1 ϕ(tk )−1 1 so that Ψ ≥ . We use the construction of Example k 4 ϕ(tk ) n 0 n 41 with r = 1 and t = tk to have a function fk ∈ Lϕ Φ (R ) ∩ Lc (R ) with

Generalized Orlicz–Morrey spaces

79

kfk kLϕΦ . 1. Note that each fk is supported on Q(4). Note that 

Z Ψ Q(4)

fk (x) 4k



 dx = Ψ


1 −1 −1 Φ ϕ(t ) |supp(fk )| & 1. k 4k

As a result, kfk kLψ & 4k . Ψ ∞ P Hence, if we set f ≡ 2−k fk , then we obtain the desired function. k=1

13.1.2

Hardy–Littlewood maximal operator in generalized Orlicz–Morrey spaces of the first kind

We discuss the boundedness property of the Hardy–Littlewood maximal operator. We obtain a characterization for the Hardy–Littlewood maximal operator to be bounded. We start with a necessity. Lemma 82. Let Φ, Ψ : [0, ∞) → [0, ∞) be bijective Young functions, and −1 −1 ) let ϕ, ψ ∈ G1 . Assume that sup Φ Ψ(ϕ(r) = ∞. Then the Hardy–Littlewood −1 (r) r>0

ψ n n maximal operator M is unbounded from Lϕ Φ (R ) to LΨ (R ).

Proof Our assumption reads that, for each k ∈ N, there exists rk > 0 such that

Φ−1 ϕ(rk )−1 > kΨ−1 ψ(rk )−1 (13.4) for each k ∈ N. Let r > 0 and x ∈ Rn . Since r 7→ ϕ(r)−1 rn is increasing, we have Z

Φ Φ−1 ϕ(rk )−1 χQ(rk ) (y) dy = ϕ(rk )−1 |Q(x, r) ∩ Q(rk )| Q(x,r)

. ϕ(r)−1 |Q(x, r)|. ∞
n Φ−1 ϕ(rk )−1 χQ(rk ) k=1 is a bounded set in Lϕ Φ (R ). Meanwhile, !
−1 −1 Φ ϕ(rk ) M χQ(rk ) (x) Ψ dx ≥ |Q(rk )| thanks to (13.4) and the k Q(rk )
∞  fact that M χQ(rk ) = χQ(rk ) on Q(rk ). Thus Φ−1 ϕ(rk )−1 M χQ(rk ) k=1 is n not a bounded set in Lψ Ψ (R ). Consequently, M fails to be bounded from ϕ ψ n n LΦ (R ) to LΨ (R ).

Thus Z



Except when sup ϕ(t) < ∞, we can completely characterize the boundt>0

edness of the Hardy–Littlewood maximal operator M in generalized Orlicz– Morrey spaces of the first kind. Theorem 83. Let Φ, Ψ : [0, ∞) → [0, ∞) be bijective Young functions, and let ϕ, ψ ∈ G1 . Assume that sup ϕ(t) = ∞. Then the Hardy–Littlewood maximal t>0

80

Morrey Spaces

ψ n n operator M is bounded from Lϕ Φ (R ) to LΨ (R ) if and only if there exists a constant A > 0 such that

Φ−1 ϕ(r)−1 ≤ AΨ−1 ψ(r)−1 (13.5)

for all r > 0 and that Z

s

Ψ(t) Φ(As) ϕ(r) dt ≤ (13.6) 2 s ψ(r) Ψ−1 (ψ(r)−1 ) t

provided s and r satisfy 2Ψ−1 ψ(r)−1 ≤ s < A−1 sup Φ−1 ϕ(u)−1 . u>0

Proof Let us suppose the sufficiency: We need to prove the boundedness n of M under conditions (13.5) and (13.6). For all cubes Q(x, r) and f ∈ Lϕ Φ (R ), 0 00 0 ϕ we let f1 ≡ χQ(x,5r) f , f2 ≡ f − f1 and λ ≡ (4A + A A )kf kLΦ , where A and A00 are constants appearing in (13.7) and (13.8) below. Then by Z t Ψ(t) = Ψ0 (s)ds 0

and the Layer-Cake-Formula, we have   Z Z ∞ M f1 (x) Ψ dx = Ψ0 (t)|Q(x, r) ∩ {M f1 > λt}|dt. λ Q(x,r) 0 By the change of variables we obtain   Z M f1 (x) dx Ψ λ Q(x,r) Z Ψ−1 (ψ(r)−1 ) = Ψ0 (t)|Q(x, r) ∩ {M f1 ≤ λt}|dt 0 Z ∞ + Ψ0 (t)|Q(x, r) ∩ {M f1 > λt}|dt Ψ−1 (ψ(r)−1 )

Z

Ψ−1 (ψ(r)−1 )

≤

0

Z

∞

Ψ0 (2t)|{M f1 > 2λt}|dt.

Ψ (t)|Q(x, r)|dt + 2 2Ψ−1 (ψ(r)−1 )

0

If we use Fubini’s theorem and the inequality |{M f1 > 2λt}| ≤ |{M [f1 χ{|f1 |>λt} ] > λt}| . then Z



1 λt

Z |f1 (y)|dy, {|f1 |>λt}

 M f1 (x) Ψ dx λ Q(x,r) ! Z ∞ Z rn 1 . + Ψ0 (2t) |f1 (y)|dy dt ψ(r) λt {|f1 |>λt} Ψ−1 (ψ(r)−1 ) !   Z Z |f1λ(y)| rn Ψ0 (2t) |f1 (y)| = + dt |f1 (y)|χ(Ψ−1 (ψ(r)−1 ),∞] dy. ψ(r) λt λ Rn Ψ−1 (ψ(r)−1 )

Generalized Orlicz–Morrey spaces

81

Since Ψ is a Young function, we have   Z M f1 (x) Ψ dx λ Q(x,r) !   Z Z |f1λ(y)| |f1 (y)| rn Ψ(4t) dt |f1 (y)|χ(Ψ−1 (ψ(r)−1 ),∞] . + dy 2 ψ(r) λ Rn Ψ−1 (ψ(r)−1 ) λt !   Z Z 4|f1λ(y)| Ψ(t) |f1 (y)| rn dt |f1 (y)|χ(Ψ−1 (ψ(r)−1 ),∞] + dy. . 2 ψ(r) λ Ψ−1 (ψ(r)−1 ) λt Rn 4|f1 (y)| If we insert (13.6) with s = and the definition of λ into the above λ expression, then we obtain     Z Z M f1 (x) rn ϕ(r) 4A|f1 (y)| rn Ψ dx . + Φ dy . . λ ψ(r) ψ(r) Rn λ ψ(r) Q(x,r) We now need to handle the second term; a geometric observation shows !   Z Z M f2 (x) 1 A0 Ψ dx ≤ |Q(r)|Ψ sup |f (z)|dz (13.7) λ λ R∈Q] (Q(x,r)) |R| R Q(x,r) for some A0 > 1. By (13.1) and H¨older’s inequality for Orlicz spaces, we obtain mR (|f |) . ϕ(`(R))kf kLϕΦ kχR kLϕ˜ ≤ A00 kf kLϕΦ Φ−1 (ϕ(`(R)))

(13.8)

Φ

for some A00 > 1. If we insert (13.5) and (13.8) into (13.7), then we obtain the boundedness of M . We already saw in Lemma 82 that (13.5) is necessary for the boundedness of M . Assume that (13.6) fails. Then, for each k ∈ N, we can find rk , sk > 0 such that Z sk Ψ(t) Φ(ksk ) ϕ(rk ) dt ≥ (13.9) · 2 sk ψ(rk ) Ψ−1 (ψ(rk )−1 ) t

1 A and that 2Ψ−1 ψ(rk )−1 < sk < sup Φ−1 ϕ(u)−1 . Let k > . Assuming k u>0 2 that (13.5) holds, we have
A
1 −1 Φ ϕ(rk )−1 ≤ Ψ−1 ψ(rk )−1 k k
< 2Ψ−1 ψ(rk )−1 < sk
1 < sup Φ−1 ϕ(u)−1 . k u>0

(13.10)

82

Morrey Spaces

Denote by D the doubling constant of ϕ. Then from (13.10) there exists tk ∈ (0, rk ) such that

1 −1 1 Φ ϕ(tk )−1 ≤ sk ≤ Φ−1 ϕ(tk )−1 . Dk k

(13.11)

By using Example 41 we construct a function fk such that; kfk kLϕΦ . 1,

(13.12)

supp(fk ) ⊂ Q(4rk ),

(13.13)

n

rk ϕ(tk ) , ϕ(rk )
ϕ(tk )−1 (x ∈ supp(f )).

|supp(fk )| ∼ fk (x) = Φ−1

(13.14) (13.15)

Let x ∈ Rn \ Q(12rk ). Notice that M fk (x) .




ϕ(rk ) −1 Φ ϕ(tk )−1 . Φ−1 ϕ(rk )−1 . Ψ−1 ψ(rk )−1 , ϕ(tk )

where we used (13.5) for the last
inequality, so that there exists D > 0 such that M fk (x) ≤
DΨ−1 ψ(rk )−1 . Consequently, when k > D and M fk (x) ≥ kΨ−1 ψ(rk )−1 , we have x ∈ Q(12rk ). Thus, we have     Z Z ∞ M fk (x) M fk 0 Ψ > t dt dx ≥ Ψ (t) Q(12rk ) ∩ k k −1 −1 Q(12rk ) Ψ (ψ(rk ) )   Z ∞ M fk 0 = Ψ (t) > t dt. k −1 −1 Ψ

(ψ(rk )

)

By the sunrise lemma (Lemma 153 in the first book), we have 

Z Ψ Q(12rk )

M fk (x) k



Z dx &

Ψ−1 (ψ(rk )−1 )

Z &

∞

sk

Ψ−1 (ψ(rk )−1 )

From (13.15), we have   Z M fk (x) Ψ dx k Q(12rk ) Z sk Z &

Z {|fk |>kt}

Z {|fk |>kt}

! |fk (y)| dy Ψ0 (t)dt kt ! |fk (y)| dy Ψ0 (t)dt. kt

!
1 −1 −1 Φ ϕ(tk ) dy Ψ0 (t)dt Ψ−1 (ψ(rk )−1 ) supp(fk ) kt Z sk
1 −1 Ψ0 (t) −1 = |supp(fk )| × Φ ϕ(tk ) dt. × k t Ψ−1 (ψ(rk )−1 )

Generalized Orlicz–Morrey spaces

83

If we insert (13.14) into the right-hand side of the above estimate and then use (13.9) and (13.11), then   Z Z sk rk n ϕ(tk ) sk Ψ0 (t) M fk (x) dx & dt Ψ k ϕ(rk ) t Ψ−1 (ψ(rk )−1 ) Q(12rk ) rk n ϕ(rk ) ψ(rk ) × Φ(ksk ) × ϕ(tk ) ϕ(rk ) n & ψ(rk )rk . ≥

Since the functions are doubling, kM fk kLψ & k. If we combine (13.12) with Ψ the above estimate, we see that M is unbounded. We give some concrete cases where M is bounded or unbounded using Theorem 83. Example 42. Let p > 1. Write Φ ≡ L log L and Φ0 = Ψ ≡ id[0,∞) . For r > 0, 1 1 rp ϕ(r) = . we let ϕ(r) = ψ0 (r) ≡ r p and ψ(r) ≡ max(1, log r) max(1, log r) ψ n n (1) We claim that M is bounded from Lϕ Φ (R ) to LΨ (R ). We need to check (13.5) and (13.6). It is easy to check Z s 1 Φ(As) ϕ(r) log s + log ψ(r) = dt ≤ (13.16) t s ψ(r) −1 ψ(r)

2 ≤ s < ∞. See Exercise 20. ψ(r) That is, keeping in mind that Ψ = id[0,∞) , we verify
Φ−1 ϕ(r)−1 . ψ(r)−1 (13.17)

with A ≥ 1, provided s and r satisfy

for all r > 0 and that for some large constant A
Condition (13.17) follows from the general formula log ra (log(e + b))b ∼ log r, r ≥ 1 for a > 0, b ∈ R. ψ n n (2) We observe that M fails to be bounded from Lϕ Φ0 (R ) to LΨ (R ) since (13.16) fails. ψ0 n n (3) We observe that M fails to be bounded from Lϕ Φ (R ) to LΨ (R ) since (13.17) fails.

As a corollary of Theorem 83, we characterize the boundedness condition n of the Hardy–Littlewood maximal operator M on Lϕ Φ (R ). It is noteworthy that ϕ does not come into play strongly. Theorem 84. Let Φ : [0, ∞) → [0, ∞) be a Young function, and let ϕ ∈ G1 . n Assume that sup ϕ(t) = ∞. Then M is bounded on Lϕ Φ (R ) if and only if t>0

Φ ∈ ∇2 . Then in particular, M is bounded from LΦ (Rn ) if and only if Φ ∈ ∇2 .

84

Morrey Spaces

Proof If Φ ∈ ∇2 , then we can check (13.5) and (13.6) with ease using n Lemma 49 in the first book. Assume instead that M is bounded on Lϕ Φ (R ). LetZ r > 0. In Theorem 83 we consider the case where Φ = Ψ and ϕ = ψ. r Φ(t) Φ(r) dt . Then using Theorem 83. Thus, by Theorem 40, we have 2 t r 0 Z r Z r Φ(r) Φ(ar) Φ(t) Φ(t) dt . for some ε > 0. Consequently, . dt . 2+ε 1+ε 1+ε 2+ε r (ar) ar t 0 t Φ(r) for any 0 < a  0. This implies that Φ ∈ ∇2 . That is, 2kΦ(kr) ≤ Φ(r) r1+ε for some k > 1. ∞ u ≡ kk{fj } We define the vector-valued norm k{fj }∞ for j=1 kLΦ j=1 k`u kLΦ u (` ) u ∞ 0 n Φ u n ∞ {fj }j=1 ⊂ L (R ). The space Lu (` , R ) is defined as the set of all {fj }j=1 ⊂ u < ∞. We prove the Fefferman–Stein vectorL0 (Rn ) for which k{fj }∞ j=1 kLΦ u (` ) valued inequality for Orlicz–Morrey spaces of the first kind.

Proposition 85. Let Φ ∈ ∆2 ∩ u < ∞, and ∈ G1 . Assume

∇2 , 1 <

let ϕ

ϕ u . {fj }∞ ϕ u for all in addition that ϕ1 ∈ Z0 . Then {M fj }∞ j=1 L (` ) j=1 L (` ) Φ

ϕ u n {fj }∞ j=1 ∈ LΦ (` , R ).

Φ

Proof We take the usual local/global strategy. For the local part, we resort to the Fefferman–Stein vector-valued inequality for LΦ (Rn ) (see Example 71 in the first book) and for the global part we use the assumption ϕ1 ∈ Z0 and argue as before.

13.1.3

Singular integral operators in generalized Orlicz–Morrey spaces of the first kind

We investigate the boundedness of singular integral operators. Note that our integral condition considered in Section 12.2.2 will be described in terms of ψ considered in Proposition 79. We ignore the issue of whether we can n define singular integral operators on the whole space Lϕ Φ (R ), although we can actually do so. Theorem 86. Let Φ ∈ ∆2 ∩ ∇2 , and let ϕ ∈ G1 . Set ψ(r) ≡ kχQ(r) kLϕΦ

(r > 0).

(1) Let T be a singular integral operator and assume n kT f kLϕΦ . kf kLϕΦ for all f ∈ L∞ c (R ). n (2) If kR1 f kWLϕΦ . kf kLϕΦ for all f ∈ L∞ c (R ), then

1 ψ

1 ψ

∈ Z0 . Then

∈ Z0 .

Proof (1) Fix a cube Q. We will use the Lerner–Hyt¨onen formula, Theorem 174 in the first book. From Proposition 85, the Fefferman–Stein vector-valued ∞ P P n inequality on Lϕ ω2−n−2 (T f ; Qjk )χQj . Φ (R ), we can easily handle j=0 k∈Kj

k

Generalized Orlicz–Morrey spaces

85

It remains to treat the median. Since T is weak-L1 bounded and T satisfies the size condition, we have kMed(T f ; Q)χQ k(ϕ,Φ);Q ≤ ψ(`(Q))|Med(T f ; Q)| ∞ X . ψ(`(Q)) m2j Q (|f |)

(13.18)

j=1

by the definition of ψ. By Proposition 79 and the fact that kMed(T f ; Q)χQ k(ϕ,Φ);Q . ψ(`(Q))

∈ Z0 ,

∞ X kf kMψ 1

j=1

. ψ(`(Q))

1 ψ

ψ(2j `(Q))

∞ X j=1

kf kLϕΦ ψ(2j `(Q))

∼ kf kLϕΦ . (2) We let V ≡ {x = (x1 , x2 , . . . , xn ) ∈ Rn : 2x1 > |x|}. Assume that 1 / Z0 , so that thanks to Theorem 37, for any m ∈ N ∩ [3, ∞) there ψ ∈ exists rm > 0 such that ψ(2m rm ) ≤ 2ψ(rm ). Then, consider fm ≡ χV ∩B(2m−1 rm )\B(2rm ) . We use (12.23) and (12.24) and the boundedness 2

n of M on Mψ 2 (R ) to obtain 1

ψ(rm ) . kχB(rm ) kMψ . kM ( 2 ) χV ∩B(rm ) kMψ . kχV ∩B(rm ) kMψ . 1

1

1

By using the inequality log m . |R1 fm (x)| for x ∈ V ∩B(rm ), the embedψ ϕ n n n ding WLϕ Φ (R ) ,→ WM1 (R ) and the boundedness of R1 from LΦ (R ) ϕ n ϕ to WLΦ (R ), we have ψ(rm ) log m . kR1 fm kWMψ . kR1 fm kWLΦ . 1 kfm kLϕΦ . By the translation invariance and the rotation invariance of n m ϕ Lϕ Φ (R ) we have ψ(rm ) log m ≤ kχB(2m rm ) kLΦ . ψ(2 rm ) ∼ ψ(rm ). This implies log m ≤ D where D is independent of m, contradictory to the fact that m ≥ 3 is arbitrary. Thus, ψ1 ∈ Z0 .

13.1.4

Generalized fractional integral operators in generalized Orlicz–Morrey spaces of the first kind

We are oriented to the extension of the Adams inequality for fractional integral operators and fractional maximal operators. Assume Z

k2 r

sup ρ(s) .

r 2 0). s

(13.19)

The function ρ will generalize the parameter α. We are interested in the following condition:

86

Morrey Spaces

Definition 17 (Quasi-submultiplicative). A function κ : [0, ∞) → [0, ∞) is quasi-submultiplicative if κ(st) . κ(s)κ(t). for all s, t ≥ 0. Example 43. Let a ∈ R and 1 < p < ∞. (1) The function κ : t ∈ [0, ∞) 7→ ta ∈ [0, ∞) is quasi-submultiplicative. (2) Let a > 0. Another example of the quasi-submultiplicative function κ is given by κ(t) ≡ ta log(e + t), t ≥ 0. (3) Let a < 0. It should be noted that κ(t) ≡ tp (log(e + t))a , t ≥ 0 is not a quasi-submultiplicative function. See Exercise 19. For θ ∈ M+ (0, ∞), write Z r θ(t) dt, θ∗ (r) ≡ t 0

Z θ∗ (r) ≡ r

∞

θ(t) dt. t

We give a sufficient condition for Iρ to be bounded on generalized Orlicz– Morrey spaces. Theorem 87. Let A > 0, Φ, Ψ, Θ : [0, ∞) → [0, ∞) be bijective Young functions and ϕ, ψ ∈ G1 . Let ρ ∈ M+ (0, ∞) satisfy (13.19). Set  

2 −1 −1 −1 −1 E ≡ (r, s) ∈ (0, ∞) : 2AΘ ψ(r) < s < sup Φ ϕ(u) . u>0

Assume

Φ−1 ϕ(r)−1 ≤ AΘ−1 ψ(r)−1 Z

s

Θ−1 (ψ(r)−1 )

Θ(t) Φ(As) ϕ(r) dt ≤ A t2 s ψ(r)

(r > 0),

(13.20)

((r, s) ∈ E)

(13.21)

and



Θ−1 ϕ(r)−1 ρ∗ (r) + Φ−1 ϕ(·)−1 · ρ ∗ (r) . Ψ−1 ϕ(r)−1

(13.22)

for all r > 0. If a ≡ lim ϕ(r) < ∞, then assume in addition that r ∈ [0, a] 7→ r→∞

n Ψ−1 (r)/Θ−1 (r) is decreasing. Then the operator Iρ is bounded from Lϕ Φ (R ) ψ n to LΨ (R ). n + n ϕ Proof We fix r > 0, x ∈ Rn and f ∈ Lϕ Φ (R ) ∩ M (R ) with kf kLΦ = 1. We distinguish three cases to prove
Ψ D−1 Iρ f (x) ≤ Θ (M f (x)) (13.23)

for some constant D > 0. Once (13.23) is proven, we can use kM f kLψ . 1 for Θ all f ∈ L0 (Rn ) since we are in the position of applying Theorem 83 thanks to (13.20) and (13.21).

Generalized Orlicz–Morrey spaces
• There exists r > 0 satisfying Θ−1 ϕ(r)−1 = M f (x).

87

• For all r > 0, Θ−1 (ϕ(r)−1 ) ≤ M f (x). • For all r > 0, Θ−1 (ϕ(r)−1 ) ≥ M f (x), or equivalently lim ϕ(r)−1 ≥ r→∞

Θ(M f (x)). For k ∈ Z, we write Z ρ(|x − y|) J≡ dy, f (y) |x − y|n B(x,r)

Z Ik ≡

f (y) B(x,2k r)\B(x,2k−1 r)

For the first case, we decompose Iρ f (x) = J +

∞ P

Ik . Since sup ρ(s) .

k=1

Z

k2 r

k1 r

ρ(|x − y|) dy. |x − y|n

r 2 0,
Iρ f (x) . Ψ−1 ϕ(r)−1 ∼ Ψ−1 (Θ (M f (x))) . We move on to the second case. We decompose Iρ f (x) =

(13.25) ∞ P

Ik . Arguing

k=−∞

as before, We obtain ∞ X



Ik . Ψ−1 ϕ(+0)−1 = lim Ψ−1 ϕ(r)−1 . r↓0

k=∞

Since M f (x) & 1 ∼ sup Θ−1 (ϕ(r)−1 ), we obtain (13.23). r>0

The third case can be handled also similarly, since
Ψ−1 ϕ(r)−1 M f (x) Iρ f (x) ≤ lim J ≤ lim . Ψ−1 (Θ (M f (x))) . r→∞ r→∞ Θ−1 (ϕ(r)−1 ) Here for the last inequality we used the assumption that r ∈ [0, a] 7→ Ψ−1 (r)/Θ−1 (r) is decreasing. We concentrate on the case of Θ = Φ, ϕ = ψ in Theorem 87. Theorem 88. Let ρ ∈ M+ (0, ∞) satisfy (13.19). Let Φ, Ψ : [0, ∞) → [0, ∞) be Young functions and ϕ ∈ G1 . Assume in addition Φ ∈ ∇2 . Assume that



Φ−1 ϕ(r)−1 ρ∗ (r) + Φ−1 ϕ(·)−1 · ρ ∗ (r) ≤ AΨ−1 ϕ(r)−1 , ϕ n n then the operator Iρ is bounded from Lϕ Φ (R ) to LΨ (R ).

Proof Condition (13.20) is automatic since Φ = Θ. Since Φ ∈ ∇2 , we have (13.21). Condition (13.22) is our assumption itself. Since Θ−1 = Ψ−1 , we can use Theorem 87 for any ϕ ∈ G1 .

13.1.5

Exercises

Exercise 19. log(e + s) log(e + t) . Consider the case where s = t. log(e + st) s,t≥0

(1) Find sup

(2) Let a < 0 and p ∈ R. Show that κ(t) ≡ tp (log(e + t))a , t ≥ 0, is not a quasi-submultiplicative function. Hint: Compare κ(t2 ) and κ(t)2 . Exercise 20. Prove (13.16) using a + b ≤ (a + 1)(b + 1) for a, b > 0 and ψ −1 (r) ∼ rp (log(e + r))p , r > 0.

Generalized Orlicz–Morrey spaces

13.2

89

Generalized Orlicz–Morrey spaces of the second kind

We have seen “a” definition of generalized Morrey spaces. However, using the Φ-average, we can also define generalized Morrey spaces in “another” way. Section 13.2.1 proposes another definition. The boundedness properties of the Hardy–Littlewood maximal operator, singular integral operators and the generalized fractional integral operators will be investigated in Sections 13.2.2, 13.2.3 and 13.2.4, respectively.

13.2.1

Generalized Orlicz–Morrey spaces of the second kind

Definition 18 (Generalized Orlicz–Morrey spaces of the second kind). Let ϕ : (0, ∞) → [0, ∞) be a function, and let Φ : [0, ∞) → [0, ∞) be a Young function. Also let f ∈ L0 (Rn ). (1) Define kf kMϕΦ ≡ sup ϕ(`(Q))kf kΦ;Q . The generalized Orlicz–Morrey Q∈Q

n space of the second kind Mϕ Φ (R ) is defined as the the set of all f ∈ 0 n L (R ) for which the norm kf kMϕΦ is finite. n (2) Define kf kWMϕΦ ≡ sup λkχ(0,λ) (|f |)kMϕΦ . The function space WMϕ Φ (R ) λ>0

is defined as the weak generalized Orlicz–Morrey space of the second kind as the set of all f ∈ L0 (Rn ) for which the norm kf kWMϕΦ is finite. n

p n n (3) When ϕ(t) = t p , t > 0, write MpΦ (Rn ) = Mϕ Φ (R ) and WMΦ (R ) = ϕ n WMΦ (R ).

Example 44. Let Φ : [0, ∞) → [0, ∞) be a bijective Young function, and n Φ n let 1 ≤ p < ∞. Then MΦ p (R ) is non-trivial, that is, Mp (R ) 6= {0} if and p Φ n only if Φ(t) . t for t ≥ 1. Sufficiency is clear since Mp (R ) ⊃ Mpp (Rn ) = n Lp (Rn ). Meanwhile, if f ∈ MΦ p (R ) \ {0}, then there exists R > 0 such that Φ n χB(R) f ∈ Mp (R ) \ {0}. By dilation, by replacing f by f (R·) we may assume that R = 1. Also, by the monotone convergence theorem and a multiplication by a constant λ, we may assume that χB(1) |f | ≥ λχE with |E| > 0, so that f = χE for some measurable set E ⊂ B(1). Let r > 1. Note that   −1 1 |B(r)| −1 p Λ ≡ |B(r)| Φ solves |E| ! 1 Z 1 |B(r)| p χE (x) Φ dx = 1. |B(r)| B(r) Λ   −1 1 1 |B(r)| |B(r)| −1 p Thus, |B(r)| Φ . 1. As a result, Φ(|B(r)| p ) . for |E| |E| all r > 1. n The class G1 is a natural one for defining Mϕ Φ (R ).

90

Morrey Spaces

Theorem 89. Let Φ : [0, ∞) → [0, ∞) be a Young function. For any function ϕ : [0, ∞) → [0, ∞), the norm, defined by kf kMϕΦ ≡ sup ϕ(`(Q))kf kΦ;Q for Q∈Q

f ∈ L0 (Rn ), is equivalent to kf kMϕΦ∗ for some ϕ∗ ∈ G1 . Proof Let f ∈ L0 (Rn ). If we let ϕ1 (t) ≡ sup ϕ(t0 ) for t > 0, then ϕ1 ∈ t0 ∈(0,t]

M↑ (0, ∞) and kf kMϕΦ ∼ kf kMϕΦ1 . Similarly, if we let ϕ∗ (t) ≡ tn sup ϕ1 (t0 )t0−n t0 ≥t

for t > 0, then ϕ∗ ∈ G1 and kf kMϕΦ1 ∼ kf kMϕΦ∗ . We expand Example 40. ϕ n n Example 45. As a special case where ϕ(t) ≡ 1, Lϕ Φ (R ) ≈ MΦ (R ) ≈ ∞ n L (R ) with equivalence of norms.

Example 46. The class of generalized Orlicz–Morrey spaces of the second kind is a good class of describing intersection spaces. Indeed, we have that ϕ2 1 n n max(ϕ1 , ϕ2 ) ∈ G1 whenever ϕ1 , ϕ2 ∈ G1 and that Mϕ Φ (R ) ∩ MΦ (R ) ≈ max(ϕ1 , ϕ2 ) n (R ) with equivalence of norms. MΦ The next remark gives inclusion of the generalized Orlicz–Morrey spaces and the weak generalized Orlicz–Morrey spaces. Remark 4. Let Φ1 , Φ2 : [0, ∞) → [0, ∞) be Young functions. If there exists a constant C1 > 0 such that Φ1 (r) ≤ Φ2 (C1 r), for all r ≥ 1, then for all f ∈ L0 (Rn ) and cubes Q, kf kΦ1 ,Q ≤ C1 (1 + Φ1 (1))kf kΦ2 ,Q . Likewise for all γ > 0, f ∈ L0 (Rn ) and cubes Q kχ{|f |>γ} kΦ1 ,Q ≤ C1 (1 + Φ1 (1))kχ{|f |>γ} kΦ2 ,Q . Moreover, if ϕ1 and ϕ2 ∈ G1 satisfy ϕ1 (r) ≤ C2 ϕ2 (r) for r > 0, then: 2 n ϕ ϕ (1) For all f ∈ Mϕ Φ2 (R ), kf kMΦ1 ≤ C1 C2 (1 + Φ1 (1))kf kMΦ2 . 1

(2) For all f ∈

2 n WMϕ Φ2 (R ),

kf k

ϕ WMΦ1 1

2

≤ C1 C2 (1 + Φ1 (1))kf kWMϕΦ2 . 2

Hence, if Φ1 (C1 −1 r) ≤ Φ2 (r) ≤ Φ1 (C1 r) and C2 −1 ϕ2 (r) ≤ ϕ1 (r) ≤ C2 ϕ2 (r), ϕ2 ϕ1 ϕ2 1 n n n n then Mϕ Φ1 (R ) = MΦ2 (R ) and WMΦ1 (R ) = WMΦ2 (R ) with equivalent norms, respectively. We collect some useful inequalities. Example 47. Let Φ be a Young function. Let  Φ0 = id[0,∞), Φ1 = Φ and t ϕ0 = ϕ1 = ϕ in Remark 4. Since Φ0 (t) ≤ Φ1 for t ≥ 1, we min(1, Φ1 (1)) n have kf kMϕ1 . kf kMϕΦ . for all f ∈ Mϕ Φ (R ). This relation carries over to the n corresponding weak spaces: kf kWMϕ1 . kf kWMϕΦ for all f ∈ WMϕ Φ (R ).

Generalized Orlicz–Morrey spaces

13.2.2

91

Hardy–Littlewood maximal operator in generalized Orlicz–Morrey spaces of the second kind

We discuss here the boundedness property of M in generalized Orlicz– Morrey spaces of the second kind. The following theorem is an orienting point for us in Section 13.2.2. n Theorem 90. Let Φ ∈ ∇2 and ϕ ∈ G1 . Then M is bounded on Mϕ Φ (R ).

Proof Since the proof is an adaptation of the classical result, we content ourselves with an outline. We take the usual local/global strategy. For the local part, we use Corollary 158 in the first book. For the global part, we combine a geometric observation and Remark 4. Among the class of Orlicz–Morrey spaces of the second kind, of interest is the case where Φ(t) = Lq logr L(t) ≡ tq log max(e, tr ) for t  1, where q ≥ 1 n and r ≥ 0. In this case when r > 0, we use the notation Mϕ Lq logr L (R ) instead n p ϕ n n n p of Mϕ Φ (R ). If in addition ϕ(t) ≡ t , MLq logr L (R ) instead of MΦ (R ). If q = 1, then we drop the power q in the notation. The following proposition shows that generalized Orlicz–Morrey spaces of the second kind also arise naturally. Proposition 91. Let 1 < p < ∞. Then M is bounded from MpL log L (Rn ) to Mp1 (Rn ) and for all f ∈ MpL log L (Rn ). Moreover, kM f kMp1 ∼ kf kMpL log L . Proof Since the proof of the boundedness of M is an adaptation of the classical result (see Chapter 10), we outline the proof. For the local part, use Proposition 156 in the first book. For the global part, we combine a geometric observation and Remark 4 again. We are thus interested in the proof of kM f kMp1 & kf kMpL log L . Let Q be a fixed cube. By Lemma 154 in the first book, we have 1

1

|Q| p kf kL log L;Q . |Q| p kM [f χQ ]k1;Q . kM f kMp1 . Since Q is arbitrary, we obtain the desired result. We can generalize Proposition 91 as follows: Proposition 92. Let ϕ ∈ G1 , and let M j be the j-fold composition of M . Then kf kMϕ j ≈ kM j f kMϕ1 for all f ∈ Mϕ (Rn ). L logj L L log

L

Proof Mimic the proof of Proposition 91. The following is an example of the inclusion relation between Orlicz– Morrey spaces of the first and the second kinds. Example 48. Let p > 1 define n

ψ(t) ≡

tp

1

log max(e, t p )

(t > 0).

92

Morrey Spaces

Also let L log L(t) ≡ t log max(e, t) for t ≥ 0. We claim that L log L is quasisubmultiplicative. A simple calculation shows that, for any a, t > 0, L log L(at) ≤ a log max(e, a) · et log max(e, et) = L log L(a) · L log L(et) . L log L(a)L log L(t). According to Theorem 83 ϕ n n Lψ L log L (R ) ⊂ ML log L (R ).

(13.1)

Theorem 95 below will prove that equality fails in (13.1). ϕ Let 0 < u ≤ ∞. We define the vector-valued norm k{fj }∞ j=1 kMΦ (`u ) ≡ ϕ u n 0 n ϕ for {f }∞ u kk{fj }∞ k (` , R ) is defined as k ⊂ L (R ). The space M j j=1 j=1 ` MΦ Φ ∞ 0 n ∞ ϕ the set of all {fj }j=1 ⊂ L (R ) for which k{fj }j=1 kMΦ (`u ) < ∞.

Example 49. Let Φ ∈ ∆2 ∩ ∇2 , 1 < u < ∞, and let ϕ ∈ G1 ∩ Z0 . 0 n (1) Let {fj }∞ j=1 ⊂ L (R ) and let Q be a cube. If we use the modular inequality of the vector-valued maximal functions, then we have



∞

k{M [χQ fj ]}∞

j=1 k`u Φ;Q . k{fj }j=1 k`u Φ;Q .

(2) If we argue similarly to Proposition 85 using (1) for the local strategy,



ϕ u . {fj }∞ ϕ u for all {fj }∞ ∈ then we have {M fj }∞ j=1 M (` ) j=1 j=1 M (` ) u n Mϕ Φ (` , R ).

13.2.3

Φ

Φ

Singular integral operators in generalized Orlicz–Morrey spaces of the second kind

We extend the boundedness property of singular integral operators to generalized Orlicz–Morrey spaces of the second kind. As it turns out, we still need 1 the condition ∈ Z0 . ϕ Theorem 93. Let Φ ∈ ∆2 ∩ ∇2 , and let ϕ ∈ G1 . (1) Let T be a singular integral operator and n kf kMϕΦ for all f ∈ L∞ c (R ).

1 ϕ

∈ Z0 . Then kT f kMϕΦ .

n (2) If kR1 f kMϕΦ . kf kMϕΦ for all f ∈ L∞ c (R ), then

1 ϕ

∈ Z0 .

n Here we ignore the issue of definability of T f for f ∈ Mϕ Φ (R ).

Generalized Orlicz–Morrey spaces

93

Proof (1) Let Q be a fixed (right-open) cube. We will use the Lerner–Hyt¨onen formula, Theorem 174 in the first book. Since Φ ∈ ∇2 , we are in the position of using Theorem 90 to see that the Hardy–Littlewood maximal n operator is bounded on Mϕ Φ (R ). See also the proof of Theorem 86. Thus we have only to treat the median as before. By Example 47 and ϕ1 ∈ Z0 ϕ(`(Q))kMed(T f ; Q)χQ kΦ;Q . ϕ(`(Q)) . ϕ(`(Q))

∞ X kf kMϕ1 j=1 ∞ X j=1

ϕ(2j `(Q)) kf kMϕΦ ϕ(2j `(Q))

∼ kf kMϕΦ . If we combine this estimate with (13.18), then the estimate for the median is valid. (2) This is similar to Theorem 86. So the proof is omitted.

13.2.4

Generalized fractional integral operators in generalized Orlicz–Morrey spaces of the second kind

We deal with generalized fractional integral operators in generalized Orlicz–Morrey spaces of the second kind. Here, the role of the “parameters” ϕ and Φ are clear. Theorem 94. Let ϕ ∈ G1 , Φ ∈ ∇2 and 0 < a ≤ 1. Set η(t) ≡ ϕ(t)a , t > 0 ˜ ∈ ∇2 and that and Ψ(t) ≡ Φ(t1/a ), t ≥ 0. Suppose that Ψ Z ∞ ρ∗ (t) 1 ρ(s) + ds . (t > 0). ϕ(t) sϕ(s) η(t) t n Then kTρ f kMηΨ . kf kMϕΦ for all f ∈ Mϕ Φ (R ). n + n 0 Proof Let f ∈ Mϕ Φ (R ) ∩ M (R ). Let Q be a fixed dyadic cube. We 0 estimate η(`(Q ))kTρ f kΨ;Q0 . We decompose X Tρ f (x) . χR (x)ρ(`(R))mLR (f ) R∈D

for some L ∈ N. We note that ∞ X ρ(`(Q0 )) l=1

ϕ(`(Q0 ))

Z

∞

. `(Q0 )

ρ(s) 1 ds . . sϕ(s) η(`(Q0 ))

Thus, we have only to show that

X



η(`(Q0 )) χ ρ(`(R))m (f ) R LR

R∈D(Q0 )

Ψ;Q0

. kf kMϕΦ .

94

Morrey Spaces

Keeping in mind Theorem 53 in the first book, choose g ∈ LΨ (Rn ) ∩ M+ (Rn ) so that kgkΨ;Q0 = 1 and that



X Z X ρ(`(R))



≤ 2 χ ρ(`(R))m (f ) m (f ) g(x)dx R LR LR

|Q0 | R

0

R∈D(Q0 ) R∈D(Q0 ) Ψ;Q Z X ρ(`(R)) . m (M f ) g(x)dx. R |Q0 | R 0 R∈D(Q )

We know that there exists a sparse family {Qjk }j∈N0 ,k∈Kj ⊂ D(Q0 ) with a level structure such that 2j mQ0 (M f ) ∼ mQj (M f ),

|Q0 | ≤ 2|E 0 |,

k

|Qjk | ≤ 2|Ekj |,

where K0 is a singleton, Q0k = Q0

[

Ekj ≡ Qjk \

(k ∈ K0 ),

Qj+1 k0

(j ∈ N0 , k ∈ Kj )

k0 ∈Kj+1

Since any cube Q ∈ D(Q0 ) is contained in Qjk , j ∈ N0 , k ∈ Kj , with the property that mQj (M f ) ∼ 2j mQ0 (M f ), we observe k Z X ρ(`(R)) m (M f ) g(x)dx R |Q0 | R j R∈D(Qk ),2j mQ0 (M f )≤mR (M f ) 0. n

n

ϕ(tn )−1 ∼ kχ[0,t]n kMϕΦ ∼ kχ[0,t]n kL2 ∩L3 ∼ max(t 2 , t 3 ).

(13.3)

We now claim that, for any j ∈ N, ϕ(κ−nj )−1 kχFj kΦ;[0,κ−j ]n ∼ kχFj kMϕΦ

(13.4)

kχEj kΦ;[0,1]n ∼ kχEj kMϕΦ .

(13.5)

and Indeed, by symmetry of the set Ej and the doubling property of ϕ, we see that max ϕ(κ−nk )−1 kχFj kΦ;[0,κ−k ]n ∼ kχFj kMϕΦ , and hence we can choose k=1,2,...,j

kj ≤ j so that kχFj kMϕΦ ∼ ϕ(κ−nkj )−1 kχFj kΦ;[0,κ−kj ]n . A simple geometric observation shows that ϕ(κ−nkj )−1 kχFj kΦ;[0,κ−kj ]n ≤ ϕ(κ−nkj )−1 kχFkj kΦ;[0,κ−kj ]n ≤ kχFkj kMϕΦ .

Generalized Orlicz–Morrey spaces

97

This yields, together with (13.2), kχFj kL2 ∩L3 . kχFkj kL2 ∩L3 . Thus, we must have jk ∼ j and we have verified (13.4). Likewise, (13.5) is achieved. It follows from (13.2), (13.3) and (13.4) that kχFj kΦ;[0,κ−j ]n ∼ ϕ(κ−nj )kχFj kL2 ∩L3 ∼ κnj/2 .

(13.6)

Likewise, it follows from (13.5) that kχEj kΦ;[0,1]n ∼ kχEj kMϕΦ ∼ kχEj kL2 ∩L3 ∼ κjn/3 .

(13.7)

Two sided inequalities (13.1), (13.6) and (13.7) contradict because 0 < κ < 1. Example 50. Let Φ(t) ≡ t2 + t3 . Let us verify n LΦ (Rn ) = L2 (Rn ) ∩ L3 (Rn ) ⊂ L3 (Rn ) ⊂ Mϕ Φ (R ),

(13.8)

when ϕ satisfies ϕ(t)2 + ϕ(t)3 = tn for t > 0. In fact, if `(Q) ≥ 1 and f ∈ L3 (Rn ), then n (3) ϕ(`(Q))kf kΦ;Q ∼ `(Q) 3 mQ (f ) ≤ kf kL3 and if `(Q) ≤ 1, then n

n

(3)

(3)

ϕ(`(Q))kf kΦ;Q ∼ `(Q) 2 mQ (f ) ≤ `(Q) 3 mQ (f ) ≤ kf kL3 .

13.3.2

The space MpL log L (Rn )

n We will discuss the special case of Mϕ Φ (R ). Recall that L log L(t) ≡ n t log max(e, t). Let ϕ(t) = t p for t > 0 and Φ(t) = L log L(t) for t ≥ 0. Then n j write MpL log L (Rn ) ≡ Mϕ Φ (R ). For j = 1, 2, . . ., let Φ(t) ≡ t(log max(e, t)) , and let M j be the j-fold composition of M . According to Lemma 155 in the (Φ) first book, we have the equivalence of norms on any cube Q ⊂ Rn ; mQ (f ) ∼ j mQ (MQ f ), where the implicit constant depends on Q and MQ g = χQ · M g for g ∈ L0 (Q). However, this equivalence fails on Rn . By the use of the generalized n Orlicz–Morrey space Mϕ Φ (R ), we can generalize this situation.

Theorem 96. Let p > 1. There is no pair of functions ϕ ∈ G1 and bijective p n n Φ ∈ Y such that Lϕ Φ (R ) is isomorphic to ML log L (R ). Proof Assume to the contrary that there exists such a pair. That is, we suppose that kf kMpL log L ∼ kf kLϕΦ (13.9) for all f ∈ L0 (Rn ). Without loss of generality we may assume that ϕ(1) = 1. n 1 Then, we must have t p = kχ[0,t]n kMpL log L ∼ kχ[0,t]n kLϕΦ ∼ −1 for all Φ (ϕ(t)−1 ) n n 1 . Let ψ(t) ≡ t p , t > 0 and let Ψ(t) ≡ t, t ≥ 0. t > 0, and hence Φ(t− p ) ∼ ϕ(t) By Proposition 92 we also have kf kLϕΦ ∼ kM f kMp1 for all f ∈ MpL log L (Rn ), p ψ n n n so that M is bounded from Lϕ Φ (R ) to M1 (R ) = LΨ (R ). Thus, we have n

n

n

Φ(t− p )t p s log(t p s) . Φ(s)

(13.10)

98

Morrey Spaces n

n

as long as t, s > 0 satisfy t p s > 1. Set a ≡ t p s. Then (13.10) reads as Φ(a−1 s)a log a . Φ(s) for all s > 0 and a > 1. Set r ≡ a−1 s. Then Φ(r)a log a . Φ(ar)

(13.11)

for any a > 1 and r > 0. Let t > 1. Letting r = 1 in (13.11),√we have first L log L(a) . Φ(a) for a > √ 1. Using this inequality with a = t, and using (13.11) again with r = a = t, we conclude that √ √ √ √ √ L log2 L(t) ∼ log t × t × t log(e + t) ∼ log t × t × Φ( t) . Φ(t). These observations allow us to assume that, for any t > 1, L log2 L(t) ≤ Φ(Kt).

(13.12)

Let us again use the characteristic function of the Cantor set. Let 0 < κ < 1 be the solution to the equation 1

κ p = 2κ.

(13.13)

n Define inductively a sequence of the collection {Ej }∞ j=0 of subsets [0, 1] by [ E0 ≡ [0, 1]n , Ej ≡ {(1 − κ)~e + κEj−1 }. ~ e∈{0,1}n

Arithmetic shows kχEj kMpL log L ∼ kχEj kL log L;[0,1]n . This gives us that kχEj kLϕΦ kχEj kMpL log L

&

kχEj k(ϕ,Φ);[0,1]n . kχEj kL log L;[0,1]n

(13.14)

If we can prove lim

j→∞

kχEj k(ϕ,Φ);[0,1]n = ∞, kχEj kL log L;[0,1]n

(13.15)

then (13.14) contradicts (13.9) and the proof of the theorem will be finished. Therefore, we need only verify (13.15). 1 , since ϕ(1) = 1. ArithWe obtain kχEj k(ϕ,Φ);[0,1]n = −1 Φ (κ−nj/p ) 1 metic shows kχEj kL log L;[0,1]n = . For t > 1 let θ(t) ≡ (L log L)−1 (κ−nj/p ) −1 Φ (t) . Then θ(t) → 0 as t → ∞ thanks to (13.12). Consequently, we (L log L)−1 (t) obtain (13.15): kχEj k(ϕ,Φ);[0,1]n 1 = lim = ∞. −nj/p j→∞ kχEj kL log L;[0,1]n j→∞ θ(κ ) lim

This completes the proof of the theorem.

Generalized Orlicz–Morrey spaces

13.3.3

99

Exercises

√ √ Exercise 23. Let Φ(t)√ ≡ t2√+ t3 for t ≥ 0. Show that Φ−1 (t) ∼ min( t, 3 t). Hint: Calculate Φ(min( t, 3 t)) for t ≥ 0. Exercise 24. Let Φ(t) ≡ t2 + t3 . Prove that Φ is quasi-submultiplicative.

13.4

Notes

Section 13.1 General remarks and textbooks in Section 13.1 According to our best knowledge, the generalized Orlicz–Morrey space n (R ), defined initially by Nakai [342], is more investigated than the generLϕ Φ n alized Orlicz–Morrey space Mϕ Φ (R ), defined initially by Sawano, Sugano and Tanaka [417]. See [84] for another class of generalized Orlicz–Morrey spaces. Our definition of Orlicz–Morrey spaces differs from that of Kokilashvil and Krbec [235, p. 2]. There are many papers dealing with generalized Orlicz–Morrey spaces of the first kind. Sawano, Sobukawa and Tanaka considered Orlicz–Morrey spaces and obtained some extrapolation result in [414], which is a natural extension of [4, Theorem 1]. Proposition 79 is from [344, Corollary 4.7]. Section 13.1.1 Generalized Orlicz–Morrey spaces of the first kind date back to 2004; see the paper [342] by Nakai. We followed [407] for Proposition 79. Section 13.1.2 n The space Lϕ Φ (R ) is investigated in [262, 286, 293, 294, 314, 315, 345]. Nakai obtained the characterization of the Hardy–Littlewood maximal operator in [344, Theorem 5.1]. See Theorem 81 for [344, Theorem 4.9]. Theorem 83 is [344, Theorem 5.1]. Theorem 84 is [344, Corollary 5.3].

Section 13.1.3 Nakai studied singular integral operators acting on generalized Orlicz– Morrey spaces of the first kind. Theorem 86 and Theorem 93 are [344, Theorem 5.1]. See also [188, Theorem 8]. We followed [407] in this section.

100

Morrey Spaces

Section 13.1.4 Nakai investigated the boundedness property of fractional integral operators acting on generalized Orlicz–Morrey spaces of the first kind; see [344, Theorem 7.1] and [344, Theorem 7.3] for Theorems 87 and Theorem 88.

Section 13.2 General remarks and textbooks in Section 13.2 Generalized Orlicz–Morrey spaces of the second kind stem from the note in 2012 by Sawano, Sugano and Tanaka [417]. Section 13.2.1 Example 45, showing that generalized Orlicz–Morrey spaces coincide with L∞ (Rn ) when ϕ ≡ 1, can be found in [417, Remark 2.4(4)]. See [501, Definition 15] for a passage of generalized Orlicz–Morrey spaces of the second kind to the weighted setting. Section 13.2.2 n The space Mϕ Φ (R ) is investigated in [120, 121, 223, 417]. Theorem 90 is [417, Corollary 2.21].

Section 13.2.3 See [501, Definition 15] for a passage of the boundedness of singular integral operators in generalized Orlicz–Morrey spaces of the second kind to the weighted setting including the weak counterpart and the analogue of commutators. We refer to [374] for more about singular integral operators acting on Orlicz–Morrey spaces of the second kind. See [374, Theorem 2] for the boundedness of commutators generated by BMO and singular integral operators in Orlicz–Morrey spaces of the second kind. Section 13.2.4 Sawano, Sugano and Tanaka investigated the boundedness of Tρ in [417, Corollary 2.11]; see Theorem 94. We refer to [374, Theorem 4] for the boundedness of commutators generated by BMO and fractional integral operators in Orlicz–Morrey spaces of the second kind. In particular, for Φ(t) = tq and a = qs where 1 < q ≤ s < ∞, we have the boundedness of Mρ on generalized Morrey spaces in [415, Lemma 2.6]. See [149, Theorem] for the boundedness of Iα acting on generalized Morrey spaces of the second kind, where ϕ depends on x as well.

Generalized Orlicz–Morrey spaces

101

Section 13.3 General remarks and textbooks in Section 13.3 See [122] for this section. Section 13.3.1 We demonstrated that L2 (Rn ) ∩ L3 (Rn ) does not fall under the scope of Orlicz–Morrey spaces of the second kind. Theorem 95 is [122, Theorem 1.4]. Using Orlicz–Morrey spaces of the second kind, we can describe the action of the iterated maximal operator M j , whose range is Mp1 (Rn ). Proposition 92 is [417, Lemma 3.5]. Section 13.3.2 n We considered an example of functions in Lϕ Φ (R ). We recorded Example 41 based on [344, Lemma 4.8]. We constructed an example of Φ, Ψ, ϕ, ψ for ψ n n which M is bounded from Lϕ Φ (R ) to LΨ (R ): Example 42 comes from [344, p Example 5.1]. As we saw in Theorem 96, ML log L (Rn ) is never realized as any Orlicz–Morrey space of the first kind; see [122, Theorem 1.6].

Chapter 14 Morrey spaces over metric measure spaces

We worked in Rn in previous chapters. However, from Morrey’s seminal work [328] in 1938, Morrey spaces should be considered over open sets (domains). Section 14.1 considers Morrey spaces over open sets. This includes Sobolev– Morrey spaces. As seen in this book, Morrey spaces are useful when considering the potential operators. Section 14.4 works on Rn where the Lebesgue measure is replaced by general Radon measures. We will see in Section 14.4.3 that a modification of the definition is necessary for general Radon measures. Surprisingly enough, the same idea works for Morrey spaces when we consider Radon measures on Rn . Section 14.5 shows the impact of metric measure spaces on this generalization.

14.1

Morrey spaces on domains

Section 14.1 is made up of two sections. We work in Rn in Section 14.1.1 as the special case of Section 14.1.2. Section 14.1.2 develops the theory of Morrey spaces over domains.

14.1.1

Morrey spaces over the half space Rn+

We work on the upper half space Rn+ = Rn−1 × (0, ∞). Accordingly the definition of Morrey spaces undergoes the following change: Definition 19 (Mpq (Rn )). Let 0 < q ≤ p < ∞. For an Lqloc (Rn+ )-function f its Morrey norm is defined by kf kMpq (Rn+ ) ≡

sup x∈Rn + ,r>0

|B(x, r)|

1 1 p−q

Z

! q1 q

|f (y)| dy

.

B(x,r)∩Rn +

Many operators we have considered in this book can act on Morrey spaces over the half space. For example, the Hardy–Littlewood maximal operator can act on Morrey spaces over the half space. In fact, if we extend any function 103

104

Morrey Spaces

Morrey spaces over the half space by zero outside the half space, M can act on the function. If we restrict the image to the half space, we obtain the definition of M acting on Morrey spaces over the half space. However, we need to take some care of the sharp-maximal operator since we must take seriously into account the fact we are working on the half space. Definition 20. For a function f ∈ L0 (Rn+ ), define Z χQ (x) ] |f (y) − mQ (F )|dy MRn f (x) ≡ sup + Q∈Q |Q| Q∩Rn + where

( f (x) F (x) ≡ 0

(x ∈ Rn ),

(x ∈ Rn+ ), (x ∈ Rn \ Rn+ ).

The operator MR] n is called the sharp maximal operator in the half space. +

Write x ˜ ≡ (x1 , . . . , xn−1 , −xn ) for x = (x1 , . . . , xn−1 , xn ). In this definition, it matters that we can control M ] F (˜ x) by MR] n f (x) for any x ∈ Rn+ . +

n

Lemma 97. For x ∈ R , denote by x ˜ the reflection of x with respect to the hyperplane xn = 0. (1) Let f ∈ L1loc (Rn ). Then M ] f (x) ∼ sup χQ (x)mQ (|f − m5Q (f )|) for any Q∈Q

x ∈ Rn+ . (2) Let f ∈ L1loc (Rn+ ). Define F as the zero extension of f as above. Then x). for any x ∈ Rn+ , MR] n f (x) & M ] F (˜ +

Proof (1) One inequality is clear since χ5Q ≥ χQ and m5Q (g) ≥ 5−n mQ (g) for all g ∈ M+ (Rn ). We show the opposite inequality. Let Q be a cube containing x ∈ Rn . Then using f (y)−mQ (f ) = f (y)−m5Q (f )−mQ (f − m5Q (f )) for all y ∈ Rn , we obtain mQ (|f −mQ (f )|) ≤ 2mQ (|f −m5Q (f )|) by the triangle inequality. Thus, we have the equivalence. Z 1 (2) It suffices to show that |F (y) − m5Q (F )|dy . M ] f (x) for any |Q| Q∩Rn+ x ∈ Rn+ , if a cube Q contains x ˜. We may assume that Q intersects Rn+ ; otherwise the left-hand side is zero. In this case, 3Q contains x. Thus, Z Z 1 1 |F (y)−m5Q (F )|dy . |F (y)−m5Q (F )|dy ≤ M ] f (x). |Q| Q∩Rn+ |5Q| 5Q We will extend the sharp maximal inequality obtained for Morrey spaces over the whole space to Morrey spaces over the half space.

Morrey spaces over metric measure spaces

105

Theorem 98. Let 1 < q ≤ p < ∞. Then kf kMpq (Rn+ ) . kMR] n f kMpq (Rn+ ) for + any f ∈ Mpq (Rn+ ). Proof We extend f by zero outside Rn+ to obtain a function F as before. Then kf kMpq (Rn+ ) = kF kMpq . kM ] F kMpq . kMR] n f kMpq (Rn+ ) thanks to the + sharp maximal inequality for Morrey spaces in the first book and Lemma 97(2). As in Example 40(2) in the Z first book, the reflected singular integral operf (y) ˜ ˜ ator K is given by Kf (x) = dy for x ∈ Rn+ . Although there is no n |˜ x − y| Rn + cancellation, we can still handle this operator. ˜ kMp (Rn ) . kf kMp (Rn ) for all Theorem 99. Let 1 < q ≤ p < ∞. Then kKf q q + + f ∈ M+ (Rn+ ). Proof Let x0 ∈ Rn+ and r > 0 be fixed. It suffices to show that Z nq ˜ (x)|q dx . kf kMp (Rn ) q r p −n |Kf q + Q(x0 ,r)∩Rn +

with the implicit constant independent of x0 and r. We decompose Z ˜ (x)|q dx |Kf

(14.1)

Q(x0 ,r)∩Rn +

Z q f (y) = dy dx n n n |˜ x − y| R+ Q(x0 ,r)∩R+ !q Z Z |f (y)| ≤ dy dx |˜ x − y|n Rn Q(f x0 ,4r)∩Rn + + Z ∞ Z X + Z

Q(x0 ,r)∩Rn +

k=2

(Q(f x0 ,2k+1 r)∩Rn x0 ,2k r) + )\Q(f

|f (y)| dy |˜ x − y|n

!q dx.

˜ we have By the Lq (Rn )-boundedness of K, )q Z (Z Z |f (y)| dy dx . |f (x)|q dx. n n |˜ n x − y| Rn Q(f x ,4r)∩R Q(f x ,r)∩R 0 0 + + + Thus, the estimate of the first term is valid. We handle the second term. We have, for k ≥ 2, Z Z 1 |f (y)| dy . k n |f (y)|dy x − y|n (2 r) Q(f (Q(f x0 ,2k+1 r)∩Rn x0 ,2k+1 r) x0 ,2k r) |˜ + )\Q(f n

. (2k r)− p kf kMpq (Rn+ ) .

106

Morrey Spaces n

n

Putting these observations, we obtain r p − q kKf kLq (Q(x0 ,r)∩Rn+ ) . kf kMpq (Rn+ ) . Thus, we complete the proof. ˜ ·] generated by a ∈ BMO(Rn+ ) and the We move on to the commutator C[a, integral operator whose kernel is |˜ x−y|−n . Let a ∈ BMO(Rn+ ) and f ∈ L0 (Rn+ ). For x ∈ Rn+ , define Z |a(x) − a(y)| ˜ f ](x) ≡ f (y)dy C[a, n |˜ x − y|n R+ as long as the integral makes sense. We will resort to the sharp maximal inequality. To this end, we need the following pointwise estimate: Lemma 100. Let 1 < q ≤ p < ∞ and 1 < r < ∞. Also let a ∈ ˜ f ]](x) . BMO(Rn+ ) and f ∈ Mpq (Rn+ ). Then for almost all x ∈ Rn+ , M ] [C[a,   (r) ˜ (r) kak∗ M [K(|f |)](x) + M f (x) . Proof Fix a cube Q(x0 , δ) containing x. We write Z |a(z) − a(y)| ˜ f ](z) − A(z) f (y)dy, B(z) ≡ C[a, A(z) ≡ |˜ z − y|n Q(x0 ,2δ) for z ∈ Rn . Using the H¨older inequality, the equality a(z) − a(y) = a(z) − mQ(x0 ,δ) (a) + mQ(x0 ,δ) (a) − a(y) and the boundedness of K (see again Example 40(2) in the first book), we obtain mQ(x0 ,δ) (A) .  ˜ |)](x) + M (r) f (x) . Let kak∗ M (r) [K(|f Z B(z) ≡ Rn \Q(x0 ,2δ)

|a(z) − a(y)| f (y)dy. |f x0 − y|n

By the use of the mean-value inequality and H¨older’s inequality, we have Z |a(x) − a(y)| mQ(x0 ,δ) (|B − B|) . δ f (y)dy x0 − y|n+1 n R \Q(x0 ,2δ) |f   ˜ |)](x) + M (r) f (x) . kak∗ M (r) [K(|f and   ˜ |)](x) + M (r) f (x) . mQ(x0 ,δ) (|B − mQ(x0 ,δ) (B)|) . kak∗ M (r) [K(|f As a result,   ˜ f ] − mQ(x ,δ) (B)|) . kak∗ M (r) [K(|f ˜ |)](x) + M (r) f (x) mQ(x0 ,δ) (|C[a, 0 for all (x0 , δ) ∈ Rn+1 as long as x ∈ Q(x0 , δ). Thus, the proof is complete. + We consider the commutator generated by reflection.

Morrey spaces over metric measure spaces

107

Theorem 101. Let 1 < q ≤ p < ∞ and a ∈ BMO(Rn+ ). Then the commutator defined above is bounded from Mpq (Rn+ ) into itself. Precisely, we have ˜ f ]kMp (Rn ) . kak∗ kf kMp (Rn ) for f ∈ Mpq (Rn+ ). kC[a, q q + + Proof Let x0 ∈ Rn+ , ρ > 0, f ∈ Mpq (Rn ), 1 < r < q < ∞. First, we may assume f ∈ M+ (Rn ). By the truncation argument we may assume f ∈ n ∞ n p n ˜ L∞ c (R ) and a ∈ L (R ). Thus, C[a, f ] ∈ L (R ). So we can simply use Theorem 98 and Lemma 100.

14.1.2

Morrey spaces over domains

Since many partial differential equations are considered over domains, it is meaningful to consider Morrey spaces over open sets Ω. We write Ω(x, r) = Ω ∩ B(x, r) for x ∈ Rn and r > 0 as before. Definition 21 (Mpq (Ω)). Let 0 < q ≤ p ≤ ∞. For an Lqloc (Ω)-function f its Morrey norm is defined by ! q1 Z kf kMpq (Ω) ≡

1

1

|B(x, r)| p − q

sup x∈Ω, 0 0 satisfy Ω ⊂ B(R). Then we have ! q1 Z 1

1

kf kMpq 1 (Ω) ≤ |B(2R)| p1 − p2 1

1

1

|B(x, r)| p2 − q

sup x∈Ω, 00

rey space Mpq (Ω) over the domain Ω is the set of all weak f ∈ L0 (Ω) for which the norm kf kWMpq is finite.

108

Morrey Spaces

The embedding WMpr (Rn ) ,→ Mpq (Rn ) readily carries over to the function spaces over domains. Lemma 103. Let Ω be an open set. Also let 0 < q < r ≤ p < ∞. Then WMpr (Ω) ,→ Mpq (Ω). Proof Simply reexamine the proof of WMpr (Rn ) ,→ Mpq (Rn ); see Exercise 27. Definition 23. Let 0 < q ≤ p < ∞. One defines the Stummel modulus ! q1 Z ηf (r) ≡

sup x∈Ω,0 M |y 0 |}. Set ρN (x) ≡ N n ρ(N x) for x ∈ Rn . Let f ∈ W l Mpq (Ω). Then ρN ∗ f (x) makes sense since x − y ∈ Ω if x ∈ Ω and y ∈ U . Furthermore, the weak derivative of ρN ∗ f is given by ∂ α [ρN ∗f ] = ρN ∗∂ α f . Thus ρN ∗f ∈ Mpq (Ω)∩C ∞ (Ω) with ∂ α [ρN ∗f ] ∈ L∞ (Ω) for any α ∈ Nn0 . Let h ∈ Cc∞ (Rn ) be arbitrary. Note that there exists a compact set Kh ⊂ Ω such that T f = 0 on supp(h) if f vanishes almost everywhere Kh . Let Θh ∈ Cc∞ (Rn ) be such that Θh = 1 on a neighborhood of Kh . Then we have Z Z α T f (x)∂ h(x)dx = T [Θh · f ](x)∂ α h(x)dx. Rn

Rn

Since Θh · f ∈ L1 (Ω), lim ρN ∗ (Θh · f ) = Θh · f in L1 (Ω). Recall that T is N →∞

bounded on L1 (Ω) thanks to Theorem 107. So we have Z Z T f (x)∂ α h(x)dx = lim T [ρN ∗ (Θh · f )](x)∂ α h(x)dx. Rn

N →∞

Rn

112

Morrey Spaces

Since ρN ∗(Θj ·f ) ∈ C ∞ (Ω) with all the partial derivatives bounded, we obtain T [ρN ∗ (Θh · f )] ∈ C l (Ω) and the partial derivatives up to order l are bounded. So we have Z Z T f (x)∂ α h(x)dx = (−1)|α| lim ∂ α (T [ρN ∗ (Θh · f )](x)) h(x)dx. N →∞

Rn

Rn

Since ρN ∗ (Θh · f ) is smooth and ∂ α h vanishes on Kh as long as α 6= 0, ∂ α (T [ρN ∗ (Θh · f )]) = T [ρN ∗ (Θh ∂ α f )]. Thus, Z Z α |α| T f (x)∂ h(x)dx = (−1) lim T [ρN ∗ (Θh ∂ α f )](x)h(x)dx. N →∞

Rn

Rn

Finally since lim ρN ∗ (Θh ∂ α f ) = Θh ∂ α f in L1 (Ω), it follows that N →∞

Z

α

T f (x)∂ h(x)dx = (−1)

|α|

Rn

Z

T [Θh ∂ α f ](x)h(x)dx

Rn

= (−1)|α|

Z

T [∂ α f ](x)h(x)dx.

Rn

Thus, ∂ α T f = T [∂ α f ], proving the claim. Thus, the theorem is proven. By the partition of unity argument, we conclude that any function f ∈ W l Mpq (Ω) extends linearly and boundedly to a function F ∈ W l Mpq (Ω). Theorem 109. Let l ∈ N0 , 1 ≤ q ≤ p < ∞. Also let Ω be a bounded Lipschitz domain. Then there exists a bounded linear mapping E : W l Mpq (Ω) → W l Mpq (Rn ) such that Ef |Ω = f for all f ∈ W l Mpq (Ω).

Exercises Exercise 28. Let 1 ≤ q ≤ p < ∞. Also let ϕ : Rn−1 → R be a Lipschitz function. For all f ∈ W 3 Mpq (Ωϕ ), using Theorem 108, show that kDf kMpq (Ωϕ ) . kD3 f kMpq (Ωϕ ) + kf kMpq (Ωϕ ) , where the implicit constant depends only on kϕkLip . Exercise 29. Show that the domain of the form Ω ≡ {(x0 , xn ) ∈ Rn : xn > ϕ(x0 )}, where ϕ ∈ Lip(Rn ), is of type (A).

14.3

Morrey–Campanato spaces on domains

We study Morrey–Campanato spaces on domains. We consider some fundamental properties in Section 14.3.1 and relation between H¨older–Zygmund spaces in Section 14.3.2.

Morrey spaces over metric measure spaces

14.3.1

113

Morrey–Campanato spaces on domains

Denote by diam(Ω) the diameter of a bounded open set Ω; diam(Ω) ≡ sup |x − y|. Here, we work on an open set Ω ⊂ Rn such that there exists Z ∈ x,y∈Ω

(0, 1) such that |Ω(x0 , r)| ≥ Z|B(x0 , r)| for all x0 ∈ Ω and r ∈ (0, diam(Ω)). Such a domain is defined to be of type (A); see Definition 25. We use the usual abbreviation: Ω(x, ρ) ≡ Ω ∩ B(x, ρ) for x ∈ Rn and ρ > 0. We cannot handle open sets that are too singular since we will need to consider difference of functions. We consider the following condition which most of important domains satisfy. Definition 25. An open set Ω ⊆ Rn is of type (A) if there exists a constant Z > 0 such that |Ω(x, r)| ≥ Z|B(x, r)| for every r ∈ (0, diam(Ω)) and for every x ∈ Ω. Example 53. Work in R2 with coordinates x and y. (1) The domain Ω1 ≡ {(x, y) ∈ R2 : x2 + y 2 < 1} is of type (A). √ (2) The domain Ω2 ≡ {(x, y) ∈ R2 : 0 < x < 1, x < y < 1} is not of type (A). Recall that the space Pk (Rn ) denotes the set of all polynomials of order k ∈ N0 and that P−1 (Rn ) is the zero linear subspace. Definition 26 (Lλ,q k (Ω)). Let Ω be a bounded domain of type (A). Let λ ∈ R, 1 ≤ q < ∞ and k ∈ N0 ∪ {−1}. Then define the (Morrey–)Campanato space q Lλ,q k (Ω) as the set of all f ∈ Lloc (Ω) for which the quantity   1 inf n ρλ |B(ρ)|− q kf − P kLq (Ω(x0 ,ρ)) kf kLλ,q (Ω) ≡ sup k

x0 ∈Ω 0 0. Decompose −λ = m + a, where m ∈ N0 and 0 < a ≤ 1. Then the function vα is H¨ older continuous of order a for all multi-indexes α with |α| = m. Proof This follows from Lemmas 111 and 112. We show that the functions in Lλ,q older continuous together with k (Ω) is H¨ derivative up to some order when λ < 0 and λ + k > 0. Theorem 117. Let k ∈ N0 ∪{−1}. Assume that λ ∈ (−∞, 0) satisfy λ+k > 0. Decompose −λ = m + a, where m ∈ N0 and 0 < a ≤ 1. If f ∈ Lλ,q k (Ω), then f ∈ C m (Ω) and ∂ α f is H¨ older continuous. Proof It suffices to show that f (x0 ) = lim a0 (x0 ; 2−m ρ, f ) = v0 (x0 ) for m→∞

(q)

any ρ ∈ (0, diam(Ω)) and for any x0 satisfying lim mB(x0 ,r) (f − f (x0 )) = 0. r↓0

We decompose (q)

(q)

|a0 (x0 ; ρ, f ) − f (x0 )| ≤ mB(x0 ,ρ) (f − f (x0 )) + mB(x0 ,ρ) (Pk (·; x0 , ρ, f ) − f ) (q)

+ mB(x0 ,ρ) (a0 (x0 ; ρ, f ) − Pk (·; x0 , ρ, f )).

118

Morrey Spaces (q)

We note that mB(x0 ,ρ) (Pk (·; x0 , ρ, f ) − f ) ≤ ρ−λ kf kLλ,p . By the use of the k Taylor expansion Pk (x; x0 , ρ, f ) − a0 (x0 ; ρ, f ) =

X 1≤|α|≤k

aα (x0 ; ρ, f ) (x − x0 )α = O(ρ) α! (q)

uniformly over x ∈ B(x0 , ρ), as ρ ↓ 0. Consequently, mB(x0 ,ρ) (a0 (x0 ; ρ, f ) − Pk (·; x0 , ρ, f )) = O(ρ) as ρ ↓ 0. As a result, |a0 (x0 ; ρ, f ) − f (x0 )| = o(1) as ρ ↓ 0, as desired.

14.3.3

Exercises

Exercise 30. Let 1 ≤ q < ∞. Suppose that k, l ∈ N0 ∪ {−1} and λ ∈ R satisfy k + 1 + λ > 0 > λ + k and l > k. Let f ∈ Lλ,q l (Ω). We fix x0 ∈ Ω and ρ > 0. Take a polynomial Pl (·; x0 , ρ, f ) ∈ Pl (Rn ) so that min

P ∈Pl (Rn )

kf − P kLq (Ω(x0 ;ρ)) = kf − Pl (·; x0 , ρ, f )kLq (Ω(x0 ;ρ)) .

Suppose that α ∈ N0 n satisfies k < |α| ≤ l. (1) We write aα (x; x0 , ρ, f ) ≡ ∂ α Pl (x; x0 , ρ, f ). Let j ≡ [log2 ρ−1 diam(Ω)] ∈ N0 . Prove that |aα (x0 ; diam(Ω), f ) − aα (x0 ; 2−j diam(Ω), f )|
−λ−|α| . kf kLλ,q 2−j diam(Ω) l

using Lemma 111. (2) By the triangle inequality show that |aα (x; ρ, f )| . ρ−λ−|α| kf kLλ,q + l |aα (x; diam(Ω), f )|. (3) Define Qj (x) ≡

X aα (x0 ; ρ, f ) (x − x0 )α for x ∈ Rn , j ∈ N0 . Verify α!

|α|≤j

that Ql = Pl (·; x0 , ρ, f ). (4) Show that |aα (x0 ; diam(Ω), f )| . kf − P kLq (Ω) for any polynomial P ∈ Pk (Rn ) and |aα (x; ρ, f )| . kf kLλ,q . Hint: For any polynomial k P ∈ Pk (Rn ), we can show that aα (x0 ; diam(Ω), f ) = ∂ α Pl (x; x0 , ρ, f −P ) using k < l. n

(5) Prove kf − Qk kLq (Ω(x0 ,ρ)) . ρ−λ+ q kf kLλ,q . k

(6) Conclude f ∈ Lλ,q l (Ω). Exercise 31. Reexamine the proof of Lemma 239 in the first book to prove Lemma 110. Hint: Decompose Pk (·; x0 , 2−j ρ, f ) − Pk (·; x0 , 2−j aρ, f ) = Pk (·; x0 , 2−j ρ, f ) − u + u − Pk (·; x0 , 2−j aρ, f ).

Morrey spaces over metric measure spaces

14.4

119

Morrey spaces for non-doubling measures on Rn

Section 14.4 works in Rn equipped with a Radon measure µ. Unlike the metric measure space (X, d, µ), we have a rich geometric property. Consequently, as it turns out, the situation differs from a general case of metric measure spaces. Here, we work in Rn , which is equipped with a Radon measure µ. We define Morrey spaces for a Radon measure µ in Section 14.4.1. We will obtain an equivalent expression in Section 14.4.2. Section 14.4.3 considers the maximal inequality as the model case of the boundedness of operators.

14.4.1

Morrey spaces for non-doubling measures on Euclidean spaces

When we work on Rn with a Radon measure, we introduce a parameter k > 0. However, as it turns out in Proposition 118, k will not be important as long as k > 1. Recall that Q(µ) = {Q ∈ Q : µ(Q) > 0}. Definition 29. Let k >n1 and 0 < q ≤ p < ∞. Define o the Morrey space q p p Mq (k, µ) by Mq (k, µ) ≡ f ∈ Lloc (µ) : kf kMpq (k,µ) < ∞ , where kf k

Mp q (k,µ)

≡ sup µ(k Q)

1 1 p−q

Q∈Q(µ)

Z

 q1 |f (z)| dµ(z) . q

(14.1)

Q

Like the classical case, Lp (µ) = Mpp (k, µ). By applying H¨older’s inequality to (14.1) we have kf kMpq1 (k,µ) ≥ kf kMpq2 (k,µ) for all p ≥ q1 ≥ q2 > 0. Thus, the following inclusions hold: Lp (µ) = Mpp (k, µ) ⊂ Mpq1 (k, µ) ⊂ Mpq2 (k, µ). As is easily seen, the space Mpq (k, µ) is a Banach space with its norm. The parameter k > 1 appearing in the definition does not affect the definition of the space. More precisely, we have the following proposition, which is a key to our arguments. Proposition 118. Let 0 < q ≤ p < ∞, and let k1 , k2 > 1. Then Mpq (k1 , µ) = Mpq (k2 , µ). That is, Mpq (k1 , µ) and Mpq (k2 , µ) coincide as a set and their norms are equivalent. Proof Let k1 ≤ k2 . Then inclusion Mpq (k1 , µ) ⊂ Mpq (k2 , µ) is obvious by the definition of the norms. Let us show the reverse inclusion. Let f ∈ Mpq (k2 , µ) and Q ∈ Q(µ). Then we have to estimate I ≡ µ(k1 Q)

1 1 p−q

Z Q

 q1 |f (x)| dµ(x) . q

120

Morrey Spaces

A simple geometric observation shows that there exist cubes Q1 , Q2 , . . ., QN with the same sidelength such that Q⊂

N [

 Qi ,

k2 Qi ⊂ k1 Q (i = 1, 2, . . . , N ) and N .

i=1

k2 − 1 k1 − 1

n .

Using this covering, we easily obtain I ≤

N X

µ(k1 Q)

1 1 p−q

≤

q

Qi

i=1 N X

 q1 |f (x)| dµ(x)

Z

µ(k2 Qi )

1 1 p−q

i=1

Z

 q1 |f (x)| dµ(x) q

Qi

≤ N kf kMpq (k2 ,µ) . Thus, the proof is complete. With Proposition 118 in mind, we occasionally omit parameter k in Mpq (k, µ) and we write Mpq (µ). Example 56. What is surprising about Mpq (κ, µ) is the independence of the parameter κ > 1. We can define Mpq (1, µ) by inserting κ = 1 into the definition of Mpq (κ, µ). However, it can happen that Mpq (1, µ) is a proper subset of Mpq (2, µ). We exhibit a counterexample showing that M21 (1, µ) is not isomorphic to M21 (2, µ). Let d = 2 and dx1 dx2 be the 2-dimensional Lebesgue measure. We set µ(x) ≡ e2x2 dx1 dx2 and f (x) ≡ e−x2 . Then let us check that f ∈ M21 (2, µ) \ M21 (1, µ). Let Q = [y − l, y + l] × [z − l, z + l] be a cube. Then s Z 1 2l(el − e−l ) p |f (w)|dµ(w) = . el + e−l µ(Q) Q Thus, taking the supremum over Q, we see that this quantity becomes ∞. As a result we conclude f ∈ / M21 (1, µ). We now will prove f ∈ M21 (2, µ). In fact a similar calculation gives s Z 2l(el − e−l ) −1/2 µ(2Q) |f (w)|dµ(w) = . 1. (14.2) 3l e + el + e−l + e−3l Q Consequently, the proof of this proposition is finished.

14.4.2

Equivalent norm of doubling type

Section 14.4.2 investigates an equivalent norm related to the doubling cubes. Although we now envisage the non-homogeneous setting, we are still

Morrey spaces over metric measure spaces

121

able to work in the setting of doubling cubes. Let k, β > 1. We say that Q ∈ Q(µ) is a (k, β)-doubling cube, if µ(kQ) ≤ β µ(Q). If β ≥ k n+1 , then for µ-almost all x ∈ Rn and for all Q ∈ Q(µ) centered at x, we can find a (k, β)-doubling cube from k −1 Q, k −2 Q, . . .. See Lemma 252 in the first book for related details; we can argue as in Lemma 252 in the first book. In what follows we denote by Q(µ; k, β) the set of all (k, β)-doubling cubes in Q(µ). We fix k, β > 1 with β ≥ k n+1 . Let 1 ≤ q ≤ p < ∞. For f ∈ L1loc (µ) define kf k

Mp q (µ),doubling

≡

sup

µ(Q)

1 1 p−q

 q1 . |f (y)| dµ(y)

Z

Q∈Q(µ;k,β)

q

(14.3)

Q

In the notation kf kMpq (µ),doubling , the parameters k and β are contained implicitly. Note that the difference between two definitions (14.1) and (14.3) are that µ(kQ) in (14.1) is replaced by µ(Q) in (14.3) and that the doubling condition on Q is newly added in (14.3). We now present the main theorem in Section 14.4.2. Theorem 119. Let µ be a Radon measure on Rn , and let 1 ≤ q < p < ∞. npq p−q If β is large enough, say β > k , then kf kMpq (µ) ' kf kMpq (µ),doubling for f ∈ L0 (µ). Proof Given k > 1, we will prove kf kMpq (µ),doubling . kf kMpq (k,µ) ,

kf kMpq (k,µ) . kf kMpq (µ),doubling .

The left inequality is obvious, so let us prove the right inequality. In view of Proposition 118, we have only to show that, for every cube Q ∈ Q(µ), I ≡ µ(2Q)

1 1 p−q

Z

 q1 |f (y)| dµ(y) . kf kMpq (µ),doubling . q

Q

Let x ∈ Q ∩ supp(µ) and Q(x) be the largest doubling cube centered at x and with sidelength k −j `(Q) for some j ∈ N. Existence of Q(x) can be ensured for µ-almost all x ∈ Rn . More precisely, in Lemma 252 in the first book, we showed that such a cube Q(x) exists for almost all x ∈ Rn . Set Q0 (j) ≡ {Q(x) : `(Q(x)) = k −j `(Q)}, j ∈ N. By virtue X of Besicovitch’s covering lemma, [ we can take Q(j) ⊂ Q0 (j) so n that χR ≤ 4 χ2Q and that x ∈ R for µ-almost all x ∈ Q with R∈Q(j)

R∈Q(j)

`(Q(x)) = k −j `(Q). A volume argument gives us that ]Q(j) ≤ 8n k jn . Since Z Q

 q1  q1 ∞ X X Z q ≤ |f (y)| dµ(y) |f (y)| dµ(y) q

j=1 R∈Q(j)

R

122

Morrey Spaces

and µ(R) ≤ β −j µ(2Q) for all R ∈ Q(j), we have I≤

∞ X

β

1 j( p − q1 )

j=1

≤

∞ X

X

µ(R)

1 1 p−q

R∈Q(j)

Z

 q1

q

|f (y)| dµ(y) R

8n k jn β j ( p − q ) kf kMpq (µ),doubling 1

1

j=1 ∞ X

     1 1 = 8 exp j n log k + − log β kf kMpq (µ),doubling p q j=1 n

≤ C kf kMpq (µ),doubling , npq

where the constant C is finite, provided β > k p−q .

14.4.3

Maximal inequalities

Section 14.4.3 investigates some maximal inequalities for Morrey spaces with non-doubling measures. In proving the maximal inequalities we do not need the growth condition on µ. For κ > 1 we defined the modified maximal operator Mκ,uc by Z χQ (x) Q Mκ,uc f (x) = Mκ,uc f (x) ≡ sup |f (y)|dµ(y) (x ∈ Rn ). Q∈Q(µ) µ(κQ) Q The following boundedness of Mκ,uc is used in the proof of the main theorem of Section 14.4.3. Section 14.4.3 extends Theorem 251 in the first book to the Morrey space Mpq (2, µ) ≈ Mpq (k, µ) with 1 < q ≤ p < ∞ and k > 1. Theorem 120. If k, κ > 1 and 1 < q ≤ p < ∞, then kMκ,uc f kMpq (k,µ) .n,p,q,κ,k kf kMpq (k,µ) for all f ∈ L0 (µ). Proof In view of Proposition 118, we will show that  p1 − q1 Z  q1  2κ(κ + 7) q I≡µ Q M f (z) dµ(z) . kf k p 0 κ,uc Mq κ2 − 1 Q0


.

2κ κ+1 ,µ

for any cube Q0 ∈ Q(µ). Set L ≡ `(Q0 )/2. Let f1 ≡ χ κ+7 Q0 f and f2 ≡ f −f1 . κ−1 Then for all y ∈ Q0 we have Mκ,uc f (y) ≤ Mκ,uc f1 (y) + Mκ,uc f2 (y). It follows from the definition of Mκ,uc that 1 Mκ,uc f2 (y) ≤ sup µ(κQ) y∈Q∈Q(µ),`(Q)≥8L/(κ−1)

Z |f (z)|dµ(z). Q

(14.4)

Morrey spaces over metric measure spaces

123

Suppose that y ∈ Q0 , y ∈ Q ∈ Q(µ) and `(Q) ≥ 8L/(κ − 1). Then a simple 1+κ calculus yields Q0 ⊂ Q. Thus, we obtain 2 Z 1   Mκ,uc f2 (y) ≤ sup |f (z)|dµ(z). (14.5) 2κ Q∈Q] (Q0 ) µ Q κ+1 Q If we use (14.4), then  q1  p1 − q1 Z 2κ(κ + 7) q Mκ,uc f1 (z) dµ(z) Q0 I≤µ κ2 − 1 Rn Z  q1 1 − q1 q p Mκ,uc f2 (z) dµ(z) + µ(Q0 ) . 

Q0

By virtue of (14.5), we obtain  I≤µ

2κ(κ + 7) Q0 κ2 − 1

 p1 − q1 Z Rn 1 p

+

sup Q∈Q] (Q0 )

 q1 Mκ,uc f1 (z) dµ(z) q

µ(Q0 )   2κ µ κ+1 Q

Z |f (z)|dµ(z). Q

We now invoke Theorem 251 in the first book. Then ! q1  p1 − q1 Z 2κ(κ + 7) µ Q0 |f (z)|q dµ(z) κ+7 κ2 − 1 Q 0 κ−1   p1 − q1 Z  q1 2κ + sup µ Q |f (z)|q dµ(z) . κ+1 Q∈Q] (Q0 ) Q 

I

.

Finally, H¨ older’s inequality yields I . kf k p 2κ
. Hence, it follows that Mq κ+1 ,µ kMκ,uc f k p 2κ(κ+7)
. kf k p 2κ
. Using Proposition 118, we obtain Mq

κ2 −1

Mq

,µ

κ+1 ,µ

the conclusion of the theorem. Furthermore, we extend Theorem 120 to the vector-valued inequality. Theorem 121. If k, κ > 1, 1 < q ≤ p < ∞ and 1 < r ≤ ∞, then



  r1  r1

∞

∞

X

X



 .n,p,q,r,κ,k  |fj |r  Mκ,uc fj r 



p

j=1

p

j=1 Mq (k,µ)

0 for all sequences {fj }∞ j=1 ⊂ L (µ).

To prove Theorem 121, we need a covering lemma.

Mq (k,µ)

.

124

Morrey Spaces

Lemma 122. For every ρ > 1 there exists an integer ν, depending only on ρ and n, which satisfies the following condition: T If {Qj }j∈J is a finite family of cubes in Rn with Qj 6= ∅, then we can j∈J

select a set J 0 ⊂ J, #J 0 ≤ ν, such that any cube Qj can be covered by some ρ Qk , k ∈ J 0 . \ Proof Let L ≡ 2−1 max `(Qj ), and let x ∈ Qj . We will choose cubes j∈J

j∈J

inductively. First, choose a cube Qj1 so that `(Qj1 ) = 2L. Suppose that Qj1 , . . . , Qjk−1 are selected. Consider the set Jk ≡ {j ∈ J : none of ρQjm , m = 1, . . . , k − 1, contains Qj }. If Jk = ∅, then we do not select cubes any more. If Jk 6= ∅, we choose a cube Qjk , jk ∈ Kj , so that Qjk maximizes `(Qj ) with j ∈ Jk . We now proceed with this step and obtain a set J 0 ≡ {j1 , j2 , . . .} ⊂ J. A simple geometric observation shows that if k, k 0 ∈ J 0 and k 6= k 0 , then ρ−1 max(`(Qk ), `(Qk0 )). |z(Qk ) − z(Qk0 )| ≥ 2 Recall that the Qj ’s contain x. This shows that the number of Qk , k ∈ J 0 , such that 2m−1 L < `(Qk ) ≤ 2m L is less than or equal to max(1, 16n (ρ−1)−n ) for all m = −1, −2, . . .. This gives the bound for ]J 0 . Noticing that `(Qk ) > (ρ − 1) L for all k ∈ J 0 , we learn that any cube Qj can be covered by some ρ Qk , k ∈ J 0 . Proof of Theorem 121 We will show that

  r1

∞

X

 r Mκ,uc fj .n,p,q,r,κ,k






j=1

p 4κ(κ+7) Mq

(3κ+1)(κ−1)

,µ

  r1

∞

X

 r |fj |



j=1

Mp q




4κ 3κ+1 ,µ

in the light of Proposition 118. Fix Q0 ∈ Q(µ) and set L ≡ `(Q0 )/2. Let fj,1 ≡ χ κ+7 Q0 fj and fj,2 ≡ fj − fj,1 . Then, in the same way as in the proof of κ−1 the boundedness of the Hardy–Littlewood maximal operator obtained in the first book, for y ∈ Q0 , Mκ,uc fj (y) ≤ Mκ,uc fj,1 (y) + Mκ,uc fj,2 (y) and Z 1   Mκ,uc fj,2 (y) ≤ sup |fj (z)|dµ(z). 2κ Q∈Q] (Q0 ) µ Q Q κ+1 Using Theorem 251 in the first book, we see that   p1 − q1 Z  q1 4κ(κ + 7) q r µ Q0 k{Mκ,uc fj (x)}∞ k dµ(x) j=1 ` (3κ + 1)(κ − 1) Q0   p1 − q1 Z  q1 4κ(κ + 7) q r . µ Q0 k{Mκ,uc fj,1 (x)}∞ k dµ(x) j=1 ` (3κ + 1)(κ − 1) Rn   r  r1 Z ∞ X  1 1  sup   +µ(Q0 ) p |fj (z)|dµ(z) . 2κ   Q∈Q] (Q0 ) µ Q Q j=1

κ+1

Morrey spaces over metric measure spaces

125

The first term of the right-hand side of the above inequality can be bounded
thanks to Proposition 118. Hence, we will conby k k{fj }∞ 4κ j=1 k`r k p Mq

3κ+1 ,µ

centrate on estimating the second term. Let {Qj }∞ j=1 be a family of cubes satisfying Qj ⊃ Q0 . Then by a simple limiting argument it suffices to verify that for any N ∈ N  r  r1  N X kfj χQj kL1 (µ)      . kk{fj }∞ µ(Q0 ) j=1 k`r kMp q 2κ   µ κ+1 Qj j=1 1 p


,

4κ 3κ+1 ,µ

(14.6)

where the implicit constant is independent of N . A duality argument reduces estimate (14.6) to proving the inequality 1 Z N X

aj µ(Q0 ) p

  I≡ |fj (z)|dµ(z) . k{fj }∞ j=1 k`r Mp q 2κ Q j Q µ j=1 j κ+1




4κ 3κ+1 ,µ

(14.7)

0

r for positive sequences {aj }∞ j=1 ∈ ` with norm 1. To prove (14.7), for each l ∈ N, we define     2κ l−1 l Jl ≡ j ∈ N ∩ [1, N ] : 2 µ(Q0 ) ≤ µ Qj < 2 µ(Q0 ) . κ+1

Then I = µ(Q0 )

1 p

∞ X X l=1 j∈Ji

Z

aj µ



2κ κ+1

Qj



|fj (z)|dµ(z).

(14.8)

Qj

3κ + 1 , 2(κ + 1) 0 0 to obtain an integer ν and a set Jl ⊂ Jl , #Jl ≤ ν, such that every cube Qj , j ∈ Jl , is covered by some ρ Qk , k ∈ Jl0 . In view of the definition of Jl , we obtain X Z 1 1 I ≤ µ(Q0 ) p i−1 aj |fj (z)|dµ(z) 2 µ(Q0 ) Qj j∈Jl Z X X 1 1 = 2 µ(Q0 ) p l aj |fj (z)|dµ(z). 2 µ(Q0 ) Qj 0 We now apply Lemma 122 to the family of cubes {Qj }j∈Jl with ρ ≡

k∈Jl j∈Jl : Qj ⊂ρ Qk

Again in view of the property of Jl , we obtain   Z X 1 X 1    I ≤ 2 µ(Q0 ) p aj |fj (z)| dµ(z). 2κ ρ Qk j∈Jl : Qj ⊂ρ Qk k∈Jl0 µ κ+1 Qk

126

Morrey Spaces

By H¨ older’s inequality, we have
1 2l−1 µ(Q0 ) p − q1 Z  q1 X  2κ r r µ × k{fj }∞ k . ρ−1 ρ Qk dµ(z) j=1 ` κ+1 ρ Qk 0

I ≤ 2−

(l−1) +1 p

k∈Jl

Finally, since Morrey spaces are nested, we obtain I≤

∞ X

21−

(l−1) p



ν k{fj }∞ j=1 k`r

Mp q

l=1


.

(14.9)

4κ 3κ+1 ,µ

It follows from (14.8) and (14.9) that we obtain the desired inequality (14.6). This completes the proof of Theorem 121.

14.4.4

Exercises

Exercise 32. [401] Let µ be a Radon measure satisfying µ(B(x, r)) . rd for some 0 < d ≤ n. Define Iα , 0 < α < n by Z f (y) Iα f (x) ≡ dµ(y) d−α Rn |x − y| for suitable functions f ∈ L0 (µ). Let 1 < q ≤ p < ∞ and 1 < t ≤ s < ∞ satisfy 1 α t q 1 = − , = . s p d s p (1) Let u ∈ [1, ∞] and f1 , f2 , . . . ∈ Mpq (µ). Then that k{Iα fj }∞ j=1 k`u ≤ u Iα [k{fj }∞ k ]. j=1 ` (2) Show that, for all u ∈ [1, ∞] and f1 , f2 , . . . ∈ Mpq (µ),



∞

k{Iα fj }∞

j=1 k`u Mp (µ) . k{fj }j=1 k`u Mp (µ) q

q

where the implicit constant in . is independent of u. It matters that we torelate the case u = 1. 0

0

Exercise 33. [424] Define Hqp0 (2, µ) as a predual space of Mpq0 (2, µ) and establish its Fatou property. Exercise 34. Let 1 < q ≤ p < ∞. Suppose that we have a Radon measure µ satisfying µ(B(x, r)) . rd for some 0 < d ≤ n. Suppose that we have an Lq (µ)-bounded linear operator T , so that kT kLq (µ)→Lq (µ) ≡ sup{kT f kLq (µ) : kf kLq (µ) = 1} < ∞.

Morrey spaces over metric measure spaces

127

Assume in addition that there exists K ∈ L0 (Rn × Rn ) such that |K(x, y)| . |x − y|−n and that

Z T f (x) =

K(x, y)f (y)dy

(a.e. x ∈ / supp(f ))

Rn

for all f ∈ Lq (Rn ) with bounded support. Then show that there exists a constant cp,q such that



  r1  r1



∞ ∞

X

X



 r r |fj | |T fj | .p,q (1 + kT kLq (µ)→Lq (µ) )





j=1

j=1

p

p Mq (µ)

Mq (µ)

for all f1 , f2 , . . . ∈ Mpq (µ) ∩ Lq (µ).

14.5

Morrey spaces on the metric measure space setting

We know that Morrey spaces can describe the local regularity and the global regularity more precisely than Lebesgue spaces. The same can be said when we have metric measure spaces. Due to a nice geometrical structure of Euclidean space, the Hardy–Littlewood maximal operators are bounded in Morrey spaces. In the general metric measure setting, we need to equip Morrey spaces with another parameter, which is not of importance in the Eucledean space. Section 14.5.1 deals with Morrey spaces with a locally doubling measure. Consequently, the underlying measure is not completely doubling but to some extent doubling. As an example of the theory developed in Section 14.5.1, we take up Gauss Morrey spaces in Section 14.5.2. Section 14.5.3 is also devoted to an example. We will show that an additional parameter k introduced in Definition 29 cannot be removed.

14.5.1

Fundamental properties of Morrey spaces on locally doubling metric measure spaces

Let (X, d, µ) be a locally doubling metric measure space as in Definition 68 in the first book. We consider the definition of Morrey spaces adapted to this triple. For 1 ≤ q ≤ p < ∞ and a > 0, the Morrey space Mp,q Ba (µ) is defined 1 to be the collection of all f ∈ Lloc (µ) such that kf kMp,q ≡ sup µ(B) B (µ) a

B∈Ba

1 1 p−q

Z B

 q1 |f (y)| dµ(y) q

(14.1)

128

Morrey Spaces

is finite. We start with the following observation: Remark 5. (1) Since the supremum in (14.1) is only taken over a class of admissible balls, we see that for all p ≥ 1 and a > 0, Lp (µ) ⊂ Mp,p Ba (µ). The inclusion turns out to be strict even when µ is taken to be the Gauss measure. (2) Applying H¨ older’s inequality easily yields the following embedding relap,q2 tions: for all 1 ≤ q1 ≤ q2 ≤ p < ∞, Lp (µ) ⊂ Mp,p Ba (µ) ⊂ MBa (µ) ⊂ p,q1 MBa (µ) in the sense of continuous embedding. The next proposition shows that the space Mp,q Ba (µ) is independent of the choices of the scale a ∈ (0, ∞). Proposition 123. For any 0 < a ≤ b < ∞ and 1 ≤ q ≤ p < ∞, the spaces p,q p,q p,q Mp,q Ba (µ) and MBb (µ) coincide with equivalent norms; k·kMBa (µ) ∼ k·kMBb (µ) with implicit constants depending only on a, b, p, q and X. Proof Note that the proof resembles that of Proposition 118. Since a ≤ b, we have Ba ⊂ Bb trivially. From this and (14.1), it follows p,q p,q p,q that f ∈ Mp,q Ba (µ) and kf kMB (µ) ≤ kf kMB (µ) whenever f ∈ MBb (µ). a

b

To establish the converse, given any f ∈ Mp,q Ba (µ), we need to prove that, for any ball B0 ∈ Bb , 1

1

µ(B0 ) p − q

Z

|f (y)|q dµ(y)

 q1 . . kf kMp,q B (µ)

(14.2)

a

B0

Observe that if rB0 ≤ am(cB0 ), then B0 ∈ Ba and (14.2) follows directly . Thus, it suffices to show that (14.2) holds from the definition of k · kMp,q Ba (µ) when rB0 > am(cB0 ). Denote by {zν }ν∈I a maximal set of points in B0 such a that for any indices ν 6= ν 0 , d(zν , zν 0 ) ≥ bΘ rB0 , where Θb is the positive b −1 constant satifying Θb m(x) ≤ m(y) ≤ Θb m(x). Here, the word “maximal ” a means that, for any x ∈ B0 , there exists ν ∈ I such that d(x, zν ) < bΘ rB0 . b The existence of such a maximal set is guaranteed by Zorn’s lemma. a Let δa,b ≡ 2bΘ . For every ν ∈ I, let Bν ≡ B(zν , δa,b rB0 ). By the choice of b {zν }ν∈I , we see that {Bν }ν∈I is pairwise disjoint and [ B0 ⊂ 2Bν . (14.3) ν∈I

For each ν ∈ I, by zν ∈ B0 ∈ Bb and the two-sided estimate m(y) ∼ m(x), we conclude that 2δa,b rB0 =

a a rB ≤ m(cB0 ) ≤ am(zν ), bΘb 0 Θb

Morrey spaces over metric measure spaces

129

which implies that 2Bν ∈ Ba . The fact that zν ∈ B0 and the locally doubling property of µ imply that for any ν ∈ I, µ(B0 ) ≤ µ(B(zν , 2rB0 )) . µ(2Bν ) and µ(2Bν ) ≤ µ (B(cB0 , (2 + 2δa,b )rB0 )) . µ(B0 ). Therefore, µ(B0 ) ∼ µ(2Bν )

(14.4)

with implicit constants depending only on a, b and X. We now show that the cardinality of I, denoted by ]I, is bounded by a positive constant independent of B0 . To see this, notice that {Bν }ν∈I is pairwise S disjoint and that Bν ⊂ B(cB0 , (1 + δa,b )rB0 ). Moreover, an argument simiν∈I

lar to the proof of (14.4) implies that for all ν ∈ I, µ(B(cB0 , (1 + δa,b )rB0 )) ∼ µ(Bν ). Therefore, X ]I µ(B(cB0 , (1 + δa,b )rB0 )) . µ(Bν ) . µ(B(cB0 , (1 + δa,b )rB0 )), ν∈I

which implies that ]I . 1.

(14.5)

By using (14.3)–(14.5), Minkowski’s inequality, and 2Bν ∈ Ba , we obtain µ(B0 )

1 1 p−q

Z

 q1 1 1 |f (y)| dµ(y) ≤ µ(B0 ) p − q

B0

ν∈I

.

X

µ(2Bν )

! q1

XZ

q

1 1 p−q

q

|f (y)| dµ(y)

2Bν

Z

 q1 |f (y)| dµ(y) q

2Bν

ν∈I

. kf kMp,q , B (µ) a

which proves (14.2) when rB0 > am(cB0 ). This concludes the proof of Proposition 123. The space Mp,q BA (γ), whose norm is given by (14.1), is a special case of defined in Definition 29. when 1 ≤ q < p < ∞.

Mpq (µ)

Theorem 124. Let A ∈ (0, ∞), a ∈ (1, ∞) and 1 ≤ q < p < ∞. Then p Mp,q BA (γ) = Mq (a, γ) with equivalent norms. The proof of Theorem 124 is based on Lemma 265 in the first book. Proof Let Cp,q,n ≡

32 7

npq

log(81+ p−q ). It suffices to establish that

kf kMp,q . kf kMpq (2,γ) , kf kMpq (µ),doubling . kf kMp,q B (γ) B a

Cp,q,n

(γ)

for all f ∈ L0 (γ) in view of Proposition 123. Here kf kMpq (µ),doubling defined by (14.3) implicitly contains parameters k and β satisfying β ≥ k n+1 . Here npq we may suppose that β = 81+ p−q and k = 8 using Theorem 119.

130

Morrey Spaces

By the local doubling property of µ, it is easy to see that kf kMp,q . Ba (γ) kf kMpq (2,γ) . It thus remains to show that 1

1

γ(B) p − q

Z

|f (x)|q dγ(x)

 q1 . kf kMp,q B

Cp,q,n

B

(14.6)

(γ)

npq

for all balls B such that γ(8B) ≤ 81+ p−q γ(B). Fix such a ball B. We distinguish the following two cases. Case 1) |c(B)| ≥ 2r(B). In this case, by Lemma 265 in the first book, B itself belongs to BCp,q,n and inequality (14.6) obviously holds. Case 2) |c(B)| < 2r(B). In this case, B ⊂ B(3r(B)) ⊂ 5B, which, together npq with assumption γ(8B) ≤ 81+ p−q γ(B), yields that γ(B(3r(B))) ≤ γ(5B) < npq γ(8B) ≤ 81+ p−q γ(B). Consequently, we have 1

1

γ(B) p − q

Z

|f (x)|q dγ(x)

 q1 (14.7)

B 1 n+ p − q1

≤8

γ(B(3r(B)))

1 1 p−q

! q1

Z

|f (x)|q dγ(x)

.

B( 3r(B))

If B(3r(B)) ∈ BCp,q,n , then it follows, from (14.7), that γ(B)

1 1 p−q

Z

q

|f (x)| dγ(x) B

 q1

1

1

≤ 8n+ p − q kf kMp,q B

Cp,q,n

(γ) .

If B(3r(B)) ∈ / BCp,q,n , then 3r(B) > Cp,q,n > 1 and γ(B(3r(B))) ≥ γ(B(1)). This, combined with (14.7), gives that 1

1

γ(B) p − q

Z B

 q1 |f (x)|q dγ(x) . kf kLq (γ) . kf kMp,q B

Cp,q,n (γ)

,

where we used q < p. Therefore, we show that (14.6) also holds when |c(B)| < 2r(B), which completes the proof of Theorem 124.

14.5.2

Gauss Morrey space 2

Equip Rn with the standard distance. Let γ(x) = e−π|x| dx be the Gauss measure. The Gauss measure space (Rn , | · |, γ) falls under the scope of the framework of locally doubling Morrey spaces. Morrey space Mp,q Ba (γ) over (Rn , | · |, γ) is called the Gauss Morrey space. We explore the structure of Gauss Morrey space. Due to Lemma 264 in the first book, we obtain the following embedding: Proposition 125. Let 1 ≤ q < p < ∞ and a > 0. Then, there exists a positive constant C, depending only on p, q, a and n, such that kf kL1 (γ) ≤

Morrey spaces over metric measure spaces

131

kf kLq (γ) . kf kMp,q for all f ∈ L0 (Rn ). In particular, the following chain Ba (γ) of inequalities holds: kf kL1 (γ) ≤ kf kLq (γ) . kf kMp,q . kf kMp,p . kf kLp (γ) . B (γ) B (γ) a

a

Obviously, kf kL1 (γ) ≤ kf kLq (γ) follows from H¨older’s inequality and γ(Rn ) = 1. Proof According to Proposition 123, we need only to show kf kLq (γ) . kf kMp,q To this end, we select a collection of balls, {Bj,i }j∈N0 ,i∈Ij , belongB1 (γ) ing to B1 and satisfying (i)–(iv) of Lemma 264 in the first book. Then, kf kLq (γ) ≤

∞ X X

kχ4Bj,i f kLq (γ)

j=1 i∈Ij

≤

XX

1

1

. γ(4Bj,i ) q − p kf kMp,q B (γ) a

j∈N i∈Ij

Applying Lemma 264(iii) in the first book, together with (13.12), we conclude that ∞ X X

1

1

[γ(Bj,i )] q − p .

j=1 i∈Ij

∞ X

2

n

1

1

]Ij [e−aj j − 2 ] q − p .

j=1

∞ X

n

1

1

j n [e−2j j − 2 ] q − p .

j=1

Recall that we are assuming that q < p. Thus, the last series converges and ∞ P P 1 1 we see that [γ(Bj,i )] q − p . 1. From this, it follows that kf kLq (γ) . j=1 i∈Ij

kf kMp,q , which completes the proof of Proposition 125. B (γ) a

We consider what happens if p = q.
Example 57. Let 1 ≤ p < ∞. Then if we set f (x) ≡ exp − p1 |x|2 for < ∞ since in the definition of x ∈ Rn , then kf kLp (γ) = ∞, while kf kMp,p Ba (γ) the Morrey norm we are considering the balls in Ba .

14.5.3

An example of Morrey spaces for non-doubling measures on metric measure spaces

We now go back to a general metric measure space (X, d, µ), where we define the modified Morrey space Mpq (k, µ) of order k > 0 by the norm; kf kMpq (k,µ) ≡

sup (x,r)∈X×(0,∞)

µ(B(x, kr))

1 1 p−q

Z

! q1 q

|f (y)| dµ(y)

.

B(x,r)

(14.8) We aim to show that in a general setting, it can happen that Mpq (2, µ) and Mpq (6, µ) are different. As we did in Chapter 6, we will work in the following set X.

132

Morrey Spaces

(1) One writes ∆(z, r) ≡ {w ∈ C : |w − z| < r}. (2) Let Ak ≡ {z ∈ C : |z| = 3−k } for k ∈ N0 . (2) Define X0 ≡ {0} ∪

∞ [

Ak ⊂ C.

k=0

(3) Let X ≡ X0 N ⊂ CN be the cross product. (4) Let O ≡ (0, 0, . . .) ∈ X. In Chapter 6 we gave its “singular” metric d on X. (1) The integer N0 is chosen so that log3 N0 is a big integer. N0 = 32a

(14.9)

for some a ∈ N large enough. Note that (14.9) is a new requirement that did not appear in Chapter 6. (2) Denote by [·] a Gauss symbol and define N (δ) ≡ max(1, [logN0 δ −1 ]) for δ > 0. ∞ (3) Let x = {xj }∞ j=1 and y = {yj }j=1 be points in X, and define the distance d(x, y) of x and y by

d(x, y) ≡ inf{δ > 0 : |xj − yj | ≤ δ for all j ≤ N (δ)}. (4) One defines a sphere Sk by Sk ≡ (Ak )N (3

−k

)

× X0 × X0 × · · ·

(14.10)

for each k ∈ N. Importantly, as long as b ∈ [1, 9] and k is an odd multiple of a, logN0 b · 3−k and logN0 3−k have the same integer part since we have (14.9). Here we seek to show the following proposition: Proposition 126. Let (X, d) be a metric measure space in Definition 69 in Chapter 6. Let 1 ≤ q < p < ∞. Then, Mpq (2, µ) is a proper subset of Mpq (6, µ). Let us show that we can move the center of the ball to the origin with the cost of enlarging the radius 10 times. Lemma 127. We have µ(B(O, r)) ≤ µ(B(x, 10r)) for all x ∈ X and r > 0. Proof We start with a simple geometric observation: For all r ∈ (0, 2), arithmetic shows r H1 ({(x, y) ∈ R2 : x2 +y 2 = 1}∩{(x, y) ∈ R2 : (x−1)2 +y 2 ≤ r2 }) = 4 sin−1 . 2

Morrey spaces over metric measure spaces

133

Let us write x = (x1 , x2 , . . . , xN , . . .) ∈ X. In view of Lemma 276 in the first book, we have N (r)

µ(B(O, r)) =

Y

N (10r)

µ0 (∆(0, r)) and µ(B(x, 10r)) =

j=1

Y

µ0 (∆(xj , 10r)).

j=1

For the definition of ∆(z, r) see Definition 69 in the first book. Now that N (r) ≥ N (10r) and µ0 is a probability measure, it suffices to prove µ0 (∆(0, r)) ≤ µ0 (∆(z, 10r))

(14.11)

for all z ∈ Ak for some k ∈ N0 and r > 0. Assume first that r > 3−k−2 . Then, since |z| = 3−k , a geometric observation shows ∆(0, r) ⊂ ∆(z, 10r). (14.12) This shows (14.11). Assume that 0 < r ≤ 3−k−2 . Then, from the equality µ0 (∆(3k z, 3k r) ∩ A0 ) µ0 (∆(1, 3k r) ∩ A0 ) µ0 (∆(z, r) ∩ Ak ) = = µ0 (Ak ) µ0 (A0 ) µ0 (A0 ) and the observation above, we deduce µ0 (∆(z, r) ∩ Ak ) 2 · 3k r 3k r ≥ 1 = . µ0 (Ak ) H (A0 ) π Hence, it follows from Lemma 277 in the first book that µ0 (∆(z, 10r)) ≥ µ0 (∆(z, r) ∩ Ak ) ≥

3k µ0 (Ak ) 2 · 3k γ 2γ r= r = k r. (14.13) π (3 · k!)k k!

It follows from the definition of Al that  µ0 (∆(0, r)) = µ0 

 [

Al  .

l>− log3 r

Let l ∈ N. Then l > − log3 r if and only if l ≥ [1 − log3 r]. Hence, from Lemma 277 in the first book, µ0 (∆(0, r)) =

X l>− log3 r

2πγ ≤ (3 · l!)l

X l≥[1−log3 r]

2πγ . 3l · ([1 − log3 r]!)[1−log3 r]

If we evaluate the geometric series, then we obtain µ0 (∆(0, r)) ≤
µ0 (∆(0, r)).

(14.15)

Combining (14.12) and (14.15) gives (14.11). Lemma 128. For all k ∈ N such that k is an odd multiple of a we have µ(B(O, 2.2 × 3−k )) . µ(Sk ). Proof Let us write B(O, 2.2 × 3−k ) out in full by using (14.3): N (2.2×3−k ) −k

B(O, 2.2 × 3

\

)=

−k {x = {xj }∞ }. j=1 ∈ X : |xj | < 2.2 × 3

j=1

Thus, from the definition of Ak , we have  B(O, 2.2 × 3−k )

= 

∞ [

N (2.2×3−k ) × X0 × X0 × · · · .

Aj 

j=k

Hence, it follows from Lemma 277 in the first book that µ(B(O, 2.2 × 3−k )) ≤



2πγ (3 · k!)k

N (2.2×3−k )  1+

1 γ · (k + 1)!

N (2.2×3−k ) .

Now that (k + 1)! grows much faster than N (2.2 × 3−k ) = max(1, [(logN0 3)k − logN0 2.2]), we have µ(B(O, 2.2 × 3

−k

 )) .

2πγ (3 · k!)k

N (2.2×3−k ) .

(14.16)

Meanwhile, from (14.10) and Lemma 277 in the first book, we deduce N (3−k ) 

µ(Sk ) =

Y

j=1

2πγ (3 · k!)k



 =

2πγ (3 · k!)k

N (3−k ) .

(14.17)

Since k/a is an odd integer, N (3−k ) = N (2.2 × 3−k ). We thus deduce the desired result from (14.16) and (14.17). We show that µSk decays rapidly keeping in mind that cos−1

2 < 0.6. 3

Morrey spaces over metric measure spaces

135

Lemma 129. Let 1 ≤ q < p < ∞. Suppose that 2a = log3 N0 . Then  lim inf k→∞

1 2 cos−1 π 3

N (3k−1 )( p1 − q1 )

µ(Sk ) = 0. µ(Sk−1 )

Proof We have  µ(Sk ) =

2πγ (3 · k!)k

N (3−k ) .

(14.18)

Recall that we are assuming N0 > 100. If k ∈ N ∩ [3a, ∞) is such that k/a is an odd multiple of positive integers, then    k k−a −k k N (3 ) = max(1, [logN0 3 ]) = max 1, , = 2a 2a    k k−a 1−k k N (3 ) = max(1, [logN0 3 − logN0 3]) = max 1, − logN0 3 = . 2a 2a This implies N (3−k+1 )( p1 − q1 ) 1 2 µ(Sk ) cos−1 k→∞ π 3 µ(Sk−1 ) N (−3(2k−1)a+1 )( p1 − q1 )  µ(S(2k−1)a ) 2 1 ≤ lim cos−1 k→∞ π 3 µ(S(2k−1)a−1 ) 

0 ≤ lim inf

 = lim

k→∞

1 2 cos−1 π 3

 p1 − q1

(3 · [(2k − 1)a − 1]!)(2k−1)a−1 (3 · [(2k − 1)a]!)(2k−1)a

!N (3−(2k−1)a +1)

= 0. Consequently, we obtain the desired result. We go back to the proof of Proposition 126. Assume that Mpq (2, µ) = Then by the closed graph theorem,

Mpq (6, µ).

kf kMpq (2,µ) . kf kMpq (6,µ)

(14.19)

for all f ∈ Mpq (2, µ) = Mpq (6, µ). See Exercise 35. Let k be fixed. We need estimates of kχSk kMpq (2,µ) and kχSk kMpq (6,µ) . The estimate we need for kχSk kMpq (2,µ) is as follows: 1

1

1

1

1

kχSk kMpq (2,µ) ≥ µ(B(o, 2.2 · 3−k )) p − q µ(B(o, 1.1 · 3−k ) ∩ Sk ) q 1

= µ(B(o, 2.2 · 3−k )) p − q µ(Sk ) q 1

≥ Cµ(Sk ) p ,

136

Morrey Spaces

where for the last inequality we have used Lemma 128. Consequently, 1

kχSk kMpq (2,µ) ≥ Cµ(Sk ) p .

(14.20)

Meanwhile, we need the estimate for kχSk kMpq (6,µ) . Let B = B(x, r) be a ball which intersects Sk . We distinguish two cases. (1) If r ≤ 2−1 · 3−k , then 6B ⊃ B ⊃ B ∩ Sk . Hence 1

1

1

1

µ(6B) p − q µ(B ∩ Sk ) q ≤ µ(B ∩ Sk ) p . Write x = (x1 , x2 , . . . , xN (3−k ) , . . .). Recall that B = ∆(x1 , r) × ∆(x2 , r) × · · · × ∆(xN (r) , r) × X0 × X0 × · · · according to Lemma 276 in the first book. Thus µ(B ∩ Sk ) 



N (3−k )

= µ 

Y





∆(xj , r) × ∆(xN (3−k ) , r) × X0 × X0 × · · ·  ∩ Sk 

j=1 N (3−k )

=

Y

µ0 (∆(xj , r) ∩ Ak ).

j=1

We will estimate µ0 (∆(xj , r) ∩ Ak ) for j = 1, 2, . . . , N (3−k ). Suppose that ∆(xj , r) and Ak meet at a point yj . Then ∆(xj , r) ⊂ ∆(yj , 2r). Hence ∆(xj , r) ∩ Ak ⊂ ∆(yj , 2r) ∩ Ak . Since r ≤ 2−1 · 3−k , it follows that µ0 (∆(xj , r)∩Ak ) ≤ µ0 (∆(yj , 2r)∩Ak ) ≤ µ0 (∆(yj , 3−k )∩Ak ) =

µ0 (Ak ) . 3

Using this geometric observation, we have  µ(B ∩ Sk ) ≤

1 µ0 (Ak ) 3

N (3−k )

= 3−N (3

−k

)

µ(Sk ).

Hence, we have 1

1

1

µ(6B) p − q µ(B ∩ Sk ) q ≤ 3−

N (3−k ) p

1

µ(Sk ) p .

(2) If r > 2−1 · 3−k , then Lemma 278 in the first book yields  µ(6B ∩ Sk−1 ) ≥

2 1 cos−1 π 3

N (3−k+1 ) µ(Sk−1 ).

(14.21)

Morrey spaces over metric measure spaces

137

Hence 1

1

1

µ(6B) p − q µ(B ∩ Sk ) q 1

1

1

≤ µ(6B ∩ Sk−1 ) p − q µ(B ∩ Sk ) q  N (3−k+1 )( p1 − q1 ) 1 1 1 1 −1 2 µ(Sk−1 ) p − q µ(Sk ) q . ≤ cos π 3 Consequently, it follows that 1

1

1

µ(6B) p − q µ(B ∩ Sk ) q  N (3−k+1 )( p1 − q1 )   p1 − q1 ! N (3−k ) 1 µ(S ) 1 2 k−1 − −1 p ≤ max 3 cos µ(Sk ) p . , π 3 µ(Sk ) Since the ball B is arbitrary, we obtain kχSk kMpq (6,µ) kχSk kMpq (2,µ) −

. max 3

N (3−k ) p

 ,

1 2 cos−1 π 3

N (3−k+1 )( p1 − q1 ) 

µ(Sk−1 ) µ(Sk )

 p1 − q1 !

thanks to (14.20). This is a contradiction to (14.19), since lim max 3

N (3−k ) − p

k→∞

 ,

1 2 cos−1 π 3

N (3−k+1 )( p1 − q1 ) 

µ(Sk−1 ) µ(Sk )

 p1 − q1 ! =0

by virtue of assumption 1 ≤ q < p < ∞, (14.18) and Lemma 129.

14.5.4

Exercises

Exercise 35. Let b > a > 0 and 1 < q ≤ p < ∞, and let (X, d, µ) be a metric measure space. Assume that a Radon measure µ satisfies Mpq (a, µ) = Mpq (b, µ). Let T be a mapping given by f ∈ Mpq (b, µ) 7→ f ∈ Mpq (a, µ). p (1) Suppose that {fj }∞ j=1 is a sequence in Mq (b, µ) which is convergent to f in Mpq (b, µ) and that {T fj }∞ is a sequence in Mpq (a, µ) which is j=1 ∞ convergent to g in Mpq (a, µ). Using the fact that {fj }∞ j=1 = {T fj }j=1 p converges to g in Mpq (a, µ), show that {fj }∞ is a sequence in M (b, µ) q j=1 which is convergent to g in Mpq (b, µ). Conclude that f = g.

(2) Show that T is a closed operator. (3) Conclude that kf kMpq (a,µ) . kf kMpq (b,µ) for all f ∈ L0 (µ). Z Exercise 36. Let γ(E) ≡ exp(−|x|)dx for a Borel set E. Show that E

γ(B(x, r)) ∼ rn exp(−|x|) for all 0 < r ≤ 1 and x ∈ Rn .

138

14.6

Morrey Spaces

Notes

Section 14.1 General remarks and textbooks in Section 14.1 See the textbook of Morrey [329] for Morrey spaces over open sets. The analysis of extension domains goes back to the works of Whitney [505, 506] and Hestenes [200] who considered spaces of continuously differentiable functions. In the case of Sobolev spaces W l,p (Ω), with l ∈ N and p ∈ [1, ∞], defined on open sets Ω in Rn with minimal boundary regularity, i.e. Ω in the Lipschitz class C 0,1 , Calderon [59], Stein [457, 458] and Burenkov [35, 36] contributed importantly to the problem. Each of them constructed different linear bounded extension operators from W l,p (Ω) to W l,p (Rn ). Compared with the classical extension operator by Hestenes [200], the main striking feature of Calderon’s, Stein’s and Burenkov’s operators consists in the fact that Ω is not required to be of class C l with l > 1. For a discussion concerning the differences between those operators, as well as for historical remarks and other references, we refer to Burenkov [36, 37], and to the earlier Stein’s book [458]. See the lecture note [387] by Rupflin for Camapanto spaces defined over domains. Section 14.1.1 We refer to [13] for applications of Morrey spaces over the half space to Navier–Stokes equations. In connection with the elliptic differential operators defined on the half space, Chiarenza, Frasca and Longo considered the commutator generated by BMO and the integral operator having the positive kernel |˜ x − y|−n . We refer to [31, 75] for Theorem 101. Section 14.1.2 As we saw in Example 51, Morrey spaces over domains can be embedded into Morrey spaces over the whole spaces by zero extension. Zorko proved this fact in the setting of generalized Morrey spaces; see [544, Proposition 2]. Chiarenza and Frasca proved Theorem 104 [74]. The notion of type (A) is due to Prato [371]; see Definition 25. Transirico, Canale, Gironimo, Troisi and Vitolo considered Morrey spaces over (un)bounded domains [67, 480]. Ragusa and Zamboni introduced the Stummel class [380]. Many authors investigated the relation of Stummel classes and Morrey spaces [99, 399, 432]. The work [34] is a passage to a generalization. See [399] for the vanishing Stummel class.

Morrey spaces over metric measure spaces

139

More is investigated on Sobolev–Morrey spaces. The embedding relation can be found in [88, 227, 334]. See [290] for Sobolev–Morrey spaces over metric measure spaces. See [366] for the profile decomposition.

Section 14.2 General remarks and textbooks in Section 14.2 Sobolev–Morrey spaces arose in the study of elliptic differential equations. Campanato considered Sobolev–Morrey spaces in [63]. See the paper by Vitolo [490] for more. Section 14.2.1 Di Fazio, Hakim and Sawano considered interpolation property of Sobolev– Morrey spaces in [113, Theorem 4.1]; see Theorem 106 for a more general result. Section 14.2.2 We followed the idea of the paper [107] to obtain Theorem 108, which is in turn based on [37]. See also [44, 251] for the extension operator on Morrey spaces. For another contribution in this field of investigation, we refer to Khidr and Yeihia [233] who obtain results radically different from ours.

Section 14.3 General remarks and textbooks in Section 14.3 See the book of Morrey [329] for Morrey spaces on domains. Section 14.3.1 We followed [64]. Lemmas 111 and 112 and Theorem 113 can be found in [64, (3.15)], [64, Lemma 3.II] and [64, Lemma 6.I], respectively. We refer to [126, 138] for general information on Morrey–Campanato spaces on domains. Section 14.3.2 We followed [64] again. See [64, Theorem 5.I] for Theorem 117. Corollary 114 and Theorems 115 and 116 can be found in [64, Lemma 3.IV], [64, Theorem 4.II] and [64, Lemma 4.I], respectively. Gorka passed the observation by Campanato to metric measure spaces [131]. Fan considered a passage of the analysis of Morrey–Campanato spaces to bounded domains [108].

140

Morrey Spaces

Section 14.4 General remarks and textbooks in Section 14.4 See the textbook [271] for the Gauss measure space. We can find an application of Morrey spaces with parabolic distance to the Schr¨odinger equation; see [22]. Section 14.4.1 As we saw in Proposition 118, the parameter k > 1 does not affect the definition of Mpq (k, µ); see [418, Proposition 1.1]. Example 56 is recorded in [404] but this comes from a personal communication with Hitoshi Tanaka. Wang, Ma and Lu considered the Littlewood-Paley gµλ -function and its commutator on non-homogeneous generalized Morrey spaces [502]. It is fundamental to calculate the norm of the indicator function of balls; see [102, Lemma D] for the case of spaces of homogeneous type. There are many concrete examples other than Gauss Morrey spaces. See [297, 390, 391]. Another useful definition of Morrey spaces for general Radon measure is given as follows: kf k†Mpq (k,µ)

1 p

≡ sup `(Q) µ(k Q) Q∈Q(µ)

− q1

Z

 q1 . |f (z)| dµ(z) q

Q

This norm is sometimes adapted to fractional integral operators considered in [322, 362, 363, 364, 408]. Section 14.4.2 Tolsa defined the notion of doubling cubes in [479]. Sawano and Tanaka obtained the doubling characterization of Mpq (2, µ) in [419, Theorem 2.1]; see Theorem 119. As we saw in Example 56 Mpq (2, µ) and Mpq (1, µ) are different in general; see [423, Example 2.3]. Section 14.4.3 Based on Proposition 118, Sawano and Tanaka investigated the boundedness property in [418, Theorem 2.3]; see Theorem 120. Sawano and Tanaka also passed Theorem 120 to the vector-valued case in [418, Theorem 2.4].

Section 14.5 General remarks and textbooks in Section 14.5 We refer to [6, Chapter 17] for an introduction to Morrey spaces over Riemannian manifolds. See the textbook [132] for the p-adic field. Geisler considered Morrey spaces over manifolds [124]. Let Qp denote the p-adic field considered in Example 106 in the first book. Then Lp,φ (X; µ) is a special case

Morrey spaces over metric measure spaces

141

of the norm dealt in [491]. Volosivets considered the case where φ depends on x as well in [491]. Section 14.5.1 Theorem 124, which gives various equivalent expressions of the Morrey norm k · kMpq (a,γ) , is due to Liu, Sawano and Yang [270, Theorem 3.8]. There are many works on Morrey spaces on metric measure spaces of various settings. Overall, there are two different directions in this field of research. One is to work on general metric measure spaces (X, d, µ). We sometimes assume that (X, d, µ) is bounded, compact or separable or that d is a quasimetric. See [102, 487, 488] for fractional integral operators adapted to Frostman measures over quasi-metric measure spaces and [401, 449] for fractional integral operators adapted to general Radon measures. Ohno and Shimomura considered Sobolev embedding of Morrey spaces [362, 363]. Stempak and Tao considered [461] for local Morrey spaces and local Morrey–Campanato spaces. Lin, Nakai and Yang defined localized Morrey–Campanato spaces over doubling metric measure spaces [266]. Another direction is to consider some special metric measure spaces. Mao, Wu and Yang proved the boundedness and the compactness of commutators generated by the Riesz transform on Morrey spaces in the Bessel setting [298]. Many authors worked in Euclidean space with general Radon measure to define generalized Morrey spaces [181, 182, 269, 403]. The work [68] can be regarded as an extension of [418] to the geometrically doubling setting. See [179, 192] for the discrete setting. Izumi and Sato worked in the unit circle [225]. Arai and Mizuhara worked in the spaces of homogeneous type [17]. See [441] for Herz–Morrey spaces, for example. Parabolic Morrey spaces can be found [72, 175]. Morrey spaces of the anisotropic setting can be found in [23, 91, 452] and singular operators acting on Morrey spaces in this setting were investigated in [453], while their local counterpart and their local Morrey-type counterpart can be found in [10] and [11, 12], respectively. Some authors used Lie groups as the underlying space; see [95] and [94, 146, 153, 164] for Morrey spaces over the Heisenberg group and Carnot groups, respectively. We can consider Morrey spaces over the padic field. See [76, 491, 509] for Morrey spaces and related function spaces over the p-adic field. Section 14.5.2 Liu, Sawano and Yang defined Gauss Morrey spaces in [270].

142

Morrey Spaces

Section 14.5.3 The example in this section is from [408]. The theory developed here is expanded in [409, 410, 411]. The notion of locally doubling metric measure spaces was first introduced in [279]. See [449, 536] for the analysis of Morrey spaces over metric measure spaces; Zhang and Liu took up singular integral operators [536]. Sawano and Sihwaningrum considered fractional integral operators [449]. See [530] for Sobolev– Morrey spaces over metric measure spaces. Sarikaya and Yildirim considered Morrey spaces over the half space Rn+ generated by a shift operator [400]. See [308, 309] for the boundedness property of the maximal operator and singular integral operators on grand Morrey spaces in metric measure spaces. Duong and Yan used generalized approximations to identity to consider the average to deal with Morrey–Campanato spaces over metric measure spaces [90]. Bilalov and Quliyeva worked in rectifiable curves in C [30]. Guliyev and Mustafayev considered singular integral operators and the maximal operators acting on generalized Morrey spaces in the anisotropic setting in [168]. Lukkassen, Lars and Samko considered Hardy type operators in local vanishing Morrey spaces over fractals [291].

Chapter 15 Weighted Morrey spaces

Chapter 15 examines the boundedness of linear and sublinear operators in weighted Morrey spaces and weighted local Morrey spaces. There are mainly two types of weighted Morrey spaces. In 2008, Natasha Samko defined weighted Morrey spaces of Samko-type and in 2009 Komori(-Furuya) and Shirai defined weighted Morrey spaces of Komori–Shirai-type. Let us describe some backgrounds and recent research on weighted Morrey spaces. Compared to generalized Morrey spaces, weighted Morrey spaces emerged a little later. Komori-Furuya and Shirai investigated weighted Morrey spaces mainly in connection with the boundedness of operators. They assumed that the weight belongs to Aq with q ∈ [1, ∞). Meanwhile, Samko considered weighted Morrey spaces in the context of the Cauchy integral. In fact, one m Q occasionally considers the weight ρ of the form ρ = ωk (| · −xk |). See [391, k=1

Section 6] for the original research of Morrey spaces equipped with the weight of the above type, whose motivation dates back to the Cauchy integral on the complex plane. Chapter 15 proves an analogy in weighted Morrey spaces. Let 0 < q ≤ p < ∞, and let u and w be weights. The weighted Morrey space Mpq (u, w) may be defined as ! q1 Z kf kMpq (u,w) ≡

sup

1

1

u(B(x, r)) p − q

|f (y)|q w(y)dx

(x,r)∈Rn+1 +

(15.1)

B(x,r)

for f ∈ L0 (Rn ). In particular, kf kMpq (1,w) ≡

|B(x, r)|

sup

1 1 p−q

(x,r)∈Rn+1 +

! q1

Z

q

|f (y)| w(y)dx

(15.2)

B(x,r)

and kf kMpq (w,w) ≡

sup (x,r)∈Rn+1 +

w(B(x, r))

1 1 p−q

Z

! q1 q

|f (y)| w(y)dx

(15.3)

B(x,r)

for 0 < q ≤ p < ∞. The norm defined by (15.2) is called the weighted Morrey norm of Samko-type (Section 15.1), while that defined by (15.3) is 143

144

Morrey Spaces

called the weighted Morrey norm of Komori–Shirai-type (Section 15.2 ). If u(x) ≡ |x|α , v(x) ≡ |x|β with α, β ∈ R, then Mpq (u, w) is a power weighted Morrey space. In this case, the symbol Mpq (|x|α , |x|β ) is used. Although the boundedness property of operators in both types cannot be completely characterized, a standard sufficient condition can be postulated. Likewise, we consider local weighted Morrey spaces or weak weighted Morrey spaces. For example, we do not define LMpq (1, w), WMpq (w, w) or WMpq (1, w). However, we can guess what they mean. We next refer to the motivation for considering weighted Morrey spaces. The weighted Morrey norm (15.3) of Komori–Shirai-type can be located as the generalized Morrey norm considered in Section 14.4. The norm (15.2) arises naturally when we consider the elliptic differential equation of the   n P ∂u ∂ a w = f, on Rn where the functions ajk satisfy form − jk ∂xj ∂xk j,k=1

n P

ajk (x)ξi ξj ≥ θ|ξ|2 for almost all x ∈ Rn and for all ξ = (ξ1 , ξ2 , . . . , ξn ) ∈

j,k=1 n

1,2 R . Let u ∈ Wloc (Rn ) be the weak solution that u satisfies n Z X j,k=1

ajk (x)

Rn

∂u ∂v (x) (x)w(x)dx = ∂xj ∂xk

Z f (x)v(x)dx Rn

for all v ∈ W 1,2 (Rn ) with compact support. The space Mpq (1, w) has a lot to do with the localized a priori estimate. Let Ω be a bounded domain and f ∈ L2 (Ω). Consider the weak solution to ( −div[A(x)∇u] = f in Ω, u=0 on ∂Ω, where A = {ajk }j,k=1,2,...,n is an n × n-matrix whose component ajk belongs to L∞ (Rn ) such that n X

ajk (x)ξj ξk ≥ θ

j,k=1

n X

|ξk |2

(x ∈ Ω, ξ1 , ξ2 , . . . , ξn ∈ R)

k=1

for some θ > 0. A weak solution to this equation is a unique element u ∈ H01 (Ω) ≡ Cc∞ (Ω)

H 1 (Ω)

such that n Z X j,k=1

Ω

Z aij (x)∂j u(x)∂k v(x)dx =

f (x)v(x)dx. Ω

Let κ ∈ C ∞ (Rn ) be a bump function χB(1) ≤ κ ≤ χB(2) . One of Morrey’s fundamental techniques is substituting v = ϕu, where ϕ is a function   proposed · − x0 for x0 ∈ Ω and r > 0 satisfying B(x, 3r) ⊂ Ω. of the form ϕ = κ r

Weighted Morrey spaces

145

After substituting this v into the above equation, we are led to a term of the form   Z 1 x − x0 × u(x)aij (x)∂i u(x)∂j κ dx. r r Ω Due to the factor 1r , we feel that the weighted Morrey norm of Samko-type seems to be suitable when we want to localize the estimate by using ϕ above. The analysis of weighted Morrey spaces seems difficult because we cannot apply the general theory of Berezhnoi [26] due to the fact that Morrey spaces do not satisfy the assumption there.

15.1

Weighted Morrey spaces of Komori–Shirai-type

We investigate weighted Morrey spaces of Komori–Shirai-type for a start. We organize Section 15.1 as follows: Section 15.1.1 investigates weighted Morrey spaces of Komori–Shirai-type. We handle maximal operators, singular integral operators and fractional maximal operators in Sections 15.1.2, 15.1.3 and 15.1.4, respectively. Section 15.1.4 also handles fractional integral operators.

15.1.1

The structure of weighted Morrey spaces of Komori–Shirai-type

We start with some preliminary observations in power weighted Morrey spaces. In the power weight case, we investigate how strong a singularity at the origin we can tolerate. Although we are mainly concerned with weighted Morrey spaces of Komori–Shirai-type, we can still consider power weighted Morrey spaces of Samko-type in this example. Example 58. Set wδ (x) ≡ |x|δ , x ∈ Rn for δ ∈ R. (1) Let α ∈ R. In order that wα ∈ L1loc (Rn ) it is necessary and sufficient that α > −n. (2) Let α, β > −n and γ ∈ R. Then wγ ∈ Mpq (|x|α , |x|β ) if and only if α+n α−β γ=− + . In fact, for any ball B(x, r), p q 1

1

1

wα (B(x, r)) p − q ∼ (max(r, |x|)α rn ) p

− q1

and 1

wβ+γq (B(x, r)) q ∼ max(r, |x|)β+γq rn so that we have   1 1 β α − + + γ ≤ 0, p q q

 (α + n)

1 1 − p q


q1

,

 + (β + γq + n)

1 = 0. q

146

Morrey Spaces Observe that the inequality above is automatic, once the equality holds. For example, we have the following: (a) 1 ∈ / Mpq (|x|α , |x|α ) for all α > −n. (b) w−n ∈ Mpq (|x|α , |x|β ) if and only if

α+n α−β − = n. p q

(3) Let α, β > −n and γ ∈ R. Thus χB(1) wγ ∈ Mpq (|x|α , |x|β ) if and only if α+n α−β α+n α−β γ≥− + . In fact, if γ ≥ − + , this is a direct conp q p q α+n α−β + , sequence of (2), since wγ χB(1) ≤ w− α+n + α−β . If γ < − p q p q then we argue as in (2). As a direct consequence of this observation, we have the following:   q q (a) χB(1) ∈ Mpq (|x|α , |x|β ) if and only if β ≥ α 1 − − n. p p In particular, χB(1) ∈ Mpq (|x|α , |x|α ) whenever α > −n, which reflects the fact that kχB(1) kMpq (|x|α ,|x|α ) = kχB(1) kLp (|x|α ) . Likeqn wise χB(1) ∈ Mpq (1, |x|β ) if and only if β ≥ − (> −n). p   q q p α β +qn− n. (b) w−n χB(1) ∈ Mq (|x| , |x| ) if and only if β ≥ α 1 − p p In particular, w−n χB(1) ∈ Mpq (|x|α , |x|α ) if and only if α ≥ n(p−1) qn while w−n χB(1) ∈ Mpq (1, |x|β ) if and only if β ≥ nq − . p The same applies to local weighted Morrey spaces. The dilation relation in Mpq (|x|α , |x|α ) is given by the following formula: For the power weight, we have the following scaling law: Example 59. Let t > 0. Denote by Dt the dilation operator given by Dt f ≡ f (t·) for f ∈ L0 (Rn ). Let 0 < q ≤ p < ∞ and α ∈ R. n+α

(1) Then for f ∈ L0 (Rn ), kf (t·)kMpq (|x|α ,|x|α ) ≡ t− p kf kMpq (|x|α ,|x|α ) , as is Z Z α n+α easily seen from the scaling law: |x| dx = t |x|α dx for {tx : x∈E}

E

any measurable set E. 0

(2) Let q > 1. Denote by Hqp0 (|x|α , |x|α ) the K¨othe dual of Mpq (|x|α , |x|α ). Then we have kDt gkHp0 (|x|α ,|x|α ) = t q0

0 Hqp0 (|x|α , |x|α ).

n+α p −n

kgkHp0 (|x|α ,|x|α ) for g ∈ q0

In fact, by a change of variables,

kDt gkHp0 (|x|α ,|x|α ) = sup{kDt g · f kL1 : kf kMpq (|x|α ,|x|α ) = 1} q0

= t−n sup{kg · Dt−1 f kL1 : kf kMpq (|x|α ,|x|α ) = 1}.

Weighted Morrey spaces

147

If we replace Dt−1 f by h, then we have kDt gkHp0 (|x|α ,|x|α ) = t−n sup{kg · hkL1 : kDt hkMpq (|x|α ,|x|α ) = 1} q0

= t−n sup{kg · hkL1 : khkMpq (|x|α ,|x|α ) = t =t =t

n+α p −n n+α p −n

n+α p

}

sup{kg · hkL1 : khkMpq (|x|α ,|x|α ) = 1} kgkHp0 (|x|α ,|x|α ) . q0

15.1.2

Maximal operators in weighted Morrey spaces of Komori–Shirai-type

When w ∈ Aq , we have the following boundedness: Theorem 130. Let 1 < q ≤ p < ∞, and let w ∈ Aq . Then kM f kMpq (w,w) . kf kMpq (w,w) for all f ∈ L0 (Rn ). Proof By the openness property, w ∈ Ar for some r ∈ (1, q). By H¨older’s  r1  Z 1 |f (y)|r w(x)dy for all x ∈ inequality, M f (x) ≤ [w]Ar sup χQ (x) w(Q) Q Q∈Q Rn . Since w is doubling, we obtain the desired result from Theorem 120. Consequently, it turned out that if the weight is in Aq , then the maximal operator is bounded. However it is important that we do not have to require that the weight belong to Aq for the boundedness of the maximal operator. Proposition 131. Let 1 < q < p < ∞ and β ∈ R. Then the following are equivalent: (1) The maximal operator M is bounded on Mpq (|x|β , |x|β ); (2) −n < β < n(p − 1); (3) | · |β ∈ Ap . We borrowed from the idea of [89] for the proof of Proposition 131. Remark that (2) is a consequence of general theory once we assume (1); see Exercise 38. However, we give a different proof of “(1) =⇒ (2)”. As it turns out, we will classify the cubes into three types according to [107] 1. Type I: cubes centered at the origin. 2. Type II: cubes Q(x, r) centered at x 6= 0 and such that 4r ≤ |x|. 3. Type III: cubes Q(x, r) centered at x 6= 0 and such that 4r > |x|.

148

Morrey Spaces

We will not have to stick the number 4r; any other number br will do as long as b  1. In the power weight case, it turns out that we do not have to take into account the cubes of type III, since the doubling property allows the cubes of type I to cover the ones of type III. Proof Assume (1). Then we have | · |β ∈ L1loc (Rn ) but χB(1) | · |−n ∈ / This is equivalent to (2) according to Example 58. A direct calculation shows (2) and (3) are equivalent. Consequently, it remains to show (1) assuming (2). Let −n < β < n(p − 1). Write wβ (x) ≡ |x|β for x ∈ Rn . Let Q be a cube and f ∈ L0 (Rn ). We need to show that Mpq (|x|β , |x|β ).

1

1

wβ (Q) p − q k(M f )χQ kLq (|x|β ) . kf kMpq (|x|β ,|x|β )

(15.1)

for all cubes Q. If −n < β < n(q − 1), then we can use Theorem 130 since wβ ∈ Aq . Hence, we assume otherwise; β ≥ n(q − 1).  Q  We may assume that q is centered at the origin or that 0 ∈ / 4Q. Let ε ∈ 0, (n(p − 1) − β) . We p distinguish four cases: (i) Assume that f is supported on 3Q and that Q is centered at the origin. Thus Q is of type I. In this case, wβ (Q) ∼ `(Q)n+β and k(M f )χQ kLq (|x|β ) . `(Q)

β−n(q−1)+ε q

k(M f )χQ kLq (|x|n(q−1)−ε ) ,

since β ≥ n(q − 1) > n(q − 1) − ε. Since | · |n(q−1)−ε ∈ Aq , we have k(M f )χQ kLq (|x|n(q−1)−ε ) . kf χ3Q kLq (|x|n(q−1)−ε ) . We next decompose 3Q dyadically to obtain kf χ3Q kLq (|x|n(q−1)−ε ) ≤

Z ∞ X l=0

! q1 q

n(q−1)−ε

|f (x)| |x|

dx

3·2−l Q\3·2−l−1 Q

∞ X n(q−1)−ε−β q ∼ (2−l `(Q)) l=0

! q1

Z

q

β

|f (x)| |x| dx 3·2−l Q\3·2−l−1 Q

By the use of the weighted Morrey norm, we obtain kf χ3Q kLq (|x|n(q−1)−ε ) .

∞ X n(q−1)−ε−β n+β n+β q (2−l `(Q)) (2−l `(Q)) q − p kf kMpq (|x|β ,|x|β ) . l=0

Arithmetic shows that n(q − 1) − ε − β n+β n+β n(p − 1) − β ε + − = − > 0. q q p p q

.

Weighted Morrey spaces

149

Thus, the series in the most right-hand side converges to obtain `(Q)n−

n+β ε p −q

kf χ3Q kLq (|x|n(q−1)−ε ) . kf kMpq (|x|β ,|x|β ) .

(15.2)

Consequently, we have (15.1). (ii) Assume that f is supported outside 3Q and that Q is centered at the origin. Thus Q is of type I. In this case, we have 1

1

wβ (Q) p − q k(M f )χQ kLq (|x|β ) . `(Q)

n+β p

sup mQ(R) (|f |) R>`(Q)

. sup `(Q)

n+β p

l∈N

1 l |2 Q|

Z |f (y)|dy. 2l Q\Q

Let l ∈ N be fixed. We decompose 2l Q \ Q dyadically to obtain Z l Z X |f (y)|dy. |f (y)|dy = 2l Q\Q

k=1

2k Q\2k−1 Q

If we use H¨ older’s inequality, then Z n+β 1 p |f (y)|dy `(Q) |2l Q| 2l Q\Q ≤ `(Q)

n+β p

l X

1 |2k Q|

2n(k−l)

k=1

! q1

Z

|f (y)|q dy

.

2k Q\2k−1 Q

Since |y| ∼ |z| for any y, z ∈ 2k Q \ 2k−1 Q, Z n+β 1 |f (y)|dy `(Q) p |2l Q| 2l Q\Q . `(Q)

n+β β p −q

l X

n(k−l)− β qk

2

k=1

.

l X

2n(k−l)−

n+β p k

2n(k−l)−

n+β p k

1 |2k Q|

! q1

Z

q

β

|f (y)| |y| dy 2k Q\2k−1 Q

kf kMpq (|x|β ,|x|β ) .

k=1

Since

l X

. 1, we obtain (15.1).

k=1

(iii) Assume that f is supported on 3Q and that 0 ∈ / 4Q. Thus Q is of type II. Then 1

1

1

1

β

1

1

β

wβ (Q) p − q k(M f )χQ kLq (|x|β ) ∼ (|c(Q)|β |Q|) p − q |c(Q)| q k(M f )χQ kLq . (|c(Q)|β |Q|) p − q |c(Q)| q kf χ3Q kLq 1

1

. wβ (Q) p − q kf χ3Q kLq (|x|β ) . kf kMpq (|x|β ,|x|β ) .

150

Morrey Spaces Consequently, we have (15.1).

(iv) Assume that f is supported outside 3Q and that 0 ∈ / 4Q. Thus Q is of type II. Then 1

1

1

β

1

wβ (Q) p − q k(M f )χQ kLq (|x|β ) ∼ (|c(Q)|β |Q|) p − q |c(Q)| q kχQ M f kLq 1

. (|c(Q)|β |Q|) p

mR (|f |)

sup R∈Q] (Q)

1

∼ (|c(Q)|β |Q|) p

mR (|f |).

sup R∈Q] (Q) c(Q)=c(R)

If 0 ∈ / 8R, c(Q) = c(R) and Q ⊂ R, then 1 (|c(Q)| |Q|) mR (|f |) . (|c(R)| |R|) wβ (R) 1 p

β

β

∼ wβ (R)

1 p

1 p

1 wβ (R)

Z

|f (x)||x|β dx

R\3Q

! q1

Z

|f (x)|q |x|β dx

R\3Q

≤ kf kLMpq (|x|β ,|x|β ) . Suppose instead that 0 ∈ 8R, c(Q) = c(R) and Q ⊂ R. Then Z  10 q ε 1 −n+ q−1 kf χR kLq (|x|n(q−1)−ε ) |x| mR (|f |) ≤ dx |R| R ε 1 ∼ `(R) q kf χR kLq (|x|n(q−1)−ε ) . |R| From (15.2) we have kf χR kLq (|x|n(q−1)−ε ) . `(R)

−β+n(q−1)−ε q

1

1

wβ (R)− p + q kf kMpq (|x|β ,|x|β ) .

As a result, 1

1

1

wβ (Q) p − q k(M f )χQ kLq (|x|β ) . wβ (R) q `(R)−

β+n q

kf kMpq (|x|β ,|x|β )

. kf kMpq (|x|β ,|x|β ) . Combining these results gives (15.1).

15.1.3

Singular integral operators acting on weighted Morrey spaces of Komori–Shirai-type

If w ∈ Aq , we can handle singular integral operators with ease. Theorem 132. Let 1 < q ≤ p < ∞ and w ∈ Aq . Also let T be a singular n integral operator. Then kT f kMpq (w,w) . kf kMpq (w,w) for all f ∈ L∞ c (R ).

Weighted Morrey spaces

151

Proof Let Q be a cube. Then we will show that 1

1

w(Q) p − q kχQ T [χ3Q f ]kLq (w) . kf kMpq (w,w) and that

1

(15.3)

1

w(Q) p − q kχQ T [χRn \3Q f ]kLq (w) . kf kMpq (w,w) .

(15.4)

q

The proof of (15.3) is simple. Use the L (w)-boundedness of T as usual. For the proof of (15.4), we use Example 124 in the first book. Since we have a size condition, kT [χRn \3Q f ]kL∞ (Q) .

∞ X

m2l Q (|f |).

(15.5)

l=1

Thus, combining Example 124 in the first book and (15.5) gives 1 p

w(Q) kT [χRn \3Q f ]kL∞ (Q) .

1 ∞ X w(Q) p 1

l=1

w(2l Q) p

kM f kMpq (w,w) . kM f kMpq (w,w) .

Since M is bounded on Mpq (w, w) thanks to Theorem 130, we obtain (15.4). As before, for the power weight, we have the following characterization: Proposition 133. Let 1 < q < p < ∞ and β ∈ R. (1) If −n < β < n(p − 1), then any singular integral operator T is bounded β β n p fpq (|x|β , |x|β ), the closure of L∞ on M c (R ) in Mq (|x| , |x| ). (2) If the first Riesz transform R1 can be extended to a bounded linear operator on Mpq (|x|β , |x|β ), then −n < β < n(p − 1). Proof (1) Argue as in Theorem 132. See Exercise 39. (2) Assume that R1 is bounded on Mpq (|x|β , |x|β ). Then as before we have | · |β ∈ L1loc (Rn ) but χB(1) | · |−n ∈ / Mpq (|x|β , |x|β ). This is equivalent to −n < β < n(p − 1) according to Example and 58. For the purpose of dealing with singular integral operators on Mpq (1, |x|α ) or Mpq (|x|α , |x|α ), consider the following definition: Definition 30. Let u, w be weights, and let 0 < q ≤ p < ∞. Let the tilde n p fpq (u, w) be the closure of L∞ closed subspace M c (R ) in Mq (u, w). Accordingly, p p p α p α β fq (w, w), M fq (1, w), M fq (1, |x| ) and M fq (|x| , |x| ) can be defined for α, β ∈ M R and a weight w. 0

Arguing as in Theorem 356 and the duality result of Hqp0 (Rn ) in the first book, we have the following theorem:

152

Morrey Spaces

Theorem 134. Let 1 < q ≤ p < ∞ and w be a weight. Then the bidual of fpq (w, w) is canonically identified with Mpq (w, w). M The proof of Theorem 134 is omitted due to similarity to the unweighted case. We can follow the line as in Chapter 10. We omit further details. See Exercise 40. Using Theorems 130 and 134, we can extend T as a bounded linear operator on Mpq (w, w) for 1 < q ≤ p < ∞ and w ∈ Aq .

15.1.4

Fractional maximal operators and fractional integral operators in weighted Morrey spaces of Komori–Shirai-type

We aim here to investigate the fractional integral operator Iα and the fractional maximal operator Mα . As we saw, if w ∈ Aq , then we can handle the maximal operator and singular integral operators easily. Likewise, if we assume that w belongs to a suitable Aq,t class, we can control fractional integral operators and fractional maximal operators. Here, instead of considering such a situation, we consider the power weight case. Proposition 135. Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞. Assume that 1 α 1 1 α 1 = − , = − . s p n t q n Then the following are equivalent: (1) The fractional maximal operator Mα is bounded from Mpq (|x|tβ , |x|qβ ) n ns to Mpq (|x|tβ , |x|tβ ) if and only if − ≤ β < 0 . t pt (2) The fractional integral operator Iα is bounded from Mpq (|x|tβ , |x|qβ ) to n ns Mpq (|x|tβ , |x|tβ ) if and only if − < β < 0 . t pt We borrow from the idea of [89] for the proof of Proposition 135. Proof (1) As before, if the fractional maximal operator Mα is bounded from n ns Mpq (|x|tβ , |x|qβ ) to Mpq (|x|tβ , |x|tβ ), then − ≤ β < 0 ; in fact, thanks t pt tβ + n to Example 58, χB(1) ∈ Mpq (|x|tβ , |x|tβ ) if and only if 0 ≥ and p tβ + n tβ − qβ | · |−n χB(1) ∈ Mpq (|x|tβ , |x|qβ ) if and only if −n ≥ − + . p q n ns If − ≤ β < 0 , then we can mimic the argument in (2) below. In t pt particular, instead of taking the summation in (b) in (2) below, we can consider the supremum. So, the details are omitted.

Weighted Morrey spaces

153

(2) If the fractional integral operator Iα is bounded from Mpq (|x|tβ , |x|qβ ) n ns to Mpq (|x|tβ , |x|tβ ), then − ≤ β < 0 since the fractional maximal t pt operator Mα is bounded from Mpq (|x|tβ , |x|qβ ) to Mpq (|x|tβ , |x|tβ ). Since n | · |−α χRn \B(1) ∈ / Mpq (|x|tβ , |x|qβ ), we have β 6= − . Conversely assume t n ns n that − < β < 0 . Let f ∈ L∞ c (R ). If β ≤ 0, then we can argue as t pt Komori and Shirai did in [243, Theorem 3.6] using a similar technique to Theorem 130; see Exercise 41. Let us assume β > 0. We distinguish four cases as before to show that 1

1

wtβ (Q) s − t kIα f kLt (|x|tβ ) . kf kMpq (|x|tβ ,|x|qβ ) ,

wtβ (x) ≡ |x|tβ . (15.6)

Take an auxiliary parameter τ slightly larger than

t β. s

(a) If Q is a cube centered at 0 and if f is supported on 4Q, then 1

1

wtβ (Q) s − t k(Iα f )χQ kLt (|x|tβ ) 1

1

. wtβ (Q) p − q `(Q)β−τ k(Iα f )χQ kLt (|x|tτ ) . By the Ap,q theory, k(Iα f )χQ kLt (|x|tτ ) . kf kLq (|x|qτ ) . We decompose ∞ X kf kLq (|x|qτ ) ≤ kχ22−j Q\21−j Q f kLq (|x|qτ ) . j=0

From the definition of the norm, kf kLq (|x|qτ ) . .

∞ X (2−j `(Q))τ −β kχ22−j Q\21−j Q f kLq (|x|qβ ) j=0 ∞ X

1

1

(2−j `(Q))τ −β wtβ (2−j Q) q − p kf kMpq (|x|tβ ,|x|qβ ) .

j=0

We observe that       1 1 1 1 t 1 1 τ −β +(n+tβ) − = τ +n − − β≥n − > 0, q p q p s q p t β < τ for the penultimate inequality. Thus, the s series is summable and we have (15.6). where we used

(b) If Q is a cube centered at 0 and if f is supported outside 4Q, then 1

1

1

wtβ (Q) s − t k(Iα f )χQ kLt (|x|tβ ) . wtβ (Q) s kf kL1 (|x|α−n )

(15.7)

154

Morrey Spaces from the expression of Iα f (x). We note that kf kL1 (|x|α−n ) .

∞ X

1 α j |2 Q|1− n

Z

`(2j Q)α

1 j `(2 Q)n

j=1

.

∞ X j=1

.

∞ X

|f (y)|dy 2j Q\2j−1 Q

t

`(2j Q)α−β+ q β

1 q

=

1 t

+α n,

1 p

=

|f (y)|q dy

2j Q\2j−1 Q

kf χ2j Q\2j−1 Q kLq (|x|qβ ) 1

.

(wtβ (2j Q)) q

j=1

Since

! q1

Z

1 s

+α n , we have t

`(2j Q)α−β+ q β 1

wtβ (2j Q) p

=

`(2j Q)α+

tαβ n

α

1

wtβ (2j Q) n wtβ (2j Q) s 1

∼ wtβ (2j Q)− s j

1

= 2− s (tβ+n) wtβ (Q)− s . Since β > 0, 1

kf kL1 (|x|α−n ) . wtβ (Q)− s kf kMpq (|x|tβ ,|x|qβ ) .

(15.8)

Combining (15.7) and (15.8) gives (15.6). (c) If Q is a cube such that 0 ∈ / 32Q and if f is supported on 4Q, then by the classical Hardy–Littlewood–Sobolev theorem 1

1

1

1

1

1

1

1

I ≡ wtβ (Q) s − t k(Iα f )χQ kLt (|x|tβ ) . wtβ (Q) s − t |c(Q)|β kIα f kLt . wtβ (Q) s − t |c(Q)|β kf kLq ∼ wtβ (Q) s − t kf kLq (|x|qβ ) . kf kMpq (|x|tβ ,|x|qβ ) , proving (15.6). (d) If Q is a cube such that 0 ∈ / 32Q and if f is supported outside 4Q, then Z 1 |f (y)|dy s I . wtβ (Q) |y − c(Q)|n−α n R from the integral expression of Iα f . Consequently I . wtβ (Q)

1 s

∞ X j=1

1 α j |2 Q|1− n

Z |f (y)|dy. 2j+1 Q\2j Q

Weighted Morrey spaces

155

Let j0 ≥ 6 be the smallest integer such that 0 ∈ 2j0 −1 Q. Since |y| ∼ `(2j0 Q) for all y ∈ 2j0 −3 Q, we have II ≡

jX 0 −4 j=1

≤

jX 0 −4

1 α |2j Q|1− n j

`(2 Q)

α

j=1

.

jX 0 −4

j

Z |f (y)|dy 2j+1 Q\2j Q

|f (y)| dy

`(2 Q) |c(Q)|

j=1

.

jX 0 −4

q

2j+1 Q\2j Q

−β

α

! q1

Z

1 |2j Q|

1 j |2 Q|

! q1

Z

q

qβ

|f (y)| |y| dy 2j+1 Q\2j Q

n

1

1

`(2j Q)α− q |c(Q)|−β wtβ (Q) q − p kf kMpq (|x|tβ ,|x|qβ ) .

j=1

j

We note `(2 Q)

α− n q

Z

 q1 − p1

tβ

|x| dx

n

tβ

∼ `(2j Q)− s |c(Q)| q

− tβ p

2j Q

for j ≤ j0 − 4, since α −

II .

jX 0 −4

n n = − . As a result, since p s tβ

n

`(2j Q)− s |c(Q)| q

− tβ p −β

1 p

− 1q =

1 s

− 1t ,

kf kMpq (|x|tβ ,|x|qβ )

j=1 1

∼ wtβ (Q)− s kf kMpq (|x|tβ ,|x|qβ ) . Consequently, wtβ (Q)

1 s

jX 0 −4 j=1

1 α |2j Q|1− n

Similarly, using β < wtβ (Q)

ns p0 t ,

Z 2j+1 Q\2j Q

we have

∞ X

1 s

j=j0

|f (y)|dy . kf kMpq (|x|tβ ,|x|qβ ) .

Z

1 j Q|1− α n |2 −3

2j+1 Q\2j Q

|f (y)|dy . kf kMpq (|x|tβ ,|x|qβ ) .

Thus, combining these observations together gives 1

wtβ (Q) s

∞ X j=1

1 α j |2 Q|1− n

Z 2j+1 Q\2j Q

|f (y)|dy . kf kMpq (|x|tβ ,|x|qβ ) .

All together then, we conclude that Iα is bounded from Mpq (|x|tβ , |x|qβ ) to Mpq (|x|tβ , |x|tβ ).

156

15.1.5

Morrey Spaces

Exercises

Exercise 37. [142, Remark 1] In Definitions 34 and 35, prove the following: (1) Let 1 ≤ q < ∞ and ϕ ∈ Gq . If w ≡ 1, then show that Mϕ q (w, w) = n Mϕ (R ). q 1

(2) Let 1 ≤ q ≤ p < ∞. If ϕ(x, r) ≡ w(B(x, r)) p for (x, r) ∈ Rn+1 + , then p show that Mϕ (w, w) = M (wdx), where the right-hand side is given by q q (14.1), the definition of Morrey spaces with general Radon measures. 1

(3) Let 1 ≤ q < ∞. If ϕ(x, r) ≡ w(B(x, r)) q for (x, r) ∈ Rn+1 + , then show q that Mϕ (w, w) = L (w). q Exercise 38. Let 1 ≤ q ≤ p < ∞, and let w be a weight. (1) Using Theorem 19 in the first book, show that Lp (w) ,→ Mpq (w, w). (2) By mimicking the proof of Proposition 286 in the first book, show that M is bounded from Lp (w) to WMpq (w, w) if and only if w ∈ Ap . (3) Show that M is bounded from Mpq (w, w) to WMpq (w, w) if w ∈ Ap . Exercise 39. Prove Proposition 133(1) by using a similar argument to Theorem 132. Exercise 40. Prove Theorem 134 mimicking the proof of Theorem 356 in the first book. Exercise 41. [243, Theorem 3.6] Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞. Assume that 1 1 α = − , s p n

q t = , p s

−

n < β ≤ 0. t

Then using the class Aq,t show that the fractional integral operator Iα is bounded from Mpq (|x|tβ , |x|qβ ) to Mpq (|x|tβ , |x|tβ ).

15.2

Weighted Morrey spaces of Samko-type

We now investigate weighted Morrey spaces of Samko-type. Let 1 ≤ q ≤ p < ∞, be a function, and let w be a weight. The weighted Morrey space Mpq (1, w) is defined as the subset of all f ∈ L0 (Rn ) satisfying kf kMpq (1,w) ≡ sup |Q| Q∈Q

1 1 p−q

Z

q

|f (x)| w(x)dx Q

 q1

1

= kf w q kMpq < ∞. (15.1)

Weighted Morrey spaces

157

We can also introduce the weak weighted Morrey space WMpq (1, w) of Samko type with the following norm: kf kWMpq (1,w)

1

1

sup |Q| p − q kf χQ kWLq (w)

≡

Q∈Q

=

sup

|Q|

1 1 p−q

 q1

Z λ

.

χ(λ,∞] (|f (x)|)w(x)dx

Q∈Q,λ>0

Q

We are interested in the case of w(x) ≡ |x|α : 1

1

kf kMpq (1,|x|α ) ≡ sup |Q| p − q

 q1 < ∞. |f (x)|q |x|α dx

Z

Q∈Q

Q

Mixing these definitions, we also introduce the weak weighted Morrey space, denoted by WMpq (1, |x|α ), by the following norm: kf kWMpq (1,|x|α )

1

≡

sup |Q| p

kf χQ kWLq (|x|α ) 1

|Q| q   q1 Z 1 1 χ(λ,∞] (|f (x)|)|x|α dx . = sup |Q| p λ |Q| Q Q∈Q,λ>0 Q∈Q

Section 15.2 is organized as follows: We define weighted Morrey spaces of Samko-type in Section 15.2.1. We introduce the class Bα in Section 15.2.2. We study the Hardy–Littlewood maximal operator, singular integral operators, fractional maximal operators and fractional integral operators in Sections 15.2.3, 15.2.4, 15.2.5 and 15.2.6, respectively. We pass some of the results above to the vector-valued case in Section 15.2.7. Section 15.2.8 explains the weighted inequality of Stein-type. Iida and the Nakamura obtained a complete characterization of the dual inequality of Stein-type in weighted Morrey spaces Mpq (1, w) of Samko-type in [220].

15.2.1

The structure of weighted Morrey spaces of Samko-type

As the following example shows, the weighted Morrey space Mpq (1, w) contains a non-zero element: 1

Example 60. We fix any Q0 ∈ D(Rn ) and calculate the norm of w− q χQ0 : 1

1

= kχQ0 kMpq = |Q0 | p . kw− q χQ0 kD Mp q (1,w) It is useful to consider the weighted dyadic Morrey norm k · kD of Mp q (1,w) 0 n Samko type, which is defined for f ∈ L (R ) by kf kD Mp q (1,w)

≡ sup |Q| Q∈D

1 p



1 |Q|

Z

q

|f (x)| w(x)dx Q

 q1 .

158

Morrey Spaces

We notice the equivalence of norms: kf kMpq (1,w) ∼ kf kD . Hence, we will Mp q (1,w) use the dyadic norm instead of the ordinary one without comment. We introduce a weight class Bp,q and investigate the boundedness of some operators on a new class of Morrey spaces. We define the class Bp,q as follows: Definition 31 (Bp,q ). Let 1 ≤ q ≤ p < ∞. One says that a weight w is in the class Bp,q if for any Q0 ∈ Q, sup

Φp,w,q (Q) . Φp,w,q (Q0 )

(15.2)

Q∈Q,Q⊂Q0 1

1

1

holds, where Φp,w,q (Q) ≡ |Q| p − q w(Q) q for Q ∈ Q. When there is no possibility of confusion, write Φ = Φp,w,q . The advantage of the restriction Bp,q is that it becomes possible to calculate the norm of χQ for Q ∈ D(Rn ). Lemma 136. Let 1 ≤ q ≤ p < ∞ and w be a weight. For any Q ∈ D(Rn ), we 1 1 1 1 1 1 = sup |E| p − q w(E) q . Thus, if (15.2) have |Q| p − q w(Q) q ≤ kχQ kD Mp q (1,w) holds, then kχQ kMpq (1,w) ∼

E∈D(Q) 1 1 1 D kχQ kMpq (1,w) ∼ |Q| p − q w(Q) q .

Proof The left inequality follows from the definition of sup, by which the right equality is a consequence of the geometric property of dyadic cubes; when two dyadic cubes intersect, one is contained in the other. Example 61. Let 1 ≤ q ≤ p < ∞ and α > −n. We investigate for which α ∈ R the power function w(x) ≡ |x|α is in Bp,q . We claim that | · |α ∈ Bp,q if q and only if α ≥ − n. p To check this, we will also use the `∞ -distance defined by |x|∞ ≡ max (|x1 |, . . . , |xn |). However, we do not distinguish between the two norms | · |∞ and | · |, explicitly, because these are comparable. Let Q(x, R) denote the cube of length 2R centered at x in Rn . Let us calculate w(Q0 ) by cases, where Q0 = Q(x0 , R0 ). When 3Q0 does Z not contain the origin, this is easy. That is, |x|α dx ∼ |x0 |α · |Q0 | from Example 112

for |x0 | ≥ 3R0 , we have w(Q0 ) = Q0

in the first book. It is more difficult to handle the case where 3Q0 contains the origin. That is, we suppose |x0 | ≤ 3R0 . Using Example 112 in the first book, we obtain the condition of | · |α ∈ Bp,q . To show Φ(Q0 ) . Φ(Q1 ) for any Q0 = Q(x0 , R0 ), Q1 = Q(x1 , R1 ) satisfying Q0 ⊂ Q1 , we have to consider four possibilities: (1) |x0 | > 3R0 and |x1 | > 3R1 ,

(2) |x0 | > 3R0 and |x1 | ≤ 3R1 ,

(3) |x0 | ≤ 3R0 and |x1 | > 3R1 ,

(4) |x0 | ≤ 3R0 and |x1 | ≤ 3R1 .

Note that Q0 ⊂ Q1 implies |x0 | + R0 ≤ |x1 | + R1 and R0 ≤ R1 . It is clear that |x0 | ≤ 3R0 and |x1 | > 3R1 never happen at the same time, since Q0 ⊂ Q1 .

Weighted Morrey spaces

159 n

α

p q and Φ(Q ) ∼ If |x 0 | >α 3R0 and |x1 | > 3R1 , we have Φ(Q0 ) ∼ R0 |x0 | 1 n p q R1 |x1 | . If α ≥ 0, then by |x0 | + R0 ≤ |x1 | + R1 and 3R1 ≤ |x1 |, it follows that   αq α α n n n 4 p q p q p |x1 | ∼ Φ(Q1 ). Φ(Q0 ) ∼ R0 |x0 | ≤ R1 (|x1 | + R1 ) ≤ R1 3

If α < 0, by observing that |x0 | ≥ |x1 | − R1 ≥ 32 |x1 |, it holds that n

α



n

Φ(Q0 ) ∼ R0 p |x0 | q ≤ R1 p

2 |x1 | 3

 αq ∼ Φ(Q1 ).

Hence, we obtain Φ(Q0 ) . Φ(Q1 ) without assuming α ≥ − pq n. If |x0 | > 3R0 α

n

n

α

and |x1 | ≤ 3R1 , we have Φ(Q0 ) ∼ R0 p |x0 | q and Φ(Q1 ) ∼ R1 p + q . If α ≥ 0, then by observing |x0 | ≤ |x1 | + R1 ≤ 4R1 , we have α

n

α

n

Φ(Q0 ) ∼ R0 p |x0 | q ≤ R1 p (4R1 ) q ∼ Φ(Q1 ). n

α

α

n

n

α

If − pq n ≤ α ≤ 0, we have Φ(Q0 ) ∼ R0 p |x0 | q ≤ R0 p (3R0 ) q . R1 p + q ∼ α

n

Φ(Q1 ). If |x0 | ≤ 3R0 and |x1 | ≤ 3R1 , we have Φ(Q0 ) ∼ R0 p R0q and Φ(Q1 ) ∼ n

α q

R1 p R1 . Thus, the condition: Φ(Q0 ) . Φ(Q1 ) is equivalent to α ≥ − pq n. All together, we conclude that | · |α ∈ Bp,q holds if and only if α ≥ − pq n. Lemma 137. Let w be a doubling weight. Then there exists a constant κ > 1 which depends only on n, p, q and w such that κ−1 Φ(Q∗ ) ≤ Φ(Q) ≤ κΦ(Q∗ ). Proof Indeed, the conditions Q ⊂ Q∗ and `(Q∗ ) = 2`(Q) imply that Φ(Q) = |Q|

1 p



w(Q) |Q|

 q1

∗

≤ |Q |

1 p



w(Q∗ ) 2−n |Q∗ |

 q1

n

= 2 q Φ(Q∗ ).

Meanwhile, Let τ be the doubling constant of w; w(2Q) ≤ τ w(Q) for all Q ∈ Q. Then we obtain ∗

Φ(Q ) ≤ |3Q|

1 p



w(3Q) |Q∗ |

 q1

n

1

1

1

n

≤ 2− q τ |Q| p − q w(Q) q = 2− q τ Φ(Q). n

n

Hence, the conclusion holds with κ ≡ max(2 q , 2− q τ ). As it turns out, the class Aq ∩ Bp,q is important for our consideration. Consequently, we investigate the necessary and sufficient conditions for the power weights to belong to this class. Example 62. Let 1 < q < p < ∞. According to Theorem 288, | · |α ∈ Aq if and only if −n < α < n(q − 1). Therefore, using Example 61, we see that q | · |α ∈ Aq ∩ Bp,q if and only if α satisfies − n ≤ α < n(q − 1). p

160

Morrey Spaces

Here, we will see that the quantity Φp,w,q (Q) plays an important role in many situations. We adjust Theorem 37 to our vector-valued and weighted setting. Lemma 138. Let 1 ≤ q ≤ p < ∞ and w ∈ Bp,q . The following are equivalent: (1) There exists a constant C0 > 0 such that the weighted integral condition Z

∞

1

1 ds C0 ≤ Φp,w,q (sQ) s Φp,w,q (Q)

(Q ∈ Q).

(15.3)

holds. (2) There exists a constant c1 > 1 such that 2Φp,w,q (Q) ≤ Φp,w,q (c1 Q)

(Q ∈ Q).

(15.4)

Moreover, we can take c1 = 2m for some m ∈ N. In addition, if w satisfies the above equivalent conditions, then
lim Φp,w,q c1 −k Q = ∞,

k→−∞


lim Φp,w,q c1 −k Q = 0,

k→∞

(15.5)

and for any ν ∈ (0, ∞) Z

∞

1

ds Cν 1 ≤ Φp,w,q (sQ)ν s Φp,w,q (Q)ν

(Q ∈ Q).

(15.6)

Proof Theorem 37 gives that (15.3) and (15.4) are equivalent. Note that (15.6) readily follows from (12.10). Finally, we will show that (15.3) or (15.4) implies (15.5). The weighted integral condition (15.3) gives us ∞ X 1 1 . < ∞, which means that lim Φp,w,q (Q) = ∞. j Q) Φ (2 Φ `(Q)→∞ p,w,q (Q) j=1 p,w,q Meanwhile, (15.4) yields

Φp,w,q c1 −1 Q ≤ 2−1 Φp,w,q (Q), . . . , Φp,w,q c1 −k Q ≤ 2−k Φp,w,q (Q). Since c1 > 1, lim c1 −k = 0. Thus, we conclude (15.5). k→∞

If we work on domains, the following example seems natural. Example 63. Let s ∈ R. Let Ω be a proper open subset of Rn . Denote by ρ the distance function from a point in Ω to its boundary. Then the weighted space normed by kρs f kMpq (Ω) is also a candidate of weighted Morrey spaces of Samko-type.

Weighted Morrey spaces

15.2.2

161

The class Bα

To investigate the boundedness properties of operators acting on weighted Morrey spaces of Samko-type, we introduce Bα . Recall that A1 is the Muckenhoupt class and that the symbol H d stands for the Hausdorff capacity of dimension d. Definition 32 (Bα ). Let 0 < α < n. Define the class Bα of weights by   Z α Bα ≡ b ∈ A1 : b(x)dH (x) ≤ 1 . Rn

We will use the following special cases. Example 64. (1) Let 0 < α < n and ε > 0. According to Theorem 281 in the first book and Example 151 in the first book, a typical example of an element in −1 α Bα is b ≡ γα,ε `(Q)−α M (( n +ε) ) χQ , where Q ∈ Q is arbitrary and the constant γα,ε is chosen so that b ∈ Bα . q 0 (2) For 1 < q ≤ p < ∞, the space H q ,n(1− p ) (Rn ) is defined by the set of all f ∈ L0 (Rn ) for which the quantity

Z kf k

q q 0 ,n 1− p H

(

) ≡ b∈Binf q n(1− ) p

q0

0

|f (x)|q b(x)− q dx

 10 q

< ∞.

Rn

Here, we present a typical example of an element in Ba . Lemma 139. Let 1 < q < p < ∞ and w(x) = wα (x) ≡ |x|α with − pq n ≤ α <   (Q ) n q − pq . Let Q0 ∈ D(Rn ). Then we can find a collection {bQ 0 }Q∈D(Q0 ) ∈ Bn(1− q ) satisfying p

1 1 1 |Q| p (mQ (w)) q |Q|

Z Q

(Q ) −q [bQ 0 (x)w(x)] q0

 10 q

dx

.p,q,w 1

(15.7)

and for any Q ∈ D(Q0 ) (Q0 )

bQ

q

q

(Q )

(x)|Q|1− p .p,q,w bQ00 (x)|Q0 |1− p

(15.8)

for a.e. x ∈ Q. Proof Fix any Q0 ∈ D(Rn ) and split the proof into the two cases. (Q )

q

q

−1

(1) Suppose 0 ∈ / 3Q0 . Then define bQ 0 ≡ γp,q,ε |Q|− p +1 M ((1− p +ε) ) χQ for each Q ∈ D(Q0 ), where 0 < ε < pq and γp,q,ε are fixed small numbers. (Q0 )

Note that bQ

∈ Bn(1− q ) . By the geometrical observation, we see that p

162

Morrey Spaces for any Q ∈ D(Q0 ), 0 ∈ / 3Q holds, so that (15.8) clearly holds for any q (Q ) Q ∈ D(Q0 ). It remains to check (15.7). We notice that bQ 0 (x) = |Q| p −1 for x ∈ Q and that wκ (Q) ∼ |c(Q)|κ |Q| for any κ ∈ R, since 3Q does not contain the origin. Hence, we have that Z  Q

− qq0

(Q ) bQ 0 (x)wα (x)

dx =

w−α q0 (Q) q

|Q|

q0

|Q| p0

0

q q 1+(1− p )q

· |Q|

∼

q0

|c(Q)|α q

.

Meanwhile, since we have wα (Q) ∼ |c(Q)|α |Q|, 1 1 |Q| p |Q|



wα (Q) |Q|

 q1

1 α 1 ∼ |Q| p |c(Q)| q |Q|

Z  Q



(Q ) bQ 0 (x)wα (x) 0

|c(Q)|

−α qq

|Q|

(Q0 )

proving (15.7). Therefore, we see that bQ

q0 p0

− qq0

! 10 q

dx

 10 q

= 1,

satisfies (15.7) and (15.8).

(2) Suppose 0 ∈ 3Q0 . In this case, for Q ∈ D(Q0 ), 3Q may contain the origin or not. When 3Q contains the origin, we define (Q0 )

bQ

q

(x) ≡ |Q|β−1+ p |x|−nβ

(x ∈ Rn ),

where we take β > 0 so that   q α < nβ + n(q − 1) < n 1 − + n(q − 1). p

(15.9)

q

(Q )

In particular, bQ00 (x) ≡ |Q0 |β−1+ p |x|−nβ . When 3Q does not contain (Q0 )

the origin, we define bQ (Q0 )

of all, bQ

q

q

≡ γp,q,ε |Q| p −1 M (1− p +ε) χQ , as before. First

∈ Bn(1− q ) no matter where Q is located. p

(Q )

q

(Q )

q

Next, we check (15.8): bQ 0 |Q|1− p .p,q,α bQ00 |Q0 |1− p . In the case of Q ∈ D(Q0 ) such that 0 ∈ 3Q, with β > 0 in mind, we have (Q0 )

bQ

q

(Q )

q

(x)|Q|1− p = |Q|β |x|−nβ ≤ |Q0 |β |x|−nβ = bQ00 (x)|Q0 |1− p .

When the case of Q ∈ D(Q0 ) such that 0 ∈ / 3Q, by observing that q √ (Q ) |x| ≤ 4 n`(Q0 ) for all x ∈ Q0 and bQ 0 (x)|Q|1− p = 1 for all x ∈ Q, we see that √ q q (Q ) bQ0 (Q0 )(x)|Q0 |1− p ≥ |Q0 |β (4 n`(Q0 ))−nβ ∼β 1 = bQ 0 (x)|Q|1− p , for all x ∈ Q, implying (15.8). Finally, we check the more complicated part of (15.7): we prove ) 10   q1 (Z  q − qq0 1 w (Q) α (Q ) −1 |Q| p bQ 0 (x)wα (x) dx .p,q,w 1. (15.10) |Q| Q

Weighted Morrey spaces

163

Since we can handle the case of 0 ∈ / 3Q similar to (1), we have only to show (15.10) in the case of 0 ∈ 3Q. Let us calculate Z  Q

(Q0 )

bQ

Z − qq0 q q0 q0 (x)wα (x) dx = |x|− q (α−nβ) dx · |Q|− q (β−1+ p ) . Q 0

Arithmetic shows that the power − qq (α − nβ) = − α−nβ q−1 > −n by the q0

choice of β > 0: α < nβ −n(q −1),. Hence, the power weight |·|− q (α−nβ) is integrable near the origin. Thus, with 0 ∈ 3Q in mind, we have that Z  Q

(Q0 )

bQ

− qq0 q0 0 β 1 1 dx ∼ |Q|− nq (α−nβ) |Q|1−q ( q − q + p ) (x)wα (x) q0 α

0

q0

= |Q|− nq +q − p . By inserting this equivalence into the left-hand side of (15.10), we obtain that ! 10   q1 Z  q − qq0 1 1 w (Q) α (Q ) 0 |Q| p bQ (x)wα (x) dx |Q| |Q| Q 1 α α 1 1 |Q| p |Q| nq |Q|− nq +1− p |Q| = 1,

∼

which implies (15.10). (Q0 )

In summary, we obtain the desired family {bQ

15.2.3

}Q∈D(Q0 ) ⊂ M+ (Rn ).

Maximal operator in weighted Morrey spaces of Samko-type

We investigate the boundedness of the maximal operator M on Mpq (1, w). Example 65. Let 1 < q < p < ∞, and let w be a weight. Suppose that M is bounded on Mpq (1, w). (1) Let us show w ∈ Aq+1 . Since we have the pointwise estimate: 1

1

χQ0 (x)mQ0 (w− q ) ≤ M [w− q χQ0 ](x)

(x ∈ Rn ),

the boundedness of M implies that 1

1

mQ0 (w− q )kχQ0 kMpq (1,w) ≤ kM kMpq (1,w)→Mpq (1,w) kw− q χQ0 kMpq (1,w) 1

∼ |Q0 | p .

164

Morrey Spaces If we notice that kχQ0 kMpq (1,w) ≥ Φp,q,w (Q0 ), then it follows that 1

1

(mQ0 (w)) q mQ0 (w− q ) ≤ kM kMpq (1,w)→Mpq (1,w) , which implies w ∈ Aq+1 with [w]Aq+1 ≤ (kM kMpq (1,w)→Mpq (1,w) )q . We can refine the conclusion: we can show that w ∈ Aq+1− pq in Exercise 45. 1

1

(2) Since χB((2,2,...,2),1) w− q ∈ Mpq (1, w) and M [χB((2,2,...,2),1) w− q ] & χB(1) , we must have χB(1) ∈ Mpq (1, w). Thus, in order that M is bounded on nq Mpq (1, |x|α ), it is necessary that α ≥ − due to Example 58(3)(b). p (3) Since M [χB(1) |·|−n ] ≡ ∞, χB(1) |·|−n ∈ / Mpq (1, w). Thus, in order that M nq p α according is bounded on Mq (1, |x| ), it is necessary that α − nq < − p to Example 58(3)(b). Later we will see that these conditions are also sufficient. The difficulty of this space is that it is not easy to calculate kχQ kMpq (1,w) , even if Q is a cube. To avoid this problem, we will use the following restriction on the weights. We will see that this restriction permits us to calculate the norm of χQ in Lemma 136. The first theorem is the boundedness of the maximal operator. Theorem 140. Let 1 ≤ q ≤ p < ∞. If w ∈ Aq ∩ Bp,q , then M is bounded from Mpq (1, w) to WMpq (1, w). If in addition q > 1, then M is bounded on Mpq (1, w). Proof Assume first that q > 1 to prove that M is bounded from Mpq (1, w) to itself. For any Q ∈ D(Rn ), we show that 1

1

|Q| p − q kχQ M f kLq (w) . kf kMpq (1,w) for all f ∈ L0 (Rn ). We may assume f ∈ M+ (Rn ). Define f1 ≡ f · χ2Q , f2 ≡ f − f1 . Write 1

1

I ≡ |Q| p − q kχQ M f1 kLq (w) ,

1

1

II ≡ |Q| p − q kχQ M f2 kLq (w) .

Then we have only to show I + II . kf kMpq (1,w) . Since 1 < q and w ∈ Aq , we can apply Theorem 290 in the first book for I to obtain   q1 Z 1 1 q p I . |Q| f1 (x) w(x)dx . kf kMpq (1,w) . |Q| Rn 1

1

1

We estimate f2 . We have II . |Q| p − q w(Q) q

sup

mQ0 (f ) from a geometric

Q0 ∈D ] (Q)

observation. By H¨ older’s inequality II . |Q|

1 1 p−q

w(Q)

1 q

sup Q0 ∈D ] (Q)

(q) mQ0 (f

1 q

·w )



1 |Q0 |

Z w(y) Q0

 10

0

− qq

q

dy

.

Weighted Morrey spaces

165

We use assumption (15.2) to have q

|Q0 | p II . sup 0 Q0 ∈D ] (Q) |Q | q

w(Q0 ) f (y) w(y)dy · |Q0 | Q0

Z

q



1 |Q0 |

Z

 q0

0

− qq

w(y)

q

dy

.

Q0

1

As a result, I + II ≤ kf kMpq (1,w) ([w]Aq ) q . We next show that the maximal operator is bounded from Mpq (1, w) to WMpq (1, w) if q = 1 ≤ p < ∞. Fix any Q ∈ D(Rn ), and let f1 ≡ f · χ2Q , f2 ≡ f − f1 again. Due to Theorem 280 in the first book, we see that
1 |Q| p −1 sup λw {x ∈ Q : M f1 (x) > 2−1 λ} . kf kMp1 (w) . λ>0

It is thus enough to show that
1 II0 ≡ |Q| p −1 sup λw {x ∈ Q : M f2 (x) > 2−1 λ} . kf kMp1 (1,w) . λ>0

Meanwhile, a geometric observation gives Z 1 −1 0 p II ≤ 2|Q| sup λ>0 1

≤ |Q| p

M f2 (x)w(x)dx

{x∈Q:M f2 (x)>2−1 λ}

2w(Q) sup mQ0 (f ) |Q| Q0 ∈D] (Q)

≤ 2Cp,q

1

|Q0 | p

sup Q0 ∈D ] (Q)

≤ 2Cp,q [w]A1

sup Q0 ∈D ] (Q)

w(Q0 ) · mQ0 (f ) |Q0 |   1 0 p inf 0 w(y) mQ0 (f ) |Q | y∈Q

≤ 2Cp,q [w]A1 kf kMp1 (1,w) . The boundedness of M from Mpq (1, w) to itself implies the weak type inequality when q > 1, and this finishes the proof. Here, we summarize the properties of the weights for which the maximal operator M is bounded on Mpq (1, w). As we will see in Proposition 142 below, it can happen that the weight w does not belong to Aq . However, under some condition together with the boundedness of the maximal operator M , we will have w ∈ Aq . Theorem 141. Let 1 < q ≤ p < ∞ and w be a weight. Assume that M is bounded on Mpq (1, w). (1) w ∈ Bp,q . 1

(2) w− q−1 ∈ Bp,q implies w ∈ Aq . 1

(3) w− q−1 ∈ A∞ is equivalent to w ∈ Aq .

166

Morrey Spaces 1

(4) Under the assumption w− q−1 ∈ A∞ ∪ Bp,q , M is bounded on Mpq (1, w) if and only if w ∈ Bp,q ∩ Aq . Proof (1) Fix Q0 ∈ Q. Let γ0 ≡ mQ0 (w), and let a  1. Define U 0 ≡ {Q0 }. We set  U 1 ≡ Q ∈ D(Q0 ) : mQ0 (w) > aj γ0 . We denote by U 1,∗ the set of all maximal cubes in U 1 . We have the maximal family U 1,∗ = {Q1k }k∈K1 . Then aj γ0 < mQj (w) ≤ 2n aj γ0 k S Q1k . We know that {Q1k }k∈K1 satfor all Q1k ∈ U 1,∗ . Set Ω1 ≡ k∈K1

isfies 2|Ω1 | ≤ |Q0 | according to Theorem 170 in the first book. Then since 2M χQ0 \Ω1 ≥ χQ0 and M is assumed bounded on Mpq (1, w), kχQ0 kMpq (1,w) . kχQ0 \Ω1 kMpq (1,w) . Since w is a doubling weight and p > q, we have 1

1

(q)

kχQ0 \Ω1 kMpq (1,w) ∼ sup |Q| p mQ (w q χQ0 \Ω1 ). Q∈U

In view of the maximality of U 1 , we conclude that 1

1

(q)

1

1

kχQ0 kMpq (1,w) . sup |Q| p mQ (w q χQ0 \Ω1 ) . |Q| p γ0 q , Q∈U

which shows that w ∈ Bp,q . 1

(2) We fix any Q ∈ D(Rn ) and write σ ≡ w− q−1 . Since σ ∈ Bp,q is the dual weight of w: σ q w = σ, we notice that 1

1

kσχQ kMpq (1,w) = kχQ kMpq (1,σ) ∼ |Q| p (mQ (σ)) q . On the other hand, if we notice that mQ (σ)χQ (x) ≤ M [σχQ ](x), then the boundedness of M on Mpq (1, w) yields that 1

1

mQ (σ)kχQ kMpq (1,w) ≤ kσχQ kMpq (1,w) ∼ |Q| p (mQ (σ)) q . 1

1

Moreover, since kχQ kMpq (1,w) ≥ |Q| p (mQ (w)) q , by dividing both terms 1

1

1

by |Q| p , it follows that mQ (σ) (mQ (w)) q . (mQ (σ)) q , or equivalently,  q−1 Z 1 1 w(x)− q−1 dx mQ (w) . 1, which implies w ∈ Aq . |Q| Q 1

(3) One implication is trivial. Indeed, if w ∈ Aq , then w− q−1 ∈ Aq0 ⊂ A∞ thanks to Lemma 287 and Example 123 in the first book. Let us show 1 w ∈ Aq assuming that w− q−1 ∈ A∞ . Fix any Q0 ∈ D(Rn ) and by the Calder´ on–Zygmund decomposition, we will construct a sparse family

Weighted Morrey spaces

167

{Qkj }j∈N0 ,k∈Kj with a level structure. We now set γ0 ≡ mQ0 (σ) and take a large constant a([σ]A∞ )  2n . Then we define U 0 ≡ {Q0 } and  U j ≡ Q ∈ D(Q0 ) : σ(Q) > a([σ]A∞ )j γ0 |Q| (j ∈ N). We denote the maximal subset of U j by U j,∗ ≡ {Qkj }k∈Kj again. In view of 2n ≤ λ0σ a([σ]A∞ ), we see that {Qkj }j∈N0 ,k∈Kj is a sparse family with a level structure. In particular, it follows from Lemma 296 that σ(Q0 \ Ω1 ) &[σ]A∞ σ(Q0 ), S S where Ω1 ≡ Q1k = R. This implies that k∈K1

χQ0 (x)

R∈U 1

σ(Q0 \ Ω1 ) σ(Q0 ) .[σ]A∞ χQ0 (x) ≤ M [χQ0 \Ω1 · σ](x). 0 |Q | |Q0 |

By taking the weighted Morrey norm of both sides and using the boundedness of M and σ q · w = σ, it follows that σ(Q0 ) kχQ0 kMpq (1,w) .[σ]A∞ kχQ0 \Ω1 kMpq (1,σ) . |Q0 |

(15.11)

By recalling that [  Ω1 = R, U 1 = R ∈ D(Q0 ) : mR (σ) > a([σ]A∞ )mQ0 (σ(Q0 ) , R∈U 1 1

1

1

we see that kχQ0 \Ω1 kMpq (1,σ) ≤ a([σ]A∞ ) q |Q0 | p (mQ0 (σ)) q . Meanwhile,
1 1 it is trivial that kχQ0 kMpq (1,w) ≥ |Q0 | p mQ0 (w) q . As a result, inserting these two estimates into (15.11), we obtain that mQ0 (σ) mQ0 (w)


q1

1

.[σ]A∞ (mQ0 (σ)) q ,

which implies w ∈ Aq . (4) If M is bounded on Mpq (1, w), then by (1), (2) and (3) we have w ∈ Aq ∩ Bp,q . Conversely, if w ∈ Aq ∩ Bp,q , then M is bounded on Mpq (1, w) by Theorem 140. We have been investigating the possibility for which the weight w belongs to Aq . However, as the following example shows, it can happen that the weight does not belong to Aq . Proposition 142. Let 1 < q < p < ∞and β ∈ R. Then M is bounded on 1 q n. Mpq (1, |x|β ) if and only if − n ≤ β < q 1 − p p

168

Morrey Spaces

Proof We can take the approach used in the proof of Proposition 131. We omit the details. Alternatively combine Lemma 139 and Theorem 149 with α = 0 to follow. One of the ways to investigate the boundedness of operators acting on Morrey spaces is combining the translation and the boundedness of operators acting on corresponding local Morrey spaces. Propositions 142 and 168 below are significant in that Proposition 142 cannot be obtained by the translation of Proposition 168 since the weight is translation invariant.

15.2.4

Singular integral operators in weighted Morrey spaces of Samko-type

Our goal here is to extend singular integral operators T to Mpq (1, w) with 1 ≤ q ≤ p < ∞. Here w ∈ Bp,q ∩ Aq , so that T can be regarded as a bounded linear operator on Lq (w). For the moment, we denote the singular integral operator on Lq (w) by T0 to avoid confusion. Then we can define the singular integral operator on Mpq (1, w) as follows: For f ∈ Mpq (1, w) and any x ∈ Rn , we define Z T f (x) ≡ T0 (f χ2Q ) (x) + K(x, y)f (y)dy, Rn \2Q

where Q is any cube containing the point x. It is easy to check that the definition is independent of the choice of Q. The following lemma ensures the well-definedness of T : Lemma 143. Let 1 ≤ q ≤ p < ∞ and w ∈ Bp,q ∩ Aq satisfy the weighted integral condition (15.3). Then the second term of the right-hand side defining T f (x) converges absolutely for any f ∈ Mpq (1, w) and almost every x ∈ Rn . Proof Write

Z I≡

|K(x, y)f (y)|dy. Rn \2Q

We need to show that I < ∞. Fix any Q containing x. By the size condition, we have ∞ Z ∞ X X |f (y)| I. dy . m2l Q (|f |). n 2l+1 Q\2l Q |x − y| l=1

l=1

We may employ (15.3) to obtain  q1  l 1 Z ∞  X |2 Q| q 1 q |f (y)| w(y)dy I. |2l Q| 2l Q w(2l Q) l=1

≤

∞ X l=1

.

1 kf kMpq (1,w) Φ(2l Q)

kf kMpq (1,w) < ∞. Φ(Q)

Weighted Morrey spaces

169

A direct consequence of Lemma 143 and its proof is the following theorem: Theorem 144. Let 1 < q ≤ p < ∞ and w ∈ Bp,q ∩ Aq satisfy the weighted integral condition (15.3). Then any singular integral operator T , defined initially on Lq (w), can be extended to a bounded linear operator on Mpq (1, w). The proof is similar to the classical case. So we omit the details. We end this section with an observation using the weighted integral condition (15.3). Lemma 145. Let x ∈ Rn , R > 0 and w be a doubling weight satisfying the weighted integral condition (15.3). Let Qx,l ∈ D(Rn ) be a maximal dyadic cube satisfying x ∈ Qx,l and Φ(Qx,l ) ≤ 2l R for each l ∈ N. Suppose we have a collection {Qj }∞ j=1 of dyadic cubes such that x ∈ Qj for all j ∈ N. Set Jl ≡ {j ∈ N : 2l−1 R ≤ Φ(Qj ) ≤ 2l } for each l ∈ N. Then, there exists k ∈ N such that Qj \ 2−k Qx,l 6= ∅ for all j ∈ Jl . Proof For a dyadic cube S, we denote by S ∗ its dyadic parent. We denote the m-th ancestor of S by S (m) . In other words, S (m) is defined by S (m) = S ∗···∗ , where ∗ appears m times. Assume that for all k ∈ N there exists jk ∈ Jl such that Qjk ⊂ 2−k Qx,l and we will obtain a contradiction. (1) (m) (m) We know that Qjk ⊂ Qjk ⊂ · · · ⊂ Qjk and `(Qjk ) = 2m `(Qjk ) by the definition of dyadic parents. Meanwhile, the assumption Qjk ⊂ 2−k Qx,l yields `(Qjk ) ≤ · · · ≤ 2m `(Qjk ) ≤ · · · ≤ 2k `(Qjk ) ≤ `(Qx,l ). (m)

(k)

We also have `(Qjk ) ≤ · · · ≤ `(Qjk ) ≤ · · · ≤ `(Qjk ) ≤ `(Qx,l ), which implies (m)

Qjk

⊂ Qx,l for m = 1, . . . , k, by x ∈ Qjk ∩ Qx,l 6= ∅. Since w ∈ Bp,q , we (m)

obtain Φ(Qjk ) ≤ Cp,q Φ(Qx,l ) for any m = 1, . . . , k. Therefore, the weighted integral condition (15.3) gives k k −1 −1 X X Cp,q kCp,q 1 C ≥ ≥ = . (m) Φ(Qjk ) m=1 Φ(Q ) m=1 Φ(Qx,l ) Φ(Qx,l )

(15.12)

jk

Here, recall that Φ(Qx,l ) ∼ 2l R and Φ(Qjk ) ∼ 2l R, since jk ∈ Jl . More precisely, we have Φ(Qx,l ) ≤ 2l R ≤ κΦ(Qx,l ) and 2l−1 R ≤ Φ(Qjk ) ≤ 2l R. This implies Φ(Qx,l ) ≤ 2l R ≤ 2Φ(Qjk ), or equivalently, Φ(Qx,l )−1 ≥ 21 Φ(Qjk )−1 . By inserting this inequality into (15.12), we see that −1 kCp,q C ≥ Φ(Qjk ) 2Φ(Qjk )

or, equivalently, c = 2C · Cp,q > k holds for any k ∈ N. However this is a contradiction since k is arbitrary. Note that the constant c which we obtained in here is independent of l ∈ N, since the argument is independent of the choice of l ∈ N. Therefore, more precisely, we obtain the contradiction by supposing that for all k ∈ N, there exist lk ∈ N and jk ∈ Jlk such that Qjk ⊂ 2−k Qx,lk . This means that there exists c < 1 such that for all l ∈ N and for all j ∈ Jl , cQx,l ⊂ Qj .

170

Morrey Spaces

15.2.5

Fractional maximal operators in weighted Morrey spaces of Samko-type

We will take the approach following Hedberg. We use an inequality similar to Lemma 181 in the first book. Lemma 146. Let 0 ≤ α < n, 1 ≤ q ≤ u, p < ∞ and w ∈ Aq . Assume that w satisfies  1 1 1− uq q 1 . (15.13) `(Q)α . Φp,w,q (Q)1− u = |Q| p − q w(Q) q q

q

Then, Mα f . (kf kMpq (1,w) )1− u (M f ) u for any f ∈ M+ (Rn ). Proof Fix x ∈ Rn . Take any Q ∈ D(Rn ) such that x ∈ Q. By assump1− uq  1 1 1 −q α p q w(Q) M f (x). Meanwhile, tion (15.13), we have `(Q) mQ (f ) . |Q| H¨ older’s inequality, (15.13) and the Aq -condition yield  1 1 − uq 1 `(Q)α mQ (f ) . |Q| p − q w(Q) q kf kMpq (1,w) . In fact, when 1 < q < ∞, we have   q1   10 Z Z q q0 1 1 `(Q)α mQ (f ) . `(Q)α f (y)q w(y)dy w(y)− q dy |Q| Q |Q| Q 1 1   [w]Aq q |Q| q α p ≤ `(Q) kf kMq (1,w) 1 w(Q) |Q| p q  1 1  −u 1 . |Q| p − q w(Q) q kf kMpq (1,w) . When q = 1, we have 1− u1 1 Z  1 f (y)dy `(Q)α mQ (f ) . |Q| p −1 w(Q) |Q| Q Z  1 − u1 1 ≤ |Q| p −1 w(Q) [w]A1 essinfz∈Q w(z) × |Q| p −1 f (y)dy Q



1 p −1

Z − u1 1 −1 p w(Q) |Q| f (y)w(y)dy



1 p −1

− u1 w(Q) kf kMp1 (1,w) .

. |Q|

Q

. |Q|

1

1

1

Hence, if we set tQ ≡ |Q| p − q w(Q) q , then we see that n o 1− q −q `(Q)α mQ (f ) . min tQ u M f (x), tQ u kf kMpq (1,w) n o q q ≤ sup min t1− u M f (x), t− u kf kMpq (1,w) t>0

=

q

q

(kf kMpq (1,w) )1− u M f (x) u .

Weighted Morrey spaces

171

Since Q ∈ D(Rn ) is arbitrary, this concludes the proof. The next theorem is on the fractional maximal operator Mα . Theorem 147. Let 0 ≤ α < n, 1 < q ≤ u, p < ∞ and w ∈ Aq ∩ Bp,q . Then the following are equivalent: (1) For any Q ∈ Q w satisfies  1 1 1− uq q 1 `(Q)α . Φp,w,q (Q)1− u = |Q| p − q w(Q) q .

(15.14)

(2) The fractional maximal operator Mα is bounded from Mpq (1, w) to q

Mupu (1, w). By using Lemma 146, we can show Theorem 147. Proof Let us show that (15.14) implies the boundedness of the generalized q

fractional maximal operator Mα from Mpq (1, w) to Mupu (1, w). Fix any Q ∈ D(Rn ). We may employ Lemma 146 and Theorem 140 to get |Q|

q pu



. |Q|

1 |Q| 

Z

Mα f (x) w(x)dx Q

1 |Q|

q pu

 u1

u

Z Q

 u1 1− q M f (x) w(x)dx kf kMpqu(1,w)

q u

q

1− q

≤ kM f kMpq (1,w) · kf kMpqu(1,w) . kf kMpq (1,w) . To prove the converse, fix any Q ∈ D(Rn ) and assume the boundedness of Mα . Fix x ∈ Q. Now, observe that Z `(Q)α Mα χ2Q (x) ≥ χ2Q (y)dy · χQ (x) = `(Q)α χQ (x). |Q| Q Hence, if we notice that kχQ k

q Mupu

q

∼ |Q| pu (1,w)



w(Q) |Q|

 u1

thanks to assump-

tion (15.2), then we may employ the boundedness of Mα to get q

|Q| pu



w(Q) |Q|

 u1

1

.

|Q| p 1 q . kMα χ2Q k pu α `(Q) `(Q)α Mu (1,w)



w(Q) |Q|

 q1 ,

which implies (15.14). We can extend Theorem 140 to the weak boundedness. Theorem 148. Let 0 ≤ α < n, 1 ≤ q ≤ u, p < ∞ and w ∈ Aq ∩ Bp,q . Then the following are equivalent:

172

Morrey Spaces

(1) w satisfies (15.14) for any Q ∈ Q, (2) the fractional maximal operator Mα is bounded from Mpq (1, w) to q

WMupu (1, w). The proof of Theorem 148 is similar. That is, we have only to combine Lemma 146 and Theorem 140, again. Thus, the details are left for the interested readers; see Exercise 44. We establish a general theorem. Theorem 149. Let 0 ≤ α < n, 1 < q ≤ p < ∞, 1 < t ≤ s < ∞ satisfying 1 α 1 = − , s p n

q t = , p s

(15.15)

and let w be a weight. (1) Assume that kMα f kMst (1,wt ) . kf kMpq (1,wq ) holds for every f ∈

Mpq (1, wq ).

(15.16)

Then

α

sup |Q| n −1 kwχQ kMst kw−1 χQ k(Mpq )0 . 1.

(15.17)

Q∈Q

(2) Assume that there exist s ≥ 1, C3 > 0 and a collection {bQ0 }Q0 ∈Q satisfying the following: 1

kf bQ0 q kLq (wq ) ≤ C3 kf kMpq (1,wq ) , sup Q∈Q(Q0 )

(t)

(q 0 )

1

(15.18) 1

1

mQ (w)mQ (w−1 bQ0 − q ) ≤ C3 `(Q0 ) q − p

for any Q0 ∈ Q and

1

[wbQ0 q ]As ≤ C3 ,

(15.19)

(15.20)

where the constant C3 is independent of the choices Q0 . If (15.17) holds, then (15.16) holds. Proof (1) For any cube Q ∈ Q and any function f with f w ∈ Mpq (Rn ), Z α −1 n |f (x)|dx ≤ kMα [f χQ ]kMst (1,wt ) . kf χQ kMpq (1,wq ) . kwχQ kMst |Q| Q

Taking the supremum over all functions f with kf χQ kMpq (1,wq ) ≤ 1, we α have |Q| n −1 kwχQ kMst kw−1 χQ k(Mpq )0 . 1 by Lemma 342 in the first book, proving (14.9).

Weighted Morrey spaces

173

n + n (2) We may assume that the function f ∈ L∞ c (R ) ∩ M (R ). Fix Q0 ∈ Q. 1 1 − We have to evaluate the quantity |Q0 | s t kMα f kLt (wt ) . Using Theorem 179, we select a sparse family S ⊂ D(Q0 ) such that (Mα f )χQ0 . P Q] (Q ) Q] (Q ) LSα f +c∞,α 0 (f ), where LSα f (x) ≡ χEQ `(Q)α m3Q and c∞,α 0 (f ) ≡ Q∈S

`(Q)α mQ (f ). Thus, using this decomposition, we obtain

sup Q∈Q] (Q0 ) 1

1

|Q0 | s − t kMα f kLt (wt ) Z  1t 1 1 1 S t t Q] (Q0 ) − 1t s . |Q0 | Lα f (x) w(x) dx + c∞,α (f )|Q0 | s − t kwkLt (Q0 ) . Q0 Q] (Q )

1

1

We first estimate A ≡ c∞,α 0 (f )|Q0 | s − t kwkLt (Q0 ) . To this end fix a cube Q which contains Q0 . Then 1

1

`(Q)α mQ (f )|Q0 | s − t kwkLt (Q0 ) Z ≤ `(Q)α−n kwχQ kMst w(x)−1 χQ (x) · f (x)w(x)dx Rn

. `(Q)α−n kwχQ kMst kw−1 χQ k(Mpq )0 kf kMpq (1,wq ) . kf kMpq (1,wq ) , where we have used (14.9). Since the cube Q ⊃ Q0 is arbitrary, we conclude A . kf kMpq (1,wq ) . Z We next estimate B ≡ LSα f (x)t w(x)t dx. Take b = b9Q0 satisfyQ0 1

ing (15.19) and (15.20) with 9Q0 replaced by Q0 and kf b q kLq (wq ) . 1 0 kf kMpq (1,wq ) . Set u ≡ V and σ ≡ [wb p ]−p . Since the sets EQ , Q ∈ S, P t are pairwise disjoint, B = 3−nq (`(Q)α m3Q (f )) u(EQ ). Q∈S

Since 1 σ(9Q)

Z

1 f (x)dx = σ(9Q) 3Q

Z

f (x)σ(x)−1 dσ(x) ≤ inf Mc,σ [f σ −1 ](y), y∈Q

3Q

where Mc,σ is the centered weighted Hardy–Littlewood maximal operator with respect to σ. Thus, we obtain q XZ X 1 Z f (x)dx σ(EQ ) ≤ Mc,σ [f σ −1 ](x)q dσ(x) σ(9Q) 3Q EQ Q∈S

Q∈S

≤ (kMc,σ [f σ −1 ]kLq (σ) )q . (kf σ −1 kLq (σ) )q 1

= (kf b q kLq (wq ) )q . (kf kMpq (1,wq ) )q .

(15.21)

174

Morrey Spaces With this in mind, we will estimate (`(Q)α m3Q (f )) this end, we write X ≡ (`(Q)α m3Q (f ))

t

t

u(EQ ) . To σ(EQ )

u(EQ ) σ(EQ ) and Y σ(EQ )

t u(EQ ) t σ(EQ )1− q , so that (`(Q)α m3Q (f )) σ(EQ ) q q X 1− t Y t . We calculate

≡

u(EQ ) σ(EQ )

t

(`(Q)α m3Q (f ))

∼

t

X ≤ u(Q) (`(Q)α m3Q (f )) !t 1 Z  (kwχQ kLt )t t 1− st α−n = |Q| f (x)dx `(Q) t |Q|1− s 3Q  t t . |Q|1− s `(Q)α−n kwχQ kMst kw−1 χ3Q k(Mpq )0 kf χ3Q kMpq (1,wq )  t t . |Q|1− s kf kMpq (1,wq ) , where we have used our assumption (14.9) in the last line. Using Q ⊂ 9Q, we obtain t

Y = σ(EQ )− q ( ≤



σ(9Q) σ(EQ )

1

`(Q)α−n u(Q) t σ(9Q)

 q1

1 t

1 σ(9Q)

`(Q)α−n u(9Q) σ(9Q)

1 q0

t

Z f (x)dx 3Q

1 σ(9Q)

)t

Z f (x)dx

.

3Q

By (15.20) together with |9Q| = 9n |Q| ≤ 2 · 9n |EQ |, Corollary 291 in the first book gives 1  σ(9Q) q . 1. σ(EQ ) Thus,  Y .

1

1

`(Q)α−n u(9Q) t σ(9Q) q0

1 σ(9Q)

t

Z f (x)dx 3Q

Arithmetic shows    1 1 n t n − +α−n=− 0 − . n 1− s q t q t

.

Weighted Morrey spaces

175

Consequently,  q 1 q t 1 (|Q|1− s )1− t `(Q)α−n u(9Q) t σ(9Q) q0  q 1 t 1 1 1 = |Q|(1− s )( q − t ) `(Q)α−n u(9Q) t σ(9Q) q0 q  1 1 −1− 1 = |Q| t q0 u(9Q) t σ(9Q) q0 ( 1   1 )q q u(9Q) t σ(9Q) q0 ' . `(Q0 )1− p , |9Q| |9Q| where we have used (15.19) for the last inequality. Consequently,  q Z 1 t u(EQ ) q−p α p . (kf kMq (1,wq ) ) f (x)dx . (`(Q) m3Q (f )) σ(EQ ) σ(9Q) 3Q Combining this with (15.21), we obtain X q −1 q−p p p `(Q0 ) B . (kf kMq (1,wq ) ) Q∈S

1 σ(9Q)

q

Z f (x)dx

σ(EQ )

3Q

. (kf kMpq (1,wq ) )q . As before, we can completely characterize the range of the parameter β for which the fractional maximal operator Mα is bounded from Mpq (1, |x|qβ ) to Mst (1, |x|tβ ). Proposition 150. Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞. Assume that 1 1 α 1 1 α = − , = − . s p n t q n Then the following are equivalent: (1) The fractional maximal operator Mα is bounded from Mpq (1, |x|qβ ) to Mst (1, |x|tβ ); (2) −

n n ≤ β < 0. s p

Proof Condition (2) is necessary for (1). In fact, once we assume (1), we have χB(1) ∈ Mst (1, |x|tβ ) and | · |−n χB(1) ∈ Mpq (1, |x|tβ ). Due to Example 58, this is equivalent to (2). n n Let − ≤ β < 0 . Thanks to Proposition 175 to follow, we see that Mα is s p bounded from LMpq (1, |x|qβ ) to LMst (1, |x|tβ ). So, we have only to show that 1 1 |Q| s − t kIα f kLt (Q) . kf kMst (1,|x|tβ ) for all cubes Q such that 0 ∈ / 5Q. We simply use the Hardy–Littlewood–Sobolev theorem if f is supported on 3Q. Assume that f is supported away from 3Q. Then for all x ∈ Q,

176

Morrey Spaces

Mα f (x) . sup Rα mB(R) (|f |) + R>0

rα mB(c(Q),r) (|f |). The second term

sup 0 γ0 aj }.

Moreover, we use Theorem 284 in the first book and Lemma 296. Now let us go back to the proof of Theorem 151. We now define D0 ≡ {Q ∈ D(Q0 ) : hhwiQ ≤ γ0 a} and, for j ∈ N and k ∈ Kj , Dkj ≡ {Q!∈ D(Qjk ) : γ0 aj < hhwiQ ≤ γ0 aj+1 }. Then D(Q0 ) = S D0 ∪ Dkj . Set j∈N,k∈Kj



I(h) a

(h)

Ib

≡ |Q0 |

 X `(Q)α Z  f (y)dy · χQ (x) h(x)w(x)dx |Q| 3Q Q0 Q∈D0   Z ∞ X X α Z X `(Q)  f (y)dy · χQ (x) h(x)w(x)dx. |Q| 3Q Q0 j j=1 Z

q 1 pu − u

q

1

≡ |Q0 | pu − u

k∈Kj Q∈D k

(h)

(h)

We can write I(h) = I(h) a + Ib . We will evaluate Ib γ0 ak+1 for Q ∈ Dkj , we have (h) Ib

= |Q0 |

q 1 pu − u

∞ X X X

α

Z

`(Q)

j=1 k∈Kj Q∈D j k q

1

≤ |Q0 | pu − u

∞ X X j=1 k∈Kj

γ0 ak+1

X j Q∈Dk

first. Since hhwiQ ≤

1 f (y)dy · |Q| 3Q

`(Q)α

Z h(x)w(x)dx Q

Z f (y)dy. 3Q

178

Morrey Spaces

Now, observe that X

`(Q)α

Z f (y)dy =

∞ X

3Q

j Q∈Dk

X

`(Q)α

Z

.

∞ X

(2−l `(Qjk ))α

l=0

∼ `(Qjk )α

f (y)dy 3Q

l=0 Q∈D j ∩Dl (Qj ) k k

Z f (y)dy 3Qjk

Z f (y)dy. 3Qjk

By γ0 aj < hhwiQj and assumption (15.22), we have k

(h)

Ib

q

1

. |Q0 | pu − u

∞ X X

1

Z

h(x)w(x)dx · `(Qjk )α

Z

f (y)dy |Qjk | Qjk 3Qjk Z ∞ X Z X q 1 j 1− u h(x)w(x)dx · Φ(Qk ) f (y)dy. j |Qjk | 3Qjk j=1 k∈Kj Qk j=1 k∈Kj

q

1

. |Q0 | pu − u

If we go through a similar argument to Lemma 146, we obtain Z q q q 1 j 1− u Φ(Qk ) f (y)dy . inf j M f (x) u kf kMpq (1,w) 1− u j j |Qk | 3Qk x∈Qk and (h)

Ib

q

1

. |Q0 | pu − u

∞ X Z X j=1 k∈Kj

∞ P l=1

Qjk

q

x∈Qk

R To proceed, we focus on Qj h(x)w(x)dx. k By Theorem 284 in the first book, if we let H M l [hw] (2kM kLu0 (σ)→Lu0 (σ) )l

1− q

h(x)w(x)dx · inf j M f (x) u kf kMpqu(1,w) .

≡

hw +

, then 1 0

[H]A1 ≤ 2kM kLu0 (σ)→Lu0 (σ) ≤ 2Cu0 [σ]Au u−1 0

(15.24)

kHkLu0 (σ) ≤ 2khwkLu0 (σ) = 2.

(15.25)

and Furthermore, since w ∈ Aq ⊂ Au and σ is the dual weight of w, we see 1 that [σ]Au0 u0 −1 = [w]Au ≤ [w]Aq thanks to Lemma 287 in the first book. This implies that [H]A∞ ≤ [H]A1 ≤ 2Cu0 [w]Aq . Here, recall that we chose n+4 a > 2n+1+2 ·2Cu0 [w]Aq in (15.23). By the above calculation, we see that n+4 a > 2n+1+2 [H]A∞ . Thus, we may apply Lemma 296 in the first book to

Weighted Morrey spaces

179

obtain H(Qjk ) . H(Ekj ). Hence, if we notice that hw ≤ H and that {Ekj }k,j partitions Q0 , then (h)

Ib

q

1

. |Q0 | pu − u

∞ X X

q

1

∞ X X

q

1

1

Z

≤ |Q0 | pu − u

Q0 q

≤ |Q0 | pu − u

q

q

H(Ekj ) · inf j M f (x) u kf kMpq (1,w) 1− u x∈Qk

j=1 k∈Kj

Z

q

x∈Qk

j=1 k∈Kj

. |Q0 | pu − u

q

H(Qjk ) · inf j M f (x) u kf kMpq (1,w) 1− u

 q 1− q M f (x) u H(x)dx kf kMpqu(1,w) M f (x)q w(x)dx

 u1

Q0 q u

1− q

kHkLu0 (σ) kf kMpqu(1,w)

1− q

≤ kM f kMpq (1,w) · 2 · kf kMpqu(1,w) . kf kMpq (1,w) . Note that the constants in the above are independent of h. We can estimate I(h) a in a similar manner. In fact, we have Z q 1 pu − u γ a · `(Q )α I(h) . |Q | f (y)dy 0 0 0 a 3Q0 Z Z q q 1 1 h(x)w(x)dx · Φ(Q0 )1− u . |Q0 | pu − u f (y)dy. |Q0 | 3Q0 Q0 By the use of the maximal operator Z q q 1 1− q pu − u . |Q | h(x)w(x)dx · inf M f (x) u kf kMpqu(1,w) I(h) 0 a x∈Q0

Q0

1− q

q u

≤ kM f kMpq (1,w) · khkLu0 (w) · kf kMpqu(1,w) . kf kMpq (1,w) . Again, the constants are independent of h. In summary, we conclude that I . kf kMpq (1,w) . We are left with evaluating II. Observe that for x ∈ Q0 , the integrand is a constant: X `(Q)α Z X `(Q)α Z f (y)dy · χQ (x) = f (y)dy. |Q| 3Q |Q| 3Q ] ] Q∈D (Q0 )

Q∈D (Q0 )

Thus, we have q

II ∼ Φ(Q0 ) u

∞ X m=0

(m)

`(Q0 )α m3Q(m) (f ). 0

180

Morrey Spaces (m)

(m)

(m)

q

Here, Q0 denotes the m-th ancestor of Q0 . Since `(Q0 )α . Φ(Q0 )1− u , we can use the weighted integral condition (15.3) to get

II . Φ(Q0 )

q u

q

≤ Φ(Q0 ) u q

∞ X

q 1 (q) (m) Φ(Q0 )1− u mQ0 (f w q )

m=0 ∞ X

1

(m) q m=0 Φ(Q0 ) u Z ∞

∼ Φ(Q0 ) u 1

(m)

|Q0 |

! q1

(m)

w(Q0 )

· kf kMpq (1,w)

1 ds · kf kMpq (1,w) q Φ(sQ0 ) u s

. kf kMpq (1,w) . Therefore, we obtain the boundedness of Iα . Finally, let us show (15.22) assuming the boundedness of Iα . Fix any Q ∈ Q q

and consider Iα χ3Q . By taking the Mupu (1, w)-norm and by using Example q q 83 in the first book, we obtain `(Q)α kχQ k pu . kIα χ3Q k pu . Mu (1,w)

kχ3Q kMpq (1,w) ∼ Φ(Q). Since kχQ k q

q Mupu

q

Mu (1,w)

∼ Φ(Q) u , this implies that (1,w)

`(Q)α . Φ(Q)1− u , which completes the proof. For the fractional integral operator Iα , we have the following. Theorem 152. Let 0 < α < n, 1 < q ≤ p < ∞, 1 < t ≤ s < ∞ satisfying (15.15), and let w be a weight. Then the following are equivalent: (1) For f ∈ M+ (Rn ), kIα f kMst (1,wt ) . kf kMpq (1,wq ) . (2) For f ∈ M+ (Rn ), kMα f kMst (1,wt ) . kf kMpq (1,wq ) and there exists κ > 1 such that 2kχQ kMst (1,wt ) ≤ kχκQ kMst (1,wt ) (15.26) holds for every Q ∈ Q. Proof Suppose that (1) holds. Then for f ∈ M+ (Rn ), kMα f kMst (1,wt ) . kf kMpq (1,wq ) since Mα . Iα . Also, if (15.26) fails, for each m ∈ N, we can find a cube Qm such that 2kχQm kMst (1,wt ) > kχmQm kMpq (1,wt ) . We have Iα fm (x) & χQm (x) log m for fm (y) ≡ |y − c(Qm )|−α χmQm \2Qm (y), y ∈ Rn . By H¨older’s α inequality, kfm kMpq (1,wq ) . kχQm kMpq (1,wq ) (log m) n . If we combine all the α observations above, then we obtain (log m)1− n . 1. This is a contradiction. Assume instead that (2) holds. For the purpose of proving (1), by the monon 0 tone convergence theorem, we may assume that f ∈ L∞ c (R ) as well. Let Q be a fixed cube. Then |Iα [χ3Q0 f ](x) − MED(Iα [χ3Q0 f ], Q)| . Mα [χ3Q0 f ](x) by the Lerner–Hyt¨ onen decomposition. Since we are assuming (2), nwe can handle Mα f with ease. Since Iα is bounded from L1 (Rn ) to WL n−α (Rn ),

Weighted Morrey spaces

181

MED(Iα [χ3Q0 f ], Q) . `(Q)α kf kL1 (3Q0 ) . Also, for all x ∈ Q0 , we have Iα [χRn \3Q0 f ](x) . . .

∞ X j=1 ∞ X j=1 ∞ X j=1

.

`(κj Q)α mκj Q (f ) inf Mα f (y)

y∈κj Q

1 kMα f kMst (1,wt ) kχ2j Q kMst (1,wt )

1 kf kMpq (1,wq ) . kχQ kMst (1,wt )

Thus, we can control Iα [χRn \3Q0 f ]. A direct consequence of Theorem 152 is: Proposition 153. Let 1 < q < p < ∞, 0 < α < n and 1 < t < s < ∞. Assume that 1 1 α q t = − , = . s p n p s n Then Iα is bounded from Mpq (1, |x|qβ ) to Mst (1, |x|tβ ) if and only if − < s n β < 0. p n n ≤ β < 0 is t q necessary for the boundedness of Iα from Mpq (1, |x|qβ ) to Mst (1, |x|tβ ), since n Mα . Iα . Since | · |− p −β ∈ Mpq (1, |x|qβ ), we can rule out the possibility of n n n n β = − . In fact, if − < β < 0 , then kχB(r) kMst (1,|x|tβ ) ∼ rβ+ s . This allows s s p n us to choose κ > 1 so that (15.26) holds, since β > − . Thus, we are in the s position of applying Proposition 150 and Theorem 152. Proof As before, due to Proposition 150, the condition −

15.2.7

Vector-valued estimates in weighted Morrey spaces of Samko-type

Here, we consider the vector-valued extension of our results. Let 0 < q ≤ p < ∞ and 0 < r ≤ ∞. Also let w be a weight. The space Lq (`r ; 1, w) is defined as a Banach space with the norm

  r1

∞

X

∞ ∞ r

 k{fj }j=1 kLq (`r ;1,w) ≡ k{fj }j=1 k`r Lq (w) = |fj | .



q

j=1 L (w)

182

Morrey Spaces

Meanwhile the space Mpq (`r ; 1, w) is a Banach space of sequences of measurable functions indexed by N with the norm

  r1

∞

X

r ∞

 p |f | . = k{fj }∞

j j=1 kMq (`r ;1,w) ≡ k{fj }j=1 k`r Mp q (1,w)

j=1

p Mq (1,w)

We also have the vector-valued maximal inequality: Theorem 154. Let 1 < q ≤ p < ∞, 1 < r < ∞ and w ∈ Aq ∩ Bp,q . Then ∞ p p w satisfies (15.3) if and only if k{M fj }∞ j=1 kMq (`r ;1,w) . k{fj }j=1 kMq (`r ;1,w) ∞ p r for any {fj }j=1 ∈ Mq (` ; 1, w). Proof Without loss of generality, we may suppose fj ≥ 0. By the monon tone convergence theorem, we may further assume that fj ∈ L∞ c (R ) and that fj = 0 if j  1. We abbreviate Φp,w,q to Φ. Assuming the weighted integral condition (15.3), let us show the vector-valued inequality. Fix any Q ∈ D(Rn ) and define fj,1 ≡ f · χ2Q and fj,2 ≡ f · χ(2Q)c . All we have to do is prove  I≡

|Q|

 q1

Z

q p −1

q k{M fj,1 (x)}∞ j=1 k`r w(x)dx

p . k{fj }∞ j=1 kMq (`r ;1,w) ,

Q

 II ≡

|Q|

 q1

Z

q p −1

q k{M fj,2 (x)}∞ j=1 k`r w(x)dx

p . k{fj }∞ j=1 kMq (`r ;1,w) .

Q

For I, we employ the weighted vector-valued inequality for M to obtain  I.

q

|Q| p −1

Z

q k{fj,1 (x)}∞ j=1 k`r w(x)dx

 q1

p . k{fj }∞ j=1 kMq (`r ;1,w) .

Q

For II, notice that M fj,2 (x) =

sup

mQ0 (fj ) holds for any x ∈ Q. Hence,

Q0 ∈D ] (Q)

we see that

II

= |Q|

1 1 p−q

w(Q)

1 q

 ∞ X 

=

Φ(Q)

∞ X j=1

aj

j=1

sup

!r  r1  sup mQ0 (fj )  Q0 ∈D ] (Q) mQ0 (fj ).

Q0 ∈D ] (Q)

∞ Here, a real sequence {aj }∞ j=1 satisfies k{aj }j=1 k`r0 = 1. We can take Qj ∈ n D(R ) such that Q ⊂ Qj and that sup mQ0 (fj ) ≤ 2mQj (fj ) holds for Q0 ∈D ] (Q)

Weighted Morrey spaces

183

each j ∈ N. Thus, we have Z ∞ X aj II . Φ(Q) fj (y)dy |Qj | Qj j=1 = Φ(Q)

∞ X

X

l=1 j∈N:2l−1 `(Q)≤`(Qj )≤2l `(Q)

. Φ(Q)

∞ X

X

l=1 j∈N,2l−1 `(Q)≤`(Qj )≤2l `(Q)

aj |Qj |

Z

aj |2l Q|

fj (y)dy Qj

Z fj (y)dy. 2l Q

By H¨ older’s inequality, II ≤ Φ(Q)

∞ X

1 (q 0 ) m2l Q (w− q )

l=1



1 l |2 Q|

Z 2l Q

q k{fj (y)}∞ j=1 k`r w(y)dy

 q1 .

By the definition of the Morrey norm and w ∈ Aq , we obtain  l  q1 ∞ X 1 1 |2 Q| p II . [w]Aq q Φ(Q) k{fj }∞ 1 j=1 kMq (`r ;1,w) l Q) l w(2 p l=1 |2 Q| ∞ X 1 1 p k{fj }∞ = [w]Aq q Φ(Q) j=1 kMq (`r ;1,w) Φ(2l Q) l=1

1 q

p . [w]Aq k{fj }∞ j=1 kMq (`r ;1,w) .

For the last step, we used the weighted integral condition (15.3). Conversely, supposing the vector-valued inequality, we show the weighted integral condition (15.3). By virtue of Lemma 138, we have only to find c1 > 0 so that 2Φ(Q) ≤ Φ(c1 Q) holds for any cube. Assume that such a constant does not exist; i.e., assume that there exists Qm ∈ Q such that Φ(Qm ) > (m) Φ(2m Qm ) for all m ∈ N. For each m ∈ N, define fj ≡ χ2j Qm \2j−1 Qm , ∞ P (m) (m) j = 1, . . . , m. When j > m, define fj ≡ 0. Then fj = χ2m Qm \Qm and j=1

(m)

that M fj ≥ (1 − 2−n )χ2j Qm for j = 1, . . . , m. See Exercise 62 in the first book. In particular, ∞ X (m) M fj (x)r & mχQm (x) (x ∈ Rn ) j=1

holds. Hence, we see that (m) ∞ }j=1 kMpq (`r ;1,w)

k{M fj

1

1

& m r kχQm kMpq (1,w) ∼ m r Φ(Qm ),

since w ∈ Bp,q . Meanwhile, the vector-valued inequality implies that (m) ∞ }j=1 kMpq (`r ;1,w)

k{M fj

(m) ∞ }j=1 kMpq (`r ;1,w)

. k{fj

= kχ2m Qm \Qm kMpq (1,w) . Φ(2m Qm ) < 2Φ(Qm ).

184

Morrey Spaces 1

In total, we obtain m r Φ(Qm ) . 2Φ(Qm ) for any m ∈ N, which is a contradiction. Thanks to the Ap theorem in the first book, T is well defined as a bounded operator on Lq (w), when w ∈ Aq with 1 < q < ∞. We can extend the results as follows: Theorem 155. Let 1 ≤ q ≤ p < ∞ and w ∈ Bp,q ∩ Aq satisfy the weighted integral condition (15.3). Then kT f kMpq (1,w) . kf kMpq (1,w) holds for any f ∈ Mpq (1, w) ∩ Lq (w). A passage to the vector-valued inequality is also available. We ignore the issue of whether T can act on weighted Morrey spaces. Theorem 156. Let 1 ≤ q ≤ p < ∞, 1 < r < ∞ and w ∈ Bp,q ∩ Aq satisfy the weighted integral condition (15.3). Then ∞ p p k{T fj }∞ j=1 kMq (`r ;1,w) . k{fj }j=1 kMq (`r ;1,w) p q n holds for any {fj }∞ j=1 ⊂ Mq (1, w) ∩ L (R ).

We will prove Theorems 155 and 156 based on Lemma 143 and Theorem 157 to follow. Theorems 155 and 156 follow once we prove the following: Theorem 157. Let 1 ≤ q ≤ p < ∞, 1 < r < ∞ and w ∈ Bp,q ∩ Aq satisfy the weighted integral condition (15.3). Then ∞ p p k{T fj }∞ j=1 kMq (`r ;1,w) . k{fj }j=1 kMq (`r ;1,w) n ∞ holds for any {fj }∞ j=1 ⊂ Lc (R ).

Proof As usual, take any Q ∈ D(Rn ), and let  I≡

|Q|

q p −1

Z

q k{T fj,1 (x)}∞ j=1 k`r w(x)dx

 q1 ,

Q

 II ≡

|Q|

q p −1

Z

q k{T fj,2 (x)}∞ j=1 k`r w(x)dx

 q1 .

Q

with fj,1 ≡ fj χ2Q and fj,2 ≡ f χRn \2Q . It suffices to show that I + II . p k{fj }∞ j=1 kMq (`r ;1,w) . The estimate of I follows by Theorem 311 in the first book. If we notice that for x ∈ Q, Z Z ∞ X 1 |T fj,2 (x)| = K(x, y)fj (y)dy . |fj (y)|dy, Rn \2Q |2l Q| 2l Q l=1

Weighted Morrey spaces

185

then thanks to Minkowski’s inequality and the weighted integral condition (15.3), we have

II . Φ(Q)

 ∞ ∞ X X 

≤ Φ(Q)

j=1

∞ X l=1

. Φ(Q) = Φ(Q)

1 l |2 Q|

∞  X l=1 ∞ X l=1

l=1

1 l |2 Q|

Z

1 |2l Q|

2l Q

Z |fj (y)|dy

!r  r1 

2l Q



  r1 ∞ X  |fj (y)|r  dy

Z 2l Q

j=1 q k{fj,2 (y)}∞ j=1 k`r w(y)dy

 q1 

|2l Q| w(2l Q)

 q1

1 p k{fj }∞ j=1 kMq (`r ;1,w) Φ(2l Q)

p ∼ k{fj }∞ j=1 kMq (`r ;1,w) .

We finish the proof. The following vector-valued inequality also holds: Theorem 158. Let 1 < q ≤ u < ∞, 1 < r < ∞ and w ∈ Aq ∩ Bp,q . If w satisfies the weighted integral condition (15.3) as well as (15.14), then ∞ p p . k{fj }∞ k{Mα fj }∞ j=1 kMq (`r ;1,w) holds for any {fj }j=1 ∈ j=1 k qu Mpq (`r ; 1, w).

Mu (`r ;1,w)

We prove Theorem 158 using a pointwise estimate below. The next proposition can be regarded as the vector-valued extension of Lemma 146: Proposition 159. Let 1 ≤ q ≤ u, p < ∞, 1 < r < ∞ and w ∈ Aq ∩ Bp,q . Suppose that w satisfies (15.14) and the weighted integral condition (15.3) for any Q ∈ Q. Then we have a point-wise estimate q

q

∞ 1− u u p k{Mα fj (x)}∞ (k{M fj (x)}∞ j=1 k`r . k{fj }j=1 kMq (`r ;1,w) j=1 k`r ) .

The proof of Theorem 158 relies upon a geometric observation (15.28). Proof Without loss of generality, we may assume fj ≥ 0. We may also assume that f1 is not zero by relabelling. In this proof, we write Φ = Φp,w,q . −1 p Fix x ∈ Rn , and let Rx ≡ (k{M fj (x)}∞ k{fj }∞ j=1 k`r ) j=1 kMq (`r ;1,w) . It suffices to show that q

q

q

q

I ≡ kMα,1 fj (x)k`r . (k{fj }`r kMpq (1,w) )1− u kM fj (x)k`ur , II ≡ kMα,2 fj (x)k`r . (k{fj }`r kMpq (1,w) )1− u kM fj (x)k`ur ,

186

Morrey Spaces

where Mα,1 fj (x) ≡ sup {`(Q)α mQ (fj )χQ (x) : Q ∈ D, Φ(Q) ≤ Rx } , Mα,2 fj (x) ≡ sup {`(Q)α mQ (fj )χQ (x) : Q ∈ D, Φ(Q) > Rx } . It is easy to evaluate I. In fact, if we notice that n o q Mα,1 fj (x) . sup Φ(Q)1− u mQ (fj )χQ (x) : Q ∈ D, Φ(Q) ≤ Rx q 1− u

≤ Rx

M fj (x),

by (15.14), then we obtain the estimate for I. To calculate II, we choose Qj ∈ D(Rn ) such that x ∈ Qj , Φ(Qj ) > Rx and that Mα,2 fj (x) ≤ 2`(Qj )α mQj (fj ) holds. We write Jl ≡ {j ∈ N : 2l−1 Rx ≤ Φ(Qj ) ≤ 2l Rx }. By using the 0 `r (N)-`r (N) duality and (15.14), we can find a sequence {aj }∞ j=1 such that k{aj }∞ j=1 k`r0 = 1 and that kMα,2 fj (x)k`r

. =

∞ X j=1 ∞ X

q

aj Φ(Qj )1− u mQ (fj ) X

q

aj Φ(Qj )1− u mQ (fj ).

(15.27)

l=1 j∈Jl

Here, we introduce the dyadic parent of Q ∈ D(Rn ), denoted by Q∗ . That is, Q∗ ∈ D] (Q) satisfies `(Q∗ ) = 2`(Q). We observe that Q∗ ⊂ 3Q. Now, return to (15.27). For the moment, we fix l ∈ N and consider j ∈ Jl . Recall that the weighted integral condition implies (15.5). Thus, we can find a maximal dyadic cube Qx,l ∈ D(Rn ) satisfying x ∈ Qx,l , Φ(Qx,l ) ≤ 2l Rx . The maximality of Qx,l implies that Qj ⊂ Qx,l holds for all j ∈ Jl and that 2l Rx < Φ(Q∗x,l ) holds. If we combine this fact and Lemma 137, then we obtain Φ(Qx,l ) ≤ 2l Rx ≤ κΦ(Qx,l ). That is, Φ(Qx,l ) ∼ 2l Rx . Furthermore, the weighted integral condition yields a constant c < 1 such that c Qx.l ⊂ Qj holds for all j ∈ Jl . In summary, we found Qx,l ∈ D(Rn ) and a constant c < 1 such that, for any j ∈ Jl , Qj \ cQx,l 6= ∅,

Qj ⊂ Qx,l ,

Φ(Qx,l ) ∼ 2l Rx .

(15.28)

Weighted Morrey spaces

187

With this observation, we turn back to (15.27). Using H¨oder’s inequality, we see that, for any x ∈ Rn , kMα,2 fj (x)k`r .

∞ X X l=1 j∈Jl ∞ X X

q

aj

Φ(Qj )1− u |Qj |

Z fj (y)dy Qj

q Z Φ(Qx,l )1− u . fj (y)dy aj |cQx,l | Qx,l l=1 j∈Jl q Z ∞ X Φ(Qx,l )1− u k{fj (y)}∞ . j=1 k`r dy. |Qx,l | Qx,l

l=1

By H¨ older’s inequality, we have

{Mα,2 fj (x)}∞

j=1 `r . .

∞ X l=1 ∞ X

q 1− u

Φ(Qx,l )

1 |Qx,l |

Z

q k{fj (y)}∞ j=1 k`r w(y)dy

Qx,l

q

p Φ(Qx,l )− u k{fj }∞ j=1 kMq (`r ;1,w) ∼

l=1

∼

! q1 

|Qx,l | w(Qx,l )

 q1

∞ X q p (2l Rx )− u k{fj }∞ j=1 kMq (`r ;1,w) l=1

q 1− u p (k{fj }∞ j=1 kMq (`r ,1,w) )

uq

{M fj (x)}∞

j=1 `r .

It completes the proof of Proposition 159. Using Proposition 159, we can show Theorem 158 as follows: To show the boundedness, we fix Q ∈ D(Rn ) and calculate |Q|

q pu



 .

|Q|

1 |Q| q p −1

Z

u k{Mα fj (x)}∞ j=1 k`r w(x)dx

 u1

Q

Z

q k{M fj (x)}∞ j=1 k`r w(x)dx

 u1

p k{fj }∞ j=1 kMq (`r ;1,w)

Q q

q

∞ 1− u u p p ≤ k{M fj }∞ j=1 kMq (`r ;1,w) k{fj }j=1 kMq (`r ;1,w) p . k{fj }∞ j=1 kMq (`r ;1,w) .

Here, we used Proposition 159 and Theorem 154. Since Iα is a linear operator, we can obtain the following directly from Theorem 151. We follow the idea of [401, Theorem 2] for the proof. Corollary 160. Let 1 < q ≤ u, p < ∞, 1 ≤ r ≤ ∞, and let w ∈ Bp,q ∩ Aq . Suppose that w satisfies the weighted integral condition (15.3). ∞ q p Then, k{Iα fj }∞ . k{fj }∞ j=1 kMq (`r ;1,w) for any {fj }j=1 ∈ j=1 k pu Mu (`r ;1,w)

Mpq (`r ; 1, w) if and only if w satisfies (15.22).

188

Morrey Spaces

15.2.8

Dual weighted estimates of Stein-type

Although it is difficult (still open) to have a complete characterization of the maximal inequality for weighted Morrey spaces of Samko type, we can characterize the weighted dual inequality of Stein-type. Theorem 161. Let 1 < q ≤ p < ∞, and let w be a weight. Then the following are equivalent: (1) The weighted dual inequality of Stein-type kM f kMpq (1,w) . kf kMpq (1,M w) holds for all f ∈ L0 (Rn ). (2) mQ (|f |)kχQ kMpq (1,w) . kf kMpq (1,M w) for all f ∈ L0 (Rn ) and Q ∈ Q(Rn ). 1

1

1

1

(3) |Q| p (mQ (w)) q . |Q0 | p (mQ0 (w)) q for all Q, Q0 ∈ Q(Rn ) such that Q ⊂ Q0 . Proof Since mQ (|f |)χQ ≤ M f , we have only to show that (2) implies (3) and that (3) implies the weighted dual inequality of Stein-type. (1) Assume that (2) holds. We will show that for any Q0 ∈ Q(Rn ), 1 1 1 kχQ0 kMpq (1,w) . |Q0 | p − q w(Q0 ) q . We let  D(Q0 ) ≡ Q ∈ D(Q0 ) : mQ (M [χQ0 w]) > amQ0 (w) . 0 0 Denote by S E(Q ) the set S of all maximal dyadic cubes in D(Q ). We write Z ≡ Q = Q. Then as long as a  1, we have |Z| = Q∈E(Q0 ) Q∈D(Q0 ) S |Q| ≤ 12 |Q0 |. Consequently, for any f ∈ L1 (Q0 ) ∩ M+ (Rn ) \ {0}, Q∈E(Q0 )

1

1 ≤ mQ0 (χQ0 \Z ) ≤ 2

kχQ0 \Z kMpq (1,M [wχQ0 ]) kχQ0 (M [wχQ0 ])− q kHp0 q0

|Q0 | 0

by the duality Mpq (Rn )-Hqp0 (Rn ). From the definition of Z, we have 1 ≤ mQ0 (χQ0 \Z ) 2

 1

1 χQ0 aw(Q0 ) q

−1 0 p p 0 \Z kMp (1,M [wχ ≤ |Q | kχ

Q q Q0 ]) q |Q0 | M [wχQ0 ]

. 0

Hp q0

If we use (2), then we obtain the desired result.

Weighted Morrey spaces

189

(2) Assume that (3) holds. Then for any fixed cube Q, we have Z q q |Q| p −1 M [χRn \3Q f ](x)q w(x)dx . |Q| p −1 w(Q) sup mR (|f |)q R∈Q] (Q)

Q

.

|R|

sup

q p −1

w(R)mR (|f |)q

R∈Q] (Q) q

. |R| p −1

Z

M f (x)q w(x)dx,

R

where we have used (3) for the second inequality. Meanwhile, |Q|

q p −1

Z

q

M [χ3Q f ](x) w(x)dx . |Q|

q p −1

Q

Z

|f (x)|q M w(x)dx

3Q

by the dual inequality of Stein-type. Thus, if we assume (3), then we have a weighted dual inequality of Stein-type. For the case of the power weights, we have the following characterization for the power weighted dual inequality of Stein-type: Corollary 162. Let 1 < q < p < ∞ and −n < α ≤ 0. Then the power weighted dual inequality of Stein-type kM f kMpq (1,|x|α ) . kf kMpq (1,|x|α ) holds q for all f ∈ L0 (Rn ) if and only if α ≥ − n. p Proof The “if” part is clear since we can easily verify Theorem 161(3) using Example 113 in the first book. Meanwhile the “only if” part follows since χB(1) . M χB(2)\B(1) ∈ / Mpq (1, |x|α ), while χB(2)\B(1) ∈ Mpq (1, |x|α ) ≈ q Mpq (1, M wα ) if α < − n. p

15.2.9

Exercises

Exercise 42. [89, Lemma 5.1] Let 1 ≤ q ≤ p < ∞. 1

(1) Show that f ∈ Mpq (1, w) if and only if f w q ∈ Mpq (Rn ). p

(2) Show that Lp (w q ) ,→ Mpq (1, w). Exercise 43. [423, Proposition 2.1] Let w(t) ≡ et for t ∈ R. Show that w−1 ∈ M21 (1, w2 ) \ L2 (w2 ). Exercise 44. [353, Theorem 4.2] Prove Theorem 148 by combining Lemma 146 and Theorem 140. Exercise 45. [89, Proposition 5.2] Let 1 ≤ q ≤ p < ∞. Assume that M : Mpq (1, w) → WMpq (1, w) is bounded. Then show that w ∈ Aq+1− pq .

190

Morrey Spaces

Exercise 46. Let 1 < q ≤ p < ∞. Let ( ) ∞ X k ∞ N 1 n F ≡ y+ R ak : {ak }k=1 ∈ {0, 1} ∩ ` (N), y ∈ [0, 1] , k=1 n

n

n

where R > 1 solves (1 + R) p − q 2 q = 1. Recall that F is the self-similar increasing sequence for Mpq (Rn ) considered in Chapter 1. Denote by Q(F ) the set of all connected components in F . X (1) Show that kχQ kMpq kχQ kHp0 = ∞. q0

Q∈Q(F )

(2) Denote by Y the set of all families of disjoint cubes. Disprove that the estimate X kf χQ kMpq kgχQ kHp0 . kf kMpq kgkHp0 q0

Q∈A

q0

holds for all f, g ∈ M+ (Rn ) and A ∈ Y.

15.3

Weighted local Morrey spaces

We move on to local Morrey spaces. We characterize the class of weights for which the Hardy–Littlewood maximal operator M is bounded. It turns out that singular integral operators are bounded if and only if M is bounded. As an application of the characterization, the power weight function | · |α is considered. The condition on α for which M is bounded can be described completely. Let 1 < q ≤ p < ∞. Following the notation in the works [71, 129, 435], we define weighted local Morrey spaces as follows: For f ∈ L0 (Rn ) and weights u and w, we write 1

1

kf kLMpq (u,w) ≡ sup u(Q(R)) p − q kf χQ(R) kLq (w) . R>0

The two-weight local Morrey space LMpq (u, w) is the set of all f ∈ L0 (Rn ) for which the norm kf kLMpq (u,w) is finite. If u = 1, then we call Mpq (1, w) the local weighted Morrey space of Samko-type [389, 390] and if u = w, then we call Mpq (w, w) the local weighted Morrey space of Komori–Shirai-type [243]. When p = q, LMpp (u, w) = Lp (w) with coincidence of norms. Hence LMpp (1, w) = LMpp (w, w) = Lp (w). Consequently, the theory of Ap applies readily in this case. We are thus interested in the case where 1 < q < p < ∞.

Weighted Morrey spaces

191

The weighted local Morrey space LMpq (u, w) is a contrast to the weighted Morrey space Mpq (u, w) which consists of all f ∈ L0 (Rn ) for which the norm 1

1

kf kMpq (u,w) ≡ sup u(Q) p − q kf χQ kLq (w) Q∈Q

is finite. If u(x) ≡ |x|α and w(x) ≡ |x|β , x ∈ Rn with α, β ∈ R, then we say that LMpq (|x|α , |x|β ) ≡ LMpq (u, w) is the power weighted local Morrey space. We investigate the boundedness property of M in Section 15.3.1, where we clarify the structure of the weighted local Morrey spaces. As an application of what we obtained in Section 15.3.1, Section 15.3.2 considers the boundedness property of singular integral operators. Fractional maximal operators and fractional integral operators will be considered in Section 15.3.3.

15.3.1

Maximal operators in weighted local Morrey spaces

Recall that D = D(Rn ) stands for the set of dyadic cubes. Denote by dist∞ α (Rn ) and the `∞ -distance on Rn . Motivated by the norm equivalence of K˙ p∞ n p LMq (R ) in the first book, we defined the base LQ of cubes by LQ ≡ {Q ∈ D : dist∞ ({0}, Q) = `(Q)}. A local cube is a cube in LQ. Definition 33. We define the weight class G by the set of all weights u with the following properties: (i) u is doubling and reverse doubling at the origin. That is, there exist β > α > 1 such that αu(Q(R)) ≤ u(Q(2R)) ≤ βu(Q(R)) for all R > 0; (ii) u is doubling with respect to LQ: u(Q(2`(Q))) . u(Q) for all Q ∈ LQ. Our results are based upon the following structure of weighted local Morrey α (Rn ) and spaces, whose motivation comes from the norm equivalence of K˙ p∞ LMpq (Rn ) in the first book. Lemma 163. Let 1 < q < p < ∞, and let u and w be weights. Assume that 1 1 u ∈ G. Then kf kLMpq (u,w) ∼ sup u(Q) p − q kf χQ kLq (w) for any f ∈ L0 (Rn ). Q∈LQ

Proof The proof consists of two auxiliary equivalences. We first claim 1

1

kf kLMpq (u,w) ∼ sup (u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) .

(15.1)

m∈Z

1

1

Clearly, kf kLMpq (u,w) ≥ sup u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) follows m∈Z

from the definition of the norm. To obtain the reverse inequality, we fix 1 1 R > 0. If we set j ≡ [1 + log2 R], then u(Q(R)) p − q kf χQ(R) kLq (w) .

192

Morrey Spaces 1

1

u(Q(2j )) p − q kf χQ(2j ) kLq (w) thanks to the doubling property of u at the origin. We decompose j

u(Q(2 ))

1 1 p−q

j

kf χQ(2j ) kLq (w) ≤ u(Q(2 ))

j−1 X

1 1 p−q

kf χQ(2l+1 )\Q(2l ) kLq (w) .

l=−∞

Since q < p and the weight u is assumed reverse doubling at the origin, we j−1 P 1 1 1 1 u(Q(2l ))− p + q ∼ 1. Hence, have u(Q(2j )) p − q l=−∞ j

u(Q(2 ))

1 1 p−q

kf χQ(2j ) kLq (w)

1

j−1 X

1

≤ u(Q(2j )) p − q

1

1

1

1

u(Q(2l ))− p + q · u(Q(2l )) p − q kf χQ(2l+1 )\Q(2l ) kLq (w)

l=−∞ 1

j−1 X

1

≤ u(Q(2j )) p − q

l=−∞ 1

1

1

sup

u(Q(2m+1 )) p − q 1

1

u(Q(2l )) p − q

m∈Z

kf χQ(2m+1 )\Q(2m ) kLq (w)

1

∼ sup u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) . m∈Z 1

1

Thus, we have kf kLMpq (u,w) . sup u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) , m∈Z

which yields (15.1). Since 1

1

u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) .

Q∈LQ∩D−m

1

1

u(Q) p − q kf χQ kLq (w) ,

sup (Rn )

we have 1

1

1

1

u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) . sup u(Q) p − q kf χQ kLq (w) (15.2) Q∈LQ 0

n

for any f ∈ L (R ) and m ∈ Z. Finally thanks to the doubling condition of u with respect to LQ, we have the reverse inequality 1

1

1

1

u(Q) p − q kf χQ kLq (w) . u(Q(2m+1 )) p − q kf χQ(2m+1 )\Q(2m ) kLq (w) for all cubes Q ∈ LQ ∩ D−m (Rn ). The proof of Lemma 163 is thus complete. We seek a characterization of the boundedness of M . We will use the K¨othe dual. Let 1 < q < p < ∞, and let u and w be weights. For g ∈ L0 (Rn ), its LMpq (u, w)-associate norm kgkLMpq (u,w)0 is defined by n o kgkLMpq (u,w)0 ≡ sup kf · gkL1 : f ∈ LMpq (u, w), kf kLMpq (u,w) ≤ 1 . The space LMpq (u, w)0 collects all g ∈ L0 (Rn ) for which the norm kgkLMpq (u,w)0 is finite. The space LMpq (u, w)0 is called the K¨ othe dual of LMpq (u, w) or the associated space of LMpq (u, w).

Weighted Morrey spaces

193

Theorem 164. Let 1 < q < p < ∞ and let u and w be weights. Assume that u ∈ G. Then the following are equivalent: (1) The Hardy–Littlewood maximal operator M is bounded on LMpq (u, w); 1 kχR kLq (w) kχR kLq (w)0 . 1 Q∈LQ R∈Q(2Q) |R| 1 and sup sup kχQ kLMpq (u,w) kχR kLMpq (u,w)0 . 1. |R| ] Q∈LQ R∈Q (Q)

(2) sup

sup

Proof Assume (1). One can deduce kχQ kLMpq (u,w) kχR kLMpq (u,w)0 . |R| for all dyadic cubes Q ∈ LQ and all cubes R ∈ Q] (Q) with ease: simply use mR (|f |)χQ ≤ inf M f (x). Meanwhile, if Q ∈ LQ, thanks to (1), x∈Q 1

1

u(Q(4`(Q))) p − q kχ2Q M [f χ2Q ]kLq (w) . kM [f χ2Q ]kLMpq (u,w) . kf χ2Q kLMpq (u,w) 1

1

∼ u(Q(`(Q)/2)) p − q kf χ2Q kLq (w) . Thus, kM [f χ2Q ]χ2Q kLq (w) . kf χ2Q kLq (w) due to the doubling property of u 1 at the origin. This is equivalent to sup sup kχR kLq (w) kχR kLq (w)0 . 1. Q∈LQ R∈Q(2Q) |R| Let us prove (2) assuming (1). We start with (M f )χQ . M [f χ2Q ]χQ + sup χQ mR (|f |) for Q ∈ LQ, so that R∈Q] (2Q) 1

1

u(Q) p − q kχQ M f kLq (w) 1

1

1

1

. u(Q) p − q kχQ M [f χ2Q ]kLq (w) + u(Q) p − q kχQ kLq (w)

sup

mR (|f |).

R∈Q] (2Q)

Together with the Aq -property at Q, Lemma 163 and conditions (i) and (ii) on u in Definition 33, we have 1

1

u(Q) p − q kχQ M f kLq (w) 1

1

. u(Q(4`(Q))) p − q kf χ2Q kLq (w) 1 + kχQ kLMpq (u,w) sup kχR kLMpq (u,w)0 kf kLMpq (u,w) R∈Q] (2Q) |R| . kf kLMpq (u,w) . Thus, the proof is complete. It seems useful to relate (LMpq )0 (Rn ) and LMpq (1, w)0 . Example 66. Let 1 < q ≤ p < ∞. Then for all f ∈ L0 (Rn ), kf kLMpq (1,w)0 = 1

1

kf w− q k(LMpq )0 , since kf kLMpq (1,w) = kf w q kLMpq .

194

Morrey Spaces

As a special case of Theorem 164, we obtain the following characterizations of LMpq (1, w) and LMpq (w, w). Corollary 165. Let 1 < q < p < ∞ and let w be a weight. Then M is bounded 1 on LMpq (1, w) if and only if sup sup kχR kLq (w) kχR kLq (w)0 . 1 |R| Q∈LQ R∈Q(2Q) 1 kχQ kLMpq (1,w) kχR kLMpq (1,w)0 . 1. and sup sup Q∈LQ R∈Q] (Q) |R| For local Morrey spaces of Komori–Shirai-type, we have the following characterization: Proposition 166. Let 1 < q < p < ∞, and let w be a weight. Then M is bounded on LMpq (w, w) if and only if w ∈ G, 1 sup sup kχR kLq (w) kχR kLq (w)0 . 1 |R| Q∈LQ R∈Q(2Q) 1 and sup sup kχQ kLMpq (w,w) kχR kLMpq (w,w)0 . 1. Q∈LQ R∈Q] (Q) |R| Proof We need only verify that w ∈ G, if (1) holds. Once this is verified, then we are in the position of using Theorem 164. For Q = Q(R), R > 0, and any measurable set E ⊂ Q, since



1 1 |E| − q1 |E| p p w(Q) = w(Q) χQ ,

|Q| |Q| Lq (w) Lp (w) ,→ LMpq (w, w) and M is assumed bounded on LMpq (w, w), 1 1 |E| w(Q) p ≤ kM χE kLMpq (w,w) . kχE kLMpq (w,w) ≤ kχE kLp (w) = w(E) p . |Q|

This A∞ -property of w at the origin is more than enough to guarantee that w ∈ G. We can also consider the boundedness of the maximal operator local weighted Morrey spaces of Komori–Shirai-type. It is noteworthy the conditions (2) and (3) in Proposition 167 below coincide with (2) and (3) in Proposition 131. Proposition 167. Let 1 < q < p < ∞ and let β ∈ R. Then the following are equivalent: (1) The maximal operator M is bounded on LMpq (|x|β , |x|β ); (2) −n < β < n(p − 1); (3) | · |β ∈ Ap .

Weighted Morrey spaces

195

Proof According to Theorem 288, (2) and (3) are equivalent. If we assume (1), then M is bounded from Lp (|x|β ) to LMpq (|x|β , |x|β ) since Lp (|x|β ) is R  p1 embedded into LMpq (|x|β , |x|β ). As a consequence, Q |x|β dx mQ (|f |) . kf k p β for all f ∈ L0 (Rn ) and all cubes Q centered at the origin. Thus, R L β(|x| ) p−1 |x| dx (mQ (σβ )) . 1 for all cubes centered at the origin, where σβ (x) ≡ Q β

|x|− p−1 , x ∈ Rn . Consequently, we have −n < β < n(p − 1). Assume −n < β < n(p − 1). As before, 1 kχR kLq (|x|β ) kχR kLq (|x|β )0 R∈D(Q) |R| sup

∼

β β 1 kχR |c(Q)| q kLq kχR |c(Q)|− q kLq0 R∈D(Q) |R|

sup

=1 for all cubes Q ∈ LQ. Next we will establish Z kχQ kLq (|x|β ) kχR k

β β 0 LMp q (|x| ,|x| )

β

|x| dx

.

 q1 − p1 |R|

(15.3)

Q

or equivalently kχQ kLMpq (|x|β ,|x|β ) kχR kLMpq (|x|β ,|x|β )0 . |R| for all cubes Q ∈ LQ and R ∈ Q] (Q). To this end, by replacing R with a larger one, say Q(0, 10`(R)), we may assume that R = Q(2r) is centered at the origin. Write R∗ = [r, 2r]n as before. Note that kχ2−l R\2−l−1 R kLMpq (|x|β ,|x|β )0 ∼ kχR∗ (2l ·)kLMpq (|x|β ,|x|β )0 thanks to the rotational invariance of the measure. By the triangle inequality, kχR kLMpq (|x|β ,|x|β )0 ≤ .

∞ X l=0 ∞ X

kχ2−l R\2−l−1 R kLMpq (|x|β ,|x|β )0 kχR∗ (2l ·)kLMpq (|x|β ,|x|β )0 .

l=0

By the dilation formula for LMpq (|x|β , |x|β )0 as in Example 59, we have kχR∗ (2l ·)kLMpq (|x|β ,|x|β )0 = 2−ln+

l(n+β) p

kχR∗ kLMpq (|x|β ,|x|β )0 .

196

Morrey Spaces

Assuming β < n(p − 1), we have kχR kLMpq (|x|β ,|x|β )0 .

∞ X

2−ln+

l(n+β) p

kχR∗ kLMpq (|x|β ,|x|β )0 .

l=0

∼ kχR∗ kLMpq (|x|β ,|x|β )0 β

β

β

1

. |R∗ | q − p |c(R∗ )|− q |R∗ | q0 β

. |R∗ |1− p . As a result, since n + β > 0, kχQ kLq (|x|β ) kχR kLMpq (|x|β ,|x|β )0 . `(Q)

n+β q n+β

`(R)n−

n+β p

n+β

. `(Q) q − p |R| Z  q1 − p1 β . |x| dx |R|, Q

proving (15.3). We can also consider the case of power weights. We investigate local weighted Morrey spaces of Samko type. It is worth comparing Proposition 142 with Proposition 168 below. Proposition 168. Let 1 < q < p < ∞, andlet β ∈ R. Then M is bounded q 1 n. on LMpq (1, |x|β ) if and only if − n ≤ β < q 1 − p p It may be interesting that Proposition 168 can be used for the proof of Proposition 142. Proof One implication is easy to prove:Assuming  that M is bounded on q 1 LMpq (1, |x|β ), we can deduce − n ≤ β < q 1 − n in the same manner as p  p  q 1 Example 65. Assume − n ≤ β < q 1 − n. We calculate p p β

β

kχR kLq (|x|β ) kχR kLq (|x|β )0 kχR |c(Q)| q kLq kχR |c(Q)|− q kLq0 sup ∼ sup |R| |R| R∈D(Q) R∈D(Q) =1 for all cubes Q ∈ LQ. 1 1 Next we will prove kχQ kLq (|x|β ) kχR kLMpq (1,|x|β )0 . |Q| q − p |R|, namely, kχQ kLMpq (1,|x|β ) kχR kLMpq (1,|x|β )0 . |R| for all cubes Q ∈ LQ and R ∈ Q] (R). To this end, by replacing R with a larger one, we may assume that R = Q(2r). Write R∗ ≡ [r, 2r]n . Write wβ (x) ≡ |x|β for x ∈ Rn . By the dilation formula

Weighted Morrey spaces

197

for (LMpq )0 (Rn ), we have lβ

1

1

kχR∗ (2l ·)wβ − q k(LMpq )0 = 2 q kχR∗ (2l ·)wβ (2l ·)− q k(LMpq )0 lβ

=2q

− ln p0

1

kχR∗ wβ − q k(LMpq )0 .

Using the triangle inequality, we have 1

kχR kLMpq (1,|x|β )0 = kχR wβ − q k(LMpq )0 ≤

∞ X

1

k(χR (2l ·) − χR (2l+1 ·))wβ − q k(LMpq )0 .

l=0

Since the weight is symmetric with respect to the origin, we have kχR kLMpq (1,|x|β )0 .

∞ X

1

kχR∗ (2l ·)wβ − q k(LMpq )0 .

l=0

Since β < q 1 −

1 p




n, we have β

1

kχR kLMpq (1,|x|β )0 . kχR∗ wβ − q k(LMpq )0 ∼ |c(R∗ )|− q kχR∗ k(LMpq )0 . Thus, since

q n + β > 0, thanks to Example 113 in the first book p β

β

kχQ kLq (|x|β ) kχR kLMpq (1,|x|β )0 . |c(Q)| q kχQ kLq |c(R∗ )|− q kχR∗ k(LMpq )0 . `(Q) . `(Q) 1

1

β

n+β q

|c(R∗ )|− q |R| p0

n+β q

`(R)n− p − q

n

β

1

. |Q| q − p |R|, as required. The ranges obtained in Propositions 167 and 168 are the same as those for weighted Morrey spaces of Komori–Shirai-type (Proposition 131) and those for weighted Morrey spaces of Samko-type (Proposition 142), respectively.

15.3.2

Singular integral operators in weighted local Morrey spaces

Our characterization can be applied to singular integral operators including the Riesz transform. For singular integral operators, we have the following characterization: Theorem 169. Let T be a singular integral operator. Also let 1 < q < p < ∞, and let u and w be weights. Assume that u ∈ G and that u and w satisfy the equivalent conditions in Theorem 164 and that there exists α > 1 such that kχ2k+1 Q0 kLMpq (u,w) ≥ αkχ2k Q0 kLMpq (u,w) for any k ∈ N. Then n kT f kLMpq (u,w) . kf kLMpq (u,w) for all f ∈ L∞ c (R ).

198

Morrey Spaces

Proof Let Q0 ∈ LQ be fixed. By decomposing T f into its real part and its imaginary part, we may assume T f is a real-valued function, so that Med(T f ; Q0 ) makes sense. Form the Lerner–Hyt¨onen decomposition of T f at Q0 . Then we obtain a sparse family of dyadic cubes {Qjk }j∈N0 , k∈Kj in Q0 ∞ P P satisfying |T f (x) − Med(T f ; Q0 )| ≤ 2 ωj,k χQj (x) for almost every k

j=0 k∈Kj

0

x ∈ Q . Write 1

ωjk

≡

ω2−n−2 (T f ; Qjk ),

1

u(Q0 ) p − q kχQ0 T f kLq (w)

Thus, we have



X ∞ X

1 − q1 0 p ωj,k χQj ≤ 2u(Q )

k

j=0 k∈Kj

Lq (w)

+ u(Q0 )

1 1 p−q

1 q

w(Q0 ) |Med(T f ; Q0 )|.

1

Set σ ≡ w− q−1 . To dualize the first term in the right-hand side, we choose ∞ P R P 0 g ∈ Lq (σ) ∩ M+ (Rn ) and consider I ≡ Rn ω(T f ; Qjk )χQj (x)g(x)dx. k

j=0 k∈Kj

Then I=

∞ X X

ω(T f ; Qjk )

j=0 k∈Kj

Z g(x)dx ≤

∞ X X

Qjk

ω(T f ; Qjk )|Qjk | inf j M g(x). x∈Qk

j=0 k∈Kj

Set Ekj ≡ Qjk \ Ωk+1 . Then 2|Ekj | ≥ |Qjk |. Thanks to Lemma 217 in the first book, I.

∞ X X j=0 k∈Kj

Since

{Ekj }j∈N0 ,k∈Kj

inf M f (x)|Ekj | inf j M g(x).

x∈Qjk

x∈Qk

0

Z

is a disjoint family contained in Q , I .

M f (x)M g(x)dx. Q0

0

If we use the Lq (σ)-boundedness of M in Q0 and kgkLq0 (σ) = 1, then



∞ X

1X

w q ωj,k χQj

k

j=0 k∈Kj

1

. kw q M f kLq (Q0 ) .

Lq

Consequently, since M is bounded on LMpq (u, w) thanks to Theorem 164,



∞ X X

1 1 1

. u(Q0 ) p1 − q1 kw q1 M f kLq (Q0 ) q j u(Q0 ) p − q w ω χ j,k Q

k

q j=0 k∈Kj L

. kM f kLMpq (u,w) . kf kLMpq (u,w) .

Weighted Morrey spaces

199

Since there exists α > 1 such that kχ2k+1 Q0 kLMpq (u,w) ≥ αkχ2k Q0 kLMpq (u,w) ∞ P for all k ∈ N0 and |Med(T f ; Q0 )| . m2l Q0 (|f |) by Lemma 217, we have l=1 1

1

1

|Med(T f ; Q)|u(Q0 ) p − q w(Q0 ) q .

1 1 ∞ X u(2l Q0 ) p − q

l=1

αl

1

w(2l Q0 ) q m2l Q0 (|f |)

. kM f kLMpq (u,w) . kf kLMpq (u,w) , as was to be shown. Under the same condition as the Hardy–Littlewood maximal operator, we will characterize the boundedness of singular integral operators. Theorem 170. Let 1 < q < p < ∞ and let w be a weight. Assume in addition n that u ∈ G and that kRj f kLMpq (u,w) . kf kLMpq (u,w) for all f ∈ L∞ c (R ) and for j = 1, 2, . . . , n. Then u and w satisfy the equivalent conditions in Theorem 164. We say that a sequence Q0 , Q1 , . . . , QK in LQ is a chain if `(Qk−1 ) = 2`(Qk ) and Qk−1 and Qk intersect at a set of Lebesgue measure zero for all k = 1, 2, . . . , K and Qj ∩ Qk = ∅ if |j − k| ≥ 2. We need a lemma. Lemma 171. Let 1 < q < p < ∞, and let w be a weight. Assume that n u ∈ G and that kRj f kLMpq (u,w) . kf kLMpq (u,w) for all f ∈ L∞ c (R ) and j = 1, 2, . . . , n. Suppose that we have a chain Q0 , Q1 , Q2 , Q3 in LQ. (1) The cubes 2Q0 and Q3 do not intersect. n + n (2) For any f ∈ L∞ c (R ) ∩ M (R ) supported on 2Q0 , we have Z n X 1 |Rj f (x)| & f (y)dy (x ∈ Q3 ). |Q0 | 2Q0 j=1

(15.4)

(3) kχ2Q0 kLMpq (u,w) ∼ kχQ3 kLMpq (u,w) . (4)

1 kχ2Q0 kLq (w) kχ2Q0 kLq (w)0 < ∞. |Q 0| Q0 ∈LQ sup

Proof We suppose that Q0 = [2m , 2m+1 )n and Q3 = [2m−3 , 2m−2 )n for the sake of simplicity. (1) Since 2Q0 = [2m−1 , 5 · 2m−1 ]n and Q3 = [2m−3 , 2m−2 ]n , so that 2Q0 and Q3 do not intersect. + n (2) Let f ∈ L∞ c (2Q0 ) ∩ M (R ). Since

−

n X xj − yj 1 ∼ n+1 |x − y| `(Q0 )n j=1

200

Morrey Spaces for any x ∈ Q3 and y ∈ 2Q0 , we have n X

|Rj f (x)| = −

j=1

n X j=1

Rj f (x) = −

n Z X j=1

Rn

xj − yj f (y)dy & m2Q0 (f ). |x − y|n+1

(3) We can show that kχQ3 kLMpq (u,w) & kχ2Q0 kLMpq (u,w) by considering f ≡ χ2Q0 in (15.4) and using kRj χ2Q0 kLMpq (u,w) . kχ2Q0 kLMpq (u,w) . We can swap the role of 2Q0 and Q3 to give kχ2Q0 kLMpq (u,w) . kχQ3 kLMpq (u,w) . (4) From (15.4), 1

1

kχQ0 kLMpq (u,w) m2Q0 (f ) . kf kLMpq (u,w) ∼ u(Q0 ) p − q kf kLq (w) for any f ∈ M+ (2Q0 ). It remains to use the duality. Going through a similar argument, we have the following corollary. Corollary 172. Let 1 < q < p < ∞ and let w be a weight. Assume that n u ∈ G and that kRj f kLMpq (u,w) . kf kLMpq (u,w) for all f ∈ L∞ c (R ) and j = 1 1, 2, . . . , n. Then sup sup kχR kLq (w) kχR kLq (w)0 < ∞. |R| Q∈LQ R∈Q(2Q) Proof Let R ∈ Q(2Q). By decomposing R or by expanding R, we can assume that R ∈ D(Rn ). Then we can choose a chain S = S1 , S2 , S3 = R. If we replace Q3 by S and Q0 by R, respectively in the proof of Lemma 171, we can argue as before. We prove Theorem 170. In view of Corollary 172, it remains to show that sup

sup

Q∈LQ R∈Q] (Q)

1 kχQ kLMpq (u,w) kχR kLMpq (u,w)0 < ∞. |R|

Let R ∈ Q] (Q). By expanding and decomposing R suitably, we can further assume that there exists a chain R = Q0 , Q1 , Q2 , Q3 such that R and 2Q3 do not intersect. n P Then mR (f ) . |Rj f (y)| for all y ∈ Q3 and for any f ∈ L∞ c (R) ∩ j=1

M+ (Rn ). As a result, assuming that kRj f kLMpq (u,w) . kf kLMpq (u,w) , we see mR (f )kχQ3 kLMpq (u,w) . kf kLMpq (u,w) . Since kχQ3 kLMpq (u,w) ∼ kχQ kLMpq (u,w) thanks to Lemma 171(3), we obtain mR (f )kχQ kLMpq (u,w) . kf kLMpq (u,w) . By passing to the K¨ othe dual, we obtain kχR kLMpq (u,w)0 kχQ kLMpq (u,w) . |R|. Thus, the proof is therefore complete. To investigate weighted Morrey spaces of Samko-type, we simply let u = 1 in Theorems 169 and 170. Meanwhile, the sufficiency of the boundedness of singular integral operators on weighted Morrey spaces of Komori–Shiraitype is clear. However the necessity is somewhat non-trivial. Consequently, we formulate it.

Weighted Morrey spaces

201

Proposition 173. Let 1 < q < p < ∞, and let w be a doubling weight with respect to LQ. Assume that kRj f kLMpq (w,w) . kf kLMpq (w,w) for all n f ∈ L∞ c (R ) and for all j = 1, 2, . . . , n. Then u and w satisfy the equivalent conditions in Theorem 164. Proof We need to prove that w ∈ G in particular, w is reverse doubling at the origin. To this end, we observe kRj f kLMpq (w,w) . kf kLp (w) for all n 0 f ∈ L∞ c (R ). Let Q be a cube centered at the origin, and let Q be a cube 0 0 such that |Q| = |Q | and ](Q ∩ Q ) = 1, where ]E stands for the cardinality of the set E ⊂ Rn . Since w is doubling with respect to LQ, we have 1

1

w(Q) p mQ (|f |) . w(Q0 ) p mQ (|f |) . kf kLp (w) 1

for all f ∈ L∞ (Rn ) with supp(f ) ⊂ Q. Let R ∈ Q(Q). If we let f = w− q χR , then we obtain w ∈ Aq+1 (Q). Thus, we see that w ∈ G and we are in the position of using Theorem 170.

15.3.3

Fractional maximal operators and fractional integral operators in weighted local Morrey spaces

The boundedness of fractional integral operators can be characterized in a similar way. Let 1 < p < s < ∞. Recall that the class Aα p,s of weights, considered in Chapter 7, turned out to be the set of all weights for which there exists a constant C > 0 satisfying kIα f · wkLs ≤ Ckf · wkLp for any f ∈ M+ (Rn ). To extend this boundedness to Morrey spaces, we can consider two types of boundedness. One is due to Spanne [454] and the other is due to Adams [2]. The following theorem corresponds to the result due to Spanne. Theorem 174. Let 1 < q < p < ∞, 1 < t < s < ∞ and 0 < α < n satisfy α=

n n n n − = − . p s q t

Let w be a weight. Also let u ∈ G. Then the following are equivalent: (1) for any Q ∈ LQ, `(R)α kχR kLt (wt ) kχR kLq (wq )0 . 1 R∈Q(2Q) |R| sup

(15.5)

and `(R)α kχQ kLMst (u,wt ) kχR kLMpq (u,wq )0 . 1. R∈Q] (Q) |R| sup

(15.6)

(2) for f ∈ M+ (Rn ) kMα f kLMst (u,wt ) . kf kLMpq (u,wq ) .

(15.7)

202

Morrey Spaces

(3) Furthermore, if there exists κ > 1 such that 2kχQ kLMst (u,wt ) ≤ kχκQ kLMst (u,wt ) for all cubes Q ∈ LQ, then these equivalent conditions are also equivalent to the following: (3) For f ∈ M+ (Rn ) kIα f kLMst (u,wt ) . kf kLMpq (u,wq ) .

(15.8)

Proof We can prove that (15.5) and (15.6) are equivalent to (15.7) similar to Theorem 164. Meanwhile, (15.8) is stronger than (15.7). It remains to show that (15.5) and (15.6) imply (15.8) if there exists κ > 1 such that 2kχQ kLMst (u,wt ) ≤ kχκQ kLMst (u,wt ) for all cubes Q ∈ LQ. Since we can use the Lerner–Hyt¨ onen decomposition, this is achieved similar to Proposition 168. Remark that the assumption in (3) is necessary for (15.8) to hold. We move on to the Spanne type inequality of weighted Morrey spaces of Samko-type. Proposition 175. Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞. Assume that 1 α 1 1 α 1 = − , = − . s p n t q n Then the following are equivalent: (1) The fractional maximal operator Mα is bounded from LMpq (1, |x|qβ ) to n n LMst (1, |x|tβ ) if and only if − ≤ β < 0 . s p (2) The fractional integral operator Iα is bounded from LMpq (1, |x|qβ ) to n n LMst (1, |x|tβ ) if and only if − < β < 0 . s p Proof (1) We have χQ(1) ∈ LMst (1, |x|tβ ) and χQ(1) | · |−n ∈ / LMpq (1, |x|qβ ), if the fractional maximal operator Mα is bounded from LMpq (1, |x|qβ ) to n n LMst (1, |x|tβ ). Due to Example 58 this is equivalent to − ≤ β < 0 . s p n n If we assume − ≤ β < 0 , then we can argue as we did in the proof of s p Proposition 168 to conclude that `(R)β kχR kLt (|x|tβ ) kχR kLq (|x|qβ )0 < ∞ Q∈LQ R∈Q(2Q) |R| sup

sup

and that `(R)β kχQ kLMst (1,|x|tβ ) kχR kLMpq (1,|x|qβ )0 < ∞. Q∈LQ R∈Q] (Q) |R| sup

sup

Weighted Morrey spaces

203

In fact, if R ∈ Q(2Q), then `(R)β `(R)β kχR kLt (|x|tβ ) kχR kLq (|x|qβ )0 ∼ kχR kLt kχR k(Lq )0 = 1. |R| |R| Meanwhile if R ∈ Q] (Q), we denote by R∗ the cube satisfying |R∗ | = 100n |R| and centered at the origin. Then 1

|Q| s |c(Q)|β kχR∗ kLMpq (1,|x|qβ )0 kχQ kLMst (1,|x|tβ ) kχR kLMpq (1,|x|qβ )0 . . `(R)n−β `(R∗ )n−β We note that n

kχR∗ kLMpq (1,|x|qβ )0 = k| · |−β χR∗ k(LMpq )0 ≤ k| · |−β χR∗ kLp0 . `(R) p0

−β

,

n to guarantee that k|·|−β χR∗ kLp0 is finite. Observe p0 n also that c(Q) ∼ `(Q). As a result, since − ≤ β, s β+ ns  `(R)β `(Q) kχQ kLMst (1,|x|tβ ) kχR kLMpq (1,|x|qβ )0 . ≤ 1. |R| `(R)

where we used β
1 s p such that 2kχQ kLMst (1,|x|tβ ) ≤ kχκQ kLMst (1,|x|tβ ) for all cubes Q ∈ LQ. Thus, we are in the position of using (1) and Theorem 174.

(2) Thanks to (1) we have −

In the case of power weighted Morrey spaces of Komori–Shirai-type, we have the following characterization of the boundedness. Proposition 176. Let 0 < α < n, β ∈ R, 1 < q < p < ∞ and 1 < t < s < ∞. Assume that 1 1 α 1 1 α = − , = − . s p n t q n Then the following are equivalent: (1) The fractional maximal operator Mα is bounded from LMpq (|x|tβ , |x|qβ ) n ns to LMpq (|x|tβ , |x|tβ ) if and only if − ≤ β < 0 . t pt (2) The fractional integral operator Iα is bounded from LMpq (|x|tβ , |x|qβ ) to n ns LMpq (|x|tβ , |x|tβ ) if and only if − < β < 0 . t pt

204

Morrey Spaces

We borrow from the idea of [89] for the proof of Proposition 176 again. Proof (1) Assume that Mα is bounded from LMpq (|x|tβ , |x|qβ ) to LMpq (|x|tβ , |x|tβ ); Then χQ(1) ∈ LMpq (|x|tβ , |x|tβ ) and χQ(1) | · |−n ∈ / LMpq (|x|tβ , |x|qβ ). n ns Using Example 58, we conclude − ≤ β < 0 . t pt n ns Assume instead that − ≤ β < 0 . We start with arithmetic. Let t pt   t t n n t n n λ≡β 1+ − + − + ε = β + − + ε. p q p q s p q Here, ε > 0 is chosen small enough to have λ
> 0. q0 p0 p q p0 t p q Arguing as we did in the proof of Theorem 167, we claim that `(R)β kχR kLt (|x|tβ ) kχR kLq (|x|qβ )0 Q∈LQ R∈Q(2Q) |R| sup

sup

`(R)β kχQ kLMpq (|x|tβ ,|x|tβ ) kχR kLMpq (|x|tβ ,|x|qβ )0 . 1. Q∈LQ R∈Q] (Q) |R|

+ sup

sup

In fact, `(R)β kχR kLt (|x|tβ ) kχR kLq (|x|qβ )0 Q∈LQ R∈Q(2Q) |R| sup

sup

`(R)β |c(R)|β kχR kLt |c(R)|−β kχR k(Lq )0 |R| Q∈LQ R∈Q(2Q)

. sup

sup

= 1. Next, we prove `(R)β kχQ kLMpq (|x|tβ ,|x|tβ ) kχR kLMpq (|x|tβ ,|x|qβ )0 . 1. Q∈LQ R∈Q] (Q) |R| sup

sup

Let Q ∈ LQ and R ∈ Q] (Q). We may assume that R is centered at the n+tβ origin. Note that kχQ kLMpq (|x|tβ ,|x|tβ ) ∼ `(Q) s . For f ∈ M+ (Rn ) Z

Z f (x)dx ≤ kχR f kLq (|x|qλ ) R

R

dx |x|q0 λ

 10 q

n

. `(R) q0

−λ

kχR f kLq (|x|qλ ) .

Weighted Morrey spaces

205

∞ P We decompose kχR f kLq (|x|qλ ) ≤ kχ21−j R\2−j R f kLq (|x|qλ ) as usual. j=1   Since qλ + (n + tβ) 1 − pq − qε = qβ,

kχR f kLq (|x|qλ ) .

∞ X `(R)ε j=1

2jε

tβ

|x| (2

−j

R)

1 1 p−q

! q1

Z

q

qβ

f (x) |x| dx 21−j R\2−j R

. `(R)ε kf kMpq (|x|tβ ,|x|qβ ) . Z

n

f (x)dx . `(R) q0

Thus,

−λ+ε

kf kMpq (|x|tβ ,|x|qβ ) . We note that β − n +

R

n n t n − λ + ε = − − β . Consequently, since β ≥ − , q0 s s t `(R)β kχQ kLMpq (|x|tβ ,|x|tβ ) kχR kLMpq (|x|tβ ,|x|qβ )0 Q∈LQ R∈Q] (Q) |R| sup

sup

. sup Q∈LQ

sup

n

t

n

t

`(Q) s +β s `(R)− s −β s

R∈Q] (Q)

= 1. Thus, Mβ is bounded from LMpq (|x|tβ , |x|qβ ) to LMpq (|x|tβ , |x|tβ ) thanks to Theorem 174. (2) Suppose that Iβ is bounded from LMpq (|x|tβ , |x|qβ ) to LMpq (|x|tβ , |x|tβ ). tβ+n tβ−qβ n ns Then we have − ≤ β < 0 , since Mβ . Iβ . Since | · |− p + q ∈ t pt tβ + n tβ − qβ Mpq (|x|tβ , |x|qβ ), we must have α < − < 0 in order p q tβ+n tβ−qβ that Iα [| · |− p + q ] be finitely valued, which is equivalent to β > n ns n − . Conversely, if − < β < 0 , then we can find κ > 1 such that t t pt 2kχQ kLMst (|x|tβ ,|x|tβ ) ≤ kχκQ kLMst (|x|tβ ,|x|tβ ) for all cubes Q ∈ LQ. Thus, we are in the position of using (1) and Theorem 174.

15.3.4

Exercises

Exercise 47. Show that

X

χQ = χRn \{0} .

Q∈LQ

Exercise 48. Show that ]{Q ∈ LQ : `(Q) = a} ≤ 2n (2n − 1).

206

15.4

Morrey Spaces

Notes

Section 15.1 General remarks and textbooks in Section 15.1 Many researchers investigated the boundedness properties of the linear operators acting on weighted Morrey spaces. See the references below. Applications of weighted Morrey spaces to partial differential equations can be found in [96, 165, 171, 436]. The technique of using the cubes of types I and II and using the dyadic decomposition to control the power weights can be found in [107, §2], which deals with weighted Morrey spaces of Komori– Shirai type and of Samko type at the same time. Section 15.1.1 Guliyev, Hasanov, Karaman, Serbetci, Zhang and Wu considered singular integral operators in [158, 229, 538]. Chen, Guliyev, Karaman, Mustafayev and Serbetci considered sublinear operators [71, 142, 163, 229]. See [142, 169, 524, 527] for commtators, [207] for pseudo-differential operators, for [130] the square functions and [513] for Toeplitz operators. Embedding relations are investigated in [191, Section 3] for power weights and [191, Section 4] for A∞ -weights. A passage of weighted Morrey spaces to metric measure spaces is done in [533, (2.2)]. D. Yang and S. Yang, considered Morrey–Campanato spaces of Komori– Shirai-type [516, 517]. Morrey–Campanato spaces are known to be equivalent to Morrey spaces in many cases. See [470, 516, 517] for weighted characterizations. Several characterizations of weighted Morrey–Campanato spaces given in Example 67 are obtained by Tang in [470]. Section 15.1.2 A fundamental result on the boundedness of the Hardy–Littlewood maximal operator is [243, Theorems 3.1 and 3.2]. Komori and Shirai investigated the boundedness property of the maximal operator acting on weighted Morrey spaces. To prove Theorem 130, we go through the same argument as [243] which is followed by [142, 281, 513, 524] and so on. Zhong, and Jia also obtained the boundedness of the Hardy–Littlewood maximal operator See [539, Theorem 1]. Iida et. al. investigated multilinear maximal operators acting on weighted Morrey spaces [214, 215, 218, 221]. Ho investigated the vector-valued maximal inequality in a wide framework in [203, Theorem 4.2]. The two-weight case for the Hardy–Littlewood maximal operator is investigated in [526]. Ye and Wang used the cube testing to get a characterization of a sufficient condition which guarantees the boundedness of the Hardy–

Weighted Morrey spaces

207

Littlewood maximal operator [526]. Sawano, Nakamura and Tanaka obtained the boundedness of M on Mpq (|x|β , |x|β ) in [357, Proposition 1.11]; see Proposition 131. Section 15.1.3 A fundamental result on the boundedness of singular integral operators is [243, Theorem 3.3]. Hu, Li and Wang, investigated multilinear singular integral operators on weighted Morrey spaces [208]. Singular integral operators with rough kernels are investigated in [434, 493, 497, 540]. Xie and Cao obtained the weighted Morrey boundedness of the commutator given by [T, b]f (x) ≡ T [b · f ](x) − b(x)T f (x). See [513, Corollary 3.3]. We refer to [503, 525] for commutators. Wang dealt with the boundedness of singular integral operators in weighted Morrey spaces of Komori–Shirai-type [501]. Aliyev applied the boundedness of singular integral operators to elliptic differential equations [15]. Persson, N. Samko and Wall considered singular integral operators acting on weighted generalized Morrey spaces [368]. Wang investigated intrinsic square functions [495, 496, 497, 498]. Nakamura, Sawano and Tanaka obtained a characterization of the power β for singular integral operators to be extended to a bounded linear operator on Mpq (|x|β , |x|β ); see [357, Theorems 1.5 and 1.6]. Finally, we make a remark regarding the definition of singular integral operators. There are several methods of the definition of singular integral operators. The definition given here is close to that in [336, (3.1)]. There is another definition by means of the predual [423, p. 483]; see [383, 484] for more recent information. Section 15.1.4 We refer to [214, 218, 219] for fractional integral operators and to [444, 495, 504] for fractional integrals associated to operators including the related commutators. Nakamura, Sawano and Tanaka considered weighted Morrey spaces of Komori–Shirai-type in [357, Proposition 1.18]; see Proposition 135.

Section 15.2 General remarks and textbooks in Section 15.2 Weighted Morrey spaces emerged a little later than generalized Morrey spaces. In fact, in 2008 and 2009, two papers on weighted Morrey spaces appeared. Komori-Furuya and Shirai investigated weighted Morrey spaces mainly in connection with the boundedness of operators [243]. They assumed that the weight belongs to Aq with q ∈ [1, ∞). Meanwhile, Samko considered weighted Morrey spaces in the context of the Cauchy integral [390]. See [389]

208

Morrey Spaces

and [390, Section 6] for her original research of Morrey spaces equipped with the weight of the above type, whose motivation dates back to the Cauchy integral on the complex plane. See her works [292, 392, 393, 395, 397, 398] as well as a recent paper [373] by Qi, Yan and Li for more about weighted Morrey spaces of Samko-type. The paper [26] explains one of the reasons why it is difficult to characterize weights for which the Hardy–Littlewood maximal operator is bounded on weighted Morrey spaces of Samko-type. According to [26, Theorem 2.1], we can characterize the weights for any pair X and Y of Banach function spaces in the class G(B)). We do not give the definition of G(B); we content ourselves with Exercise 46(2) showing that Morrey spaces do not belong to G(B). Iida, Sato, Sawano and Tanaka investigated multilinear maximal operators and multilinear fractional integral operators on weighted Morrey spaces in [221]. In Theorem 154, we obtained the vector-valued maximal inequality. Ho investigated the vector-valued maximal inequality in a wide framework in [203, Theorem 4.2]. Komori-Furuya considered a local good-λ inequality and gave the characterization of weighted Morrey spaces by using the sharp-maximal operators in [240]. It seems that the weighted Morrey norm of Samko-type is connected to the Morrey embedding result as is seen from [106, Theorem 2.3.12]; see [489] for applications of this idea to the Schr¨odinger equation. Section 15.2 N. Samko investigated the boundedness on generalized Morrey spaces of fractional integral operators and Hardy operators in [392, Sections 4 and 5]. Guliyev, Aliyev and Karaman considered the boundedness of various operators in generalized Morrey spaces [145]. Section 15.2.1 The definition of Bp,q goes back to [353, Definition 1.1]. Section 15.2.2 The definition of Bα goes back to the work of Adams and Xiao [9]. After the solution of the Kato problem by Auscher, Hofmann, Lacey, McIntosh and Tchamitchian [19, 206], more and more attention has been paid to the semigroup generated by elliptic operators. We refer to [444, 495] for the boundedness of operators generated by the generator of such semigroup on weighted Morrey spaces. Section 15.2.3 Nakamura investigated weighted Morrey spaces of Samko type in [353]. See [353, Theorem 1.3] for Theorem 140. See [355, Lemma 2] for Theorem 141.

Weighted Morrey spaces

209

Tanaka, Duoandikoetxea and Rosenthal specified the range of β for which M is bounded on Mpq (1, |x|β ) in [466, Proposition 4.2] and [89, Proposition 5.3]; see Proposition 142. The case of q = 1 is also covered in [89, Proposition 5.7]. The idea of using the Hausdorff capacity for the purpose of investigating the boundedness of M goes back to the work of Tanaka; see [466, Proposition 4.2] for Lemma 139. We refer to [240, 421, 485]. The weighted Morrey boundedness of the sharp maximal operator given is investigated in [513, Lemma 3.1]. Shi considered auxiliary weighted sharp maximal estimates [435, Proposition 3.1]. Tran considered the Hardy operator in [476]. Section 15.2.4 Wang, Gao and Wu investigated intrinsic square functions on weighted Morrey spaces together with commutator estimates [123, 494]. Guliyev considered the boundedness of higher commutators in [142, Theorem 3.8], [163, Theorem 1]. See [172] for more about commutators. A fundamental result on the boundedness of fractional integral operators is [243, Theorems 3.6 and 3.7]. The fractional integral operator Iα , 0 < α < n, on weighted Morrey spaces is investigated in [219, 221, 486]. Izumi, KomoriFuruya and Sato also considered the boundedness of Iα on weighted Morrey spaces whose norm is given by (15.1). Ye investigated the boundedness of multicommutators on weighted Morrey spaces in [524, Theorem 1.1]. Ye used the weighted Orlicz-type inequalities to obtain the weak-type estimate [524, Theorem 1.2]. We refer to [504] for more about commutators. The Toeplitz-type operator is given by Tb =

M X

Tj,1 Mb Tj,2 ,

(15.1)

j=1

where Tj,1 and Tj,2 are singular integral operators and b is a function. Xie and Cao obtained the boundedness of Toeplitz-type operators given by (15.1) in weighted Morrey spaces [513, Theorem 3.2]. Liu considered the sharp-maximal inequality in weighted Morrey spaces of Samko-type; see [275, Lemma 7] and [276, Lemma 2.2]. Zhang and Li dealt with the case of generalized Morrey spaces [534], where they investigated commutators. Liu also considered the boundedness of operators such as Littlewood–Paley operators and so on; see [275, 276]. He considered Toeplitz operators with non-smooth kernels; see [198]. See [353, Theorem 1.10] for Theorem 144. Section 15.2.5 Nakamura characterized the condition on α for Mα to be bounded from q Mpq (1, w) to Mupu (1, w); see [353, Theorem 1.7] for Theorem 147. Nakamura,

210

Morrey Spaces

Sawano and Tanaka obtained the necesssary and sufficient condition on β for Mα to be bounded on Mpq (1, |x|β ) in [356, Proposition 4.1]; see Proposition 150. See [436] for applications to the elliptic differential operators, which can be viewed as a weighted version of the result in [111]. Section 15.2.6 Most of the results in Section 15.2.6 came from [353]. See [353, Theorem 1.4], [353, Theorem 1.10] and [353, Theorem 1.7] for Theorems 154, 155, and 151. We employed the argument used in [465] for the proof of Theorem 151. See [353, Theorem 1.11] for Theorems 156 and 157 as well as [418, Corollary 6.7]. We complemented the power weight case in [356]; see [356, Proposition 4.2] for Proposition 153. The relation between Iα and Mα acting on weighted Morrey spaces can be found in [356, Theorem 1.5]; see Theorem 152. We remark that Iida [216] also considered fractional integral operators acting on weighted Morrey spaces. Section 15.2.7 See [353] for Theorem 158, the vector-valued extension of the boundedness of singular integral operators on weighted Morrey spaces. Shi considered the vector-valued boundedness of commutators in Morrey spaces [435, Theorem 1.3] Shi applied this result to the equation ut − (−1)m P (D)u = f, where P is a homogeneous polynomial of degree 2m. Thanks to the kernel expression obtained in [385], we can apply the result of singular integral operators. In fact, Shi used weighted Morrey spaces whose norm is given by (15.1) and applied the results to parabolic equations in [435, Section 4.3]. Section 15.2.8 Iida and Nakamura obtained the necessary and sufficient conditions in [220, Theorem 3]; see Theorem 161.

Section 15.3 General remarks and textbooks in Section 15.3 See [357] for the exhaustive approach. Section 15.3.1 Samko obtained conditions for the Hardy–Littlewood maximal operator to be bounded in generalized local Morrey spaces [396, Theorem 3.1]. See [357] for theorems in Section 15.3.1. Nakmaura, Sawano and Tanaka obtained the necessary and sufficient condition on β for M to be bounded on Mpq (|x|β , |x|β )

Weighted Morrey spaces

211

in [357, Proposition 1.9]; see Proposition 167. Likewise Nakmaura, Sawano and Tanaka obtained the necessary and sufficient condition on β for M to be bounded on Mpq (1, |x|β ) in [357, Proposition 1.9]; see Proposition 168. Section 15.3.2 Nakamura, Sawano and Tanaka also considered the boundedness of singular integral operators acting on local weighted Morrey spaces: Theorems 169 and 170 and Proposition 168 are [357, Theorem 1.5], [357, Theorem 1.6] and [357, Proposition 1.7], respectively. See [356, Theorem 1.4] for Theorem 149. Section 15.3.3 We can define generalized weighted Morrey spaces of Komori–Shirai-type based on the work [243]. Definition 34. [142, Definition 1] Let 1 ≤ q < ∞, ϕ ∈ M+ (Rn × (0, ∞)) and 0 n w ∈ M+ (Rn ). Denote by Mϕ q (w, w) the set of all f ∈ L (R ) with finite norm kf kMϕq (w,w) ≡

1 w(B(x, r))

ϕ(x, r)

sup (x,r)∈Rn+1 +

! q1

Z

q

|f (y)| w(x)dx

.

B(x,r) 1

Of interest is the case where ϕ(x, r) ≡ w(B(x, r)) p , x ∈ Rn and r > 0, with q ≤ p < ∞, in which case we have Mpq (w, w) instead of Mϕ q (w, w). Similarly one can define weak weighted Morrey spaces. Definition 35. [142, Definition 1] Let 1 ≤ q < ∞, ϕ ∈ M+ (Rn ) and w ∈ 0 n M+ (Rn ). Denote by Mϕ q (w, w) the set of all f ∈ L (R ) with finite norm kf kMϕq (w,w) ≡

1

ϕ(x, r)w(B(x, r))− q λkχ{|f |>λ}∩B(x,r) kLq (w) .

sup ,λ>0 (x,r)∈Rn+1 +

We can also consider the weighted Morrey–Campanato spaces. Here, we content ourselves with the definition. Example 67. [470] Let β > 0 and 1 < p < ∞, and also let w be a weight. The space L(β, p, w) is defined to the set of all f ∈ L1loc (Rn ) for which the (semi)norm sup B

1 w(B)β



1 w(B)

Z

 p1 |f (x) − fB |p w(x)1−p dx

B

is finite, where B moves over all balls in Rn . As for weighted Morrey spaces of Samko-type, the boundedness property of the sharp maximal operator, the maximal operator, singular integral operators and fractional operators including the multilinear setting are investigated in

212

Morrey Spaces

[170, 214, 215, 353, 355, 356]. We can find its application to singular integral equations in [292]. Liu considered the boundedness of the pseudo-differential operators in the setting of generalized Morrey spaces [273]. There are many attempts at obtaining a necessary and sufficient condition for the weighted norm inequality. See [221] for a characterization of a sufficient condition which guarantees the boundedness of the Hardy–Littlewood maximal operator. Two-weighted Morrey spaces of the type Mpq (u, v) can be found in [203, 330, 374, 435, 535] including generalized Morrey spaces [82, 123, 142, 154, 172, 190, 208] and their closed subspaces [20]. Note that Theorem 158 extends [188, Theorem 8]. Komori-Furuya and Sato investigated the boundedness of fractional integral operators on weighted local Morrey spaces [242]. We have a complete characterization for Iα and Mα to be bounded; Theorem 174 is [357, Theorem 1.12].

Chapter 16 Morrey-type spaces

As a continuation of what we did in Chapters 12 and 13, we consider another generalization of the definition of Morrey spaces. Let 0 < q ≤ p < ∞. Let the Morrey space Mpq (Rn ) be the space defined by the norm kf kMpq ≡

sup

1

1

|B(x, r)| p − q kf kLq (B(x,r)) for f ∈ L0 (Rn ). Likewise,

(x,r)∈Rn+1 +

let the local Morrey space LMpq (Rn ) be the space defined by the norm 1

1

kf kLMpq ≡ sup |B(r)| p − q kf kLq (B(r)) for f ∈ L0 (Rn ); see Chapter 1. Section r>0

16.1 changes the supremum with respect to r by other norms and considers the weighted Lθ (0, ∞)-norm. This seems to be a mere quest to generalization. However, this type of generalization arises naturally in connection with real interpolations. Hence, the boundedness property is also investigated since the boundedness of the Hardy operator can be referred to in the weighted norm spaces. Section 16.2 further generalizes the parameter p to a function: Section 16.2 considers the ϕ-generalization. In the remainder of this section, we consider the boundedness of various operators from local general Morrey-type spaces to other local general Morrey-type spaces. We will change the target space from the initial space since we would like to make full use of the fact that p underwent the ϕ-generalization. Initially, Chapter 16 may seem a mere quest to generalization. However, Chapter 18, which considers the real interpolation, convincingly shows that general Morrey-type spaces are natural objects.

16.1

Morrey-type spaces

Here, as a model case we consider changing the supremum into the weighted Lθ (0, ∞)-norm; so we are heading for a generalization that differs from ϕ or Φ-generalizations and is called the θ-generalization. We investigate the boundedness property of operators we have been investigating in this book. The plan of Section 16.1 is similar to other sections. Section 16.1.1 starts with the definition of local and global Morrey-type spaces. We deal with the Hardy–Littlewood maximal operator, singular integral operators including commutators and fractional integral operators including fractional maximal operators in Sections 16.1.2, 16.1.3 and 16.1.4, respectively. 213

214

16.1.1

Morrey Spaces

Local and global Morrey-type spaces

We solidify our idea above and we define two spaces as follows: Definition 36. Let 0 < p, q, θ ≤ ∞. (1) The local Morrey-type space LMpqθ (Rn ) collects all f ∈ Lqloc (Rn ) for Z ∞  θ dr  θ1 n n − p q r which kf kLMpqθ ≡ < ∞. kf kLq (B(r)) r 0 (2) The global Morrey-type space GMpqθ (Rn ) = Mpqθ (Rn ) collects all f ∈ Lqloc (Rn ) for which Z kf k

Mp qθ

≡ sup x∈Rn

0

∞



r

n n p−q

θ dr  θ1 kf kLq (B(x,r)) r

= sup kf (· − x)kLMpqθ < ∞. x∈Rn

Remark that Ragusa used the same symbol as Mpqθ (Rn ) to indicate Morrey–Lorentz spaces [379]. If f ∈ LMpqθ (Rn ), then we obtain certain information about the behavior of f . However, the most important stress is the one about the behavior of f in a neighbourhood of the origin. To simplify the notaton, it is convenient to use the space Φλ,q (0, ∞) defined in the first book for λ ∈ R and 0 < q ≤ ∞. Recall that Φλ,q (0, ∞) denotes the Z ∞  q1 −λ q dt + set of all ϕ ∈ M (0, ∞) for which kϕkΦλ,q (0,∞) ≡ (t ϕ(t)) 0 for some r0 ∈ (0, ∞). Hence, it folZ ∞  θ dr  θ1 n −n p q p r lows that kf kLMqθ ≥ = ∞, implying kf kLq (B(r0 )) r r0 p that f ∈ / LMqθ (Rn ). Conversely, if q < p, then χB(v,1) ∈ LMpqθ (Rn ) for any v ∈ Rn with kvk = 2. (2) If θ < p = ∞ and f 6= 0, then we claim that f ∈ / Mpqθ (Rn ). In fact, we may assume that f is bounded. By translation we may also assume that n0 is the Lebesgue point of |f |p and that f (0) 6= 0, so that kf kLq (B(r)) ∼ r q for r ∈ (0, 1) with the implicit constant dependent on f . From this estimate, we learn that Z kf kLMpqθ &

1



0

r

−n q

θ dr  θ1 = ∞. kf kLq (B(r)) r

Thus in view of the criterion for LMpqθ (Rn ), we learn that Mpqθ (Rn ) is non-trivial only if q < p < ∞. If q < p < ∞, then χB(1) ∈ Mpqθ (Rn ), n since kχB(1) kLq (B(x,r)) . min(1, r q ) for r > 0. We clarify the nesting structure of local and global Morrey type spaces. The first one concerns the monotonicity in θ. We content ourselves with the case θ = ∞. Proposition 178. Let 0 < q < p ≤ ∞ and 0 < θ < ∞. Then LMpqθ (Rn ) ⊂ LMpq∞ (Rn ) = LMpq (Rn ). Hence, Mpqθ (Rn ) ⊂ Mpq∞ (Rn ) = Mpq (Rn ) whenever 0 < q < p < ∞ and 0 < θ < ∞. Proof Let f ∈ L0 (Rn ). Then simply observe that   θ1 ∞   X   θ jn jn kf kLMpqθ ∼ 2 p − q kf kLq (B(2j )) .  

(16.1)

j=−∞

The right-hand being monotone in θ, we obtain the desired result. We consider the monotonicity in q. Again we rule out the possibility θ = ∞, since this is already covered as the result in Morrey and local Morrey spaces. Proposition 179. Let 0 < q1 < q2 < p ≤ ∞ and 0 < θ < ∞. Then LMpq2 θ (Rn ) ⊂ LMpq1 θ (Rn ). Hence Mpq2 θ (Rn ) ⊂ Mpq1 θ (Rn ) provided 0 < q1 < q2 < p < ∞ and 0 < θ < ∞. Proof Simply use H¨ older’s inequality as we did for classical Morrey spaces in Theorem 19.

216

16.1.2

Morrey Spaces

Maximal and sharp maximal inequalities for local Morrey-type spaces

Having clarified the structure of Morrey-type and local Morrey-type spaces, let us investigate the boundedness property of M . We do not consider the case where θ = ∞, since this case is already handled by a result in Chapter 10. Theorem 180. Let 1 < q < p ≤ ∞ and 1 < θ < ∞. Then M is bounded on LMpqθ (Rn ). Proof We need to show kkM f kLq (B(·)) kΦ n − n ,θ . kf kLMpqθ . We will take q

p

the local/global strategy in this case as before. We decompose the estimate as follows: kkM [χB(2·) f ]kLq (B(·)) kΦ n − n ,θ . kf kLMpqθ , q

p

kkM [Rn \ χB(2·) f ]kLq (B(·)) kΦ n − n ,θ . kf kLMpqθ . q

p

In the first estimate, we simply use the fact that the Hardy–Littlewood maximal operator is bounded on Lq (Rn ): kkM [χB(2·) f ]kLq (B(·)) kΦ n − n ,θ ≤ kkM [χB(2·) f ]kLq kΦ n − n ,θ q

p

q

p

. kkf kLq (B(2·)) kΦ n − n ,θ q

p

' kf kLMpqθ . For the second estimate, from a geometric observation, we deduce Z ∞ n M [χRn \B(2r) f ](x) . kf kLq (B(s)) s−1− q ds (x ∈ B(r)). r

Thus, matters are reduced to the proof of the following estimate: (Z  Z ) θ1 θ ∞ ∞ n dr −1− n p q r kf kLq (B(s)) s ds . kf kLMpqθ , r 0 r which is a consequence of Hardy’s inequality. Thus, we obtain the desired estimate. We can transplant this result to the global Morrey-type spaces. Theorem 181. Let 1 < q < p < ∞ and 1 < θ < ∞. Then kM f kMpqθ .p,q,θ kf kMpqθ for all f ∈ Mpqθ (Rn ). Proof We leave the proof for the interested readers. See Exercise 50. To investigate the boundedness property of singular integral operators, we will use the sharp-maximal operator approach. Similar to the sharp-maximal inequality for Morrey spaces in the first book, we can prove the following theorem:

Morrey-type spaces

217

Theorem 182 (Sharp-maximal inequality for local Morrey-type spaces). Let p +kf k 0 < r < q ≤ p < ∞ and 1 < θ < ∞. Then kM2],D −n−2 f kLM LMqrθ ∼p,q,r,θ qθ 0 n p kf kLMqθ for all f ∈ L (R ). Proof We leave the proof for the interested readers. See Exercise 50.

16.1.3

Singular integral operators in Morrey-type spaces

Let T be a singular operator. We discussed how to define T on LMpq (Rn ) in the first book. Keeping in mind that LMpqθ (Rn ) is a smaller function space than LMpq (Rn ), as is seen from Proposition 178, we consider the boundedness of T on LMpq (Rn ). Theorem 183. Let 1 < q < p < ∞ and 1 < θ < ∞, and let T be a singular integral operator, which is initially defined in LMpq (Rn ). If we restrict T to LMpqθ (Rn ), then kT f kLMpqθ .q kf kLMpqθ for all f ∈ LMpqθ (Rn ). Proof Mimic the proof of Theorem 180. As before, by translation we can transplant the result above into global Morrey-type spaces. Theorem 184. Let 1 < q < p < ∞ and 1 < θ < ∞, and let T be a singular integral operator, which is initially defined in Mpq (Rn ). If we restrict T to Mpqθ (Rn ), then kT f kMpqθ .q kf kMpqθ for all f ∈ Mpqθ (Rn ). Proof Simply use Theorem 183 and the fact that f 7→ T [f (· − y)](· + y) is a singular integral operator for all y ∈ Rn with the constants the same as those of T . We move on to commutators. Let T be a singular operator, and let a ∈ BMO(Rn ). We discussed how to define [a, T ] on LMpq (Rn ) in the first book. Keeping in mind that LMpqθ (Rn ) is a smaller function space than LMpq (Rn ), we consider the boundedness of [a, T ] on LMpq (Rn ). Theorem 185. Let 1 < q < p < ∞, 1 < θ < ∞, and let T be a singular integral operator. Also let a ∈ BMO(Rn ) and T be a singular integral operator. Then the commutator [a, T ], which is initially defined in LMpq (Rn ), is bounded on LMpqθ (Rn ) if T is restricted to LMpqθ (Rn ). Proof Mimic the proof of Theorem 184. As before, we can globalize Theorem 185. Theorem 186. Let 1 < q < p < ∞, 1 < θ < ∞, and let T be a singular integral operator. Also let a ∈ BMO(Rn ) and T be a singular integral operator. Then the commutator [a, T ], which is initially defined in Mpq (Rn ), is bounded on Mpqθ (Rn ) if T is restricted to Mpqθ (Rn ). Proof Mimic the proof of Theorem 184.

218

Morrey Spaces

16.1.4

Fractional integral operators in Morrey-type spaces

Here, we deal with fractional integral operators. We consider the boundedness of Spanne type. Theorem 187. Let 0 < α < n, 1 < θ < ∞, 1 < q < p < ∞ and 1 < t < s < ∞ satisfy 1 1 α 1 1 α = − , = − . s p n t q n Then Iα is bounded from LMpqθ (Rn ) to LMstθ (Rn ). Proof Let f ∈ M+ (Rn ). Then n

n

n

n

n

n

n

r s − t kIα f kLt (B(r)) . r p − q kf kLq (2B) + r p − q + t

Z

∞

kf kL1 (B(s)) r

ds . sn+1−α

The first term can be handled with ease by the change of variables, while the estimate of the second term simply uses Hardy’s inequality. Hence Iα is bounded from LMpqθ (Rn ) to LMstθ (Rn ). As a corollary of the above result, we obtain the boundedness of Iα of Spanne-type. We globalize Theorem 187. Corollary 188. Let 0 < α < n, 1 < θ < ∞, 1 < q < p < ∞ and 1 < t < s < ∞ satisfy 1 1 α 1 1 α = − , = − . s p n t q n Then Iα is bounded from Mpqθ (Rn ) to Mstθ (Rn ). Proof Simply observe that Iα commutes with translation. We can even obtain the Adams type theorem by using a different method. Theorem 189. Let 0 < α < n, 1 < θ, τ < ∞, 1 < q < p < ∞ and 1 < t < s < ∞ satisfy 1 α 1 = − , s p n

t q τ = = . s p θ

Then Iα is bounded from Mpqθ (Rn ) to Mstτ (Rn ). Proof Simply use the Hedberg inequality keeping in mind that Mpqθ (Rn ) is a subset of Mpq (Rn ).

16.1.5

Exercises

Exercise 49. Let 0 < q < p ≤ ∞ and 0 < θ1 < θ2 < ∞. (1) Show that LMpqθ1 (Rn ) ⊂ LMpqθ2 (Rn ) using (16.1).

Morrey-type spaces

219

(2) Suppose p < ∞. Show that Mpqθ1 (Rn ) ⊂ Mpqθ2 (Rn ) using (1). Exercise 50. (1) Prove Theorem 181 by mimicking the proof of the Morrey boundedness of the Hardy–Littlewood maximal operator in Chapter 10. (2) Prove Theorem 182 by mimicking the proof of the sharp maximal inequality for Morrey spaces in the first book. Exercise 51. [38, p. 18] Let 0 < q < p ≤ ∞ and 0 < θ < ∞, and let f ∈ L0 (Rn ) and t > 0. n

(1) Show that kf (t·)kLMpqθ = t− p kf kLMpqθ . n

(2) Show that kf (t·)kMpqθ = t− p kf kMpqθ if p < ∞.

16.2

General Morrey-type spaces

We have considered generalized Morrey spaces by passing the parameter p to a function ϕ. We will do the same thing for Morrey-type spaces. Section 16.2 consists of four sections. We define general Morrey-type spaces in Section 16.2.1; we are oriented to the ϕ-generalization. We investigate the maximal operator, singular integral operators and fractional maximal operators in Sections 16.2.2, 16.2.3 and 16.2.4, respectively.

16.2.1

Elementary properties of general Morrey-type spaces

We investigate the structure of general Morrey-type spaces. We occasionally generalize the parameter p as follows: Definition 37. Let 0 < q ≤ ∞ and 0 < θ < ∞, and let ϕ ∈ M+ (0, ∞). n 0 n (1) The local Morrey-type space LMϕ qθ (R ) is the space of all f ∈ L (R ) Z ∞  θ1 n dr with finite quasi-norm kf kLMϕqθ ≡ (ϕ(r)r− q kf kLq (B(r)) )θ . r 0 n 0 n (2) The local Morrey-type space LMϕ q∞ (R ) is the space of all f ∈ L (R ) n with finite quasi-norm kf kLMϕq∞ ≡ sup ϕ(r)r− q kf kLq (B(r)) , that is, r>0

n ϕ n LMϕ q∞ (R ) ≈ LMq (R ) with equivalence of norms.

Later on, we will discuss what condition on ϕ we need. Here we content ourselves with obtaining rough information on the norm kχB(2t)\B(t) kLMϕqθ .

220

Morrey Spaces

Example 69. Let 0 < q, θ ≤ ∞, and let ϕ ∈ M+ (0, ∞). Then Z ∞  θ1 1 Z ∞ n dr θ −n −n − θ dr θ q q q (ϕ(r)r ) . t kχB(2t)\B(t) kLMϕqθ . (ϕ(r)r ) r r 2t t for all t > 0. n The general Morrey-type space Mϕ qθ (R ) below is also called a global general Morrey-type space in comparison with local generalized Morrey spaces. n Definition 38. Let 0 < q, θ ≤ ∞, and let ϕ ∈ M+ (0, ∞). Denote by Mϕ qθ (R ) the general Morrey-type space, the space of all functions f ∈ L0 (Rn ) with finite quasi-norm kf kMϕqθ ≡ sup kf (· + x)kLMϕqθ . x∈Rn

ϕ n n The spaces LMϕ qθ (R ) and Mqθ (R ) are mostly aimed at describing the behavior of kf kLq (B(r)) and kf kLq (B(x,r)) , for small r > 0 and x ∈ Rn , in a very general setting. n

Example 70. Let 0 < q ≤ p ≤ ∞. For u ∈ (0, ∞), we write ϕu (r) ≡ r u . n n ϕ (1) For ϕ ∈ M+ (0, ∞), Mϕ q∞ (R ) = Mq (R ) with coincidence of norms. ϕ

ϕ

q q (Rn ) = Lq (Rn ). (Rn ) = Mq∞ (2) With coincidence of norms LMq∞

ϕ

p (Rn ) = Mpq (Rn ). (3) With coincidence of norms Mq∞

ϕ

(4) Let 0 < q < p < ∞ and 0 < θ < ∞. Then Mqθp (Rn ) = Mpqθ (Rn ) with coincidence of norms. n We now discuss some conditions under which the space LMϕ qθ (R ) or is non-trivial. For the time being, we seek sufficient conditions but it turns out that they are also necessary. n Mϕ qθ (R )

Definition 39. Let 0 < q, θ ≤ ∞. (1) Z Denote by LΩqθ the set of all functions ϕ ∈ M+ (0, ∞) \ {0} such that ∞ n dr (ϕ(r)r− q )θ < ∞ for some t > 0. r t (2) Denote by Ωqθ the set of all functions ϕ ∈ M+ (0, ∞) \ {0} such that Z ∞ n dr (max(1, r)− q ϕ(r))θ < ∞. (16.1) r 0 Intuitively, we have to be sensitive to the behavior of functions near the origin. So, once we disregard this behavior by truncating functions at zero, then we do not have to be concerned with this issue. The condition on t in LΩqθ seems to reflect this aspect. It should be noted that the class LΩqθ involves two parameters q and θ because of the different notation of local general Morrey spaces. In fact, the notation of local general Morrey spaces differs from the one in [46].

Morrey-type spaces Example 71. Let 0 < q, θ ≤ ∞, and let ϕ ∈ LΩqθ . Choose β ∈



221  n , ∞ . q

n Define f (x) ≡ χRn \B(1) (x)|x|−β for x ∈ Rn . Then f ∈ LMϕ qθ (R ), since

kf kLMϕ ≤ kχ(1,∞) ϕk qθ

Lθ (r

−1− nθ q

)

kf kLq < ∞.

ϕ n n We first check that LMϕ qθ (R ) and Mqθ (R ) is non-trivial under some conditions proposed in Definition 39.

Proposition 190. Let 0 < q, θ ≤ ∞, and let ϕ ∈ M+ (0, ∞) \ {0}. (1) Let ϕ ∈ LΩqθ , and write  Z τ ≡ inf s > 0 :

∞

n

(ϕ(r)r− q )θ

s

 dr < ∞ (< ∞). r

n q n Then f ∈ LMϕ qθ (R ) for any f ∈ L (R ) supported away from B(t) for some t > τ . n (2) If ϕ ∈ Ωqθ , then Lq (Rn ) ∩ L∞ (Rn ) ⊂ Mϕ qθ (R ).

Proof (1) Using B(r) ⊂ Rn for any r > 0, we have Z kf kLMϕqθ =

∞

(ϕ(r)r

−n q

θ dr

kf kLq (B(r)) )

r

t

≤ kf kLq kχ(t,∞) ϕk

−1− nθ q Lθ (r

 θ1

)

< ∞. (2) Let f ∈ Lq (Rn ) ∩ L∞ (Rn ). Then we have Z kf kMϕqθ . sup

x∈Rn

1

(ϕ(r)r

kf kLq (B(x,r)) )

0 ∞

Z + sup x∈Rn

−n q

(ϕ(r)r

−n q

θ dr

r

kf kLq (B(x,r)) )

θ dr

≤ kf kL∞ 0

1

ϕ(r)θ

dr r

 θ1

 θ1

r

1

Z

 θ1

Z + kf kLq

∞

n

(ϕ(r)r− q )θ

1

dr r

 θ1

< ∞. ϕ n n We check when LMϕ qθ (R ) and Mqθ (R ) contain non-zero functions.

Theorem 191. Let 0 < q ≤ ∞ and 0 < θ < ∞, and let w ∈ M+ (0, ∞) \ {0}. n (1) The space LMϕ qθ (R ) is non-trivial if and only if ϕ ∈ LΩqθ .

222

Morrey Spaces

n (2) The space Mϕ qθ (R ) is non-trivial if and only if ϕ ∈ Ωqθ .

Before we come to the proof, we remark the structure of the proof. We can say more for the “if” part according to Proposition 190. Thus, we have only to concentrate on the “only if” part. Proof n (1) Let f ∈ LMϕ qθ (R ) \ {0}. Then there exists t0 > 0 such that kf kLp (B(t0 )) > 0. Hence

kf kMϕqθ ≥ kf kLMϕqθ 1 Z ∞ n dr θ ≥ (ϕ(r)r− q kf kLq (B(r)) )θ r t0 Z ∞ 1 n dr θ ≥ (ϕ(r)r− q kf kLq (B(t0 )) )θ r t0 Z ∞ 1 n dr θ = kf kLp (B(t0 )) (ϕ(r)r− q )θ r t0 = ∞, ∞

Z

dr = ∞. This implies ϕ ∈ / LΩqθ . r t0 Z ∞ n dr (2) We may assume that there exists t0 > 0 such that < (ϕ(r)r− q )θ r t0 ϕ n ∞ in view of (1). Assume that Mqθ (R ) is not trivial, so that we can n take f ∈ Mϕ qθ (R ) \ {0}. By the Lebesgue differentiation theorem and the fact that f ∈ Lqloc (Rn ), it follows that there exist G ⊂ Rn such that |Rn \ G| = 0 and that lim

n

(ϕ(r)r− q )θ

showing that

kf kLq (B(x,r))

r↓0

1

|B(x, r)| q

= lim r↓0

1 |B(x, r)|

! q1

Z

q

|f (y)| dy

= |f (x)|

B(x,r)

for all x ∈ G. Suppose |f (x)| > 0 at some point x ∈ G. Then 1 |B(x, r)|− q kf kLq (B(x,r)) &f 1 for all 0 < r < t0 . Hence, Z kf k

Mϕ qθ

≥

∞ θ

(ϕ(r)kf kLp (B(x,r)) ) 0

 θ1

dr n

r1+θ q

Z

t0

ϕ(r)

& 0

θ dr

r

 θ1 ,

so that ϕ ∈ / Ωqθ . Keeping in mind Theorem 191, we will always assume that ϕ ∈ LΩqθ for ϕ n n LMϕ qθ (R ) and ϕ ∈ Ωqθ for Mqθ (R ). n We investigate the local integrability property of functions in LMϕ qθ (R ) ϕ and Mqθ (Rn ).

Morrey-type spaces

223

n Corollary 192. Let 0 < q, θ ≤ ∞, and let ϕ ∈ LΩqθ . Then LMϕ qθ (R ) ⊂ q ϕ q n n n Lloc (R ). In particular, Mqθ (R ) ⊂ Lloc (R ).

Proof Simply reexamine the proof of Theorem 191(1). n We compare LMϕ qθ (R ) by fixing q and θ and by varying ϕ. 1 n Lemma 193. Let 0 < q, θ ≤ ∞ and ϕ1 , ϕ2 ∈ LΩqθ . Then LMϕ qθ (R ) ⊂ ϕ2 LMqθ (Rn ) as a set if and only if

kχ(t,∞) ϕ2 k

Lθ (r

−1− nθ q

)

. kχ(t,∞) ϕ1 k

Lθ (r

−1− nθ q

)

(t > 0).

(16.2)

ϕ2 1 n n In particular, LMϕ qθ (R ) = LMqθ (R ) as a set if and only if

kχ(t,∞) ϕ2 k

Lθ (r

−1− nθ q

)

∼ kχ(t,∞) ϕ1 k

Lθ (r

−1− nθ q

)

(t > 0).

(16.3)

We learn from (16.2) that ϕ2 . ϕ1 by considering functions of the form ϕ2 1 n n χB(R)\B(r) , 0 < r < R < ∞, since LMϕ qθ1 (R ) is a subset of LMqθ2 (R ). ϕ2 1 n n Proof To check that (16.3) holds if and only if LMϕ qθ (R ) = LMqθ (R ), we have only to use (16.2) twice. So let us concentrate on (16.2). ϕ2 1 n n The embedding LMϕ qθ (R ) ⊂ LMqθ (R ) is equivalent to the estimates: Z ∞ Z ∞ n dr −n θ dr q (ϕ1 (r)r kf kLq (B(r)) ) & (ϕ2 (r)r− q kf kLq (B(r)) )θ r r 0 0

for all simple functions f . Since only kf kLq (B(r)) comes into play in the above estimate, we may restrict f to be radial. Since kf kLq (B(r)) is non-decreasing in r > 0 and any non-decreasing function can be represented as kf kLq (B(r)) for some f ∈ L0 (Rn ), this inequality in its turn is equivalent to Z ∞ Z ∞ n dr −n θ dr q (ϕ1 (r)r ψ(r)) & (16.4) (ϕ2 (r)r− q ψ(r))θ r r 0 0 for all ψ ∈ M↑ (0, ∞), such that lim ψ(r) = 0 if q < ∞. It suffices to note that r↓0

inequality (16.4) is equivalent to (16.2). Local Morrey-type spaces and weighted Lebesgue spaces are related as follows: Lemma 194. Let 0 < q, θ ≤ ∞ and ϕ ∈ LΩqθ . For x ∈ Rn , we define (Z

∞

W (x) ≡

(ϕ(r)r |x|

dr −n q θ )

r

) θ1

Z = 0

∞

n

χB(r) (x)(ϕ(r)r− q )θ

dr r

 θ1 .

n ϕ (1) If q ≤ θ, then Lq (W q ) ,→ LMϕ qθ (R ). More precisely, kf kLMqθ ≤ kf kLq (W q ) for all f ∈ Lq (W q ).

224

Morrey Spaces

n ϕ (2) If q ≥ θ, then Lq (W q ) ←- LMϕ qθ (R ). More precisely, kf kLMqθ ≥ ϕ n kf kLq (W q ) for all f ∈ LMqθ (R ). n q q In particular, LMϕ qq (R ) = L (W ) with coincidence of norms.

Proof Both assertions follow by the Minkowski inequality.

16.2.2

Maximal operator in general Morrey-type spaces

Here, we are interested in general necessary conditions for the maximal operators to be bounded in general Morrey-type spaces. We also collect auxiliary estimates. We start with the following easy observation using the embedding relation: Corollary 195. Let 1 < q < ∞, 0 < θ1 ≤ θ2 < ∞, ϕ1 ∈ LΩqθ1 , ϕ2 ∈ LΩqθ2 . ϕ2 ϕ1 1 n n n Then if M is bounded from LMϕ qθ1 (R ) to LMqθ2 (R ), then LMqθ1 (R ) ⊂ ϕ2 n LMqθ2 (R ). In particular, kχ(t,∞) ϕ2 k

Lθ2 (r

−1−

nθ2 q

)

. kχ(t,∞) ϕ1 k

Lθ1 (r

−1−

nθ1 q

(16.5) )

for all t > 0. 1 n Proof Let f ∈ LMϕ qθ1 (R ). Thanks to the Lebesgue differentiation theorem M f (x) ≥ |f (x)| for almost all x ∈ Rn . Consequently, kf kLMϕ2 ≤ qθ2

ϕ2 1 n n kM f kLMϕ2 . kf kLMϕ1 . Thus LMϕ qθ1 (R ) ⊂ LMqθ2 (R ). Thus, we are in qθ2 qθ1 the position of using Lemma 193.

We recall that the Hardy–Littlewood maximal operator fills the hole of the annulus B(2t) \ B(t) in the sense that (M χB(2t)\B(t) )χB(2t) ∼ χB(2t) . We can strengthen Corollary 195 using this fact. Lemma 196. Let 1 < q < ∞, 0 < θ1 ≤ θ2 < ∞, ϕ1 ∈ LΩqθ1 , ϕ2 ∈ LΩqθ2 . Then the condition Z ∞ 1 n dr θ2 . kχ(t,∞) ϕ1 k (t > 0) (16.6) (ϕ2 (r)(t + r)− q )θ2 nθ1 −1− q ) r Lθ1 (r 0 ϕ2 1 n n is necessary for the boundedness of M from LMϕ qθ1 (R ) to LMqθ2 (R ).

Proof Fix t > 0, and let f ≡ χB(2t)\B(t) . Then (M f )χB(2t) ∼ χB(2t) . Thus, our assumption Z ∞  θ1 2 −n θ2 dr q (ϕ2 (r)r kM f kLq (B(r)) ) . kf kMϕ1 qθ1 r 0 n

implies kχ(0,t) ϕ2 kLθ2 (r−1 ) . kt q χ(t,∞) ϕ1 k kt

−n q

χ(0,t) ϕ2 kLθ2 (r−1 ) . kχ(t,∞) ϕ1 k

Lθ1 (r

Lθ1 (r −1−

−1−

with Corollary 195 gives the desired result.

nθ1 q

nθ1 r

)

. As a result, we obtain

. Combining this inequality )

Morrey-type spaces

225

Before we prove the boundedness of M , we clarify the embedding structure of local general Morrey-type spaces based on (16.6). Remark 6. Assume (16.6). Then it follows that ∞

Z

n

[ϕ2 (2r)(t + r)− q ]θ2

0

dr r

 θ1

2

. kχ(t,∞) ϕ1 k

Lθ1 (r

ϕ (2·)

2 1 n by a change of variables. Thus, Mϕ qθ1 (R ) ,→ Mqθ2

−1−

nθ1 q

)

(Rn ).

We are now oriented to a sufficient condition. Recall that the Hardy operZ 1 r g(t)dt for r > 0 and g ∈ M+ (0, ∞). Since we ator H is given by Hg(r) ≡ r 0 are interested in the boundedness property of the Hardy–Littlewood maximal operator, we suppose 1 < q < ∞. Since kM f kLMϕ2 contains kM f kLq (B(r)) qθ2

in the definition, it is natural to start with estimates for kM f kLq (B(r)) . We start by using the change of variables twice to the local estimate we obtained in Theorem 144 in the first book. Lemma Z 197. Let 1 < q < ∞, 0 < θ ≤ ∞, and let ϕ ∈ LΩqθ . We write g(t) ≡ |f (y)|q dy for t > 0 for f ∈ L0 (Rn ). Then for all f ∈ L1loc (Rn ), B(t−1/n )

∞

Z kM f k

LMϕ qθ

.



ϕ(R

 θq dR  θ1 . ) RHg(R) R

1 −n q

0

Proof By Theorem 144 in the first book,

kM f kLMϕqθ .

  Z

∞



Z

|f (x)|q dx

ϕ(r)

0

 

∞

Z r

B(t)

dt

! q1 θ

tn+1



 θ1  dr  r  

.

Change variables τ = t−n and then insert the definition of g into our inequality. Then   θ1  ! q1 θ   −n Z Z Z  ∞  r q ϕ(r)  dr kM f kLMϕqθ . |f (x)| dxdτ √ n  r  0 B( τ −1 )  0  .

 Z 

∞

" ϕ(r)q

!# θq

r −n

Z

g(τ )dτ

0

0

 θ1 dr  . r 

Next, change variables R = r−n . Then Z kM f kLMϕqθ .

0

∞



ϕ(R

 θq dR  θ1 ) RHg(R) . R

1 −n q

226

Morrey Spaces

n Remark 7. In a similar spirit to the above, for f ∈ LMϕ qθ (R ), we can prove

Z kf kLMϕqθ ∼

∞



1

ϕ(R− n )q Rg(R)

0

 θq dR  θ1 . R

Using the Hardy operator H we discuss the boundedness property of the Hardy–Littlewood maximal operator in generalized local Morrey-type spaces. Theorem 198. Fix q > 1. Also let 0 < q2 ≤ q ≤ q1 < ∞ and 0 < θ1 , θ2 ≤ ∞, 1

θk

−1

and let ϕ1 ∈ LΩq1 θ1 and ϕ2 ∈ LΩq2 θ2 We set wk (R) ≡ ϕk (R− n )θk R qk for R > 0 and k = 1, 2. Assume that the Hardy operator H is bounded from ϕ2 1 n n Lq1 (w1 ) to Lq2 (w2 ). Then M is bounded from LMϕ q1 θ1 (R ) to LMq2 θ2 (R ). Proof From our assumption, (Z )1 ∞  θq2 dR θ2 1 ϕ2 (R− n )q RHg(R) R 0 (Z )1 ∞  θq1 dR θ1 1 q −n . ϕ1 (R ) Rg(R) R 0 for all g ∈ M↓ (0, ∞). Thus, it remains to combine Lemma 197 and Remark 7. We can globalize Theorem 198 as we did in Section 16.2.2. Theorem 199. In addition to the assumption of Theorem 198, suppose 1 n that ϕ1 ∈ Ωq1 θ1 and ϕ2 ∈ Ωq2 θ2 . Then M is bounded from Mϕ q1 θ1 (R ) to ϕ2 Mq2 θ2 (Rn ). We omit the proof since this is a routine. We handle the case where θ1 ≤ min(θ2 , q). We have the following nec1 n essary and sufficient conditions for M to be bounded from LMϕ qθ1 (R ) to ϕ2 n LMqθ2 (R ): Theorem 200. Let 1 < q < ∞, and 0 < θ1 ≤ θ2 ≤ ∞. Also let ϕ1 ∈ LΩqθ1 and ϕ2 ∈ LΩqθ2 . Assume in addition that θ1 ≤ q and that H is bounded from θ1

θ1

θ2

θ2

L q (ϕ1 (·−1 )· q −1 ) to L q (ϕ2 (·−1 )· q −1 ). Then condition (16.6) is necessary ϕ2 1 n n and sufficient for the boundedness of M from LMϕ qθ1 (R ) to LMqθ2 (R ). Proof Necessity of condition (16.6) is already considered in Lemma 196; here we are interested in the sufficiency. Assume (16.6). We have to show that Z ∞  θ1 2 −n θ2 dr q (ϕ2 (r)r kM f kLq (B(r)) ) . kf kMϕ1 , qθ1 r 0 or equivalently, we have to show that  θ1 Z ∞ 2 dr −n θ 2 (ϕ2 (r)r q kM [χB(2r) f ]kLq (B(r)) ) . kf kMϕ1 I≡ qθ1 r 0

(16.7)

Morrey-type spaces and that Z II ≡

∞

(ϕ2 (r)r

−n q

kM [χ

0

Rn \B(2r)

f ]k

227

Lq (B(r))

dr ) r θ2

 θ1

2

. kf kMϕ1 . (16.8) qθ1

Since q > 1, we have Z

∞

n

(ϕ2 (r)r− q kf kLq (B(2r)) )θ2

I. 0

dr r

 θ1

2

= kf kMϕ2 (2·) qθ2

by the Lq (Rn )-boundedness of the Hardy–Littlewood maximal operator. Consequently, thanks to Remark 6, (16.7) holds. Meanwhile, from Theorem 144 in the first book, we have II . kϕ2 kf kLq (·−n M χB(·) ) kLθ2 (r−1 ) . Consequently, thanks to Corollary 162 in the first book, we have  ! # q1 θ2 "Z Z Z ∞  dr ∞ 1 q |f (z)| dz ds . II . ϕ2 (r)  r  sn+1 B(s) 0 r We calculate the right-hand side by a change of variables:  "Z −1 # q1 θ2 Z ∞  dr r
II . ϕ2 (r) sn−1 k|f |q kL1 (B(s−1 )) ds  r  0 0 Z =

∞

( ϕ2 (r

−1

0

 Z r  q1 )θ2
1 dr n−1 q . )r s k|f | kL1 (B(s−1 )) ds r 0 r 1 q

We abbreviate F (s) ≡ sn−1 k|f |q kL1 (B(s−1 )) for s > 0. Then Z ∞ θ2 dr 1 1 II . . ϕ2 (r−1 )r q (HF (r)) q r 0 By our assumption, we have  θ1 h i 1 1 θ1 dr −1 q1 q II . ϕ1 (r )r (F (r)) r 0 Z ∞ h 1
q1 iθ1 dr θ1 −1 q1 n−1 q 1 −1 = ϕ1 (r )r r k|f | kL (B(r ) r 0 1    θ1 ! q1 θ1   Z Z ∞  dr ϕ1 (r) r−n . |f (z)|q dz   r  B(r)  0  Z

∞

= kf kMϕ1 . qθ1

228

Morrey Spaces

Consequently, (16.8) holds. Thus, Theorem 200 is proven. ϕ∗

Using Theorem 200, we can look for the maximal space LMqθ11 (Rn ) for ϕ∗

2 n which M is bounded from LMqθ11 (Rn ) to LMϕ qθ2 (R ).

Corollary 201. Let 1 < q < ∞, 0 < θ1 ≤ θ2 < ∞, ϕ1 ∈ LΩqθ1 and ϕ2 ∈ LΩqθ2 . Assume in addition that θ1 ≤ q that H is bounded θ1

θ1

θ2

θ2

from L q (ϕ1 (·−1 )· q −1 ) to L q (ϕ2 (·−1 )· q −1 ). Then M is bounded from ϕ∗ 2 n ∗ LMqθ11 (Rn ) to LMϕ qθ2 (R ), where ϕ1 satisfies Z

∞

dr ) r

−n q θ2

(ϕ2 (r)(t + r) 0

 θ1

Z

2

=

∞

(r

−n q

ϕ∗1 (r))θ1

t

dr r

 θ1

1

(16.9)

for all t > 0. ϕ∗

This means that the space LMqθ11 (Rn ) is the maximal among the spaces ϕ1 1 n n LMϕ qθ1 (R ) such that ϕ1 ∈ LΩqθ1 and that M is bounded from LMqθ1 (R ) ϕ2 n to LMqθ2 (R ). Proof This is clear from Theorem 200, since condition (16.6) is automatically satisfied. In terms of the Hardy operator, we can manage to describe the sufficient condition as above. So, we are interested in the criterion which does not use the Hardy operator. Based on the weighted boundedness of the Hardy–Littlewood maximal operator, we will prove its boundedness of general local Morrey-type spaces. Theorem 202. (1) Let 1 < q1 ≤ θ ≤ q2 < ∞, and let ϕ1 ∈ LΩq1 θ and ϕ2 ∈ LΩq2 θ . Set (Z

∞

Wk (x) ≡

(ϕk (r)r

θ dr k ) r

− qn

|x|

) θ1

for x ∈ Rn and k = 1, 2. If for all balls B kM (χB W1

− q10

1

Z )kLq2 (W2 q2 ) .

− q10 1

B ϕ2 1 n n then M is bounded from LMϕ q1 θ (R ) to LMq2 θ (R ).

(2) Let 1 < q < ∞, and let ϕ1 , ϕ2 ∈ Ωqq . Set (Z

∞

Wk (x) ≡

(ϕk (r)r |x|

 q1

q

W1 (x)

dr −n q q )

r

) q1

1

dx

,

Morrey-type spaces

229

n 1 for x ∈ Rn and k = 1, 2. Then M is bounded from LMϕ qq (R ) to ϕ2 n LMqq (R ) if and only if for all balls B,

kM (χB W1

− q10

Z )kLq (W2 q ) .

W1 (x)

− qq0

 q1 dx

.

B

Proof (1) Simply use the cube testing for Mα with α = 0 in the first book and the ϕ2 1 n q1 q1 q2 q2 n embeddings LMϕ q1 θ (R ) ,→ L (W1 ) and L (W2 ) ,→ LMq2 θ (R ). (2) Letting q1 = q2 = q, we learn that the embeddings above are actual equalities. Consequently, the result is once again clear from the cube testing for Mα with α = 0 in the first book.

16.2.3

Singular integral operators in general Morrey-type spaces

We prove some boundedness assertions of singular integral operators under some mild conditions (16.10) or (16.11) postulated on ϕ1 and ϕ2 . Theorem 203. Let 1 < q < ∞, 0 < θ1 , θ2 ≤ ∞, ϕ1 ∈ LΩqθ1 and ϕ2 ∈ LΩqθ2 . For x ∈ Rn and for k = 1, 2, we let (Z

∞

Wk (x) ≡

(r

−n q

|x|

dr ϕk (r))θk r

) θ1

k

(x ∈ Rn ).

Assume sup y∈B(4|x|)\B( 41 |x|)

W2 (y) . W1 (x)

(x ∈ Rn )

(16.10)

inf

(x ∈ Rn )

(16.11)

or that W2 (x) .

y∈B(4|x|)\B( 14 |x|)

W1 (y)

(1) Let θ1 ≤ q ≤ θ2 . If sup r−n kW2 kLq (B(r)) k(M χB(r) )W1 −1 kLq0 (Rn \B(2r)) < ∞

(16.12)

sup r−n kW1 −1 kLq0 (B(r)) k(M χB(r) )W2 kLq (Rn \B(2r)) < ∞,

(16.13)

r>0

and r>0

n then kT f kLMϕ2 . kf kLMϕ1 for f ∈ L∞ c (R ). qθ2

qθ1

(2) Let θ2 ≤ q ≤ θ1 . Then conditions (16.12) and (16.13) are necessary 1 n for the boundedness of the 1st Riesz transform R1 from LMϕ qθ1 (R ) to ϕ2 n ∞ n LMqθ2 (R ). That is, kR1 f kLMϕ2 . kf kLMϕ1 for f ∈ Lc (R ). qθ2

qθ1

230

Morrey Spaces

(3) Let θ1 = θ2 = q. Then conditions (16.12) and (16.13) hold if and only n if kT f kLMϕ2 . kf kLMϕ1 for f ∈ L∞ c (R ) qθ2

qθ1

Proof We depend on Theorem 320 in the first book, which guarantees the boundedness of T under conditions (16.10) or (16.11). (1) The sufficiency of (16.12) and (16.13) for the boundedness of T yields kT f kLMϕ2 ≤ kT f kLq (W2 q ) . kf kLq (W1 q ) ≤ kf kLMϕ1 . qθ1

qθ2

(2) Remark that the role of θ1 and θ2 is swapped compared with (1). If n kR1 f kLMϕ2 . kf kLMϕ1 for any f ∈ L∞ c (R ), then by the embedding qθ2

qθ1

n we have kR1 f kLq (W2 q ) . kf kLq (W1 q ) for f ∈ L∞ c (R ). We may apply the necessity of (16.12) and (16.13).

(3) Combine (1) and (2). We globalize Theorem 203. Theorem 204. Let 1 < q < ∞, 0 < θ1 , θ2 ≤ ∞, ϕ1 ∈ Ωqθ1 and ϕ2 ∈ Ωqθ2 . satisfy θ1 ≤ q ≤ θ2 and either (16.10) or (16.11). If (16.12) and (16.13) are ϕ2 1 n n satisfied, then the operator T is bounded from Mϕ qθ1 (R ) to Mqθ2 (R ). For necessary and sufficient conditions on ϕ1 and ϕ2 ensuring the boundedness of T for other values of the parameters we will reduce the problem of the boundedness of T in local Morrey-type spaces to the boundedness of the Hardy operator in weighted Lp (0, ∞)-spaces over the cone M↑ (0, ∞). The application of Corollary 220 in the first book immediately implies the following result for the case of local Morrey-type spaces. Lemma 205. Let 1 < q < ∞, 0 < θ ≤ ∞, 0 ≤ δ < any singular integral operator T , Z kT f kLMϕqθ .

∞



ϕ(r

1 δq−n

n q − δq−n

) r

0

n for all f ∈ L∞ ˆδ (t) ≡ c (R ), where g

Z

n q

and ϕ ∈ Ωqθ . Then for

 θq dr  θ1 Hˆ gδ (r) r

(16.14)

|f (y)|q dy for t > 0.

B(tn−δq )

Proof We need to estimate Z ∞ 1 n dr θ kT f kLMϕqθ = (ϕ(r)r− q kT f kLq (B(r)) )θ . r 0 By Corollary 220 in the first book  "Z Z ∞ ∞ ϕ(r) (kT f kLMϕqθ )θ .  rδ 0 r

Z B(t)

! |f (z)|q dz

# q1 θ  dr dt . tn−δq+1  r

Morrey-type spaces

231

By the change of variables, we obtain  ! q1 θ Z rδq−n Z ∞  dr ϕ(r) (kT f kLMϕqθ )θ . g ˆ (t)dt δ  rδ  r 0 0 Z

∞

'

( ϕ(r

1 δq−n

)r

δ δq−n

Z

0

Z = 0

 q1 )θ

r

gˆδ (t)dt 0

∞

dr r

n o θq dr 1 n ϕ(r δq−n )q r− δq−n Hˆ gδ (r) . r

We have the following variant to Lemma 205. Lemma 206. Let 1 < q < ∞, 0 < θ ≤ ∞ and ϕ ∈ LΩqθ . Also let T be a singular integral operator. Then Z ∞  θ dr  θ1 q q − − n n kT f kLMϕqθ . g (r) ϕ(r )r Hˆ r 0 n ˆ(t) ≡ kf kLq B t− nq  , t > 0. for all f ∈ L∞ c (R ), where g

Proof Using Theorem 219 in the first book instead of Corollary 220 in the first book, we calculate θ Z ∞ Z ∞ n dr (kT f kLMϕqθ )θ . ϕ(r) t− q −1 kf kLq (B(t)) dt r 0 r  θ n − Z ∞ Z r q dr ϕ(r) kf kLq (B(t− nq )) dt ∼ r 0 0 Z ∞ θ dr n ∼ ϕ(r)rHˆ g (r− q ) r Z0 ∞   θ dr q q . g (r) ∼ ϕ(r− n )r− n Hˆ r 0 In the next theorem, we leave the problem of the domain of T untouched: We consider it to the minimum. Theorem 207. Let 1 < q < ∞, 0 < θ1 , θ2 ≤ ∞, ϕ1 ∈ Ωqθ1 and ϕ2 ∈ Ωqθ2 . Assume that for all gˆ ∈ M↑ (0, ∞) Z ∞  θ2 dr  θ12 q q −n −n ϕ2 (r )r Hˆ g (r) r 0 Z ∞  θ1 dr  θ11 q q −n −n . ϕ1 (r )r gˆ(r) . (16.15) r 0 Then any singular integral operator T satisfies kT f kLMϕ2 . kf kLMϕ1 for n all f ∈ L∞ c (R ).

qθ2

qθ1

232

Morrey Spaces Proof We combine Lemma 205 and the assumption for gˆ = gˆδ . Then θ1 dr  θ11 ϕ1 (r )r kf kLq (B(r− nq )) . r 0 Z ∞  θ1 n dr 1 (ϕ1 (r)r− q kf kLq (B(r)) )θ1 ∼ . r 0 Z

kT f kLMϕqθ

∞



q −n

q −n

Thus, the proof is complete. We pass Theorem 207 to the global general Morrey-type spaces. Corollary 208. Let 1 < q < ∞, 0 < θ1 , θ2 ≤ ∞, ϕ1 ∈ Ωqθ1 , ϕ2 ∈ Ωqθ2 . Assume that (16.15) for all gˆ ∈ M↑ (0, ∞). Then any singular integral operator n T satisfies kT f kMϕ2 . kf kMϕ1 for all f ∈ L∞ c (R ). qθ2

qθ1

Proof Simply globalize Theorem 207. By the use of Hardy’s inequality, we obtain the following conclusion: Theorem 209. Let 1 < q < ∞ and 0 < θ1 , θ2 ≤ ∞. If ϕ1 ∈ LΩqθ1 and ϕ2 ∈ LΩqθ2 satisfy any one of the following conditions, then kT f kLMϕ2 . qθ2

n kf kLMϕ1 for all f ∈ L∞ c (R ). qθ1

(1) Let θ1 > q and suppose that Z 0

∞

 

t

n n−δq

ϕ2 (t

1 δq−n

)q

t

Z

s

nθ1 (δq−n)(θ1 −q)

ϕ1 (s

1 δq−n

)

0



qθ1 q−θ1

 θ2  θ1θ−q  q 1 ds dt 

  is finite for some δ ∈ 0, nq .   (2) Let θ1 ≤ q and suppose that there exists δ ∈ 0, nq satisfying ∞

Z

 θq2  n 1 n 1 q −q n−δq δq−n δq−n δq−n t ϕ2 (t ) sup s ϕ1 (s ) dt < ∞. 0 1 and suppose that Z

∞

( ϕ2 (t

q −n

Z

t

)

0

s

0 qθ1 n

q −n −θ10

ϕ1 (s

)

 θ10 )θ2 1 ds dt < ∞.

0

(4) Let θ1 ≤ 1 and suppose that Z

!θ2

∞

ϕ2 (t 0

q −n

q −n

) sup ϕ1 (s s∈(0,t)

)−1 s

q n

dt < ∞.

Morrey-type spaces

233

Proof (1) Simply use Theorem 318(1) in the first book. (2) Simply use Theorem 318(2) in the first book. (3) Simply use Theorem 219 and Theorem 318(1) in the first book. (4) Simply use Theorem 219 and Theorem 318(2) in the first book.

16.2.4

Fractional maximal operators in general Morrey-type spaces

Here, we investigate the fractional maximal operator Mα , 0 ≤ α < n. We start with a necessary condition which excludes a “bad” function f (x) ≡ χRn \B(1) (x)|x|−β for β small enough. Lemma 210. Let 1 < q1 ≤ ∞, 0 < q2 ≤ ∞, 0 ≤ α < n, 0 < θ1 , θ2 ≤ ∞, and 1 n let ϕ1 ∈ LΩqθ1 and ϕ2 ∈ LΩqθ2 . Then if Mα is bounded from LMϕ q1 θ1 (R ) to ϕ2 LMq2 θ2 (Rn ), then α ≤ qn1 .   Proof Assume otherwise; assume that α > qn1 . Choose β ∈ qn1 , α . Let 1 n f (x) ≡ χRn \B(1) (x)|x|−β for x ∈ Rn . Then f ∈ LMϕ q1 θ1 (R ) according to n Example 71. Meanwhile for all x ∈ R , since β < α, Mα f (x) ≡ ∞. Thus, Mα ϕ2 1 n n fails to be bounded from LMϕ q1 θ1 (R ) to LMq2 θ2 (R ).

We seek another necessary condition using the dilation of functions. Lemma 211. Let 1 ≤ q1 ≤ ∞, 0 < q2 ≤ ∞, 0 ≤ α < n, 0 < θ1 , θ2 ≤ ∞, θ1

ϕ1 ∈ LΩqθ1 ∩ Lθ1 (r−1− q1 ) and ϕ2 ∈ LΩ qθ2 . Then  if Mα is bounded from 1 1 ϕ1 ϕ LMq1 θ1 (Rn ) to LMq22θ2 (Rn ), then α ≥ n . − q1 q2 + Remark that Lemma 211 is significant only if q2 < q1 . We prove it by the use of dilation. Let f ∈ L0 (Rn ) and write δt f ≡ f (t·), t > 0. Then note that n kδt f kLq2 (B(r)) = t− q2 kf kLq2 (B(tr)) and that Mα [δt f ] = t−α Mα f (t·). ϕ2 1 n n Proof Assume that Mα is bounded from LMϕ q1 θ1 (R ) to LMq2 θ2 (R ), so that 1 Z ∞ n dr θ2 kMα f kLMϕ2 = (r− q2 ϕ2 (r)kMα f kLq2 (B(r)) )θ2 . kf kLMϕ1 q2 θ2 q1 θ1 r 0 1 n q1 n for all f ∈ LMϕ q1 θ1 (R ) ∩ L (R ). ϕ1 n We let f ∈ LMq1 θ1 (R ) ∩ Lq1 (Rn ) \ {0}. Let t ≥ 1. By the change of variables, we have 1 Z ∞ dr θ2 − qn − qn θ 2 (t 2 r 2 ϕ2 (tr)kMα f kLq2 (B(tr)) ) . kMα f kLMϕ2 = q2 θ2 r 0

234

Morrey Spaces n

Since kMα (δt f ) kLq2 (B(r)) = t−α− q2 kMα f kLq2 (B(tr)) , we have Z ∞  θ1 2 − qn α θ2 dr ϕ 2 (r kMα f kLM 2 = ϕ2 (tr)t kMα (δt f ) kLq2 (B(r)) ) . q2 θ2 r 0 Therefore since kMα (δt f ) kLMϕ2

. kδt f kLMϕ1

q2 θ2

kMα f kLMϕ2

q 2 θ2

n

≤ tα+ q2

∞

Z

q1 θ1

and t ≥ 1,

n

(r− q2 ϕ2 (r)tα kMα (δt f ) kLq2 (B(r)) )θ2

0

dr r

 θ1

2

n

. tα+ q2 kδt f kMϕ1

q1 θ1

.t

α+ qn − qn 2

1

kf kLq1 kϕ1 k

Lθ1 (r

−1−

nθ1 q1

. )

If α < qn1 − qn2 , then by passing to the limit as t → ∞ we obtain that kMα f kLMϕ2 = 0, which is impossible since f 6= 0. q2 θ2    n n n − , From Lemmas 210 and 211, it is natural to assume α ∈ q1 q2 q1 . +

As usual we will take the Hardy operator approach. Lemma 212. Let 1 < q1 < ∞, 0 < q2 < ∞ and 0 < θ ≤ ∞ satisfy   ≤ α < qn1 . Also let ϕ1 ∈ LΩq1 θ and ϕ2 ∈ LΩq1 θ . Write n q11 − q12 + Z g(t) ≡  1  |f (y)|q1 dy (t > 0), B t αq1 −n

given f ∈ L0 (Rn ). Then (Z

∞



kMα f kLMϕ2 .

ϕ2 (t

q2 θ

1 αq1 −n

)q1

0

Z

 qθ

t

1

g(v)dv 0

dt t

) θ1

and 1

(Z

∞

kf kLMϕ1 ∼ q1 θ

0

)θ  qθ  Z t 1 dt 1 g(v)dv . ϕ1 (t αq1 −n )q1 t 0

Proof We estimate (kMα f kLMϕ2 )θ = q2 θ

Z 0

∞

n

(ϕ2 (r)r− q2 kMα f kLq2 (B(r)) )θ

dr . r

By Theorem 178 in the first book we have Z  q1 )θ Z ∞( 1 |f (x)|q1 dr θ (kMα f kLMϕ2 ) . ϕ2 (r) dx n−αq 1 q2 θ r 0 Rn (|x| + r) ( Z ∞  q1 )θ Z ∞ 1 ds dr q1 (kf kLq1 (B(s)) ) 1+n−αq1 . ϕ2 (r) . s r 0 r

Morrey-type spaces

235

We change variables: t = s−n+αq1 keeping in mind that αq1 − n < 0. Then (kMα f kLMϕ2 )θ .

∞

Z

 Z ϕ2 (r)

q2 θ

1

g(t)dt

0

0 ∞

Z

! q1 θ

r αq1 −n

∼ 0

 dr r

  qθ Z t 1 dt 1 q ϕ2 (t αq1 −n ) 1 g(v)dv . t 0

For the second formula, we also calculate Z

∞

kf kLMϕ1 ∼

(ϕ1 (t

q1 θ

1 αq1 −n

− qn

)t

1 1 αq1 −n

Z ∼

∞

1

kf k

1

Lq1 (B(t αq1 −n ))

0 n

θ

(ϕ1 (t αq1 −n )q1 t− αq1 −n g(t)) q1

0

dt t

)

θ dt

 θ1

t

 θ1 .

Thus, the proof is complete. Thus, in terms of the Hardy operator H, we are led to the following conclusion: Theorem Let 1 0 be fixed. Let us choose a ball B(a, r) such that (q )

2 kgkMϕq 2 ≤ (1 + ε)ϕ2 (r)mB(a,r) (g).

(17.12)

2

q2

We test the multiplier condition on f ≡ |g| q0 χB(a,r) . Let α be the almost increasing constant of ϕ0 q0 ϕ1 −q1 . Then we have ϕ0 (s) ϕ2 (s)

q2 q0

q2

=

ϕ0 (s) q1 ϕ1 (s)

q2 q0

q2

≤α

ϕ0 (r) q1 ϕ1 (r)

q2 q0

=α

ϕ0 (r)

(17.13)

q2

ϕ2 (r) q0

for all 0 < s ≤ r < ∞. n 0 From the scaling law, we can say that f ∈ Mϕ q0 (R ) and that   qq2 q2 (q2 ) 0 kf kMϕq 0 . kgkMϕq 2 q0 . (1 + ε)ϕ2 (r)mB(a,r) (g) ≤ ϕ0 (r)kf kLq0 (B(a,r)) . 0

2

Consequently, from (17.12) and |f · g|q1 = |g|q2 , we deduce (q )

1 kf kMϕq 0 kgkMϕq 2 . ϕ1 (r)mB(a,r) (f · g) . kf kMϕq 0 kgkPWM(Mϕq 0 ,Mϕq 1 ) . 0

2

0

0

1

260

Morrey Spaces

So, we obtain kgkMϕq 2 . kgkPWM(Mϕq 0 ,Mϕq 1 ) . In the estimate; 2

0

1

kf · gkMϕq 1 ≤ kgkPWM(Mϕq 0 ,Mϕq 1 ) kf kMϕq 0 , 1

0

1

0

we can look for a function which attains equality if ϕ0 q2 set f ≡ |g| q0 .

q0

= ϕ1 q1 . In fact, simply

In general equality can fail in (17.7). Example 82. Let 0 < q0 , q1 < ∞, ϕ0 ∈ Gq0 and ϕ1 ∈ Gq1 . Assume that ϕ0 −q0 ϕ1 q1 is not almost decreasing, so that in addition to the assumption on (sk , rk ) in Example 81, we may assume that 4k ϕ0 (sk )−q0 ϕ1 (sk )q1 ≤ ϕ0 (rk )−q0 ϕ1 (rk )q1 . Assume 0 < q1 ≤ q0 , q2 < ∞ and that ϕ2 = ϕ1 /ϕ0 ∈ Gq2 , where Let g, gk , k ∈ N be the same functions as in Example 81. Then

1 q1

=

1 q2

+ q10 .

n

k (q2 ) ϕ2 (rk )mQ(r (2− q0 gk ) k)

≥

ϕ2 (rk )(mk sk ) q2 n

k

2 q0 ϕ2 (sk )(rk ) q2 n

≥

ϕ1 (rk )ϕ0 (sk )(mk sk ) q2 k

n

2 q0 ϕ1 (sk )ϕ0 (rk )(rk ) q2 q1

k

≥

n

4 q0 ϕ1 (rk )1− q0 (mk sk ) q2 k

q1

n

2 q0 ϕ1 (sk )1− q0 (rk ) q2 k

& 2 q0 , n n ϕ2 2 / Mϕ showing that {gk }∞ q2 (R ). k=1 is an unbounded set in Mq2 (R ). Thus g ∈ ϕ0 n ϕ1 n If we use Example 81, then we see that g ∈ PWM(Mq0 (R ), Mq1 (R )).

We characterize the condition for both sides in (17.7) to coincide in the sense of equivalence of norms. Theorem 244. Suppose that we have three parameters q0 , q1 and q2 in (0, ∞) and functions ϕ0 ∈ Gq0 , ϕ1 ∈ Gq1 and ϕ2 ∈ Gq2 . Assume q11 = q12 + q10 and ϕ1 = ϕ0 ϕ2 . Then the following conditions are equivalent: (A) The function ϕ0 −q0 ϕ1 q1 is almost increasing. (B) Both sides in (17.7) coincide in the sense of equivalence of norms. (C) Both sides in (17.7) coincide in the sense of coincidence of norms. Proof We know that (A) is sufficient for (B) and (C) to hold; see Theorem 243. Suppose now that (A) fails. Then we are in the position of using Examples 81 and 82 to see that (B) and (C) fail.

Pointwise product

17.2.5

261

Pointwise multipliers for BMOϕ (Rn )

In analogy to Morrey spaces we define BMOϕ (Rn ) to generalize BMO(Rn ). Set kf kBMOϕ ≡ sup ϕ(c(Q), `(Q))mQ (|f − mQ (f )|) for f ∈ L1loc (Rn ) and for Q∈Q

a function ϕ : Rn × (0, ∞) → (0, ∞). It sometimes matters when ϕ depends on the position. So, this further generalizes the following approach: For f ∈ L1loc (Rn ) and for a function ϕ : (0, ∞) → (0, ∞), set kf kBMOϕ ≡ sup ϕ(`(Q))mQ (|f − mQ (f )|). Q∈Q

Unlike generalized Morrey spaces, it is important that ϕ also takes into account the position of the cubes; this is not a mere quest for the generalization. As we will see the pointwise multiplier for BMO(Rn ) can be described in terms of this class once we make BMO(Rn ) into a normed space BMO+ (Rn ) by adding an extra modification term kf kL1 (Q(1)) as we did in Remark 10 in n the first book. Likewise, we defined BMO+ ϕ (R ) to be the set of all functions 1 n f ∈ Lloc (R ) for which kf kBMO+ ≡ kf kBMOϕ + kf kL1 (Q(1)) < ∞. ϕ n The same applies when ϕ depends on the position. Note that BMO+ ϕ (R ) and n BMOϕ (R ) are the same but that these spaces are equipped with different functionals.

Example 83. The function χ(0,∞) is not PWM(BMO+ (R), BMO+ (R)) since it maps log | · | ∈ BMO+ (R) to χ(0,∞) log | · | ∈ / BMO+ (R). n Since BMO+ ϕ (R ) is a normed space, we can consider its pointwise multiplier space.

Definition 42. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . A function g ∈ L0 (Rn ) is n . kf kBMO+ said to be a pointwise multiplier for BMO+ ϕ (R ) if kf · gkBMO+ ϕ ϕ + n for all f ∈ BMOϕ (R ). Let us start with the general observation on the pointwise multipliers. Intuitively, Lemma 245 says that we will need something like kg · hkE . khkE for some ball Banach function space E. Lemma 245. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Also let g ∈ L1loc (Rn ). If n n kf · gkBMO·ϕ . kf kBMO·ϕ + kf kL1 (Q(1)) for all f ∈ L∞ c (R ) ∩ BMOϕ (R ), then ∞ n g ∈ L (R ). We recall that Φ∗ and Φ∗ are defined by (12.2). Proof We may suppose ϕ(1) = 1 by multiplying a constant if necessary. Let a ∈ Rn and r ∈ (0, 1). We first employ the function h in Example 36, so n that h ∈ BMO+ ϕ (R ) is supported on a ball B(a, r). Recall also that the norm

262

Morrey Spaces

of h is bounded by a constant independent of a and r. Thus, kg · hkBMOϕ . khkBMOϕ + khkL1 (Q(1)) . 1. We set     1 1 Ar ≡ B a, r \ B a, r ⊂ B(a, r) ⊂ Q(a, r). 2 4 In view of the definition of Φ∗ used to define h, the doubling property of ϕ and the fact that ϕ(r) ≥ ϕ(1) for all 0 < r < 1, for x ∈ Ar , we have Z r Z r dt dt 1 . ≤ Φ∗ (|x − a|) − Φ∗ (r) ≤ ≤ log 4 < π. (17.14) ϕ(r) ϕ(t)t ϕ(t)t r/2 r/4 Thus, the inequality |eiθ − 1| = 2 sin θ2 ≥ 2θ π , 0 ≤ θ ≤ π, implies that π π |h(x)| = | exp(iΦ∗ (|x − a|) − iΦ∗ (r)) − 1| ≥ Φ∗ (|x − a|) − Φ∗ (r) (17.15) 2 2 for x ∈ Ar . Since kg · hkBMOϕ . 1 by assumption and g · h vanishes outside Q(a, r), Z rn & |g(x)h(x) − mQ(a,r) (g · h)|dx ϕ(4r) Q(a,4r) Z ≥ |g(x)h(x) − mQ(a,r) (g · h)|dx (Q(a,4r)\Q(a,2r))∪Ar Z ≥ |mQ(a,r) (g · h)| · |Q(a, r)| + |g(x)h(x) − mQ(a,r) (g · h)|dx. Ar

Since |Q(a, r)| ≥ |Ar |, we deduce Z rn 1 ≥ |g(x)h(x)|dx & kgkL1 (Ar ) ϕ(4r) ϕ(r) Ar from (17.14) and (17.15). Thus kgkL1 (Ar ) . rn , since ϕ is doubling. Consequently, by the Lebesgue differentiation theorem, we see g ∈ L∞ (Rn ). It is a little bit tough to handle mQ (|f ·g−mQ (f ·g)|) directly although this term arises naturally in the context of the pointwise multipliers of generalized Morrey–Campanato spaces. So, we use the following estimate to transform this term into one which is a little easier to handle. Lemma 246. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . for 0 < t0 < t < ∞. Then for any f ∈ BMOϕ (Rn ), g ∈ L∞ (Rn ) and Q = Q(a, r), |mQ (|f · g − mQ (f · g)|) − |mQ (f )| · mQ (|g − mQ (g)|)| . ϕ(r)−1 kf kBMOϕ . Proof We calculate |mQ (|f · g − mQ (f · g)|) − |mQ (f )| · mQ (|g − mQ (g)|)| = |mQ (|f · g − mQ (f · g)|) − mQ (|mQ (f )g − mQ (f )mQ (g)|)| ≤ mQ (|f · g − mQ (f )g + mQ (f )mQ (g) − mQ (f · g)|) ≤ mQ (|f · g − mQ (f )g|) + |mQ (f )mQ (g) − mQ (f · g)|

Pointwise product

263

by using the triangle inequality twice. Since mQ (1) = 1, we obtain |mQ (|f · g − mQ (f · g)|) − |mQ (f )| · mQ (|g − mQ (g)|)| ≤ mQ (|f · g − mQ (f )g|) + |mQ (f )mQ (g) − mQ (f · g)| ≤ mQ (|f · g − mQ (f )g|) + |mQ (mQ (f )g) − mQ (f · g)| . mQ (|f − mQ (f )|) . ϕ(r)−1 kf kBMOϕ . Thus, the proof is complete. We consider the multiplier result. As we will see, Theorem 247 is the thrust into considering ϕ which also depends on x ∈ Rn not only on r > 0. n We characterize the pointwise multiplier theorem for BMO+ ϕ (R ). The transform ϕ 7→ wϕ is called the Nakai–Yabuta transform. Theorem 247. Let ϕ ∈ M↓ (0, ∞) satisfy Z wϕ (x, r) ≡ ϕ(r)

r

1

Z 2+|x| dt + ϕ(t)t 1

1 ϕ

∈ Gn . Set ! dt ((x, r) ∈ Rn+1 + ). ϕ(t)t

(17.16)

n Then a function g ∈ L1loc (Rn ) is a pointwise multiplier for BMO+ ϕ (R ) if and n ∞ n only if g ∈ BMOwϕ (R ) ∩ L (R ).

Recall again that Φ∗ and Φ∗ are defined by (12.2). We note that wϕ (x, r) ∼ ϕ(r)(Φ∗ (r) + Φ∗ (|x|) + Φ∗ (r))

((x, r) ∈ Rn+1 + ).

(17.17)

Proof We have g ∈ L∞ (Rn ) from Lemma 245 in any case. We let f ∈ n ∞ n BMO+ ϕ (R ) and g ∈ L (R ). n (1) Necessity: Suppose that g is a pointwise multiplier for BMO+ ϕ (R ). Let Q = Q(a, r). Then by the definition of the pointwise multipliers

. . ϕ(r)−1 kf kBMO+ mQ (|f · g − mQ (f · g)|) ≤ ϕ(r)−1 kf · gkBMO+ ϕ ϕ Consequently, for all cubes Q(a, r), we deduce from Lemma 246 |mQ(a,r) (f )| · mQ(a,r) (|g − mQ(a,r) (g)|) . ϕ(r)−1 kf kBMO+ . ϕ It remains to choose f suitably by using Proposition 74. (2) Sufficiency: Let g ∈ BMOwϕ (Rn ) ∩ L∞ (Rn ). If we use Lemma 73, then |mQ(a,r) (f )| · mQ(a,r) (|g − mQ(a,r) (g)|) . mQ(1) (|f |) · mQ(a,r) (|g − mQ(a,r) (g)|) + kf kBMO+ (Φ∗ (r) + Φ∗ (r) + Φ∗ (|a|))mQ(a,r) (|g − mQ(a,r) (g)|) ϕ . ϕ(r)−1 kf kBMO+ kgkBMOwϕ . ϕ Thus, we obtain the desired result from Lemma 246.

264

Morrey Spaces

Example 84. Let 0 < α < 1. Let Lipα (Rn ) be the Lipschitz space defined in Chapter 5. (1) Let w(x, r) ≡ | log r| + log(2 + |x|) for r > 0 and x ∈ Rn . Then thanks to Theorem 247, a function g ∈ L1loc (Rn ) is a pointwise multiplier for BMO+ (Rn ) if and only if g ∈ BMOw (Rn ) ∩ L∞ (Rn ). (2) Denote by Lipα,+ (Rn ) the set of all continuous functions f satisfying kf kLipα,+ = kf kL1 (Q(1)) + kf kLipα . Let w(x, r) ≡ r−α (rα + (2 + |x|)α ) for r > 0 and x ∈ Rn . Then thanks to Theorem 247, a function g ∈ L1loc (Rn ) is a pointwise multiplier for Lipα,+ (Rn ) if and only if g ∈ BMOw (Rn ) ∩ L∞ (Rn ). Once we take the intersection with Lp (Rn ) with 1 ≤ p < ∞, there is no need to consider the case where ψ depends on x ∈ Rn . Theorem 248. Let 1 ≤ p < ∞ and let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Define Z 2 dt = ϕ(r)Φ∗ (r) for r > 0. Then g ∈ L1loc (Rn ) is a ψ(r) ≡ ϕ(r) ϕ(t)t min(1,r) n pointwise multiplier from BMOϕ (Rn ) ∩ Lp (Rn ) to BMO+ ϕ (R ) if and only if n ∞ n g ∈ BMOψ (R ) ∩ L (R ). Note that ψ is independent of the position because we consider Lp (Rn ) instead of L1 (Q(1)). Proof Once again in any case g ∈ L∞ (Rn ) from Lemma 245. We also have Lemma 246. Let Q = Q(a, r) be a cube. Note that Lemma 246 is still valid because g ∈ L∞ (Rn ). (1) Necessity: Since g is a pointwise multiplier from BMOϕ (Rn ) ∩ Lp (Rn ) n to BMO+ ϕ (R ), we have |mQ(a,r) (f )|mQ(a,r) (|g − mQ(a,r) (g)|) . ϕ(r)−1 kf kBMO+ ϕ from Lemma 246. If we take f ≡ Φ∗ (min(| · −a|, 1)) − Φ∗ (1), then we obtain sup 1 0 2nr and that r > 10−n . In this case, we have wϕ (a, r) ∼ ϕ(r)(Φ∗ (|a|) + Φ∗ (r)) ∼ ϕ(r)Φ∗ (|a|). Then for x ∈ Q(a, r) 1 and x0 ∈ Q(a, 3r) \ Q(a, r), d ≤ 4nr and D ≥ |a| − nr ≥ |a| we have 2 |g(x) − g(x0 )| .

1 Φ∗ (D)

So, in this case our claim is true.

.

1 1 ∼ . ∗ ϕ(r)Φ (|a|) wϕ (a, r)

Pointwise product

267

(3) Assume that r ≥ min(|a|/2n, 10−n ). In this case wϕ (a, r) ∼ ϕ(r)Φ∗ (r). Meanwhile thanks to Lemma 69, Z 1 dx mB(a,r) (|g − mB(a,r) (g)|) . |B(a, r)| B(a,r) Φ∗ (|x|) Z 3nr n−1 1 t . n dt r 0 Φ∗ (t) Z 1 3nr 1 . dt r 0 Φ∗ (t) 1 . . ϕ(r)Φ∗ (r) So, in this case our claim is true. We present an example of the pointwise multiplier. Proposition 251. Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Suppose in addition that ϕ(+0) = lim ϕ(ε) = ∞. Let g ∈ L∞ (Rn ) satisfy ε↓0

|g(x + y) − g(x)| ≤

1 ϕ(|y|)Φ∗ (|y|)

((x, y) ∈ Rn × B(1)).

(17.19)

+ n n p n Then g is a pointwise multiplier from BMO+ ϕ (R ) ∩ L (R ) to BMOϕ (R ).

Proof We have to show that g ∈ BMOϕ (Rn ) according to Theorem 249. We let B(a, r) be any ball and estimate ϕ(r)mB(a,r) (|g − mB(a,r) (g)|). If 1 < r < ∞, then since ϕ ∈ M↓ (0, ∞), we have ϕ(r)mB(a,r) (|g − mB(a,r) (g)|) . 1. If 0 < r < 1, then ϕ(r)mB(a,r) (|g − mB(a,r) (g)|) ≤ sup Φ∗ (|y|)−1 . 1 by our y∈B(r)

assumption. n As a result, we obtain some examples of multipliers of BMO+ ϕ (R ).

Corollary 252. Suppose that g1 and g2 are Lipschitz continuous functions such that |g2 (x)| ≥ Φ∗ (|x|) for all x ∈ Rn . Assume in addition that g1 is n bounded. Then g ≡ g1 /g2 is a pointwise multiplier for BMO+ ϕ (R ). Proof We calculate g1 (x + y)g2 (x) − g1 (x + y)g2 (x + y) |g(x + y) − g(x)| ≤ g2 (x + y)g2 (x) g1 (x) − g1 (x + y) + g2 (x) |y| . ∗ Φ (|x|)

268

Morrey Spaces

for all x ∈ Rn and y ∈ B(1). Since Z 2 |y|Φ∗ (|y|) = |y| |y|

dt 2 1 ≤ = , ϕ(t)t ϕ(|y|) ϕ(|y|)

(17.20)

|y| 1 1 . . . Φ∗ (|x|) ϕ(|y|)Φ∗ (|x|)Φ∗ (|y|) ϕ(|y|)(Φ∗ (|x|) + Φ∗ (|y|)) Combining these observations with Proposition 250, we obtain the desired result.

it follows that

Example 87. Let ϕ ∈ M↓ (0, ∞) satisfy n of multipliers on BMO+ ϕ (R ).

1 ϕ

∈ Gn . Corollary 252 gives examples

sin | · | sin(Φ∗ (| · |)) and are multipliers on ∗ · |) Φ (| · |) Φ∗ (| · |) n ∗ BMO+ ϕ (R ). In fact, choose g1 = 1, g1 = sin | · | or g1 = sin(Φ (| · |)) and ∗ g2 = Φ (| · |).

(1) The functions

1

Φ∗ (|

,

1 n is a multiplier on BMO+ ϕ (R ). In fact, since 1+|·| Φ∗ (r) . 1 + r, we have Φ∗ (|x|) & 1 + |x| for all x ∈ Rn .

(2) The function

We consider the intersection with other Lebesgue spaces. Proposition 253. If g ∈ L∞ (Rn ) such that |g(x + y) − g(y)| . Φϕ(y) for x ∈ ∗ (|y|) + n n R and y ∈ B(1), then g is a pointwise multiplier from BMOϕ (R ) ∩ Lp (Rn ) n to BMO+ ϕ (R ). Proof The proof is akin to Proposition 250. The proof is left for the interested readers; see Exercise 68.

17.2.6

Exercises

Exercise 59. Let (X, B, µ) be a measure space and 1 ≤ p ≤ ∞. Let f ∈ L0 (µ) 0 be such that f · g ∈ L1 (µ) for all g ∈ Lp (µ). Show that f ∈ Lp (µ). Exercise 60. Let (X, B, µ) be a measure space. Let E(µ) and F (µ) be Banach lattices on a measure space (X, B, µ). (1) Show that PWM(E(µ), F (µ)) is a Banach space. (2) Let g ∈ L0 (µ) such that the pointwise product f · g belongs to F (µ) for each f ∈ E(µ). Then use the closed graph theorem to show that (17.1) holds. Exercise 61. [335, 337] Let (X, B, µ) be a σ-finite measure space. More precisely, let X be expressed as a union of {Xj }∞ j=1 with µ(Xj ) < ∞, j ∈ N. Let E1 (µ), E2 (µ), E3 (µ) be linear spaces of L0 (µ) with the following properties for all f, g ∈ L0 (µ):

Pointwise product

269

(a) χXj ∈ E2 (µ) for each j ∈ N. (b) f ∈ E2 (µ) whenever |f (x)| ≤ |g(x)| for µ-a.e. and g ∈ E2 (µ). (c) E3 (µ) ⊂ PWM(E1 (µ), E2 (µ)). (d) kgkE3 (µ) . kgkPWM(E1 (µ),E2 (µ)) for all g ∈ E3 (µ). (1) Show that E3 (µ) ≈ PWM(E1 (µ), E2 (µ)). (2) Show that L∞ (µ) ≈ PWM(E2 (µ), E2 (µ)). Exercise 62. Let 0 < q1 ≤ p1 < ∞ and 0 < q2 ≤ p2 < ∞. Define p and q by 1 1 1 1 1 1 = + , = + . p p1 p2 q q1 q2 Then show that kf · gkMpq ≤ kf kMpq 1 kgkMpq 2 by using the H¨older inequality 1 2 for Lebesgue spaces. Exercise 63. [117, 370] Let 0 < q0 ≤ p0 < ∞ and 0 < q1 ≤ p1 < ∞. (1) Let Q be a cube. Then using Theorems 18 and 19, show that the operator f ∈ Mpq00 (Rn ) 7→ f χQ ∈ Mpq11 (Rn ) is well defined if and only if q0 ≤ q1 and p1 ≤ p0 . (2) Show that Mpq00 (Rn ) ⊂ Mpq11 (Rn ) if and only if q1 ≤ q0 ≤ p0 = p1 . Exercise 64. Let g ∈ L∞ (Rn ). Using the Lebesgue differentiation theorem, show that the operator norm f ∈ Mpq (Rn ) 7→ g · f ∈ Mpq (Rn ) is kgkL∞ . That is, sup{kg · f kMpq : f ∈ Mpq (Rn ), kf kMpq = 1} = kgkL∞ . Exercise 65. [339, Theorem 2.1] If ϕ1 q1 = ϕ2 q2 , then show that (17.10) holds with coincidence of norms in Theorem 243. Exercise 66. Let q1 , q2 , q3 > 0 and ϕ1 , ϕ2 , ϕ3 ∈ L0 (0, ∞) satisfy 1 1 1 + = , q1 q2 q3

ϕ3 = ϕ1 ϕ2 .

Then show that kf · gkMϕq 3 ≤ kf kMϕq 1 kgkMϕq 2 for all f, g ∈ L0 (Rn ). 3

1

2

Exercise 67. [351, Corollary 5.2] Let ϕ ∈ M↓ (0, ∞) satisfy ϕ1 ∈ Gn . Use cos | · | cos(Φ∗ (| · |)) Corollary 252 to show that ∗ and are multipliers on Φ (| · |) Φ∗ (| · |) + n BMOϕ (R ). Exercise 68. [351, Proposition 5.3] Suppose that g1 and g2 are Lipschitz continuous functions such that |g2 (x)| ≥ Φ∗ (|x|) for all x ∈ Rn . Assume in addition that g1 is bounded. Set g = g1 /g2 . 1 (1) Show that |g(x + y) − g(x)| . for x ∈ Rn and y ∈ B(1) ϕ(|y|)Φ∗ (|y|) using (17.20). + n p n n (2) Prove that g ∈ PWM(BMO+ ϕ (R ) ∩ L (R ), BMOϕ (R )).

270

17.3

Morrey Spaces

Olsen’s inequalities

One of the fundamental estimates in the theory of partial differential equations takes the form kg ·f kL2 . kgkX k∇f kL2 for f ∈ Cc∞ (Rn ) and g ∈ X(Rn ), where X(Rn ) is a suitable Banach lattice. It is desirable that X(Rn ) contains power functions | · |α for some α ∈ R, especially α = −1. See Example 88. For this purpose we cannot use the Lebesgue space Lp (Rn ): Lp (Rn ) is not a pleasant candidate of X(Rn ). We will show that we can use Morrey spaces non-trivially. We will prove the Olsen inequality. This is a bilinear estimate of Iα , which is nowadays called an Olsen inequality [365]. The original Olsen inequality is the inequality of the form kg · Iα f kZ . kf kX kgkY , where X, Y, Z are suitable quasi-Banach spaces. Keeping in mind that Mα and Iα behave similarly, by “Olsen’s inequality” we mean an inequality of the form kg · Mα f kZ . kf kX kgkY (Section 17.3.1) or kg · Iα f kZ . kf kX kgkY (Section 17.3.2).

17.3.1

Olsen’s inequality for fractional maximal operators

H¨ older’s inequality yields Lp0 (Rn ) = Mpp00 (Rn ) ,→ Mpp01 (Rn ) ,→ Mpp02 (Rn )

(17.1)

for all p0 ≥ p1 ≥ p2 ≥ 1. Hence, the parameter q can be used to measure how precise the result is. We state our result in full generality for fractional maximal operators keeping this fact in mind. Theorem 254. Let 0 < α < n, 1 < p ≤ p0 < ∞, 1 < q ≤ q0 < ∞ and 1 < r ≤ r0 < ∞. Suppose that   1 1 p α α > , ≤ , r ≤ min q, r0 , (17.2) p0 n q0 n p0 and that

1 1 α 1 = + − . r0 q0 p0 n

(17.3)

Then kg·Mα f kMrr0 . kgkMqq0 kf kMpp0 for all f ∈ Mpp0 (Rn ) and g ∈ Mqq0 (Rn ). The case where p0 = p = q0 = q is known as the Fefferman–Phong inequality and applied to partial differential equations. n + n Proof We can assume that f, g ∈ L∞ c (R ) ∩ M (R ) by a simple limiting argument. We fix a cube Q0 ∈ D(Rn ). We choose a sparse family S for the purpose of using Theorem 179. For P each Q ∈ S, denote by EQ the kernel of Q as usual. We define F ] ≡ `(Q)α m3Q (f )χEQ and Q∈D ] (Q0 )∩S

Pointwise product F[ ≡

271

`(Q)α m3Q (f )χEQ . We will estimate kg · F ∗ kLr (Q0 ) . As for

P Q∈D [ (Q0 )∩S

F [ , keeping in mind that the set of the nutshells {EQ }Q∈S is disjoint, we insert the definition of F [ : Z  Z r X g(x)F [ (x) dx = `(Q)rα m3Q (f )r g(x)r dx. Q0

EQ

Q∈D ∗ (Q0 )∩S

From the definition of the Morrey norm, we obtain Z 1 1 1 1 g(x)r dx ≤ (|Q| r − q0 kgkMqr0 )r ≤ (|Q| r − q0 kgkMqq0 )r . EQ

Thus, X

kg · F [ kLr (Q0 ) r ≤

1

1

`(Q)rα m3Q (f )r (|Q| r − q0 kgkMqq0 )r

Q∈D(Q0 )

X

.

1

|EQ |`(Q)rα m3Q (f )r (|Q|− q0 kgkMqq0 )r

Q∈D(Q0 )

Z . Q0

Mα− qn f (x)r dx · kgkMqq0 r . 0

Thus, it remains to use the Adams inequality, Corollary 392 in the first book. We deal with F ] . A cruder estimate suffices in this case. By a property of the dyadic cubes, for all x ∈ Q0 we have X X n F ] (x) ≤ `(Q)α m3Q (f ) ≤ `(Q)α− p0 kf kMpp0 . Q∈D ] (Q0 )

Q∈D ] (Q0 )

In view of the definition of D] (Q0 ), we have F ] (x) ≤

∞ X

  j α− pn

2

0

n

n

`(Q0 )α− p0 kf kMpp0 ∼ `(Q0 )α− p0 kf kMpp0 .

j=0 n

Thus, for all x ∈ Q0 we obtain F ] (x) . kf kMpp0 `(Q0 )α− p0 . Hence n

n

n

n

kg · F ] kLr (Q0 ) . mQ0 (g)kf kMpp0 `(Q0 )α− p0 + r ≤ kf kMpp0 kgkMqq0 `(Q0 ) r − r0 . (q)

This is our desired inequality. It should be noted that Theorem 255 is not obtained by a mere combination of H¨ older’s inequality and the Adams inequality. Remark 9. Using naively the Adams theorem and H¨older’s inequality (see Proposition 220 above), one can prove a “minor” part of q in Theorem 254 as well as Theorem 255 to follow. When we consider Theorem 255, we use the

272

Morrey Spaces

Adams theorem the modified fractional integral operator Iα in the first book. p That is, the proof of Theorem 254 is fundamental provided q0 ≤ q ≤ q0 . p0 Indeed, by virtue of the Adams theorem we have, for any cube Q ∈ Q, 1

(s)

|Q| s0 mQ (Iα f ) . kf kMpp0 where

(f ∈ Mpp0 (Rn )),

(17.4)

p0 1 1 α 1 = , = − . Condition (17.3) reads s ps0 s0 p0 n   1 p0 1 1 α p0 1 = + − = + . r p q0 p0 n pq0 s

1 1 r p (r) = = Proposition 220 yields |Q| q0 + s0 mQ (g · Iα f ) . kgkMqq0 kf kMpp0 if r0 p0 q p . In view of inclusion (17.1), the same can be said if q0 ≤ q ≤ q0 . Also q0 p0 1 1 1 α 1 = + − > . Hence, we have q0 > r0 . Thus, since observe that r0 q0 p0 n q0 p p q > r, Theorem 255 is significant only when r0 < q < q0 . See also p0 p0 Theorem 263 below for the original assertion due to Olsen [365].

17.3.2

Olsen’s inequality for fractional integral operators

As we witnessed in the previous section, we have non-trivial bilinear estimates for the operator (f, g) 7→ g · Mα f . Since Iα is close to Mα , it is natural to ask ourselves how different the counterpart to fractional maximal operators is. Theorem 255. Let 0 < α < n, 1 < p ≤ p0 < ∞, 1 < q ≤ q0 < ∞ and 1 < r ≤ r0 < ∞. Suppose that q > r, and that

1 α > , p0 n

1 1 α 1 = + − , r0 q0 p0 n

1 α ≤ , q0 n r p = . r0 p0

(17.5)

Then kg · Iα f kMrr0 . kgkMqq0 kf kMpp0 for all f ∈ Mpp0 (Rn ) and g ∈ Mqq0 (Rn ). Unlike Theorem 254, an example will exclude the case where q = r. Proof Let ε > 0 be sufficiently small. Suppose that r0 (±ε) and r(±ε) satisfy α 1 α 1 > , ≤ , q > r(ε), p0 n q0 n and that 1 1 1 α±ε r(±ε) p = + − , = . r0 (±ε) q0 p0 n r0 (±ε) p0

Pointwise product

273

Then we have kg · Mα±ε f kMr0 (±ε) . kgkMqq0 kf kMpp0 . Since |Iα f | . r(±ε) p Mα+ε f · Mα−ε f thanks to Lemma 183 in the first book, we can use H¨older’s inequality and Theorem 254 to have the desired result. We have the following variant for Theorem 255. Theorem 256. Let 0 < α < n, 1 < p ≤ p0 < ∞, 1 < q0 < ∞ and 1 < r0 < ∞. Suppose that 1 α 1 < < q0 n p0 and that

1 1 α 1 = + − . r0 q0 p0 n

(17.6)

Then kg · Iα f kMrp0 . kgkMqp0 kf kMpp0 for all f ∈ Mpp0 (Rn ) and g ∈ Mqp0 (Rn ). Proof Let ε > 0 be sufficiently small. Suppose that r0 (±ε) and r(±ε) 1 1 1 α±ε (−ε) satisfy r0 > p0 , and that = . Then kg · + − r0 (±ε) q0 pp0 n Mα±ε f kMr0 (±ε) . kgkMqp0 kf kMpp0 . Since |Iα f | . Mα+ε f · Mα−ε f thanks p to Lemma 183 in the first book, we can use H¨older’s inequality to have the desired result. As is mentioned in the beginning, we have an application to our result to PDEs. Example 88. Let n ≥ 3. Suppose that 2 < q ≤ n. Then kg · f kL2 . kg · I1 [|∇f |]kL2 . kgkMnq k∇f k(L2 )n for all f ∈ Cc∞ (Rn ) and g ∈ L2 (Rn ) according to Theorem 255. In particular, k | · |−1 · f kL2 . k∇f k(L2 )n for all f ∈ Cc∞ (Rn ). This inequality shows that u 7→ −∆u+κ|·|−2 u is a positive operator if |κ|  1. Finally we will show by an example that q > r in Theorem 255 cannot be replaced by q = r. n . Then, for any c > 0 we can find Proposition 257. Let 1 < r ≤ r0 < α + n f, g ∈ M (R ) such that ∞ > kg · Iα f kMrr0 > ckgk αn kf kMrr0 > 0. Mr

Proof Let R > 1 solve (1 + R) increasing sequence {Fj }∞ j=1 given by ( Fj ≡

y+

j X

α− n r

2

n r

= 1. Consider the self-similar

) Rk ak : {ak }jk=1 ∈ {0, 1}j , y ∈ [0, 1]n

.

k=1 n

n

So we have R > 1 satisfying (1 + R)α− r 2 r = 1. If the estimate kg · n Iα f kMrr0 . kgk αn kf kMrr0 were true for all f ∈ Mrr0 (Rn ) and g ∈ Mrα (Rn ), Mr

274

Morrey Spaces

then the duality argument would give kIα [hχFj ]k 0

(Iα χFj )r −1 χ[0,(1+R)j ]n = (Iα χFj ) (Iα χFj )χ[0,(1+R)j ]n ∼

j X

1 r−1

r0

Hr00

. khk

r0

Hr00

. Let f ≡

χ[0,(1+R)j ]n . We notice

2kn (1 + R)kα−kn χF ((1 + R)−k ·)χ[0,(1+R)j ]n .

k=0 0

0

0

Since 2knr (1 + R)kαr −knr 2(j−k)n (1 + R)kn = 2jn , we have Z 0 Iα χFj (x)r dx ∼ j2jn . [0,(1+R)j ]n 1

jn

1

Meanwhile, |[0, (1 + R)j ]n | r − r0 kχFj kLr0 = 2jn (1 + R)jα− r0 , implying that kχFj k

jn

r0

Hr00

≤ 2jn (1 + R)jα− r0 . Let j, l ∈ N satisfy l ≤ j. We observe

|[0, (1 + R)l ]n |

1 r0

− r1

! r1

Z

0

Iα χFj (x)r dx

1

ln

∼ j r (1 + R) r0 −lα ,

[0,(1+R)l ]n jn

which implies that kf kMrr0 . j r (1 + R) r0 −jα . Putting our observations all together, we obtain the desired contradiction. 1

17.3.3

Exercises

Exercise 69. [252], [254, Theorem 2] Let 1 < r ≤ r0 < ∞ and r < K > 0. Show that if g ∈ M+ (Rn ) satisfies kg · Iα f kMrr0 ≤ Kkf kMrr0

(f ∈ Mpq (Rn )),

n α

and

(17.7)

n/α

then show that g ∈ Mr (Rn ). The characterization of g satisfying (17.7) is called the comparison theorem for Iα . Exercise 70. Let 0 < α < n, 1 < q ≤ p < ∞ and 1 < r ≤ r0 < ∞ satisfy 1 n r 0 n , r0 = pq and p1 − α r p,

X

λjm χ3Qjm

n m∈Z

Lq

− jn q

.2

X m∈Zn

|λjm |p

m∈Z

.

Lp

To see that Morrey spaces arise in the context of the pointwise multiplis ers, the homogeneous Besov space B˙ p1 (Rn ) is enough. Since the pointwise

Pointwise product

277

multipliers can give many other generalizations, we consider microlocal Besov spaces. We will adopt [230, Theorem 1] with p = 1, s = 0, K = [α2 + 1] and L = [α1 + 1] as a definition of microlocal Besov spaces. To formulate the definition of Besov spaces, we will employ atoms again. n Definition 44. Let w ≡ {wj }∞ j=−∞ be a positive sequence, and let F (R ) be a Banach function lattice satisfying

kχQjm kF . wj |Qj0 | (j ∈ Z, m ∈ Zn ) and 2−α1 wj ≤ wj+1 ≤ 2α2 wj

(j ∈ Z),

where α1 and α2 are fixed parameters. One defines the microlocal homogew neous Besov space B˙ 11 (Rn ) by the set of all f ∈ F (Rn ) for which it can be written as ∞ X X f (x) = λjm ajm (x) j=−∞ m∈Zn

for almost every x ∈ Rn , where for all j ∈ Z and m ∈ Zn , we have a collection ⊥ {ajm }j∈Z,m∈Zn ⊂ C ∞ (Rn ) ∩ P[α (Rn ) and a collection {λjm }j∈Z,m∈Zn of 1 +1] complex constants satisfying ! ∞ X X α j|α| −jn |∂ ajm | ≤ 2 χ3Qjm , 2 wj |λjm | < ∞ m∈Zn

j=−∞

for all multi-indexes α with |α| ≤ [α2 + 1]. Then the microlocal Besov norm kf kB˙ w is defined as the infimum of 11

∞ X j=−∞

! −jn

2

wj

X

|λjm |

m∈Zn

where Λ = {λjm }j∈Z,m∈Zn moves over all possible expressions. In the above, the convergence in S 0 (Rn ) is guaranteed thanks to the condition on Λ. Each function aij is called the atom.

17.4.2

s The pointwise multipliers from B˙ p1 (Rn ) to Lp (Rn ) n with 0 < s ≤ p

s Although B˙ p1 (Rn ) is not a Banach function space, we can still define the s pointwise multiplier space PWM(B˙ p1 (Rn ), Lp (Rn )) similar to the definition of n n PWM(E(R ), F (R )) as we did for Banach function spaces E(Rn ) and F (Rn ). We show that Morrey spaces arise naturally when we consider the pointwise n s multipliers from B˙ p1 (Rn ) to Lp (Rn ) with 0 < s ≤ . p

278

Morrey Spaces

Theorem 258. Let 1 ≤ p < ∞ and 0 < s ≤ n norms PWM(B˙ s (Rn ), Lp (Rn )) ≈ Mps (Rn ).

n p.

Then with equivalence of

p1

s Proof Let f ∈ PWM(B˙ p1 (Rn ), Lp (Rn )). From Example 89, n

kf kLp (Qjm ) . kf exp(−|2j · −m|2 )kLp . 2js−j p kf kPWM(B˙ s

p1 ,L

p)

.

n

Consequently, f ∈ Mps (Rn ). To show the opposite estimate, it suffices to show that





X

X



js λjm ajm . 2 kf k ns λjm χQjm

f Mp

n n m∈Z

m∈Z

Lp

Lp

for all sequences {ajm }j∈Z,m∈Zn of C ∞ (Rn )-functions satisfying |∂ α ajm | ≤ 2j|α| χ3Qjm with |α| ≤ [s + 1]; once this is achieved, we have only to add this estimate over j ∈ Z. We calculate

! p1

X

X

p p λjm ajm . |λjm | kf χ3Qjm kLp

f

n n m∈Z

Lp

m∈Z

! p1 ≤ sup kf χ3Qjm kLp m∈Zn js

p

|λjm |

m∈Zn

. 2 kf k

n

Mps

17.4.3

X



X

λjm χQjm

n m∈Z

.

Lp

Ho’s vector-valued Morrey spaces and pointwise multiplier spaces

Our goal of Section 17.4.3 is to show that Ho’s vector-valued Morrey spaces can be realized as a special case of pointwise multiplier spaces. In Section 17.4.3, let E(Rn ) be a Banach lattice be such that kf (· − x)kE = kf kE for all f ∈ E(Rn ) and x ∈ Rn . A direct consequence is that kχ3Q kE . kχQ kE for all cubes Q. These assumptions are postulated so as to simplify matters. Nevertheless, as our examples show, we have many function spaces satisfying the above assumption. Here we define Ho’s vector-valued Morrey spaces. Definition 45. Let E(Rn ) be a Banach lattice, and let ϕ : (0, ∞) → (0, ∞) n be a function. Then the E-based vector-valued Morrey space Mϕ E (R ) is the 0 n set of all f ∈ L (R ) for which kf kMϕE ≡ sup ϕ(`(Q)) Q∈Q

n

kχQ f kE kχQ kE

n is finite. If ϕ(t) = t p , t > 0, then we write MpE (Rn ) instead of Mϕ E (R ). One also calls MpE (Rn ) the Ho vector-valued Morrey space.

Pointwise product

279

From our assumption, we learn that the following dyadic Morrey norm is equivalent to the original norm   1 1 p ≡ sup |Q| p kf kMpE ∼ kf kD kf χ k Q E . ME kχQ kE Q∈D We note that vector-valued Morrey spaces realize mixed Morrey spaces [360], Morrey–Lorentz spaces [379] and Orlicz–Morrey spaces (of the third kind) [84]. Theorem 259. Let E(Rn ) and F (Rn ) be Banach lattices such that 1

kχQjm kF . kχQjm kE |Qj0 |− p

(j ∈ Z, m ∈ Zn ).

1 w (Rn ) is continuously embedSet wj ≡ kχQj0 kE |Qj0 |−1− p for j ∈ Z. Then B˙ 11 n ded into F (R ) and

w PWM(B˙ 11 (Rn ), E(Rn )) ≈ MpE (Rn )

with equivalence of norms. We will establish the former half of Theorem 259 to make sure that w PWM(B˙ 11 (Rn ), X(Rn )) is well defined. Lemma 260. Let F (Rn ) be a Banach lattice satisfying kχQjm kF . 2−jn wj

(j ∈ Z, m ∈ Zn ).

w Then B˙ 11 (Rn ) ,→ F (Rn ). w Proof Let f ∈ B˙ 11 (Rn ) be a decomposition as in Definition 44. By the triangle inequality, we have



X ∞ X X

∞ X

kλjm ajm kF λjm ajm ≤

j=−∞ m∈Zn

j=−∞ m∈Zn F

.

∞ X

X

λjm χ3Qjm F

j=−∞ m∈Zn

.

∞ X j=−∞

! 2

−jn

wj

X

|λjm | .

m∈Zn

We prove the latter half of Theorem 259. Let f ∈ MpE (Rn ). Also let g ∈ w B˙ 11 (Rn ), so that there exists a collection {ajm }j∈Z,m∈Zn of C ∞ -functions and a collection {λjm }j∈Z,m∈Zn of complex constants satisfying ! ! ∞ ∞ X X X X −jn g= λjm ajm , 2 wj |λjm | < ∞ j=−∞

m∈Zn

j=−∞

m∈Zn

280

Morrey Spaces

and |∂ α ajm | ≤ χ3Qjm for all j ∈ Z and m ∈ Zn . Then kf ajm kE ≤ kf χ3Qjm kE 1

= |Qjm | p

1 1 kf χ3Qjm kE kχ3Qj0 kE |Qjm |− p kχ3Qjm kE 1

. kf kMpE kχQj0 kE |Qj0 |− p . Consequently, kf · gkE ≤

∞ X

X

j=−∞

m∈Zn

!

. kf kMpE

|λjm |kf ajm kE

∞ X

! kχQj0 kE |Qj0 |

1 −p

X

|λjm | .

m∈Zn

j=−∞

If we take the infimum over all possible expressions of g, we obtain kf · gkE . kf kMpE kgkB˙ w . 11

w Thus, f ∈ PWM(B˙ 11 (Rn ), E(Rn )) and kf kPWM(B˙ w ,E) . kf kMpE . 11 w Conversely, we let f ∈ PWM(B˙ 11 (Rn ), E(Rn )). Choose κ ∈ Cc∞ (Rn ) ∩ ⊥ P[α (Rn ) so that χQ00 ≤ κ ≤ χRn \3Q00 for each j ∈ Z and m ∈ Zn . Then 1 +1] 1

since 2−jn wj kf kPWM(B˙ w ,E) = kχQj0 kE |Qj0 |− p kf kPWM(B˙ w ,E) , we have 11

11

kf χQjm kE ≤ kf κjm kE ≤ kf kPWM(B˙ w ,E) kκjm kB˙ w 11

. kχQj0 kE |Qj0 |

11

1 −p

kf kPWM(B˙ w ,E) . 11

MpE (Rn ).

Thus, f ∈ As an example of E(Rn ), we first take up the case of E(Rn ) = LΦ (Rn ). Let Φ be a Young function and let p > 1. Assume that there exists a Young 1 function Ψ satisfying Ψ(t) ≡ Φ(t)t− p for t > 0. For example, if q0 , q1 and p satisfy 1 < q0 < q1 < p, then Φ(t) = max(tq0 , tq1 ) for t ≥ 0 satisfies the requirement. In fact, if we define u0 and u1 by 1 < u0 < u1 < ∞,

1 1 1 = − , u0 q0 p

1 1 1 = − , u1 q1 p 1

then Ψ(t) = max(tu0 , tu1 ) satisfies Ψ(t) ≡ Φ(t)t− p for t > 0. As a next example of E(Rn ), we take up the case of E(Rn ) = Lq,r (Rn ). We define kf kMpLq,r ≡

sup (x,r)∈Rn+1 +

1

1

|Q(x, r)| p − q kf χQ(x,r) kLq,r

Pointwise product

281

for 0 < q ≤ p < ∞, 0 < r ≤ ∞. The Morrey–Lorentz space MpLq,r (Rn ) denotes the set of all f ∈ L0 (Rn ) for which kf kMpLq,r is finite. This space is introduced by Ragusa in [379]. However, note that the notation is different from the one used in [379]. In her paper, to denote Morrey–Lorentz spaces, she used MpLq,r (Rn ). However, this leads to confusion since we defined the global Morrey-type space Mpqθ (Rn ) in Chapter 16. Theorem 261. Let 1 < q ≤ p < ∞, 1 ≤ r, v ≤ ∞, 1 < u ≤ ∞. Assume that 1 1 1 = + . q p u n

n

p+ 0 Then B˙ 11 q (Rn ) is continuously embedded into Lu,v (Rn ) and n

n

p+ 0 PWM(B˙ 11 q (Rn ), Lq,r (Rn )) ≈ MpLq,r (Rn )

with equivalence of norms. In particular, n n p + q0

PWM(B˙ 11

(Rn ), WLq (Rn )) ≈ WMpq (Rn )

with equivalence of norms. We move on to the case of mixed Lebesgue spaces. Let q = (q1 , . . . , qn ) ∈ (0, ∞]n and p ∈ (0, ∞] satisfy n X 1 n ≥ . q p j=1 j

We define the mixed Morrey space Mpq (Rn ) as a set of all measurable functions f satisfying the following norm kf kMpq is finite: !   n 1   p1 − n1 P qj j=1 kf χQ kq : Q is a cube in Rn . kf kMpq ≡ sup |Q|   In particular, let p satisfy

n p

=

n P j=1

1 qj ,

so that Mpq (Rn ) = Lq (Rn ).

Keeping this remark in mind, we apply Theorem 259 for this space.   n Theorem 262. Let 1 ≤ q1 , . . . , qn < ∞. Then PWM B˙ 11 (Rn ), Lq (Rn ) = Mpq (Rn ).

17.4.4

Exercises

Exercise 71. Let 0 < q1 , q2 , . . . , qm < ∞. Suppose that 0 < p < ∞ satisfies m

1 X 1 ≤ . p j=1 qj

282

Morrey Spaces

Then show that k(f1 , f2 , . . . , fm )kMp(q

1 ,q2 ,...,qm )



≤ ⊗m j=1 fj Mp

q ~

for all f1 , f2 , . . . , fm ∈ L0 (Rn ), where the left-hand side denotes the multiMorrey norm of the m-dimensional vector field (f1 , f2 , . . . , fm ). Exercise 72. Let 1 ≤ p < ∞, and let 0 < s ≤ s PWM(Bp1 (Rn ), Lp (Rn ))

n s

n p.

Then establish that

n

≈ mp (R ) with equivalence of norms, where the right-hand side denotes the small Morrey space. Exercise 73. [189] Let 1 ≤ p < ∞ and 1 ≤ q1 , . . . , qn < ∞ satisfy

n P j=1

1 qj

≥

n p.

Then show that MpMpq (Rn ) = Mpq (Rn ).

17.5

Notes

Section 17.1 General remarks and textbooks in Section 17.1 See the exhaustive textbook [304] for Sobolev pointwise multipliers. Section 17.1.1 Lozanovski calculated the K¨othe dual of the Calder´on product and as a byproduct Lozanovski decomposed L2 into the product of a Banach lattice and its dual; see [284, Theorem 2] for Theorems 217 and 218. Shestakov clarified the relation between the Calder´on product of Banach lattices and the one for their closed subspaces; see [428] for Theorem 219. Section 17.1.2 H¨ older’s inequality for Morrey spaces, Proposition 220, can be found in [338] (general dimension) and [365, Lemma 11] (n = 3). See also [213] for H¨ older’s inequality for Morrey spaces.

Section 17.2 General remarks and textbooks in Section 17.2 Since pointwise multipliers appear in many branches of mathematics, it does not seem to make sense to discuss the priority. However, let us dare say that the paper [296] is a self-contained introduction to this field.

Pointwise product

283

Section 17.2.1 We followed [296, 339] in Section 17.2.1. See [296, Proposition 1] for Example 75, which summarizes H¨older’s inequality. Maligranda and Persson developed Example 78(1) there. We refer to [296, Theorem 1] for Proposition 225(1), asserting that PWM(E(Rn ), F (Rn )) is a Banach function space. Proposition 225(3), dealing with PWM(E(Rn ), E(Rn )), is [296, Theorem 1]. Section 17.2.2 The main contributors are Nakai and Lemari´e-Riuesset. Nakai proved Theorem 223 in [338] See Nakai’s works [339, Corollary 2.4] for Propositions 226, 228, 227, 232 and 233. Later on Lemari´e-Riuesset reinforced them; see [254, Theorem 1] for Propositions 226, 227, 228, 229 and 230. In particular, we borrowed the idea of mollification from [254, p. 747] to prove Proposition 227 and borrowed (17.5) in Proposition 232 from [254, p. 747]. As an application of the closed subspaces, we can discuss the compactness of the multiplier. See [70, Corollary 5.4]. Section 17.2.3 It seems that there is no literature on the pointwise multiplier spaces of local Morrey spaces. Section 17.2.4 Nakai established the theory of the pointwise multiplier spaces; see [339, Theorem 2.1], [339, Lemma 2.3] and [339, Theorem 2.2], for Example 81 and Propositions 241 and 242, respectively. See [338, Theorem 1.1] and [339, Theorem 2.1] for Theorem 243. See [339, Theorem 1.1] for Example 82 and Theorem 244. Section 17.2.5 We followed [351]; we refer to [351, Theorem 1], [351, Proposition 5.1] and [351, Proposition 5.3] for Theorem 247, Propositions 250 and 251, respectively. See [351, Theorem 2] for Theorems 248 and 249. Lin and Yang worked in the setting of RD-spaces (reverse doubling). See [260] for the pointwise multipliers for localized Morrey-Campanato spaces on RD-spaces. We refer to [515] for the characterization of the pointwise multipliers in BMO with Gauss measure.

Section 17.3 General remarks and textbooks in Section 17.3 Many authors discuessed Olsen’s inequality from the viewpoint of the trace inequality; see [6, Chapter 10] and [303, Chapter 8]. See the textbook [303] for the multiplier spaces. We also refer to [388, §10.3.7].

284

Morrey Spaces

Olsen’s inequality goes back to [365]. Olsen’s inequality is called the Fefferman inequality when Mpq (Rn ) = L2 (Rn ); see [114]. Section 17.3.1 Sawano, Sugano and Tanaka considered Olsen’s inequality for generalized fractional maximal operator [415, Theorem 1.7] (Theorem 254). Gunawan and Tanaka considered Olsen’s inequality for fractional maximal operators in the setting of measures; see [467, Theorem 1.2]. Section 17.3.2 Here is a precise result by Olsen. Theorem 263. [365, Theorem 2] Let 0 < α < n, 1 < p ≤ p0 < ∞, 1 < q ≤ q0 < ∞ and 1 < r ≤ r0 < ∞. Suppose that q > r, and that

α 1 > , p0 n

1 α ≤ , q0 n

1 1 1 1 α 1 1 α = + − , = + − . r0 q0 p0 n r q0 p n

Then kg · Iα f kMrr0 . kgkMqq0 kf kMpp0 , for all f ∈ Mpp0 (Rn ) and g ∈ Mqq0 (Rn ). There is a vast amount of literature on Olsen inequalities; see [105, 417, 415, 416, 450, 462, 463, 465, 487] for theoretical aspects and [66, 112, 119, 120, 121, 480]. See [70, Theorem 5.1] for a different type of Olsen’s inequality. In particular, we can find a prototype of Theorem 256 in [70, Theorem 5.1]. An example showing that the condition on the local integrability parameter of g is sharp can be found in [415, Proposition 4.1] and [254, Theorem 2]; see Proposition 257. We considered the property of the function g · ∇f . The idea of using fractional integral operators I1 is similar to [331, Theorem 9]. We can characterize the function g for which kg · Iα f kMpq ≤ Ckf kMpq is true for 0 < α < n and 1 < q ≤ p < ∞ by the use of wavelets; see [522, Theorem 1]. See [116, Lemma 2.3] for Theorem 256.

Section 17.4 General remarks and textbooks in Section 17.4 See standard textbooks [25, Chapter 5, §4], [136, Chapter 2], [234], [405, §2.1], [458, Chapter V, §5] and [481] for non-homogeneous Besov spaces. We refer to standard textbooks [125], [136, Chapter 2], [234], [288], [459] and [405, §2.4] for homogeneous Besov spaces. In particular, [481] deals exhaustively with these spaces.

Pointwise product

285

See standard textbooks [7, Chapter 4], [125], [136, Chapter 2], [234], [288] and [405, §4.1] for atomic decomposition of Besov spaces. In particular, [405, 481] deals exhaustively with this topic. Section 17.4.1 Besov spaces go back to the paper by Besov [29]. Section 17.4.2 Since the coefficients of partial differential equations are often functions, the pointwise multiplier spaces are quite useful when we consider partial differential equations. See [253, Lemma 6] for Theorem 258. See [280, 537, 541, 542] for applications of Theorem 258 to partial differential equations. Morrey spaces can arise from other contexts; see the works by Adams [1], Maz’ya [302], Dahlberg [81] and Kerman and Sawyer [232] as well as the textbook [305] by Maz’ya and Shaposhnikova for the characterization of Morrey spaces using capacity. Lemari´e-Rieusset obtained Theorem 258 for n = 3 and p = 2 [253, Lemma 6]. Section 17.4.3 We followed [189] in Section 17.4.3. We can consider the grand Lebesgue spaces as an analogy above. Let (X, B, µ) be a finite measure space, and let 1 < p < ∞. Then the grand Lebesgue space Lp) (µ) is the set of all f ∈ L0 (µ) for which kf kLp) (µ) ≡ sup

1

ε p−ε kf kLp−ε (µ) is finite. See the textbook [239, Chapters 14, 15 and

1 0, K(t, x) ∼ min(1, t)kxkX . If X0 = X1 = X is a Banach space, then for x ∈ X and t > 0, K(t, x) = min(1, t)kxkX . Lemma 264. Let X = (X0 , X1 ) be a compatible couple of quasi-Banach t spaces. Then for any x ∈ X0 + X1 and 0 < s < t < ∞, K(t, x) ≤ K(s, x). s Proof If we write out the left-hand side, then we obtain (kx0 kX0 + tkx1 kX1 ) s  t = inf kx0 kX0 + skx1 kX1 s x=x0 +x1 t t inf (kx0 kX0 + skx1 kX1 ) ≤ s x=x0 +x1 t = K(s, x). s

K(t, x) =

inf

x=x0 +x1

This completes the proof. Clearly, K(·, x) ∈ M↑ (0, ∞) for all x ∈ X0 + X1 , so that this is measurable. More quantitatively, we have: Lemma 265. Let X = (X0 , X1 ) be a compatible couple of quasi-Banach spaces. Then for any x ∈ X0 + X1 K(t, x) is a concave function on (0, ∞). That is, for any t1 , t2 > 0 and any α1 , α2 ≥ 0 satisfying α1 + α2 = 1 K(α1 t1 + α2 t2 , x) ≥ α1 K(t1 , x) + α2 K(t2 , x). Proof Abbreviate to

inf

x=x0 +x1

· · · the infimum of · · · over the decomposi-

tion of x into the sum x = x0 + x1 with x0 ∈ X0 and x1 ∈ X1 . We calculate

Real interpolation of Morrey spaces

289

K(α1 t1 + α2 t2 , x) = =

inf

(kx0 kX0 + (α1 t1 + α2 t2 )kx1 kX1 )

inf

α1 (kx0 kX0 + t1 kx1 kX1 ) + α2 (kx0 kX0 + t2 kx1 kX1 )

x=x0 +x1 x=x0 +x1

≥ α1

inf

x=x0 +x1

(kx0 kX0 + t1 kx1 kX1 ) + α2

inf

x=x0 +x1

(kx0 kX0 + t2 kx1 kX1 )

= α1 K(t1 , x) + α2 K(t2 , x). This completes the proof. We considered the space Φλ,q (0, ∞) of all ϕ ∈ M+ (0, ∞) for which ∞

Z kϕkΦλ,q (0,∞) =

(t 0

−λ

ϕ(t))

q dt

t

 q1 0

is finite. More precisely, the space X θ,q is called the intermediate space between X0 and X1 or the interpolation space between X0 and X1 . For θ and q above, the operation X = (X0 , X1 ) 7→ X θ,q is called the real interpolation functor. The method of creating interpolation spaces via the K-functional is called the K-method. In other words, define kxkX θ,q ≡ kK(·, x)kΦθ,q . Example 91. If X0 = X1 = X is a quasi-Banach space, then X θ,q ≈ X with equivalence of norms. We give another example of real interpolation, which motivates the definition of the weak function spaces. For a quasi-Banach lattice E(µ) over a mea1 sure space (X, B, µ) and a > 0, we define E a (µ) = {f ∈ L0 (µ) : |f | a ∈ E(µ)}. 1 The norm of E a (µ) is given by kf kE a (µ) ≡ (k |f | a kE(µ) )a .

290

Morrey Spaces

Theorem 266. Let E(µ) be a Banach function space over a measure space 1 (X, B, µ). Then [E(µ), L∞ (µ)]θ,∞ = WE 1−θ (µ) with equivalence of norms. 1

Proof Let f ∈ WE 1−θ (µ). Then K(t, f ; E(µ), L∞ (µ)) ≤

∞ X

2l+1 K(t, χ(2l ,∞] (|f |); E(µ), L∞ (µ)).

l=−∞

Since K(t, χ(2l ,∞] (|f |); E(µ), L∞ (µ)) ≤ min(kχ(2l ,∞] (|f |)kE(µ) , t), we obtain ∞ P 2l+1 min(kχ(2l ,∞] (|f |)kE(µ) , t). Since f ∈ K(t, f ; E(µ), L∞ (µ)) . l=−∞ 1

WE 1−θ (µ), we have kχ(2l ,∞] (|f |)kE(µ) ≤ (2−l kf k

1

WE

1 1−θ

(µ)

) 1−θ .

If we insert this relation, then we obtain t−θ K(t, f ; E(µ), L∞ (µ)) . kf k

1

WE 1−θ (µ)

for any t > 0. If f ∈ (E(µ), L∞ (µ))θ,∞ ∩M+ (Rn ), then we claim that f χ(y,∞] (|f |) ∈ E(µ) and that kf χ(y,∞] (|f |)kE(µ) ≤ K(kχ(y,∞] (|f |)kE(µ) , f ; E(µ), L∞ (µ)) In fact, for any t > 0, we have a decomposition f = g + h with g ∈ E(µ) and h ∈ L∞ (µ) such that K(t, f ; E(µ), L∞ (µ)) ≤ kgkE(µ) + tkhkL∞ (µ) ≤ 2K(t, f ; E(µ), L∞ (µ)). Note that khkL∞ (µ) ≤ 2t−1 K(t, f ; E(µ), L∞ (µ)) → 0

(t → ∞).

Consequently, for any y > 0, there exists a decomposition f = g + h with g ∈ E(µ) and h ∈ L∞ (µ) such that khkL∞ (µ) ≤ 12 y. Then, since |f − h| = |g|, by the triangle inequality, y χ(y,∞] (|f |) ≤ (|f | − khkL∞ (µ) )χ(y,∞] (|f |) 2 ≤ |f − h|χ(y,∞] (|f |) = |g|χ(y,∞] (|f |) ∈ E(µ). Thus, χ(y,∞] (|f |) ∈ E(µ) and |f |χ(y,∞] (|f |) ≤ |g|χ(y,∞] (|f |) + yχ(y,∞] (|f |) ∈ E(µ).

Real interpolation of Morrey spaces

291

Finally, if we consider any decomposition f = f0 + f1 with f0 ∈ E(µ) and f1 ∈ L∞ (µ), then we obtain kf χ(y,∞] (|f |)kE(µ) ≤ kf0 χ(y,∞] (|f |)kE(µ) + kf1 χ(y,∞] (|f |)kE(µ) ≤ kf0 kE(µ) + kf1 kL∞ (µ) kχ(y,∞] (|f |)kE(µ) ≤ K(kχ(y,∞] (|f |)kE(µ) , f ; E(µ), L∞ (µ)). Consequently, ykχ(y,∞] (|f |)kE(µ) ≤ kf χ(y,∞] (|f |)kE(µ) ≤ K(kχ(y,∞] (|f |)kE(µ) , f ; E(µ), L∞ (µ)) ≤ (kχ(y,∞] (|f |)kE(µ) )θ kf k(E(µ),L∞ (µ))θ,∞ . It remains to arrange this inequality. Before we go further, a helpful remark on the range of the parameters may be in order. Remark 12. This definition does not make sense if θ = 0 of θ ≥ 1 for q < ∞ or if θ > 1 for q = ∞. More precisely, for θ = 0 or θ ≥ 1 and q < ∞ if K(t, x) > 0 for t > 0, kxkX θ,q = ∞ for any x ∈ X0 + X1 . Indeed, if θ = 0, then ∞

Z kxkX θ,q =

K(t, x)q

0

dt t

 q1

Z

∞

≥ K(1, x) 1

dt t

 q1 = ∞.

Let θ ≥ 1. Observe that K(t, x) ≥ tK(1, x) for all t ∈ (0, 1). Therefore, Z kxkX θ,q ≥

∞

(t−θ K(t, x))q

0

Z ≥ K(1, x)

dt t

1 q(1−θ)−1

t

 q1  q1 dt

0

= ∞. If q = ∞ and θ > 1, then kxkX θ,∞ ≥ sup t−θ K(t, x) ≥ sup t1−θ K(1, x) = ∞. 0 0, 0 < η < 1 and 0 < r ≤ ∞. Define ρ, θ and q by ρ ≡ (1 − η)ρ0 + ηρ1 ,

θ≡

(18.2)

ρ1 η , ρ

(18.3)

q ≡ ρr.

(18.4)

0 1 Then (E0 (µ)ρpower , E1 (µ)ρpower )η,r ≈ (E(µ)θ,q )ρpower with equivalence of quasinorms.

We can calculate the J-functional of the powered quasi-Banach couple 1 0 ) as follows: , E1 (µ)ρpower (E0 (µ)ρpower Lemma 271. Let ρ0 , ρ1 > 0. Then for any x ∈ E0 (µ) + E1 (µ) and t > 0, 0 1 J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ) = J(t, x; E(µ))ρ0 ,

where s ≡ tρ1 J(t, x; E(µ))ρ0 −ρ1 . Proof Let x ∈ E0 (µ) + E1 (µ). 1 0 ) ≥ J(t, x; E(µ))ρ0 . For any x0 ∈ , E1 (µ)ρpower (1) We prove J(s, x; E0 (µ)ρpower E0 (µ), x1 ∈ E1 (µ) satisfying x = x0 + x1

J(t, x; E(µ)) ≤ max(kx0 kE0 (µ) , tkx1 kE1 (µ) ), or equivalently,  1 ≤ max

kx0 kE0 (µ)

,

tkx1 kE1 (µ)

J(t, x; E(µ)) J(t, x; E(µ))

 .

that is, at least one of kx0 kE0 (µ) tkx1 kE1 (µ) , J(t, x; E(µ)) J(t, x; E(µ)) is greater than or equal to 1. Therefore,  ρ0  ρ1  kx0 kE0 (µ) tkx1 kE1 (µ) 1 ≤ max , J(t, x; E(µ)) J(t, x; E(µ))   max kx0 kE0 (µ)ρpower , tρ1 J(t, x; E(µ))ρ0 −ρ1 kx1 kE1 (µ)ρpower 0 1 = . J(t, x; E(µ))ρ0

Real interpolation of Morrey spaces

295

Thus J(t, x; E(µ))ρ0 o n  ≤ inf max kx0 kE0 (µ)ρpower , tρ1 J(t, x; E(µ))ρ0 −ρ1 kx1 kE1 (µ)ρpower 1 0 0 1 = J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ).

(18.5)

where inf in (18.5) runs over all expressions x = x0 + x1 satisfying x0 ∈ E0 (µ) and x1 ∈ E1 (µ). 0 1 (2) We prove J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ) ≤ J(t, x; E(µ))ρ0 . For given κ > 1, let x0κ ∈ E0 (µ) and x1κ ∈ E1 (µ) satisfy x = x0κ + x1κ . Then

kx0κ kE0 (µ) , tkx1κ kE1 (µ) ≤ max(kx0κ kE0 (µ) , tkx1κ kE1 (µ) ) ≤ κJ(t, x; E(µ)). Hence, kx0κ kE0 (µ)ρpower ≤ κρ0 J(t, x; E(µ))ρ0 , 0 tρ1 kx1κ kE1 (µ)ρpower ≤ κρ1 J(t, x; E(µ))ρ1 . 1 Write ρ ≡ max(ρ0 , ρ1 ). Then  ρ0  ρ1  kx1κ kE1 (µ) kx0κ kE0 (µ) , ≤ κρ . max J(t, x; E(µ)) J(t, x; E(µ)) Thus, we have max(kx0κ kE0 (µ)ρpower , skx1κ kE1 (µ)ρpower ) ≤ κρ J(t, x; E(µ))ρ0 . 0 1 Hence 0 1 J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ) ≤ κρ J(t, x; E(µ))ρ0

for all κ > 0. By letting κ ↓ 0, we obtain 0 1 J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ) ≤ J(t, x; E(µ))ρ0 ,

as desired. We now prove Theorem 270. (1) Let us consider the case q = ∞; in this case r = ∞. From (18.1), we have −η 0 1 ρ1 kxk(E0 (µ)ρpower K(s, x; E0 (µ)ρpower , E1 (µ)ρpower ) 0 ,E1 (µ)power )θ,∞ = sup s s>0

0 1 ≤ 2 sup s−η J(s, x; E0 (µ)ρpower , E1 (µ)ρpower ).

s>0

296

Morrey Spaces Due to Lemma 271, we have ρ1 kxk(E0 (µ)ρpower 0 ,E1 (µ)power )θ,∞

≤ 2 sup t>0

0 1 J(tρ1 J(t, x; E(µ))ρ0 −ρ1 , x; E0 (µ)ρpower , E1 (µ)ρpower )

(tρ1 J(t, x; E(µ))ρ0 −ρ1 )η

.

From Lemma 271, we conclude ρ1 kxk(E0 (µ)ρpower 0 ,E1 (µ)power )θ,∞ ≤ 2 sup

t>0

J(t, x; E(µ))ρ0 (tρ1 J(t, x; E(µ))ρ0 −ρ1 )η

= 2 sup t−ρθ J(t, x; E(µ))ρ t>0

≤ 2 sup t−ρθ K(t, x; E(µ))ρ t>0

= 2kxkρ(E0 (µ),E1 (µ))θ,∞ . In a similar way, it can be proven that ρ ρ1 kxk(E0 (µ)ρpower 0 ,E1 (µ)power )θ,∞ ≥ (kxk(E0 (µ),E1 (µ))θ,∞ )

(18.6)

(2) Let q < ∞ hence r < ∞. We may assume ρ0 > ρ1 by symmetry. Then ρ1 kxk(E0 (µ)ρpower 0 ,E1 (µ)power )η,r rpower Z ∞ ds 1 0 )r , E1 (µ)ρpower = s−ηr K(s, x; E0 (µ)ρpower s 0 Z ∞ ds 1 0 )r 1+ηr ≤ 2r , E1 (µ)ρpower J(s, x; E0 (µ)ρpower s 0

from Lemma 271. We decompose the integral as follows: Z ∞ ds 1 0 s−ηr J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r s 0 (k+1)ρ1 k+1 ρ0 −ρ1 Z ∞ 2 J(2 ,x;E(µ)) 0 1 X J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r ds = . s1+ηr 2kρ1 J(2k ,x;E(µ))ρ0 −ρ1 k=−∞

Since 2ρ1 ≤ Z

∞

0

.

2(k+1)ρ1 J(2k+1 , x; E(µ))ρ0 −ρ1 ≤ 2ρ0 , we have 2kρ1 J(2k , x; E(µ))ρ0 −ρ1

ds 0 1 s−ηr J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r s ( )r ∞ kρ1 k ρ0 −ρ1 0 1 X J(2 J(2 , x; E(µ)) , x; E0 (µ)ρpower , E1 (µ)ρpower )

k=−∞

(2kρ1 J(2k , x; E(µ))ρ0 −ρ1 )η

.

Real interpolation of Morrey spaces

297

Thanks to Lemma 271, we have Z ∞ ds 0 1 s−ηr J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r s 0 r ∞  ρ0 k X J(2 , x; E(µ)) . . kρ 1 (2 J(2k , x; E(µ))ρ0 −ρ1 )η k=−∞ By using (18.2), (18.3) and (18.4), we can simplify the right-hand side to obtain Z ∞ ∞ X ds J(2k , x; E(µ))ρr 0 1 s−ηr J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r . . s 2θqr 0 k=−∞

Hence, it follows that ρ1 kxk(E0 (µ)ρpower 0 ,E1 (µ)power )η,r rpower Z ∞ ds 0 1 . s−θr J(s, x; E0 (µ)ρpower , E1 (µ)ρpower )r s 0 . kxk(E(µ)θ,q )rρ power ρ1 and we can prove kxk(E(µ)θ,q )ρpower . kxk(E0 (µ)ρpower 0 ,E1 (µ)power )η,r similarly, as was to be shown.

18.1.3

Exercises

Exercise 74. Let (X0 , X1 ) be a compatible couple of quasi-Banach spaces. Write out the norm kxk(X0 ,X1 )1/2,∞ in full for x ∈ (X0 , X1 )1/2,∞ . Exercise 75. Verify Remark 12 by using Lemma 28 in the first book.

18.2

Description of interpolation spaces

We are now interested in explicit formulas of real interpolation. As shown, the notion of generalized local Morrey spaces will play a key role. Section 18.2.1 gives a general theorem, which is applied to Section 18.2.2 for a more concrete setting.

18.2.1

Weighted Lebesgue space of monotonic functions

We can considerably easily handle the real interpolation of monotone functions. Let 0 < q ≤ ∞ and λ ≥ 0. We considered the space Φλ,q (0, ∞) in the

298

Morrey Spaces

first book. We are now going to define important operators for s > 0. Write Φ↑λ,q (0, ∞) ≡ Φ↑λ,q (0, ∞) ∩ M↑ (0, ∞). For ϕ ∈ M↑ (0, ∞), let As ϕ ≡ ϕ(min(·, s)),

Bs ϕ ≡ χ(s,∞) · ϕ.

(18.1)

For any ϕ ∈ M↑ (0, ∞) note that Bs ϕ = ϕχ(s,∞) and that (I − As )ϕ(t) ≤ Bs ϕ(t) = (I − As )ϕ(t) + ϕ(s)

(18.2)

for t > 0, so ϕ(t) = As ϕ(t) + (I − As )ϕ(t) ≤ As ϕ(t) + Bs ϕ(t). The spaces Φ↑λ0 ,q0 (0, ∞) and Φ↑λ1 ,q1 (0, ∞) interpolate well as the following theorem implies: Furthermore, we can calculate the K-functional. We are interested in the interpolation of Φ↑λ,p (0, ∞). Since Φ↑λ,p (0, ∞) is not a normed space, we modify the definition of the K-functional as follows: Definition 50. Let λ0 , λ1 ∈ R, and let 0 < q0 , q1 ≤ ∞. One defines Φ↑λ0 ,q0 (0, ∞) + Φ↑λ1 ,q1 (0, ∞) to be the set of all increasing functions that can be written as the sum of functions in Φ↑λ0 ,q0 (0, ∞) and Φ↑λ1 ,q1 (0, ∞). For f ∈ Φ↑λ0 ,q0 (0, ∞) + Φ↑λ1 ,q1 (0, ∞) one also defines K(t, f ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)) ≡

ϕ0 ∈Φ↑ λ

inf 0 ,q0

(kϕ0 kΦλ0 ,q0 (0,∞) + tkϕ1 kΦλ1 ,q1 (0,∞) ).

(0,∞),

ϕ1 ∈Φ↑ λ1 ,q1 (0,∞), f =ϕ0 +ϕ1

For 0 < q ≤ ∞ and 0 < θ < 1, the space (Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞))θ,q is defined as the set of all f ∈ Φ↑λ0 ,q0 (0, ∞) + Φ↑λ1 ,q1 (0, ∞) for which kf k(Φ↑

↑ λ0 ,q0 (0,∞),Φλ1 ,q1 (0,∞))θ,q

≡ kK(·, f ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)kΦθ,q

is finite. We start with an elementary estimate: Lemma 272. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 , λ1 > 0 and λ0 6= λ1 . Define λ ≡ (1 − θ)λ0 + θλ1 . Then for all ϕ ∈ Φ↑λ0 ,q0 (0, ∞) + Φ↑λ1 ,q1 (0, ∞), t−λ ϕ(t) . t−θ(λ1 −λ0 ) K(tλ1 −λ0 , ϕ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)). Before the proof, we remark that arithmetic shows that −λ + λ0 = −θ(λ1 − λ0 ),

−λ + λ1 = (1 − θ)(λ1 − λ0 ).

Proof Let ϕ ∈ Φ↑λ0 ,q0 (0, ∞) + Φ↑λ1 ,q1 (0, ∞), and let ϕ0 ∈ Φ↑λ0 ,q0 (0, ∞), ϕ1 ∈ Φ↑λ1 ,q1 (0, ∞) be any functions satisfying ϕ = ϕ0 + ϕ1 . Then, for t > 0,   −λ λ0 −λ −λ0 t ϕ(t) ≤ t sup η ϕ0 (η) + tλ1 −λ sup η −λ1 ϕ1 (η) η>0

η>0

= tλ0 −λ kϕ0 kΦλ0 ,∞ (0,∞) + tλ1 −λ kϕ1 kΦλ1 ,∞ (0,∞) ,

Real interpolation of Morrey spaces

299

since t−λ ϕ(t) = tλ0 −λ (t−λ0 ϕ0 (t)) + tλ1 −λ (t−λ1 ϕ1 (t)). Thanks to Lemma 37 in the first book, we have t−λ ϕ(t) . tλ0 −λ kϕ0 kΦλ0 ,q0 (0,∞) + tλ1 −λ kϕ1 kΦλ1 ,q1 (0,∞) . Hence, t−λ ϕ(t) . tθ(λ0 −λ1 ) kϕ0 kΦλ0 ,q0 (0,∞) + t(1−θ)(λ1 −λ0 ) kϕ1 kΦλ1 ,q1 (0,∞) = tθ(λ0 −λ1 ) (kϕ0 kΦλ0 ,q0 (0,∞) + tλ1 −λ0 kϕ1 kΦλ1 ,q1 (0,∞) ). Therefore, since ϕ0 and ϕ are arbitrary, t−λ ϕ(t) . tθ(λ0 −λ1 )

(kϕ0 kΦλ0 ,q0 (0,∞) + tλ1 −λ0 kϕ − ϕ0 kΦλ1 ,q1 (0,∞) )

inf

ϕ0 ∈Φλ0 ,q0 (0,∞), ϕ−ϕ0 ∈Φλ1 ,q1 (0,∞)

= t−θ(λ1 −λ0 ) K(tλ1 −λ0 , ϕ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)). For later consideration, we need the following estimates of operators As and Bs given by (18.1). Lemma 273. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 , λ1 > 0, λ0 6= λ1 and define λ ≡ (1 − θ)λ0 + θλ1 . Also let ϕ ∈ M↑ (0, ∞). We set J1 (s) ≡ kχ(0,s) ϕkΦλ0 ,q0 (0,∞) , J2 (s) ≡ kχ(s,∞) ϕkΦλ1 ,q1 (0,∞) = kBs ϕkΦλ1 ,q1 (0,∞) .

(18.3)

Then kAs ϕkΦλ0 ,q0 (0,∞) . J1 (s) + sλ1 −λ0 J2 (s) for s > 0. Proof Let s > 0 be fixed. Then  q1 Z s Z ∞ 0 −λ0 q0 dη −λ0 q0 dη (η ϕ(s)) + . kAs ϕkΦλ0 ,q0 (0,∞) = (η ϕ(η)) η η s 0 Since λ0 > 0, Z kAs ϕkΦλ0 ,q0 (0,∞) .

s

(η

−λ0

0

dη ϕ(η)) η q0

 q1

0

+ s−λ0 ϕ(s) ∼ J1 (s) + s−λ0 ϕ(s).

Since ϕ is monotonically increasing, −λ0

s

ϕ(s) = (λ1 q1 )

1 q1

λ1 −λ0

∞

Z

ϕ(s)s

η

−λ1 q1

s

≤ (λ1 q1 )

1 q1

λ1 −λ0

Z

∞

s

η s

∼ sλ1 −λ0 J2 (s). We thus proved Lemma 273. We will obtain the estimate of J1 and J2 .

−λ1 q1

dη η

 q1

dη ϕ(η) η q1

1

 q1

1

300

Morrey Spaces

Lemma 274. Suppose that the positive parameters q0 , q1 , q, θ, λ0 and λ1 satisfy q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 6= λ1 and define λ ≡ (1 − θ)λ0 + θλ1 . Furthermore, assume that q0 , q1 ≤ q, λ0 < λ1 . Let ϕ ∈ Φλ,q (0, ∞). (1) Define J1 by (18.3). Then kJ1 kΦθ(λ1 −λ0 ),q (0,∞) . kϕkΦλ,q (0,∞) . (2) Define J2 by (18.3). Then kJ2 kΦ(1−θ)(λ0 −λ1 ),q (0,∞) . kϕkΦλ,q (0,∞) . Proof (1) By virtue of the boundedness of the Hardy operator H and Lemma 36 in the first book,

kJ1 kΦθ(λ1 −λ0 ),q (0,∞) q0 = H[·−λ0 q0 −1 ϕq0 ] Φ (0,∞) θ(λ1 −λ0 )−1,q

. k ·−λ0 q0 −1 ϕq0 kΦθ(λ

q 1 −λ0 )q0 −1, q 0

(0,∞)

q

= kϕkΦλ,q (0,∞) . (2) Similar to (1); see Exercise 77. Using Lemmas 272, 273 and 274, we can prove the following interpolation result: Theorem 275. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 , λ1 > 0, λ0 6= λ1 and define λ ≡ (1 − θ)λ0 + θλ1 . Then (Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞))θ,q ≈ Φ↑λ,q (0, ∞) with equivalence of norms. Moreover, if λ0 < λ1 , kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q λ ,q λ1 ,q1

0 0 

∼ inf kAs ϕkΦλ0 ,q0 (0,∞) + ·kBs ϕkΦλ1 ,q1 (0,∞)

s>0

Φθ,q (0,∞)

∼ kϕkΦθ,q (0,∞) and if λ0 > λ1 , kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q λ ,q λ1 ,q1

0 0 

∼ inf kA ϕk + ·kB ϕk s s Φλ1 ,q1 (0,∞) Φλ0 ,q0 (0,∞)

s>0

Φ1−θ,q (0,∞)

∼ kϕkΦθ,q (0,∞) for all functions ϕ ∈ Φ↑λ,q (0, ∞), where the implicit constants depend only on q0 , q1 , q, θ, λ0 , λ1 . Proof The proof is made up of three steps. The first step is the proof of the inequality: kϕkΦλ,q (0,∞) . kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q . The second step λ0 ,q0

λ1 ,q1

Real interpolation of Morrey spaces

301

is establishing kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q λ ,q λ1 ,q1

0 0

inf (kA ϕk + ·kB ϕk ) ≤ s s Φλ0 ,q0 (0,∞) Φλ1 ,q1 (0,∞)

s>0

Φθ,q (0,∞)

and kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q λ ,q λ1 ,q1

0 0

≤ inf (kAs ϕkΦλ1 ,q1 (0,∞) + ·kBs ϕkΦλ0 ,q0 (0,∞) )

s>0

.

Φ1−θ,q (0,∞)

The final step requires us to distinguish two cases: Assuming that λ0 < λ1 , we prove



inf (kAs ϕkΦ

+ ·kB ϕk . kϕkΦλ,q (0,∞) ) s Φλ1 ,q1 (0,∞) λ0 ,q0 (0,∞)

s>0

Φθ,q (0,∞)

and assuming that λ1 < λ0 , we prove



inf (kAs ϕkΦ + ·kBs ϕkΦλ0 ,q0 (0,∞) ) λ1 ,q1 (0,∞)

s>0

. kϕkΦλ,q (0,∞) .

Φ1−θ,q (0,∞)

(1) We calculate Z kϕkΦλ,q (0,∞) =

∞

(t

−λ

ϕ(t))

0

q dt

t

 q1 .

Thanks to Lemma 272, we have kϕkΦλ,q (0,∞) Z ∞  q1 ↑ ↑ −θ(λ1 −λ0 ) λ1 −λ0 q dt . (t K(t , ϕ; Φλ0 ,q0 (0, ∞), Φλ1 ,q1 (0, ∞))) . t 0 By the change of variables, Z kϕkΦλ,q (0,∞) .

∞ −θ

(t 0

dt K(t, ϕ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)))q

 q1

t

= kϕk(Φ↑

↑ λ0 ,q0 (0,∞),Φλ1 ,q1 (0,∞))θ,q

.

(2) Recall that K(t, f ; Φ↑λ0 ,q0 (0, ∞), Φ↑λ1 ,q1 (0, ∞)) =

inf

ϕ0 ∈Φ↑ λ0 ,q0 (0,∞) ϕ−ϕ0 ∈Φλ1 ,q1 (0,∞)

kϕ0 kΦλ0 ,q0 (0,∞) + tkϕ − ϕ0 kΦλ1 ,q1 (0,∞) .

302

Morrey Spaces Thus, kϕk(Φ↑ (0,∞),Φ↑ (0,∞))θ,q λ ,q λ1 ,q1

0 0

≤ inf (kAs ϕkΦλ0 ,q0 (0,∞) + ·k(I − As )ϕkΦλ1 ,q1 (0,∞) )

s>0

Φθ,q (0,∞)



inf (kA ϕk + ·kB ϕk ) ≤ s s Φλ0 ,q0 (0,∞) Φλ1 ,q1 (0,∞)

s>0

Φθ,q (0,∞)

thanks to (18.2). Likewise, by using an analogy to Lemma 268 we can prove kϕk(Φ↑

↑ λ0 ,q0 (0,∞),Φλ1 ,q1 (0,∞))θ,q

= kϕk(Φ↑ (0,∞),Φ↑ (0,∞))1−θ,q λ1 ,q1 λ0 ,q0



≤ inf (kAs ϕkΦλ1 ,q1 (0,∞) + ·kBs ϕkΦλ0 ,q0 (0,∞) )

s>0

.

Φθ,q (0,∞)

(3) It remains to prove the right-inequality. We can assume λ1 > λ0 without any loss of generality due to symmetry. Furthermore, in view of the monotonicity of the scale Φθ,q in q, we may assume that q0 , q1 ≤ q. In fact, in the general case 0 < q0 , q1 , q ≤ ∞ we set τ0 ≡ min(q0 , q) and τ1 ≡ min(q1 , q), so that τ0 , τ1 ≤ q. Once we show the right-inequality for the case q ≥ q0 , q1 , we have

 

inf kAs ϕkΦ + ·kBs ϕkΦλ1 ,q1 (0,∞) λ0 ,q0 (0,∞)

s>0

Φ

θ,q

 

inf kAs ϕkΦλ0 ,τ0 (0,∞) + ·kBs ϕkΦλ1 ,τ1 (0,∞) .

s>0

(0,∞)

Φθ,q (0,∞)

. kϕkΦλ,q (0,∞) , which justifies that we may assume q ≥ q0 , q1 , By a change of variables, we obtain



inf (kAs ϕkΦ + ·kBs ϕkΦλ1 ,q1 (0,∞) ) λ0 ,q0 (0,∞)

s>0

Φ

(0,∞)

θ,q



λ1 −λ0

∼ inf (kAs ϕkΦλ0 ,q0 (0,∞) + · kBs ϕkΦλ1 ,q1 (0,∞) )

s>0 Φθ(λ0 −λ1 ),q



. kA· ϕkΦλ0 ,q0 (0,∞) + ·λ1 −λ0 kB· ϕkΦλ1 ,q1 (0,∞) .

Φθ(λ0 −λ1 ),q

Real interpolation of Morrey spaces Define I1 and I2 by



, I1 ≡ kA· ϕkΦλ0 ,q0 (0,∞) Φθ(λ0 −λ1 ),q (0,∞)



I2 ≡ ·λ1 −λ0 kB· ϕkΦλ1 ,q1 (0,∞)

303

(18.4) ,

Φθ(λ0 −λ1 ),q (0,∞)

so that kϕk(Φ↑

↑ λ0 ,q0 (0,∞),Φλ1 ,q1 (0,∞))θ,q

. I1 + I2 . Define J1 (τ ) and J2 (τ )

by (18.3) for τ > 0. We deduce I2 ≤ kJ2 kΦ(1−θ)(λ1 −λ0 ),q (0,∞) from (18.3). Meanwhile, from Lemma 273 I1 . kJ1 kΦθ(λ1 −λ0 ),q (0,∞) + k ·λ1 −λ0 J2 kΦθ(λ1 −λ0 ),q (0,∞) = kJ1 kΦθ(λ1 −λ0 ),q (0,∞) + kJ2 kΦ(1−θ)(λ0 −λ1 ),q (0,∞) . Thus, we have kϕk(Φ↑

↑ λ0 ,q0 (0,∞),Φλ1 ,q1 (0,∞))θ,q

. kJ1 kΦθ(λ1 −λ0 ),q (0,∞) + kJ2 kΦ(1−θ)(λ0 −λ1 ),q (0,∞) . Keeping this in mind, assume that q0 , q1 ≤ q, λ0 < λ1 . Then we can use Lemma 274.

18.2.2

Spaces defined via M↑ (0, ∞)

We generalize the notion of the K-functional slightly. Let X be a linear space and let Z0 and Z1 be linear subspaces of X on which functionals k · kZ0 : Z0 → [0, ∞), k · kZ1 : Z1 → [0, ∞) act. For 0 < q ≤ ∞ and 0 < θ < 1, let (Z0 , Z1 )θ,q denote the interpolation space of all f ∈ Z0 + Z1 for which the following norm kf k(Z0 ,Z1 )θ,q ≡ kK(·, f ; Z0 , Z1 )kΦθ,q (0,∞) is finite, where as before, for t > 0 and f ∈ Z0 + Z1 , K(t, f ; Z0 , Z1 ) ≡ inf{kf0 kZ0 + tkf1 kZ1 : f0 ∈ Z0 , f1 ∈ Z1 , f = f0 + f1 }. Motivated by the theorem on powers, Theorem 270, we define the powered mapping and a linear space. Definition 51 (Φλ,q (F ), F σ ). Let Z be a subset of a linear subspace X , and let F : Z → M↑ (0, ∞) be a functional. (1) Let σ > 0. Define a functional F σ : Z → M↑ (0, ∞) by F σ f ≡ (F f )σ for f ∈ Z. (2) For λ > 0 and 0 < q ≤ ∞, Φλ,q (F ) is the space of all f ∈ Z for which kf kΦλ,q (F ) ≡ kF f kΦλ,q (0,∞) < ∞.

304

Morrey Spaces We describe the scaling relation in this scale as follows:

Example 94. Let Z0 , Z1 , Z ⊂ X satisfy Z ⊂ Z0 + Z1 , and let F0 : Z0 → M↑ (0, ∞), F1 : Z1 → M↑ (0, ∞), F : Z → M↑ (0, ∞) be functionals. Our aim is the description of the interpolation space (Φλ0 ,p0 (F0 ), Φλ1 ,p1 (F1 ))θ,q and finding the assumptions on F0 , F1 , F ensuring that this space coincides with Φλ,p (F ). First, we want something to control F f in terms of the K-functional. For this purpose we give the following notion: Definition 52. Let the linear subspaces Z0 , Z1 , Z in a linear space X satisfy Z ⊂ Z0 + Z1 , and let F0 : Z0 → M↑ (0, ∞), F1 : Z1 → M↑ (0, ∞), F : Z → M↑ (0, ∞) be any mappings. One says that (F, F0 , F1 ) is a triple of weak quasiadditive type if there exists α ≥ 1 such that for each f0 ∈ Z0 and f1 ∈ Z1 such that f0 + f1 ∈ Z F [f0 + f1 ] . F0 f0 (α·) + F1 f1 (α·).

(18.5)

The notion of the weak quasi-additive type is abstract. Here, we remark that it is a natural generalization of the quasi-additivity. Example 95. If F0 = F1 = F , Z0 = Z1 = Z and α = 1, then (18.5) boils down to quasi-additivity: F [f0 + f1 ] . F f0 + F f1 (f0 , f1 ∈ Z). An important consequence of the weak quasi-additivity is the following control of F σ in terms of the K-functional: Lemma 276. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, 0 < σ0 , σ1 < ∞ and λ0 , λ1 > 0. Set λ ≡ (1 − θ)λ0 + θλ1 . Define σ by 1−θ θ 1 = + . σ σ0 σ1 Assume λ0 σ0 6= λ1 σ1 .

(18.6)

If (F σ , F σ0 , F σ1 ) is a triple of weak quasi-additive type, then t−λσ F σ f (t) . tη(τ0 −τ1 ) K(tτ1 −τ0 , f ; Φλ0 σ0 , σq0 (F0 σ0 ), Φλ1 σ1 , σq1 (F1 σ1 )) 0

1

for all t > 0 and f ∈ Z. Proof Write τ0 ≡ λ0 σ0 −λσ, τ1 ≡ λ1 σ1 −λσ, so that τ1 −τ0 = λ1 σ1 −λ0 σ0 . Arithmetic shows τ0 = −η(τ1 − τ0 ) and that τ1 = (1 − η)(τ1 − τ0 ). Let t > 0 and f ∈ Z be fixed. Choose any decompsotion f = f0 + f1 with f0 ∈ Z0 and

Real interpolation of Morrey spaces

305

f1 ∈ Z1 . Then for all t > 0, F σ f (t) = F σ [f0 +f1 ](t) . F0 σ0 f0 (αt)+F1 σ1 f1 (αt) by quasi-additivity. Hence t−λσ F σ f (t) . t−λσ (tλ0 σ0 sup s−λ0 σ F0 σ0 f0 (s) + tλ1 σ1 sup s−λ1 σ F1 σ1 f1 (s)) s>0 s>0
−λσ λ0 σ0 σ0 λ 1 σ1 =t t kF0 f0 kΦλ0 σ0 ,∞ + t kF1 σ1 f1 kΦλ1 σ1 ,∞ . tτ0 kF0 σ0 f0 kΦλ

q0 0 σ0 , σ 0

+ tτ1 kF1 σ1 f1 kΦλ

q1 1 σ1 , σ 1

.

Consequently, for all t > 0, t−λσ F σ f (t) . tη(τ0 −τ1 ) kf0 kΦλ σ , q0 (F0 σ0 ) + t(1−η)(τ1 −τ0 ) kf1 kΦλ σ , q1 (F1 σ1 ) 1 1 σ 0 0 σ 0 1   η(τ0 −τ1 ) τ1 −τ0 =t kf0 kΦλ σ , q0 (F0 σ0 ) + t kf1 kΦλ σ , q1 (F1 σ1 ) . 0 0 σ 0

1 1 σ 1

Hence we obtain the desired result. A direct consequence of the weak quasi-additivity is the following embedding of interpolation spaces. Theorem 277. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1 and 0 < λ0 , λ1 , σ0 , σ1 < ∞. −1  θ 1−θ + . If (F σ , F σ0 , F σ1 ) is a Set λ ≡ (1 − θ)λ0 + θλ1 and σ ≡ σ0 σ1 triple of weak quasi-additive type, then (Φλ0 ,q0 (F0 ), Φλ1 ,q1 (F1 ))θ,q ,→ Φλ,q (F ). In Theorem 277, the parameters σ0 , σ1 and σ play an auxiliary role. Proof Write τ0 ≡ λ0 σ0 −λσ, τ1 ≡ λ1 σ1 −λσ, so that τ1 −τ0 = λ1 σ1 −λ0 σ0 as before. Let η ≡ θσ σ1 . Take the norm k · kΦλ,q (0,∞) in Lemma 276 to obtain (kf kΦλσ, q (F σ ) )q Z ∞ nσ o σq dt . t−η(τ1 −τ0 ) K(tτ1 −τ0 , f ; Φλ0 σ0 , σq0 (F0 σ0 ), Φλ1 σ1 , σq1 (F1 σ1 )) 0 1 t Z0 ∞ n o σq dt = tη(τ0 −τ1 ) K(tτ1 −τ0 , f ; Φλ0 ,q0 (F0 )σ0 , Φλ1 ,q1 (F1 )σ1 ) . t 0 Recall that τ0 6= τ1 since λ0 σ0 6= λ1 σ1 . By the change of variables, we have q

(kf kΦλσ, q (F σ ) ) . σ

 σq

 kf k(Φλ0 ,q0 (F0 )σ0 ,Φλ1 ,q1 (F1 )σ1 ) θσ , q σ1

σ

q

. (kf k(Φλ0 ,q0 (F0 ),Φλ1 ,q1 (F1 ))θ,q ) , as was to be shown. Next, we will need something to control the K-functional. Recall that As and Bs are defined by (18.1) for s > 0. To this end, we give the following definition:

306

Morrey Spaces

Definition 53. The triple (F0 , F1 , F ) admits a weakly A-B majorizable decomposition if there exists α ≥ 1 such that for any s > 0 there exist a decomposition f = f0,s + f1,s with f0,s ∈ Z0 and f1,s ∈ Z1 satisfying F0 f0,s . As F f (α·),

F1 f1,s . Bs F f (α·).

The above notion is natural from what we have been doing. Example 96. In view of (18.2) the triple (idZ0 , idZ1 , idZ ) admits a weakly A-B majorizable decomposition. As the next result shows, the weakly A-B majorizable decomposition can be used to take advantage of the K-functional: Theorem 278. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 , λ1 > 0 and −1  θ 1−θ + . Sup0 < σ0 , σ1 < ∞. Set λ ≡ (1 − θ)λ0 + θλ1 and σ ≡ σ0 σ1 pose that (18.6) holds and that the triple (F0 σ0 , F1 σ1 , F σ ) admits a weak A-B majorizable decomposition. Then Φλ,q (F ) ,→ (Φλ0 ,q0 (F0 ), Φλ1 ,q1 (F1 ))θ,q , In Theorem 278, the parameters σ0 , σ1 and σ play an auxiliary role. Proof Write τ0 ≡ λ0 σ0 −λσ, τ1 ≡ λ1 σ1 −λσ, so that τ1 −τ0 = λ1 σ1 −λ0 σ0 as before. We may assume λ0 σ0 < λ1 σ1 without loss of generality. Then kf k(Φλ0 ,q0 (F0 ),Φλ1 ,q1 (F1 )) ∼ kf k

σ1

σ0

σ σ ,Φλ1 ,q1 (F1 )power ) θσ ,q (Φλ0 ,q0 (F0 )power σ

1

σ0 σ1

σ σ = K(·, f ; Φλ0 ,q0 (F0 )power , Φλ1 ,q1 (F1 )power )

Φ θσ ,q σ1

by virtue of Theorem 270. σ1 σ0 σ σ , Φλ1 ,q1 (F1 )power ). We write out We abbreviate K ≡ K(t, f ; Φλ0 ,q0 (F0 )power the definition of the K-functional in full: For t > 0   σ0 σ1 K= inf kf0 k + tkf k . 1 σ σ f =f0 +f1 ,

σ0 σ , σ1 f1 ∈Φλ1 ,q1 (F1 ) σ

Φλ0 ,q0 (F0 )power

Φλ1 ,q1 (F1 )power

f0 ∈Φλ0 ,q0 (F0 )

For each s > 0 we choose a decomposition f = f0,s + f1,s as in Definition 53 to obtain K ≤ kf0,s k

σ0

σ Φλ0 ,q0 (F0 )power

Since s > 0 is arbitrary, we have  K ≤ inf kf0,s k s>0

+ tkf1,s k

σ0 σ Φλ0 ,q0 (F0 )power

σ1

σ Φλ1 ,q1 (F1 )power

.

 σ1 + tkf1,s kΦσλ power . ,q (F1 ) 1

1

Real interpolation of Morrey spaces

307

We write out the definition of the power norms fully to obtain   σ σ K ≤ inf kF0 f0,s k + tkF1 f1,s k 0 1 σ σ s>0 Φλ0 ,q0 (0,∞)power Φλ1 ,q1 (0,∞)power (Z  qσ0σ  qσ0σ ) Z ∞ ∞ 0 1 dρ dρ q0 q1 F0 f0,s (ρ) 1+λ0 q0 F1 f1,s (ρ) 1+λ1 q1 = inf +t . s>0 ρ ρ 0 0 We use the quasi-majorization property. By the quasi-triangle inequality, we have (Z  qσ0σ ∞ 0  q0 −λ0 σ0 σ σ0 dρ ρ (As F f )(αρ) K ≤ inf s>0 ρ 0 Z ∞  qσ1σ ) 1  −λ1 σ1
q1 σ σ1 dρ +t ρ (Bs F f )(αρ) ρ 0   ∼ inf kAs F f kΦ λ0 σ0 q0 σ + tkBs F f kΦ λ1 σ1 q1 σ . s>0

,

σ

σ0

σ

,

σ1

It remains to use Theorem 275 and a trivial estimate

 

inf kAs F f kΦ

+ ·kB F f k s Φ λ1 σ1 q1 σ λ0 σ0 q0 σ

s>0 , , σ σ σ σ 0

Φ θσ ,q

1

σ1

≤

kA· F f kΦ λ0 σ0 σ

q σ , 0 σ0

+ ·kB· F f kΦ λ1 σ1 q1 σ

, σ σ 1

.

Φ θσ ,q σ1

We are thus led to the following conclusion: Theorem 279. Let 0 < q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 , λ1 , λ > 0 be as in Theorem 277. Suppose further that the triple (F0 σ0 , F1 σ1 , F σ ) admits a weak A-B majorizable decomposition and that (F σ , F σ0 , F σ1 ) is a triple of weak quasi-additive type for some 0 < σ0 , σ1 , σ < ∞. Then Φλ,q (F ) ≈ (Φλ0 ,p0 (F0 ), Φλ1 ,p1 (F1 ))θ,q with equivalence of norms. Proof Simply combine Theorems 277 and 278. We will look for many examples to which Theorem 279 is applicable.

18.2.3

Exercises

Exercise 76. Let Z be a linear subspace of a linear space X . Also let λ, σ > 0 and 0 < q ≤ ∞. Then show that   σ1 Φλ,q (F ) = Φλσ, σq (F σ ) by using a scaling law for Φλ,θ (0, ∞).

308

Morrey Spaces

Exercise 77. Suppose that the positive parameters q0 , q1 , q, θ, λ0 and λ1 satisfy q0 , q1 , q ≤ ∞, 0 < θ < 1, λ0 6= λ1 and define λ ≡ (1 − θ)λ0 + θλ1 . Furthermore, assume that q0 , q1 ≤ q, λ0 < λ1 . Let ϕ ∈ Φλ,q (0, ∞). Define J2 by (18.3). Abbreviate Θ ≡ (1 − θ)(λ0 − λ1 ). We write Z 1 ∞ f (s)ds. Hf (t) = t t

Z

(1) Show that kJ2 kΦΘ,q (0,∞) =

∞

(η −λ1 ϕ(η))q1

·

q1 1 dη

η Φ

Θq1 ,

1 (2) Verify that kϕkΦλ,q (0,∞) = ·−λ1 q1 −1 ϕq1 Φq1

q Θq1 −1, q1

. q q1

(0,∞)

(0,∞) .

(3) Using the boundedness of the Hardy operator H, conclude the proof of Lemma 274.

18.3

Real interpolation of Lebesgue spaces

We consider the real interpolation of Lebesgue spaces. What is important here is that Lorentz spaces come into play. Hence, we are convinced that Lorentz spaces are natural function spaces. As an example of real interpolation, we will deal with the real interpolation of Lebesgue spaces in Section 18.3.1. Section 18.3.2 gives a general theorem on the real interpolation.

18.3.1

Description of real interpolation spaces for a pair of Lebesgue spaces

Let (X, B, µ) be a measure space. Let us describe the interpolation space (Lp0 (µ), Lp1 (µ))θ,q for 0 < p0 , p1 , q ≤ ∞ and 0 < θ < 1. We can do this because we can calculate the K-functional explicitly. Lemma 280 (Peetre). For any f ∈ L1 (µ) + L∞ (µ) and for any t > 0 K(t, f ; L1 (µ), L∞ (µ)) = kf ∗ kL1 (0,t) . Proof We will first prove K(t, f ; L1 (µ), L∞ (µ)) ≤ kf ∗ kL1 (0,t) and then K(t, f ; L1 (µ), L∞ (µ)) ≥ kf ∗ kL1 (0,t) . (1) Let f = f0 + f1 , f0 ∈ L1 (µ) and f1 ∈ L∞ (µ), where, for x ∈ X,  (|f (x)| − f ∗ (t)) f (x) if |f (x)| > f ∗ (t), |f (x)| f0 (x) ≡  0 otherwise

Real interpolation of Morrey spaces

309

and f1 (x) ≡ f (x) − f0 (x). We write E ≡ {x ∈ X : f0 (x) 6= 0} = {x ∈ X : |f (x)| > f ∗ (t)}. Then µ(E) = µ{x ∈ X : |f (x)| > f ∗ (t)} = λf (f ∗ (t)) ≤ t by Lemma 29 in the first book. Meanwhile kχE gkL1 (µ) ≤ kg ∗ kL1 (0,|E|) thanks to Example 34 in the first book. Thus, it follows that K(t, f ; L1 (µ), L∞ (µ)) ≤ kf0 kL1 (µ) + tkf1 kL∞ (µ) Z ≤ (|f (x)| − f ∗ (t))dµ(x) + tf ∗ (t) E |E|

Z ≤

(|f | − f ∗ (t))∗ (s)ds + tf ∗ (t).

0

As a result, K(t, f ; L1 (µ), L∞ (µ)) ≤

Z

t

(|f | − f ∗ (t))∗ (s)ds + tf ∗ (t) = kf ∗ kL1 (0,t) .

0

(2) Let f ∈ L1 (µ) + L∞ (µ), and let f0 ∈ L1 (µ) and f1 ∈ L∞ (µ) be any functions satisfying f = f0 + f1 . Then f ∗ (s) ≤ f0∗ (s) + kf1 kL∞ for any s > 0. Hence, Z t Z t ∗ ∗ kf kL1 (0,t) ≤ f0 (s)ds + kf1 kL∞ (µ) ds 0 0 Z ∞ Z t ≤ f0∗ (s)ds + kf1 kL∞ (µ) ds 0

0

= kf0 kL1 (µ) + tkf1 kL∞ (µ) . Since the decomposition f = f0 + f1 is arbitrary, kf ∗ kL1 (0,t)  ≤ inf kf0 kL1 (µ) + tkf1 kL∞ (µ) : f0 ∈ L1 (µ), f1 ∈ L∞ (µ), f = f0 + f1 = K(t, f ; L1 (µ), L∞ (µ)). This completes the proof. In a similar way, we can prove the following: Lemma 281 (Kree 1967). Let 0 < p < ∞. Then K(t, f ; Lp (µ), L∞ (µ)) ∼p kf ∗ kLp (0,tp ) for any f ∈ Lp (µ) + L∞ (µ) and t > 0. Proof The proof is omitted due to similarity to Lemma 280; see Exercise 78. Using the explicit formula on the K-functional, we will obtain a description of the real interpolation of Lebesgue spaces.

310

Morrey Spaces

Theorem 282. Let 0 < p0 < ∞, and let 0 < θ < 1. We define p by p ≡ p0 . Then (Lp0 (µ), L∞ (µ))θ,q ≈ Lp,q (µ) with equivalence of norms for any 1−θ q ∈ [p0 , ∞]. In particular, (Lp0 (µ), L∞ (µ))θ,p = Lp (µ) in the diagonal case p = q. Moreover, kf k(Lp0 (µ),L∞ (µ))θ,q ∼ kf kLp,q (µ) for any f ∈ Lp0 (µ)+L∞ (µ) with the implicit constants depending only on p0 , q and θ. Proof We distinguish two cases. 1

(1) Assume p0 = 1: We will prove (L1 (µ), L∞ (µ))θ,q = L 1−θ ,q (µ) using Lemma 280. Let H be the Hardy operator given by Hf (t) = Z 1 t f (s)ds. For f ∈ L1 (µ) + L∞ (µ), we calculate t 0 Z kf k

(L1 (µ),L∞ (µ))θ,q

∞

=

(t

−θ

q dt

∞

1

K(t, f ; L (µ), L (µ)))

0

Z

∞

t−θ kf ∗ kL1 (0,t)

= 0


q dt t

 q1

t

 q1

= kH[f ∗ ]kΦθ−1,q . Using Hardy’s inequality, we obtain kf k(L1 (µ),L∞ (µ))θ,q ≤ θ−1 kf k ∗

1

L 1−θ

↓

,q

.

Meanwhile, since f ∈ M (0, ∞), we have ∞

Z


q dt t  q1
q dt t1−θ f ∗ (t) t

t−θ kf ∗ kL1 (0,t)

kf k(L1 (µ),L∞ (µ))θ,q = 0 ∞

Z ≥ 0

 q1

= kf k(L1 (µ),L∞ (µ))θ,q . 1 ,q ≤ kf k(L1 (µ),L∞ (µ)) 1 ,q . When Consequently, kf k 1−θ ≤ θ−1 kf k 1−θ θ,q L L q = 1, we learn that equality takes place if we reexamine the above 1 ,1 ; see Exercise 79 for more. estimates: kf k(L1 (µ),L∞ (µ))θ,1 = θ−1 kf k 1−θ

L

(2) If p0 6= 1, use Lemma 271 instead and argue as above.

18.3.2

Application of the A-B majorization to real interpolation of Lebesgue spaces

We will apply what we have obtained so far to the previous section. We work in a measure space (X, B, µ). Our results are rather general. Let F : L0 (µ) → M↑ (0, ∞) be a mapping, and let λ ∈ R and 0 < q ≤ ∞. Recall that the norm of Φλ,q (F ) is given by Z kf kΦλ,q (F ) = kF f kΦλ,q (0,∞) = 0

∞

(t−λ F f (t))q

dt t

 q1

(f ∈ L0 (µ)).

Real interpolation of Morrey spaces

311

Let f ∈ L0 (µ), and let t > 0. Set Gt (f ) ≡ {x ∈ X : t|f (x)| > 1} .  −1 1−θ θ Let 0 < p0 , p1 ≤ ∞ and define p ≡ . Write + p0 p1   1 Hf (t) ≡ λf (t > 0) t and define

1

F0 ≡ H p0 ,

1

F1 ≡ H p1 ,

1

F ≡ Hp.

Then kf kΦ1,p0 (F0 ) = kf k

Φ1,p0 (H

1 p0

)

= kH

1 p0

Z f kΦ1,p0 =

∞ −p0

t 0

−1

λf (t

dt ) t

 p1

0

.

By the change of variables, we obtain 1 Z ∞  p1 Z ∞ 0 dt p0 p0 −1 p0 = t λf (t)dt ' kf kLp0 (µ) . kf kΦ1,p0 (F0 ) = t λf (t) t 0 0 We can do a similar observation for other parameters. We summarize our observations: Lemma 283. Let p, p0 , p1 , F, F0 , F1 be as above. Then Lp0 (µ) ≈ Φ1,p0 (F0 ), Lp1 (µ) ≈ Φ1,p1 (F1 ) and Lp (µ) ≈ Φ1,p (F ) with equivalence of norms. With α = 2 and c = 1, we have a weak quasi-additive type estimate of (F p , F0 p0 ,
F1 p1 ). In fact, keeping in mind that F p (f0 + f1 ) = H(f0 + f1 ) = λf0 +f1 1t , we obtain     1 1 p F (f0 + f1 ) ≤ λf0 + λf1 2t 2t = Hf0 (2t) + Hf1 (2t) = F0 p0 f0 (2t) + F1 p1 f1 (2t). We claim that (F p , F0 p0 , F1 p1 ) admits the weakly A-B majorizable decomposition. For s > 0, we set f0,s ≡ f · χGs (f ) , f1,s ≡ f · χX\Gs (f ) Note that Gt (f0,s ) = Gmin(t,s) (f ). Therefore, F0 p0 (f0,s ) = Hf0,s = |Gmin(t,s) (f )| = As Hf (t) = As [F p f ](t). Likewise, ( Gt (f1,s ) ⊂

∅ t ≤ s, Gs (f ) t > s.

Hence F0 p0 (f0,s ) ≤ Bs [F p f ](t). With this in mind, we calculate the real interpolation space with our interpolation machinery:

312

Morrey Spaces

Theorem 284. Let 0 < p0 , p1 < ∞, p0 6= p1 and 0 < θ < 1. Define p by 1 1−θ θ p0 p1 p,q (µ) with equivalence of norms. p = p0 + p1 . Then (L (µ), L (µ))θ,q ≈ L Proof We calculate: 1

(Lp0 (µ), Lp1 (µ))θ,q = (Φ1,p0 (F0 ), Φ1,p1 (F1 ))θ,q = Φ1,q (F ) = Φ1,q (H p ) thanks to Lemma 283. Meanwhile for any simple function f ∈ L0 (µ) by the change of variables v = f ∗ (t), (Z  ) q1  pq ∞ −1 1 λ (t ) dt f,µ 1 kf k ' kf kLp,q (µ) , = k(Hf ) p kΦ1,q = Φ1,q (H p ) tp t 0 as was to be shown.

18.3.3

Exercises

Exercise 78. Prove Lemma 281. Hint: In the proof of Lemma 280 take  (|f (x)| − f ∗ (tp )) f (x) if |f (x)| > f ∗ (tp ), |f (x)| f0 (x) ≡  0 otherwise for x ∈ X. Exercise 79. Let f ∈ L1 (µ) + L∞ (µ) and 0 < θ < 1. Then establish that 1 ,1 mimicking the proof of Theorem 191. kf k(L1 (µ),L∞ (µ))θ,1 = θ1 kf k 1−θ L

18.4

Real interpolation of Morrey spaces

We consider the real interpolation of Morrey spaces. nRecall that the Morn rey norm Mpq (Rn ) is given by kf kMpq = sup t p − q kf kLq (B(x,t)) for (x,t)∈Rn+1 +

0

n

f ∈ L (R ). Here, we investigate the real interpolation. However, as we will see the output is not the Morrey space. Consequently, the interpolation of Morrey spaces is more difficult than that of Lebesgue spaces. This is a highlight of Chapter 18. We show that Morrey-type spaces interpolate well in Section 18.4.1 and that we can interpolate weighted Lebesgue spaces in Section 18.4.2.

18.4.1

Real interpolation of Morrey(-type) spaces

We start with a counterexample obtained from Propositions 235, 236 and 237. By letting δ=

7 , 8

γ=

15 × 1.001, 8

p = q1 = 8,

q2 =

50 , 49

q3 =

99 , 98

Real interpolation of Morrey spaces

313

we disprove that Morrey spaces do not interpolate well by the real interpolation functor. Theorem 285. There exists a bounded linear operator T : M81 (R) → L1 (R) such that T , restricted to M81+1/49 (R), maps M81+1/49 (R) to L1+1/49 (R) but 1

that T does not map T : M87 (R) to L1+ 98 (R). For f ∈ L0 (Rn ) we let F f (t) ≡ sup kf kLp (B(x,r)) for t > 0. Then x∈Rn

Φλ,∞ (F ) = Mpq (Rn ). We claim that the weak quasi-additivity is satisfied. For t > 0 F [f0 + f1 ](t) = sup kf0 + f1 kLp (B(x,t)) x∈Rn

. sup kf0 kLp (B(x,t)) + kf1 kLp (B(x,t)) x∈Rn

≤ F f0 (t) + F f1 (t). Hence, by Theorem 277, we have (Mpq 0 (Rn ), Mpq 1 (Rn ))θ,∞ ⊂ Mpq (Rn ). As for Morrey spaces, we have the following partial result: Theorem 286. Suppose that we have positive real parameters p0 , p1 , q and θ satisfying q ≤ p0 , p1 , p0 6= p1 and 0 < θ < 1. We define p ∈ (0, ∞) by 1 1−θ θ = + . Then (Mpq 0 (Rn ), Mpq 1 (Rn ))θ,∞ ( Mpq (Rn ). p p0 p1 Proof By symmetry we may assume that p0 < p1 . In view of the observation above, we have only to show that equality fails. Let F be the set defined in Example 9 in the first book. Then K(t, χF ; Mpq 0 (Rn ), Mpq 1 (Rn )) ∼ t. Thus, (Mpq 0 (Rn ), Mpq 1 (Rn ))θ,∞ is a proper subset of Mpq (Rn ). Keeping in mind that Morrey spaces do not interpolate well, let us consider local Morrey-type spaces. Let 0 < q ≤ p < ∞ and 0 < u ≤ ∞. Recall that we defined the local Morrey-type norm k · kLMpqu by ∞

Z

1

1

(|Q(r)| p − q kf kLq (Q(r)) )u

kf kLMpqu ≡ 0 0

dr r

 u1

n

for f ∈ L (R ). The local Morrey space LMpq (Rn ) is the set of all f ∈ L0 (Rn ) for which kf kLMpq is finite. Let 0 < q ≤ p < ∞ and f ∈ L0 (Rn ). Set F f (t) ≡ kf kLq (B(t))

(t > 0).

Then Z kf kΦ n − n ,u (0,∞) = q

p

∞

n

n

(t p − q kf kLq (B(t)) )u

0 n

dt t

 u1 = kf kLMpqu

(18.1)

n

when u < ∞. Likewise, kf kΦ n − n ,∞ = sup t p − q kf kLq (B(t)) = kf kLMpq . In q

p

t>0

contrast to this statement we have the following conclusion, which is a model case of our generalization.

314

Morrey Spaces

Theorem 287. Let 1 ≤ q ≤ p0 , p1 < ∞ and u0 , u1 , u > 0. Define p by 1 1−θ θ = + with some θ ∈ (0, 1). Assume in addition that p0 6= p1 . Then p p0 p1 n p0 (LMqu0 (R ), LMpqu1 1 (Rn ))θ,u = LMpqu (Rn ). Proof We need to check that Theorem 279 is applicable with F0 = F1 = 1 1 1 1 1 1 F , λ0 = − , λ1 = − , λ = − and σ0 = σ1 = σ = q defined in the q p0 q p1 q p above. Then (18.6) reads p0 6= p1 . By the quasi-triangle inequality, we have F [f0 + f1 ](t) = kf0 + f1 kLq (B(t)) . kf0 kLq (B(t)) + kf1 kLq (B(t)) = F f0 (t) + F f1 (t), so that (F q , F q , F q ) is a triple of quasi-additive type. For f ∈ L0 (Rn ) and s > 0, we let f0,s ≡ f χB(s) and f1,s ≡ f − f0,s . For each t > 0 note that F f0,s (t) = kf χB(s) kLq (B(t)) = kf χB(min(s,t)) kLp (B(t)) = As F f (t) and that F f1,s (t) = kf χRn \B(s) kLq (B(t)) ≤ χ(s,∞) (t)kf kLq (B(t)) = Bs F f (t), which verifies the weak majorization property of (F q , F q , F q ). Consequently we are in the position of using Theorem 279 and (18.1) to calculate: (LMpqu0 0 (Rn ), LMpqu1 1 (Rn ))θ,u = (Φ nq − pn ,u0 (F ), Φ nq − pn ,u1 (F ))θ,u 0

1

= Φ nq − np ,u (F ) = LMpqu (Rn ), which completes the proof.

18.4.2

Interpolation of generalized local Morrey spaces with the family G

Let (X, B, µ) be a σ-finite measure space. Let 0 < q < p < ∞ and 0 < u < ∞ or 0 < q ≤ p < u = ∞. Moreover, let G = (Gt )t>0 where the Gt ’s are µ-measurable subsets for which [ Gt 6= X for some t > 0, Gt1 ⊂ Gt2 if t1 < t2 , Gt = X. t>0

For f ∈ L0 (µ) and t > 0 we define Z F f (t) ≡ kf kLq (Gt ,µ) =

q

|f (x)| dµ(x) Gt

 q1 .

Real interpolation of Morrey spaces

315

and ∞

Z kf kLMpqu (G,µ) ≡ kf kΦ 1 − 1 ,u (F ) = q

(t

p

1 1 p−q

u dt

kf kLq (Gt ,µ) )

0

t

 u1 .

We define LMpqu (G, µ) to be the set of all f ∈ L0 (µ) for which the quantity kf kLMpqu (G,µ) is finite. If 0 < p ≤ ∞, then we can check that LMpp∞ (G, µ) = Lp (µ) from sup kf kLp (Gt ) = kf kLp (µ) . We can apply our theorem with F0 = t>0

F1 = F given above, since we can check that (F0 , F1 , F ) satisfies the weak quasi-additivity and admits the weakly A-B majorizable decomposition. Theorem 288. Suppose that we have G = (Gt )t>0 such that the Gt ’s are µ-measurable subsets satisfying Gt 6= X for some t > 0, Gt1 ⊂ Gt2 if t1 < t2 [ and Gt = X. Let 0 < θ < 1, 0 < q ≤ p0 , p1 < ∞ and 0 < u < ∞ or t>0

 0 < q ≤ p0 , p1 < u = ∞. Assume p0 6= p1 and define p ≡ Then

p1 (LMpqu0 0 (G, µ), LMqu (G, µ))θ,u 1

=

LMpqu (G, µ).

θ 1−θ + p0 p1

−1 .

Proof Analogous to Theorem 287. We will provide some concrete examples. The local Morrey-type space n 0 n LMϕ qθ (R ) is the space of all functions f ∈ L (R ) with finite quasi-norm Z kf kLMϕqθ ≡

∞

(ϕ(r)r

−n q

θ dr

kf kLq (B(r)) )

0

r

 θ1 .

Theorem 289. Let 0 < θ < 1, 0 < q, u0 , u1 , u ≤ ∞. Also let v be a differentiable positive locally absolutely continuous strictly increasing function on (0, ∞) such that lim v(t) = 0, lim v(t) = ∞. Assume that λ0 , λ1 satt↓0

t↓∞

isfy 0 < λ0 , λ1 < ∞ if u < ∞ and 0 ≤ λ0 , λ1 < ∞ if u = ∞. Set λ ≡ (1 − θ)λ0 + θλ1 . Assume in addition that λ0 6= λ1 . For r > 0 and k = 0, 1, let 1

ϕk (r) ≡ r uk ϕ(r) ≡ r

+n 0 q

1 n u+ q

1

v (r) uk v(r) 1 u

− u1 −λk k

1 −u −λ

v 0 (r) v(r)

.

n ϕ1 n ϕ n 0 Then (LMϕ qu0 (R ), LMqu1 (R ))θ,u = LMqu (R ) with equivalence of norms.

Proof We use Theorem 288. Define Gt ≡ B(v −1 (t)) ⊂ Rn for t > 0. Then F f (t) = kf kLq (B(v−1 (t))) for any t > 0, so that F satisfies the weak quasiadditive inequality and admits the weakly A-B majorizable decomposition.

316

Morrey Spaces

By a change of variables Z ∞ 1
u dt u t−λ kf kLq (B(v−1 (t))) kf kΦλ,q (F ) = t 0 u Z ∞   u1 kf kLq (B(r)) dv(r) = λ v(r) v(r) 0 Z ∞  u dr  u1 1 n n 1 1 r u + q v 0 (r) u v(r)− u −λ r− q kf kLq (B(r)) = r 0 = kf kLMϕqu . We can calculate other norms. Thus, we can use Theorem 288. As we mentioned, in this book, the notion of local Morrey-type spaces is not a mere quest to generalization. This remark is solidified with the real interpolation. Until the end of Section 18.4.2, we work in a σ-finite measure space (X, B, µ) again. Let 1 ≤ p < ∞ and w be a weight function. We now are going to consider the weighted Lebesgue space Lp (X, wdµ) given by Z  p1 p p kf kL (X,wdµ) = |f (x)| w(x)dµ(x) . X p

Then as a special case we have L (µ) = Lp (1dµ). Let G = {Gt }t>0 be as above. We recall that LMpqu (G, µ) is the set of all f ∈ L0 (µ) for which the quantity Z ∞ 1 1 1 dt u kf kLMpqu (G,µ) ≡ (t p − q kf kLq (Gt ,µ) )u < ∞. t 0 We first see that this is realized as a special case of the space LMα pp (G, µ), Z ∞  p1 1 1 dt whose norm is given by kf kLMαpp (G,µ) = (t α − p kf kLp (Gt ) p ) . t 0 Lemma 290. Let w ∈ M+ (µ). We define G = {Gt }t>0 , where Gt ≡ {x ∈ X : tw(x) > 1} for each t > 0. Let 1 ≤ p < ∞ and τ ∈ (0, p1 ). p

1−pτ Then Lp (X, wτ dµ) = LMpp (G, µ) with equivalence of norms. In particup

1−pτ lar, Lp (X, wτ dµ) = LMpp (G, µ).

Proof In fact, for f ∈ L0 (µ) Z p kf k = 1−pτ LMpp

(G,µ)

∞ −τ

(t

kf kLp (Gt ,µ) )

p dt

0

 p1

t

p  p1 dt = |f (x)| dµ(x) t 0 Gt Z  p1 ' |f (x)|p w(x)τ dµ(x) . Z

∞

X

 Z t−τ

p

Real interpolation of Morrey spaces

317

Our result is applicable to (Lp (X, wτ0 dµ), Lp (X, wτ1 dµ))θ,q , which recovers a classical result. Theorem 291. Let 0 < p < ∞, 0 < q ≤ ∞, 0 < θ < 1. Assume that τ0 , τ1 , τ ∈ (0, ∞) satisfy τ0 6= τ1 , τ0 , τ1 , τ ∈ (0, p1 ) and τ = (1 − θ)τ0 + θτ1 . p

1−pτ Then (Lp (X, wτ0 dµ), Lp (X, wτ1 dµ))θ,q = LMpq (G, µ). In particular we can recover the Stein-Weiss theorem:

(Lp (X, wτ0 dµ), Lp (X, wτ1 dµ))θ,p = Lp (X, wτ dµ). Proof This is a special case of Theorem 288 in view of Lemma 290, since p0 6= p1 in Theorem 288 corresponds to τ0 6= τ1 . Remark 13. If w ≡ 1, then (Lp (X, wτ0 dµ), Lp (X, wτ1 dµ))θ,q = Lp (µ), while 1 1 − pτ − = Gt in Lemma 290 equals ∅ for 0 < t ≤ 1 and X for t > 1. Since p p −τ < 0, we have Z ∞ dt t−τ < ∞. t 1 Thus, the right-hand side is Lp (µ) with coincidence of norms. We work in the setting of two weights. Let 0 < p < ∞ be fixed. Let w0 , w1 ∈ L0 (µ) be positive and 0 < τ0 , τ1 < ∞. Write α0 ≡

1 , τ0 − τ1

α1 ≡

1 , τ1 − τ0

β0 ≡

τ1 , τ1 − τ0

β1 ≡

τ0 . τ0 − τ1

Let the family Gτ0 ,τ1 = {Gt,τ0 ,τ1 }t>0 be defined by Gt,τ0 ,τ1 ≡ {x ∈ X : w0 (x)α0 w1 (x)α1 < t}

(t > 0).

Also define dντ0 ,τ1 (x) ≡ w0 (x)β0 w1 (x)β1 dµ(x). Observe that Lemma 290 yields p 1−pτk

LMpp and

(Gτ0 ,τ1 , ντ0 ,τ1 ) = Lp (X, wk dµ)

p 1−pτ (Gτ0 ,τ1 , ντ0 ,τ1 ) ≈ Lp (X, w0 1−θ w1 θ dµ) LMpp

for k = 0, 1 with equivalence of norms, since (w0 (x)α0 w1 (x)α1 )τk w0 (x)β0 w1 (x)β1 = wk (x)

(x ∈ X)

and (w0 (x)α0 w1 (x)α1 )τ w0 (x)β0 w1 (x)β1 = w0 (x)1−θ w1 (x)θ

(x ∈ X).

The following result gives general information on the interpolation of weighted Lebesgue spaces.

318

Morrey Spaces

Theorem 292. Let 0 < p, q ≤ ∞, 0 < τ0 , τ1 < ∞ and 0 < θ < 1. Assume τ0 6= τ1 . Set 1 , τ0 − τ1

α0 ≡

α1 ≡

1 , τ1 − τ0

β0 ≡

τ1 , τ1 − τ0

β1 ≡

τ0 τ0 − τ1

and τ ≡ (1 − θ)τ0 + θτ1 . Define Gτ0 ,τ1 ≡ {Gt,τ0 ,τ1 }t>0 by Gt,τ0 ,τ1 ≡ {x ∈ X : w0 (x)α0 w1 (x)α1 < t} β0

(t > 0).

β1

Also define dντ0 ,τ1 (x) ≡ w0 (x) w1 (x) dµ(x). Then p 1−pτ (Lp (X, w0 dµ), Lp (X, w1 dµ))θ,q = LMpq (Gτ0 ,τ1 , ντ0 ,τ1 ).

In particular, (Lp (X, w0 dµ), Lp (X, w1 dµ))θ,p = Lp (X, w0 1−θ w1 θ dµ). Proof This is a special case of Theorem 287. We seek a generalization of the above results to the case of different weights and different integrability indices. Let w0 , w1 ∈ M+ (µ) on X, and let 0 < p0 , p1 < ∞ with p0 6= p1 . Let t > 0 and f ∈ L0 (µ). Define   p0 p1  1  w0 (x) p1 −p0 w0 (x)p0 p1 −p0 h1 (x) ≡ , h2 (x) ≡ (x ∈ X) w1 (x) w1 (x)p1 and

Z Hf (t) ≡

h1 (x)dµ(x),

(18.2)

Gt (f )

where Gt (f ) ≡ {x ∈ X : t|f (x)| > h2 (x)} . Lemma 293. Let 0 < p0 , p1 , p ≤ ∞ and w0 , w1 be weights. Assume (p0 − 1−θ 1 θ = p)(p1 − p) ≤ 0. Define θ by + . For H given above, define p p0 p1 1 1 1 F0 ≡ H p0 , F1 ≡ H p1 , F ≡ H p . Then for k = 0, 1 Φ1,pk (Fk ) = Lpk (X, wk pk dµ),

Φ1,p (F ) = Lp (X, w0 p(1−θ) w1 pθ dµ).

Proof Write u ≡ p1 − p0 . Arithmetic shows p0 − p p = −θ , u p1

p − p1 p = θ − 1. u p1

Let f ∈ L0 (µ), and let k = 0, 1. Then kf kΦ1,pk (Fk ) (Z ∞

=

t

−pk

0

(Z

!

Z

h1 (x)dµ(x) Gt (f )

∞

) p1

k

Z

=

χn 0

dt t

X

x∈X

p1 : t|f (x)|w1 (x) u

p0 >w0 (x) u

w (x) o 0 w1 (x)

p0 p1 u p0 p1 u

! dµ(x)

dt tpk +1

) p1

k

.

Real interpolation of Morrey spaces

319

By Fubini’s theorem, kf kΦ1,pk (Fk ) ' kf kLpk (wk pk ) . Likewise, kf kΦ1,p (F ) (Z ∞ Z = χn 0

X

x∈X : t|f (x)|w1 (x)

p1 u

>w0 (x)

p0 u

w (x) o 0 w1 (x)

p0 p1 u p0 p1 u

! dµ(x)

dt

) p1

tp+1

 p1 |f (x)|p w0 (x)p(1−θ) w1 (x)pθ dµ(x) ,

Z ∼ X

We can thus describe (Lp0 (X, w0 p0 dµ), Lp1 (X, w1 p1 dµ))θ,q for any pair of weights. Theorem 294. Let 0 < p0 , p1 < ∞, p0 6= p1 , 0 < q ≤ ∞ and 0 < θ < 1. Define p by 1−θ θ 1 = + . p p0 p1 1

Then (Lp0 (X, w0 p0 dµ), Lp1 (X, w1 p1 dµ))θ,q = Φ1,q (H p ) with H as in (18.2). In particular, if q = p, then we can recover the Stein-Weiss theorem: (Lp0 (X, w0 p0 dµ), Lp1 (X, w1 p1 dµ))θ,p = Lp (X, w0 p(1−θ) w1 pθ dµ). Proof Simply combine Theorem 279 and Lemma 293.

18.4.3

Exercises

Exercise 80. [300, Theorem 5.2(ii) and (iii)] Let 1 ≤ q0 < p0 < ∞, 1 ≤ θ q1 < p1 < ∞ and 0 < θ < 1. Suppose p and q satisfy p1 = 1−θ p0 + p1 and 1 θ 1−θ q = q0 + q1 . (1) Show that (LMpq 0 (Rn ), LMpq 1 (Rn ))θ,∞ = LMpq (Rn ) if q0 = q1 . (2) Show that (LMpq00 (Rn ), LMpq11 (Rn ))θ,∞ = WLMpq (Rn ) if q0 6= q1 .

18.5

Notes

Section 18.1 General remarks and textbooks in Section 18.1 Lions and Peetre initially considered the real interpolation in the works [267, 268]. The textbook [28, Chapter 3] covers the topic exhausitively. See also standard textbooks [25, Chapters 5], [135, Chapter 1], [234], [288, §1.3], [405, §4.2.2] and [460, Chapter V].

320

Morrey Spaces

Section 18.1.1 See [79] in particular, [79, Theorem 4] for Theorem 266. Section 18.1.2 Theorem 270 attributes to Peetre.

Section 18.2 General remarks and textbooks in Section 18.2 See standard textbooks [25] and [28]. Section 18.2.1 Burenkov, Nursultanov and Chigambayeva obtained the interpolation of Φ↑λ,q (0, ∞); see [57, Theorem 2] for Theorem 275. Section 18.2.2 Burenkov, Nursultanov and Chigambayeva applied Theorem 275 to obtain various interpolation formulae; see [57, Theorem 4] for Theorems 277, 278 and 279.

Section 18.3 General remarks and textbooks in Section 18.3 The textbooks [25, Chapter 5] and [28, §5.1] cover the topic. Section 18.3.1 To obtain a description of the real interpolation of Lebesgue spaces in Theorem 282, we followed [28, 5.2.1. Theorem]. Section 18.3.2 Interpolation of Lorentz spaces, Theorem 284, can be found in [61] and [28, 5. 3. 1. Theorem].

Section 18.4 General remarks and textbooks in Section 18.4 See [6, Chapter 11] for interpolations of Morrey spaces. See [18, Corollary 4.3] for the real interpolation of L1 and the Morrey space based on a capacity, which is based on the equivalence of the K-functional [18, Theorem 5.1].

Real interpolation of Morrey spaces

321

Section 18.4.1 The interpolation of Morrey spaces dates back to the 1960’s. Campanato and Murthy [65], Spanne [454] and Peetre [369] obtained some results on the boundedness of operators in Morrey spaces and interpolation spaces. It should be noted that Blasco, Ruiz and Vega constructed an example similar to Theorem 285 [32, Theorem 2] and [386, Theorem]. See the works by Burenkov and Nursultanov [56], Nakai and Sobukawa [349] and by Mastylo and Sawano [300] for real interpolation of local Morrey spaces. We refer to [56, Theorem] and [300, Theorem 5.2(ii)] for Theorem 287. Theorem 286 is due to Lemari´eRieusset [254, Theorem 3(iv) and (v)]. Section 18.4.2 Burenkov, Nursultanov and Chigambayeva used general Morrey-type spaces; see [57, Theorem 5], [57, Corollary 3], [57, Theorem 7], [57, Theorem 7] and [57, Theorem 8] for Theorems 288, 289, 291, 292 and 294, respectively. See [40, 300] for more about the interpolation of local Morrey spaces and Morrey-type spaces.

Chapter 19 Complex interpolation of Morrey spaces

We present another interpolation method. One of the important aspects in the interpolation theory is that the interpolation method is a tool to obtain the boundedness of operators. Interpolation of Banach spaces is a technique to create another Banach space, which is written as H(X0 , X1 ). Here, H indicates a certain method. Usually, it is related to probability theory, theory of function spaces and complex analysis. Section 19.1 formulates complex interpolation of Banach spaces and Section 19.2 moves on to Morrey spaces.

19.1

Complex interpolation

The complex interpolation method is a method to create a Banach space H(X0 , X1 ) based on complex analysis. When it comes to complex analysis, we think of functions whose value assumes C. Here, generally speaking, this is insufficient: We have to consider functions defined in a complex domain Ω ⊂ C and taking their values in a Banach space X . Despite this generalization, everything undergoes a minor change. For example, let Ω be a domain in C. A function f : Ω → X is holomorphic if for all z ∈ Ω we can find k(z) ∈ X such that

f (z + h) − f (z)

= 0. lim − k(z)

h→0 h X We collect the estimate of the remainder term of the Taylor expansion in Section 19.1.1 to be able to argue based on the above definition of the differentiation. Apart from a general fact on interpolation, let us review the complex interpolations defined by Calder´on in 1964. The main motivation is the three-line lemma proved in Section 19.1.2. We deal with the estimate for the holomorphic functions defined in the strap in Section 19.1.2. In 1964, Calder´on defined two complex interpolation functors, whose precise definition is given later. The first method is widely known. It turns out that the outputs by the first and the second interpolation functors, which will be defined in Section 19.1.3, of Lebesgue spaces are the same. Therefore, many people forgot the 323

324

Morrey Spaces

existence of the second complex interpolation functor. Recently, some people committed a mistake for the use of the first interpolation functor. One of the reasons can be explained by the results in Section 19.1.4. However as is pointed out by Lemari´e-Rieusset, the output of Morrey spaces by the first complex interpolation functor is not what many people expected. Lemari´e-Rieusset also pointed out that it is the second functor that creates the desired space. We show the relation between the first and the second complex interpolation functors in Section 19.1.5 and Section 19.1.6.

19.1.1

Taylor expansion of exp

We deal with the complex function z 7→ ez here. Later we will need to keep track of the reminder term of a Taylor expansion. We summarize the observation that we need later. Lemma 295. Let w ∈ C, A > 0, ε > 0 and τ > 0. Assume that ε > 2|w|τ . Write ε

ε

| log A|A 2 , 2 0 1, we have −ε exp(wτ log A) − 1 A − 1 ≤ C∞ (ε)τ |w|. wτ log A Proof By the fundamental theorem of calculus, we have  Z wτ log A Z z1 exp(wτ log A) − 1 − wτ log A = exp(z2 )dz2 dz1 . 0

0

Thus, we have exp(wτ log A) − 1 − 1 wτ log A  Z |wτ log A| Z t1 1 exp(|wτ log A|)dt2 dt1 ≤ |wτ log A| 0 0 1 = |wτ log A| exp(|wτ log A|). 2

(19.1)

(19.2)

Complex interpolation of Morrey spaces

325

Since |w|τ < 2ε , we have (

exp(−|w|τ log A), 0 < A ≤ 1, exp(|w|τ log A), A > 1,

(

exp(− 2ε log A), 0 < A ≤ 1, exp( 2ε log A), A > 1.

exp(|wτ log A|) = ≤

We thus have (19.1) and (19.2). The following is also an auxiliary estimate we will need: Lemma 296. Let 1 ≤ q1 < q < q0 ≤ ∞ and f ∈ L0 (Rn ). Define functions q : S → C, F : S → L0 (Rn ) and G : S → L0 (Rn ) by 1 1−z z = + , q(z) q0 q1  q log |f | q(z)

(z ∈ S),

(19.4)

F (θ + (z − θ)t)dt

(z ∈ S),

(19.5)

 F (z) = sgn(f ) exp and Z G(z) = (z − θ)

(19.3)

1

0

respectively. Define F0 (z) ≡ F (z)χ[0,1] (|f |),

F1 (z) ≡ F (z)χ(1,∞] (|f |),

(19.6)

G0 (z) ≡ G(z)χ[0,1] (|f |),

G1 (z) ≡ G(z)χ(1,∞] (|f |).

(19.7)

and (1) For any z ∈ S, we have q

q

|G(z)| ≤ (1 + |z|)(|f | q0 + |f | q1 ). (2) For any z ∈ C with ε < q1 . By Theorem 316, we have n n ϕ1 n ϕ0 n ϕ1 0 [U Mϕ q0 (R ), U Mq1 (R )]θ ⊆ [Mq0 (R ), Mq1 (R )]θ   ϕ n ϕ = f ∈ Mq (R ) : lim kχ[0,a) (|f |)f kMq = 0 . a↓0

n n ϕ0 n ϕ1 0 Let g ∈ [U Mϕ q0 (R ), U Mq1 (R )]θ and ε > 0. Choose gε ∈ U Mq0 (R ) ∩ n ϕ1 n ϕ U Mq1 (R ) such that kg − gε k[U Mϕq 0 ,U Mϕq 1 ]θ < ε. Since U Mq00 (R ) ∩ 0 1 n n ϕ n ϕ 1 U Mϕ q1 (R ) ⊆ U Mq (R ), we have gε ∈ U Mq (R ). From n ϕ1 n ϕ0 n ϕ1 n ϕ n 0 [U Mϕ q0 (R ), U Mq1 (R )]θ ⊆ [Mq0 (R ), Mq1 (R )]θ ⊆ Mq (R ), n it follows that kg − gε kMϕq . ε. Hence, g ∈ U Mϕ q (R ). ϕ n ϕ n Conversely, let f ∈ Mq (R ) ∩ U Mq (R ) be such that

lim kχ[0,a) (|f |)f kMϕq = 0.

(19.11)

lim kχ(a−1 ,∞] (|f |)f kMϕq = 0.

(19.12)

a↓0

We have a↓0

Thus, combining (19.11) and (19.12), we obtain kf − χ[a,a−1 ] (|f |)f kMϕq ≤ kχ[0,a) (|f |)f kMϕq + kχ(a−1 ,∞] (|f |)f kMϕq ↓ 0

350

Morrey Spaces

n ϕ n as a ↓ 0. Observe also that χ[a,a−1 ] (|f |)f ∈ Mϕ q (R ) ∩ U Mq (R ) thanks to the lattice property of U . As a result, we may assume that f = χ{a 0. By Lemma 304, Hk (θ) ∈ [U Mϕ q0 (R ), U Mq1 (R )]θ . ϕ0 n ϕ1 n ϕ0 n ϕ1 n Since U Mq0 (R ) ∩ U Mq1 (R ) is dense in [U Mq0 (R ), U Mq1 (R )]θ thanks n ϕ1 n 0 to Theorem 309, we can find Jk (θ) ∈ U Mϕ q0 (R ) ∩ U Mq1 (R ) such that

kHk (θ) − Jk (θ)k[U Mϕq 0 (Rn ),U Mϕq 1 (Rn )]θ < ε. 0

Since have

n n ϕ1 0 [U Mϕ q0 (R ), U Mq1 (R )]θ

⊆

1

n ⊆ Mϕ q (R ), we

n n ϕ1 0 [Mϕ q0 (R ), Mq1 (R )]θ

kHk (θ) − Jk (θ)kMϕq . kHk (θ) − Jk (θ)k[U Mϕq 0 ,U Mϕq 1 ]θ < ε. 0

This proves Hk (θ) ∈

U Mqϕ00 (Rn )

∩

1

n Mϕ q (R ) 1 n U Mϕ . q1 (R )

This is a general interpolation theorem on the complex interpolation of the second kind for a class of closed subspace. Theorem 325. Suppose that we have 3 positive real parameters θ, q0 and q1 as well as 2 functions ϕ0 and ϕ1 satisfying θ ∈ (0, 1), q0 , q1 ≥ 1, ϕ0 q0 = ϕ1 q1 and q0 6= q1 . Define ϕ and q so that ϕ = ϕ0 1−θ ϕ1 θ

and

1−θ θ 1 = + . q q0 q1

Assume also that a linear subspace U (Rn ) ⊂ L0 (Rn ) enjoys the lattice propn n θ ϕ n ϕ1 0 erty. Then [U Mϕ q0 (R ), U Mq1 (R )] = U ./ Mq (R ). Proof In view of Lemma 321, we only need to show n ϕ1 n θ ϕ n 0 [U Mϕ q0 (R ), U Mq1 (R )] ⊆ U ./ Mq (R ).

(19.17)

n ϕ1 n θ 0 0 Let f ∈ [U Mϕ q0 (R ), U Mq1 (R )] . Then f is realized as G (θ) = f for some n ϕ1 n 0 G ∈ G(U Mϕ q0 (R ), U Mq1 (R )). Consider Hk (z) in Lemma 19.20 for z ∈ S n and k ∈ N. By Lemmas 322 and 324, we have Hk (θ) ∈ U Mϕ q (R ). Since Hk (θ) 0 ϕ0 n ϕ1 n converges to G (θ) = f in Mq0 (R ) + Mq1 (R ), by Lemma 323, it follows n that f ∈ U ./ Mϕ q (R ). n As a particular case U (Rn ) = L∞ c (R ), we have the following result:

Corollary 326. Suppose that θ ∈ (0, 1), 1 ≤ q0 < ∞, 1 ≤ q1 < ∞, and θ ϕ0 q0 = ϕ1 q1 . Define ϕ ≡ ϕ1−θ ϕθ1 and 1q ≡ 1−θ 0 q0 + q1 . Then \ n n 0 fϕ fϕ1 n θ fϕ n [M {f ∈ Mϕ q0 (R ), Mq1 (R )] = q (R ) : χ[a,b] (|f |)f ∈ Mq (R )}. 0 0. Choose gε ∈ L0c (Rn ) ∩ Mϕ q (R ) such that kg − gε kMq < ε. f n Define fε ≡ gε . Then show that fε ∈ L0c (Rn ) ∩ Mϕ q (R ) and that g kf − fε kMϕq ≤ kg − gε kMϕq < ε. ∗

n (2) Show that f ∈ Mϕ q (R ).

Exercise 86. Use Theorem 313 to prove Theorem 314.

19.3

Notes

Section 19.1 General remarks and textbooks in Section 19.1 See the textbooks [25, Chapters 5], [28, Chapter 4] and [405, §4.2.3] for fundamental facts on complex interpolation. The interpolation of operators go back to Schur [427] in 1911. The Riesz–Thorin theorem goes back to [382, 478]. These works concern the interpolation for linear operators. See the papers by Calder´ on and Zygmund [62] and by Stein [456]. The textbook [28, Chapter 3] covers the topic exhausitively. See [28, §1.1] for the three-line lemma. For the interpolation of Orlicz spaces, see the textbook of Maligranda [295, Chapters 14 and 15]. Section 19.1.1 The results in Section 19.1.1 are auxiliary and standard. Here we content ourselves with [186] for this section. Section 19.1.2 See the textbook [543] of Zhu for the generalized three-line lemma; see [543, Corollary 2.3] for Lemma 302. Lemma 300 is known as the Hirschman lemma; see See also [136, Lemma 1.3.8, Exercise 1.3.8.].

360

Morrey Spaces

Section 19.1.3 For complex interpolation, we refer to [28]. Gustavsson, Peetre, Berezhnoi, Shestakov, Nilsson [359] and Ovchinikov established that X0 1−θ X1 θ ,→ hX0 , X1 , θi ,→ (X0 1−θ X1 θ )∼ , where for X X ∼ denotes the Gagliardo closure defined in Definition 49. See [300, Corollary 3.10] for Theorem 305. By the use of the K¨ othe dual, it is not so hard to see that the θ−1 convexification, considered in Example 100, is a Banach space; see [296, Proposition 1]. The three-line theorem is a variant of the three-circle theorem by Hadamard [183]. Since it initially appeared explicitly in Doetsch [87], it is sometimes called the Doetsch three-line theorem; see Theorem 303. Section 19.1.4 We refer to [60, §9.3]. See [60, §32.1] for Theorem 308. The theory of complex interpolation of Morrey spaces went to a new stage by the discovery of Lemari´e-Rieusset. He found out that the first and second complex interpolation spaces are different [254, 255]. See [255, Theorem 2] for Example 102. Section 19.1.5 Bergh showed that X0 ∩ X1 is dense in [X0 , X1 ]θ under [X0 , X1 ]θ [27]. See Theorem 309. Section 19.1.6 See [300, Theorem 3.4] for Theorem 310. Remark 15 is due to [428]. The textbook [28, §5.1] covers the topic.

Section 19.2 General remarks and textbooks in Section 19.2 Calder´ on defined two complex interpolation spaces in [60, §5, §6]. See the textbook [6, Chapter 11] of Adams for the introduction of interpolations of Morrey spaces. Section 19.2.2 Lemari´e-Rieusset proved the following [254]: If 0 < θ < 1,

1−θ θ 1 = + , p p0 p1

1 1−θ θ = + , q q0 q1

q0 q1 6= , p0 p1

Complex interpolation of Morrey spaces

361

then [Mpq00 (Rn ), Mpq11 (Rn )]θ 6= Mpu (Rn ). See [255, Theorem 2] for Corollary 317. Based on the observation by Lemari´eRieusset, Lu, Yang, Yuan and Sickel [254, 255, 289, 532]. Hakim and Sawano investigated the interpolation spaces of the closed subspaces of Morrey spaces. Hakim and Sawano proved Theorem 314 in [185]. See [186] for Theorem 316. See [187] for the complex interpolation of diamond spaces. See [53] for the complex interpolation of predual spaces. Section 19.2.1 The interpolation of Morrey spaces dates back to the 1960’s. See Stampacchia [455], Campanato and Murthy [65], Spanne [431] and Peetre [369]. In particular, the results by Stampacchia and Peetre can be found in [6, Chapter 11]. Despite a counterexample by Blasco, Ruiz and Vega [32, 386], the interpolation theory of Morrey spaces progressed very much. As for the real interpolation results, Burenkov and Nursultanov obtained an interpolation result in local Morrey spaces [56]. Nakai and Sobukawa generalized their results to Buw setting [349], where Buw denotes the weighted Bσ space. We made significant progress in the complex interpolation theory of Morrey spaces. Cobos, Peetre and Persson pointed out that [Mpq00 (Rn ), Mpq11 (Rn )]θ ⊂ Mpq (Rn ) as long as 1 ≤ q0 ≤ p0 < ∞, 1 ≤ q1 ≤ p1 < ∞ and 1 ≤ q ≤ p < ∞ satisfy 1−θ 1 θ = + , p p0 p1

1−θ 1 θ = + . q q0 q1

(19.1)

See in [77, p. 35]. As is shown in [254, Theorem 3(ii)], when an interpolation functor F satisfies F [Mpq00 (Rn ), Mpq11 (Rn )] = Mpq (Rn ) under the condition (19.1), then q0 q1 = p0 p1

(19.2)

holds. Lemari´e-Rieusset showed this assertion by using the counterexample by Ruiz and Vega [386, p. 408]. Lemari´e-Rieusset also established that we can choose the second complex interpolation functor by Calder´on in [60] in 1964. Meanwhile, as for the interpolation result under (19.1) and (19.2) by using the first complex interpolation functor by Calder´on [60], Lu, Yang and Yuan obtained the following description: n Mp q (R )

[Mpq00 (Rn ), Mpq11 (Rn )]θ = Mpq00 (Rn ) ∩ Mpq11 (Rn )

362

Morrey Spaces

in [289, Theorem 1.2]. They also extended this result by placing themselves in the setting of a metric measure space. Their technique is calculating the Calder´ on product. Let 0 < θ < 1. Spanne and Peetre established that if F is an interpolation functor of type θ, namely, kT kF (X0 ,Y0 )→F (X1 ,Y1 ) ≤ C(kT kX0 →Y0 )1−θ (kT kX1 →Y1 )θ , then T : F (X0 , X1 ) → Mpq (Rn ) is bounded, where 1−θ θ 1 ≡ + , p p0 p1

1 1−θ θ ≡ + . q q0 q1

We are led to the following problem: Let T ∈ B(Mpq00 (Rn ), Y0 ) ∩ B(Mpq11 (Rn ), Y1 ). Then do we have T ∈ B(Mpq (Rn ), F (Y0 , Y1 ))? This was disproven by Ruiz, Vega and Blasco. In this connection, we also have; [Mpq00 (Rn )]1−θ [Mpq11 (Rn )]θ ⊂ Mpq (Rn ). In fact, in the general case [Mpq00 (Rn ), Mpq11 (Rn )]θ ( Mpu (Rn ) except in two trivial cases; p0 = q0 and p1 = q1 (Lebegsue space case) and p0 = p1 and q0 = q1 (no intermediate space case); see [254, 255] by Lemari´eRieusset and [447] by Sickel, Yuan and Yang. See [301] for Theorem 313, which gives a general formula of complex interpolation of Morrey spaces. Meskhi, Rafeiro and Zaighum worked in the setting of general metric spaces in [310, 311]. Section 19.2.3 We compare Theorem 325 with our previous result. Assume that inf ϕ > 0. According to [185, Theorem 5.12], n 0 fϕ fϕ1 n θ [M q0 (R ), Mq1 (R )] \ n e∞ n fϕ n = {f ∈ Mϕ q (R ) ∩ L (R ) : χ(a,b) (|f |)f ∈ Mq (R )}.

(19.3)

0