Function Spaces, 1 [2nd rev. and ext. ed.] 9783110250428, 9783110250411

Table of contents :
Preface
1 Preliminaries
1.1 Vector space
1.2 Topological spaces
1.3 Metric, metric space
1.4 Norm, normed linear space
1.5 Modular spaces
1.6 Inner product, inner product space
1.7 Convergence, Cauchy sequences
1.8 Density, separability
1.9 Completeness
1.10 Subspaces
1.11 Products of spaces
1.12 Schauder bases
1.13 Compactness
1.14 Operators (mappings)
1.15 Isomorphism, embeddings
1.16 Continuous linear functionals
1.17 Dual space, weak convergence
1.18 The principle of uniform boundedness
1.19 Reflexivity
1.20 Measure spaces: general extension theory
1.21 The Lebesgue measure and integral
1.22 Modes of convergence
1.23 Systems of seminorms, Hahn-Saks theorem
2 Spaces of smooth functions
2.1 Multiindices and derivatives
2.2 Classes of continuous and smooth functions
2.3 Completeness
2.4 Separability, bases
2.5 Compactness
2.6 Continuous linear functionals
2.7 Extension of functions
3 Lebesgue spaces
3.1 Lp-classes
3.2 Lebesgue spaces
3.3 Mean continuity
3.4 Mollifiers
3.5 Density of smooth functions
3.6 Separability
3.7 Completeness
3.8 The dual space
3.9 Reflexivity
3.10 The space L8
3.11 Hardy inequalities
3.12 Sequence spaces
3.13 Modes of convergence
3.14 Compact subsets
3.15 Weak convergence
3.16 Isomorphism of Lp(O) and Lp(0, µ(O))
3.17 Schauder bases
3.18 Weak Lebesgue spaces
3.19 Remarks
4 Orlicz spaces
4.1 Introduction
4.2 Young function, Jensen inequality
4.3 Complementary functions
4.4 The Δ2-condition
4.5 Comparison of Orlicz classes
4.6 Orlicz spaces
4.7 Hölder inequality in Orlicz spaces
4.8 The Luxemburg norm
4.9 Completeness of Orlicz spaces
4.10 Convergence in Orlicz spaces
4.11 Separability
4.12 The space EΦ(Ω)
4.13 Continuous linear functionals
4.14 Compact subsets of Orlicz spaces
4.15 Further properties of Orlicz spaces
4.16 Isomorphism properties, Schauder bases
4.17 Comparison of Orlicz spaces
5 Morrey and Campanato spaces
5.1 Introduction
5.2 Marcinkiewicz spaces
5.3 Morrey and Campanato spaces
5.4 Completeness
5.5 Relations to Lebesgue spaces
5.6 Some lemmas
5.7 Embeddings
5.8 The John-Nirenberg space
5.9 Another definition of the space JN(Q)
5.10 Spaces Np;λ(Q)
5.11 Miscellaneous remarks
6 Banach function spaces
6.1 Banach function spaces
6.2 Associate space
6.3 Absolute continuity of the norm
6.4 Reflexivity of Banach function spaces
6.5 Separability in Banach function spaces
7 Rearrangement-invariant spaces
7.1 Nonincreasing rearrangements
7.2 Hardy-Littlewood inequality
7.3 Resonant measure spaces
7.4 Maximal nonincreasing rearrangement
7.5 Hardy lemma
7.6 Rearrangement-invariant spaces
7.7 Hardy-Littlewood-Pólya principle
7.8 Luxemburg representation theorem
7.9 Fundamental function
7.10 Endpoint spaces
7.11 Almost-compact embeddings
7.12 Gould space
8 Lorentz spaces
8.1 Definition and basic properties
8.2 Embeddings between Lorentz spaces
8.3 The associate space
8.4 The fundamental function
8.5 Absolute continuity of norm
8.6 Remarks on || · ||1;∞
9 Generalized Lorentz-Zygmund spaces
9.1 Measure-preserving transformations
9.2 Basic properties
9.3 Nontriviality
9.4 Fundamental function
9.5 Embeddings between Generalized Lorentz-Zygmund spaces
9.6 The associate space
9.7 When Generalized Lorentz-Zygmund space is Banach function space
9.8 Generalized Lorentz-Zygmund spaces and Orlicz spaces
9.9 Absolute continuity of norm
9.10 Lorentz-Zygmund spaces
9.11 Lorentz-Karamata spaces
10 Classical Lorentz spaces
10.1 Definition and basic properties
10.2 Functional properties
10.3 Embeddings
10.3.1 Embeddings of type Λ ➥ Λ
10.3.2 Embeddings of type Λ ➥ Γ
10.3.3 Embeddings of type Γ ➥ Λ
10.3.4 Embeddings of type Γ ➥ Γ
10.3.5 The Halperin level function
10.3.6 Embeddings of type Γp,∞ (v) ➥ Λ^ (w)
10.3.7 The single-weight case Γ 1,∞(v) ➥ Λ1(ν)
10.4 Associate spaces
10.5 Lorentz and Orlicz spaces
10.6 Spaces measuring oscillation
10.7 The missing case
10.8 Embeddings
10.8.1 Embeddings of type S ➥ S
10.8.2 Embeddings of type Γ ➥ S and S ➥ Γ
10.8.3 Embeddings of type Λ ➥ S and S ➥ Λ
11 Variable-exponent Lebesgue spaces
11.1 Introduction
11.2 Basic properties
11.3 Embedding relations
11.4 Density of smooth functions
11.5 Reflexivity and uniform convexity
11.6 Radon-Nikodým property
11.7 Daugavet property
Bibliography
Index

Citation preview

De Gruyter Series in Nonlinear Analysis and Applications 14 Editor in Chief Jürgen Appell, Würzburg, Germany Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Karl-Heinz Hoffmann, Munich, Germany Mikio Kato, Kitakyushu, Japan Umberto Mosco, Worcester, Massachusetts, USA Louis Nirenberg, New York, USA Boris N. Sadovsky, Voronezh, Russia Alfonso Vignoli, Rome, Italy Katrin Wendland, Freiburg, Germany

Luboš Pick Alois Kufner Oldˇrich John Svatopluk Fuˇcík

Function Spaces Volume 1

2nd Revised and Extended Edition

De Gruyter

Mathematics Subject Classification 2010: 46E30, 46E35, 46E05, 47G10, 26D10, 26D15, 46B70, 46B42, 46B10.

ISBN 978-3-11-025041-1 e-ISBN 978-3-11-025042-8 ISSN 0941-813X

Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress.

Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

© 2013 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen 1 Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

The book “Function Spaces” [126], published in 1977 by Academia Publishing House of the Czechoslovak Academy of Sciences in Prague and the Noordhoff International Publishing in Leyden, proved over several decades to be a useful tool for specialists working in many different areas of mathematics and its applications. It has, though, for quite some time been unavailable. Since the 1970s, many other books dedicated to the study of function spaces and related topics have appeared. Nevertheless, we saw signs that a new edition of this book could be useful, which of course would be revised according to the rapid development in the field of function spaces over the past 35 years and upgraded in part by a number of new results. The current book is an attempt to make a step in this direction. Thanks to the effort spent by the de Gruyter Publishing House, the three authors signed below took upon the task. They used as their point of departure the initial book, thus, the current version now has four authors. It turned out during the preparation of the material for the new edition that the upgraded text is too long for a single monograph. Consequently, we decided to split the material into two volumes. The first volume is devoted to the study of function spaces, based on intrinsic properties of a function such as its size, continuity, smoothness, various forms of control over the mean oscillation, and so on. The second volume will be dedicated to the study of function spaces of Sobolev type, in which the key notion is the weak derivative of a function of several variables. During almost a century of their existence, Lebesgue spaces have constantly played a primary role in analysis. However, it has been known almost from the very beginning that the Lebesgue scale is not sufficiently general to provide a satisfactory description of fine properties of functions required by practical tasks. This was noted during the early 1920s by Kolmogorov, Zygmund, Titchmarsh and others, mostly in connection with research of properties of operators on function spaces. Thus, naturally, during the first half of the twentieth century, new fine scales of function spaces have been introduced. The efforts of Young, Orlicz, Hardy, Littlewood, Zygmund, Halperin, Köthe, Marcinkiewicz, Lorentz, Luxemburg, Morrey, Campanato and many others resulted in the development of a powerful and qualitatively new mathematical discipline of function spaces. This text is intended to be a motivated introduction to the subject of function spaces. It contains important basic information on various kinds of function spaces such as their functional-analytic or measure-theoretic properties, as well as their important

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characteristics such as mutual embeddings, duality relations and so on. Hence, it can be considered as a reference book and a pointer to other sources. Summary of the text The text opens with Chapter 1, which has a purely preliminary character. It contains basic information about vector spaces, topological, metric, normed and modular spaces as well as important ingredients from classical functional analysis. In comparison with the first edition, this chapter was enlarged considerably. In particular, a thorough treatment of the theory of measure and integral was added. There are of course many possible sources available for this purpose. We mostly use the book of Rana [185], where the interested reader will find detailed proofs and many further details. In addition, some new material was added on topological spaces and modular spaces. In Chapter 2 we present just a very short elementary discussion about spaces of continuous and smooth functions, including Hölder and Lipschitz spaces. Chapter 3 contains the material concerning Lebesgue spaces from the first edition, only slightly modified and upgraded. Some parts (for instance Section 3.11, devoted to the study of weighted Hardy inequalities) are completely new. A few basic facts about sequence spaces were added, and the topic of modes of convergence was reworked. In Chapter 4 we present the study of basic properties of Orlicz spaces. Again, this chapter as compared to the first edition is only slightly modified. For traditional reasons, the functions studied in Chapters 3 and 4 are assumed to act on an open subset of the Euclidean space. The main reason for this restriction is the further use of these spaces in Sobolev-type spaces, which will be dealt with in the second volume. For the study of these spaces themselves, of course, such restriction is not necessary. In fact, it is later reduced in the frame of the study of general Banach function spaces from Chapter 6 onwards, noting that Lebesgue and Orlicz spaces are particular examples of rearrangement-invariant (r.i.) Banach function spaces. Chapter 5 contains elementary properties of Morrey- and Campanato-type spaces in which the mean oscillation of a function is measured. The functions in this chapter are assumed to act on nice domains in the Euclidean space. This chapter is practically unchanged compared to the first edition. In Chapter 6 we develop the basic theory of Banach function norms and Banach function spaces, which was in some sense a culmination of efforts to cover Orlicz spaces with other types of spaces under a common theme, performed from the 1930s to the 1950s by Orlicz, Lorentz, Luxemburg, Zaanen, Köthe, Halperin and others. This material gradually appeared mostly in works of the mentioned authors. The systematic treatment of this topic can also be found in the following books: Luxemburg and Zaanen [142] or Bennett and Sharpley [14]. Here (except for some minor additions and changes) we follow the excellent exposition of this subject from [14, Chapter 1] almost verbatim (even though our ultimate goal is slightly different, as we are not so much aimed towards interpolation theory).

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In Chapter 7 we turn our attention to function spaces in which the norm is purely determined by the size of a given function. Hence, we study the distribution function and the nonincreasing rearrangement, and develop the resulting structure, the so-called rearrangement-invariant (r.i.) function spaces. This stuff can be in some sense traced back to as far as to the 1880s results of Steiner [214], but its first systematic treatment was done only in the 1930s in the work of Hardy, Littlewood and Pólya [101]. Here we once again mostly follow the book by Bennett and Sharpley [14]. The only significant addition is Section 7.11 in which more recent material concerning the important relation between function spaces called almost-compact embedding is studied in great detail. Our main source in this section is [206]. The knowledge of the notion of the nonincreasing rearrangement of a function now enables us to construct new scales of function spaces that could not exist without it. We have seen that, for example, Lebesgue and Orlicz spaces just happen to be rearrangement-invariant Banach function spaces, but for their initial definition we did not need to know this at all. In subsequent chapters however we study function spaces for whose definition the nonincreasing rearrangement is indispensable. First such class of function spaces is that of two-parameter Lorentz spaces, whose elementary properties are studied in Chapter 8. These spaces gradually emerged during the 1950s and 1960s through the efforts of Lorentz, Calderón, Hunt, Peetre, O’Neil, Weiss, Oaklander and others, mostly in connection with some kind of interpolation. We cover some of their elementary properties, embedding characteristics and duality relations. In the subsequent two chapters we turn our attention to some of the most interesting generalizations of the two-parameter Lorentz spaces. In Chapter 9, we study the important scale of the so-called Lorentz–Zygmund spaces, invented in the 1980s by Bennett, Rudnick and Sharpley, and their generalization to four-parameter spaces, studied in the 1990s by Edmunds, Gurka, Opic and others in connection with various limiting or critical-state problems concerning the action of operators on function spaces. These spaces turned out to be extremely useful in various extremal problems concerning Sobolev inequalities as well as limiting properties of operators. They cover important previously known function classes such as Lebesgue and Lorentz spaces, Zygmund classes of both logarithmic and exponential type, and also the space L1;nI1 , which appeared (under various different symbols) in connection with the optimal target space for a limiting Sobolev inequality in works by Maz’ya, Hansson, Brézis–Wainger and others. As we know very well from our own research, these spaces often arise in various practical tasks. We thus study them here in great detail, concentrating on their embedding and duality characteristics and basic functional properties. We mostly follow [172] and [73]. In Chapter 10, we investigate the so-called classical Lorentz spaces. These spaces are currently known to be of three different types (denoted as of type ƒ,  and S ) and have been widely studied by many authors. We first concentrate on their basic functional properties such as nontriviality, linearity, normability, quasi-normability, lattice

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property and so on, and then we focus on their embedding relations. These questions are highly nontrivial and they require the development of certain new methods some of which we present in detail. This is the case for instance of the inequalities involving suprema, which we just quote, or of the modification of the Hardy inequality which concerns two integral operators rather than just one, which we present in detail. The spaces of type S are known to be of great interest because of the gradient inequalities they govern and their connection to Sobolev and Besov spaces that will be studied in the second volume of this book. In particular, they contain functions with controlled nonincreasing rearrangement of mean oscillation. Here we mix classical results with recent ones scattered in many papers. We follow [33, 34, 35, 56, 89]. At some occasions we do not provide all the details of the proofs because this would increase the length of the text enormously, and restrict ourselves to the hints and references. Finally, Chapter 11 is devoted to a brief account of the generalized Lebesgue spaces with variable exponent. This topic has become very fashionable in recent years and there exist entire schools of top scientists investigating all kinds of variable exponent spaces and their generalizations. We restrict ourselves to some basic information about these spaces, following mostly [122] and [140], and we refer the reader interested in their deep study to the recent book by Diening, Hästö, Harjulehto and R˚užiˇcka [63]. Acknowledgments The main objective of this book is to provide pure mathematicians as well as applied scientists with a handbook containing a summary of results concerning various types of function spaces that might be useful for a broad variety of applications. Therefore, naturally, in most cases we do not claim any originality. We took great effort to give full credit for all the results appearing in the text to its discoverers but, obviously, it is almost an impossible task to trace the origins of every detail. It would be impossible to list all the authors, colleagues and friends who have influenced us in preparation of the second edition of the text. The exposition was partly inspired by important books in the field such as those by Bennett and Sharpley [14], Maz’ya [149], Krasnosel’skii and Rutitskii [123], Rana [185], Diening, Hästö, Harjulehto and R˚užiˇcka [63] and others. Luboš Pick wishes to express his special deep gratitude to Ms. Lenka Slavíková for many stimulating discussions and suggestions that led to great improvement of some parts of the text. We thank Mr. Komil Kuliev, Ms. Guli Kulieva and Ms. Eva Ritterová for their help with the preparation of the manuscript in LATEX. We would like to thank Ms. Anja Möbius from the publishing house De Gruyter for her collaboration and mostly for her patience.

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Finally, the authors would be grateful for critical comments and suggestions for later improvements. The three authors below would like to dedicate this second edition to the memory of the fourth author, the late Professor Svatopluk Fuˇcík, who passed away prematurely not long after the appearance of the first edition of the book [126]. Prague, September 2012

Luboš Pick, Alois Kufner and Oldˇrich John

Contents

Preface

v

1

Preliminaries

1

1.1 Vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3 Metric, metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4 Norm, normed linear space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.5 Modular spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.6 Inner product, inner product space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Convergence, Cauchy sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 Density, separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.10 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.11 Products of spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.12 Schauder bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.13 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.14 Operators (mappings) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.15 Isomorphism, embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.16 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.17 Dual space, weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.18 The principle of uniform boundedness . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.19 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.20 Measure spaces: general extension theory . . . . . . . . . . . . . . . . . . . . . . 22 1.21 The Lebesgue measure and integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.22 Modes of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.23 Systems of seminorms, Hahn–Saks theorem . . . . . . . . . . . . . . . . . . . . . 36

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Spaces of smooth functions

38

2.1 Multiindices and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 Classes of continuous and smooth functions . . . . . . . . . . . . . . . . . . . . . 39 2.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 Separability, bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.6 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 Extension of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3

Lebesgue spaces 3.1

Lp -classes

62

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Mean continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Mollifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5 Density of smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.7 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.8 The dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.9 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.10 The space L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.11 Hardy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.12 Sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.13 Modes of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.14 Compact subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.15 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.16 Isomorphism of Lp ./ and Lp .0; .// . . . . . . . . . . . . . . . . . . . . . . 96 3.17 Schauder bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.18 Weak Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.19 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4

Orlicz spaces

108

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2 Young function, Jensen inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Complementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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4.4 The 2 -condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.5 Comparison of Orlicz classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.6 Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.7 Hölder inequality in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.8 The Luxemburg norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.9 Completeness of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.10 Convergence in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.11 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.12 The space E ˆ ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.13 Continuous linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.14 Compact subsets of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.15 Further properties of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.16 Isomorphism properties, Schauder bases . . . . . . . . . . . . . . . . . . . . . . . . 163 4.17 Comparison of Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 5

Morrey and Campanato spaces

173

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Marcinkiewicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.3 Morrey and Campanato spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.4 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.5 Relations to Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.6 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.7 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.8 The John–Nirenberg space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.9 Another definition of the space JN.Q/ . . . . . . . . . . . . . . . . . . . . . . . . 194 5.10 Spaces Np; .Q/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.11 Miscellaneous remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6

Banach function spaces

203

6.1 Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.2 Associate space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3 Absolute continuity of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.4 Reflexivity of Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.5 Separability in Banach function spaces . . . . . . . . . . . . . . . . . . . . . . . . . 228

xiv 7

Contents

Rearrangement-invariant spaces

237

7.1 Nonincreasing rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.2 Hardy–Littlewood inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.3 Resonant measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.4 Maximal nonincreasing rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . 249 7.5 Hardy lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.6 Rearrangement-invariant spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.7 Hardy–Littlewood–Pólya principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.8 Luxemburg representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 7.9 Fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.10 Endpoint spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.11 Almost-compact embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.12 Gould space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8

Lorentz spaces

301

8.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.2 Embeddings between Lorentz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.3 The associate space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.4 The fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.5 Absolute continuity of norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.6 Remarks on k  k1;1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9

Generalized Lorentz–Zygmund spaces

313

9.1 Measure-preserving transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.3 Nontriviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.4 Fundamental function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.5 Embeddings between Generalized Lorentz–Zygmund spaces . . . . . . . 320 9.6 The associate space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 9.7 When Generalized Lorentz–Zygmund space is Banach function space 353 9.8 Generalized Lorentz–Zygmund spaces and Orlicz spaces . . . . . . . . . . 356 9.9 Absolute continuity of norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 9.10 Lorentz–Zygmund spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 9.11 Lorentz–Karamata spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Contents

10 Classical Lorentz spaces

xv 375

10.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 10.2 Functional properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 10.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Embeddings of type ƒ ,! ƒ . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Embeddings of type ƒ ,!  . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Embeddings of type  ,! ƒ . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Embeddings of type  ,!  . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 The Halperin level function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6 Embeddings of type  p;1 .v/ ,! ƒq .w/ . . . . . . . . . . . . . . . . . 10.3.7 The single-weight case  1;1 .v/ ,! ƒ1 .v/ . . . . . . . . . . . . . . .

388 392 393 396 399 401 404 406

10.4 Associate spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.5 Lorentz and Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 10.6 Spaces measuring oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 10.7 The missing case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 10.8 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.1 Embeddings of type S ,! S . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8.2 Embeddings of type  ,! S and S ,!  . . . . . . . . . . . . . . . . . 10.8.3 Embeddings of type ƒ ,! S and S ,! ƒ . . . . . . . . . . . . . . . . 11 Variable-exponent Lebesgue spaces

427 429 431 434 437

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 11.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 11.3 Embedding relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.4 Density of smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 11.5 Reflexivity and uniform convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 11.6 Radon–Nikodým property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 11.7 Daugavet property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Bibliography

459

Index

472

Chapter 1

Preliminaries

In this chapter we give a survey of concepts and results from functional analysis that will be used in the text. All results are stated without proofs, which can be found in standard monographs.

1.1 Vector space Let X be a set of elements denoted by u; v; w; : : : . Definition 1.1.1. Let addition in X be defined, i.e. to every pair u 2 X, v 2 X there corresponds an element w 2 X called the sum of u and v and denoted by u C v: w D u C v: Definition 1.1.2. Let multiplication by scalars be defined in X, i.e. to every real number  (called a scalar) and every u 2 X there corresponds an element w 2 X called the -multiple of u and denoted by u (or   u): w D u: Definition 1.1.3. The set X with addition and multiplication by scalars defined in it is called a real vector space if the following axioms are satisfied: (i) u C v D v C u

(symmetry);

(ii) u C .v C w/ D .u C v/ C w

(associativity);

(iii) in X there exists a uniquely determined element denoted by  and called the zero element such that uC Du for every u 2 X; (iv) for each u 2 X there exists a uniquely determined element in X denoted by u such that u C .u/ D  I (v) .u C v/ D u C v,  2 R; (vi) . C /u D u C u, ;  2 R;

2

Chapter 1 Preliminaries

(vii) .u/ D ./u, ;  2 R; (viii) 1u D u; (ix) 0u D  . Definition 1.1.4. In a vector space X, the difference (or subtraction) u  v of two elements u; v 2 X is defined by u  v WD u C .v/: Definition 1.1.5. Let M be a subset of a vector space X. Denote \ ŒM  WD Y; where the intersection is taken over all vector spaces Y  X containing M . Then ŒM  is called the linear hull of M . Obviously, ŒM  is again a vector space. Example 1.1.6. In what follows, the elements of a vector space X will usually be real-valued functions defined on a certain set R: u D u.x/; x 2 R: In this case addition and scalar multiplication are defined as usual: .u C v/.x/ WD u.x/ C v.x/; .u/.x/ WD u.x/: Remark 1.1.7. We shall speak frequently of a linear space or linear set instead of a vector space. Definition 1.1.8. A vector space X is called an algebra if for every ordered pair u 2 X, v 2 X, a product uv is defined as an element of X, which satisfies the following axioms for every w 2 X and all scalars  and : (i) .uv/w D u.vw/; (ii) u.v C w/ D uv C uw; (iii) .u C v/w D uw C vw; (iv) .u/.v/ D ./.uv/.

1.2 Topological spaces Notation 1.2.1. Let X be a nonempty set. Then by exp X we denote the set of all subsets of set X. If A  X, we denote by Ac the complement of A with respect to X, that is, X n A.

Section 1.2 Topological spaces

3

Definition 1.2.2. We say that a couple .X; T / is a topological space if X is a nonempty set and T is a system of subsets of X satisfying the following three conditions: (i) ; 2 T and X 2 T (where ; denotes the empty set). (ii) If G1 2 T and G2 2 T , then G1 \ G2 2 T . (iii) S If A is an index set of arbitrary cardinality and A˛ 2 T for every ˛ 2 A, then ˛2A A˛ 2 T . The subsets of X belonging to T are called open sets in the space X, and the family T is called a topology on X. Definition 1.2.3. Let .X; T / be a topological space. If x 2 X, B 2 T and x 2 B, then we say that B is a neighborhood of x. Definition 1.2.4. Let .X; T / be a topological space. A family B  exp X is called a base for topology T on X if every nonempty open subset of X can be represented as a union over sets from B. Remark 1.2.5. Let .X; T / be a topological space. Then every base B of the topology T has the following properties: (i) For every G1 ; G2 2 B and every point x 2 G1 \ G2 there exists a set G 2 B such that x 2 G  G1 \ G2 . (ii) For every x 2 X there exists a set G 2 B such that x 2 G. Proposition 1.2.6. Let X be a nonempty set. Let B be a collection of subsets of X, which has properties (i) and (ii) of Remark 1.2.5. We denote by T the collection of all those subsets of X, which can be represented as unions of sets from some subcollection of B. Then T is a topology on X and B is a base for this topology. Definition 1.2.7. Let X, B and T be as in Proposition 1.2.6. Then we say that the topology T is generated by the base B. Definition 1.2.8. Let .X; T / be a topological space and let F 2 exp X. We say that F is closed if F c 2 T . Remark 1.2.9. Let .X; T / be a topological space. Denote by F the system of all closed subsets of X. Then it follows from Definition 1.2.2 and De Morgan laws that (i) ; 2 F and X 2 F . (ii) If F1 2 F and F2 2 F , then F1 [ F2 2 F . (iii) T If A is an index set of arbitrary cardinality and F˛ 2 F for every ˛ 2 A, then ˛2A F˛ 2 F .

4

Chapter 1 Preliminaries

Definition 1.2.10. Let .X; T / be a topological space and let A  X. Then the set \ A WD ¹F 2 F I F  Aº is called the closure of A in X (with respect to the topology T ). Theorem 1.2.11. Let .X; T / be a topological space. The closure operator has the following properties: (i) ; D ; (ii) A  A (iii) A [ B D A [ B (iv) .A/ D A Proposition 1.2.12. Let X be a nonempty set. Assume that cl W exp X ! exp X is an operator assigning to every set A 2 exp X some set cl.A/ D A 2 exp X such that the properties (i)–(iv) of Theorem 1.2.11 hold. Then the family T WD ¹X n AI A D cl.A/º

(1.2.1)

is a topology on X. Moreover, the set cl.A/ D A is the closure of A in X with respect to T . Definition 1.2.13. Let X be a nonempty set and let T1 , T2 be two topologies on X. We say that the topology T1 is weaker than T2 (that is, T2 is stronger than T1 ) if T1  T2 . Definition 1.2.14. Let .X; T / and .Y; T 0 / be two topological spaces. A mapping f W X ! Y is called continuous if f 1 .G/ 2 T for every G 2 T 0 . Example 1.2.15. Let X D R, the set of all real numbers. Then the system ´1 μ [ T WD .an ; bn / ; nD1

where .an ; bn / are pairwise disjoint nontrivial open intervals, complemented with ; and R itself, defines a natural topology on R. Notation 1.2.16. Let .X; T / and .Y; T 0 / be two topological spaces. Then by C.X; Y / we denote the set of all continuous mappings from X to Y .

5

Section 1.2 Topological spaces

Definition 1.2.17. Let .X; T / be a topological space and let ¹fn º1 nD1 be a sequence of functions from X to R. Let f W X ! R be a function. We say that ¹fn º1 nD1 is uniformly convergent to f if, for every " > 0, there exists an n0 2 N such that for every n 2 N, n  n0 and for every x 2 X, one has jfn .x/  f .x/j < ". We write f D lim fn : n!1

Our next aim is to develop some natural topologies on C.X; Y /. We begin with the special case when Y D R. Definition 1.2.18. Let .X; T / be a topological space. We then define the operator cl on exp C.X; R/ in the following way: given a set A  C.X; R/, we set ° ± cl.A/ WD f 2 XI f D lim fn for some sequence ¹fn º1 (1.2.2) nD1  A : n!1

Proposition 1.2.19. Let .X; T / be a topological space and let cl be the operator on exp C.X; R/ defined by (1.2.2). Then cl satisfies the properties (i)–(iv) of Theorem 1.2.11. Definition 1.2.20. Let .X; T / be a topological space and let cl be the operator on exp C.X; R/ defined by (1.2.2). Then the topology generated by this closure operator on C.X; R/ through the formula (1.2.1), is called the topology of uniform convergence. Now we shall define a reasonable topology on the set C.X; Y / for an arbitrary pair of topological spaces. Definition 1.2.21. Let X and Y be topological spaces. For every pair of sets A 2 exp X and B 2 exp Y , we denote M.A; B/ WD ¹f 2 C.X; Y /I f .A/  Bº : Let TY be the topology of Y . Further, denote by FX the family of all finite subsets of X. Define next the system 9 8 k = 0 such that the set Br .x/ WD ¹y 2 XI %.x; y/ < rº satisfies Br .x/  G. The set Br .x/ is called an open ball centered at x with radius r. Remark 1.3.3. If .X; %/ is a metric space, then the metric % automatically generates a topology on X, in which open sets are those that are open with respect to %. It is an easy exercise to verify that the axioms of Definition 1.2.2 are satisfied.

1.4 Norm, normed linear space Definition 1.4.1. Let X be a vector space. A nonnegative function defined on X whose value at u 2 X is denoted by kuk is called a norm on X if it satisfies the following axioms: (i) kuk D 0 if and only if u D  ; (ii) kuk D jjkuk for every u 2 X and all scalars  (iii) ku C vk 5 kuk C kvk for every u; v 2 X

(the homogeneity axiom);

(the triangle inequality).

A vector space X endowed with a norm k  k is called a normed linear space; the number kuk is called the norm of u 2 X. Remark 1.4.2. If it is necessary to specify the vector space X on which the norm is defined we use kukX instead of kuk. Sometimes the normed linear space X will be denoted by .X; k  kX /. Definition 1.4.3. If the function k  k satisfies only the axioms (ii), (iii) from Definition 1.4.1 then it is called a seminorm (or sometimes a pseudonorm).

7

Section 1.5 Modular spaces

If the function kk satisfies the axioms (i), (ii) from Definition 1.4.1 and there exists a constant C > 1 such that (iii) becomes (iii0 ) ku C vk 5 C.kuk C kvk/

for every u; v 2 X,

then it is called a quasinorm. Proposition 1.4.4. Every normed linear space is a metric space with the metric % defined by %.u; v/ WD ku  vk; u; v 2 X: Hence, owing to Remark 1.3.3, it is also a topological space. Definition 1.4.5. Let X be a normed linear space, u0 2 X, r > 0. The set B.u0 ; r/ WD ¹u 2 XI ku  u0 k < rº is called an open ball (with center u0 and radius r). A subset M  X is called an open set in X if for every u0 2 M there exists an r D r.u0 / > 0 such that B.u0 ; r/  M . A subset M  X is called a closed set in X if X n M is an open set in X.

1.5 Modular spaces Definition 1.5.1. Let X be a vector space. A function % W X ! Œ0; 1 is called left-continuous if the mapping  7! %.x/ is continuous on Œ0; 1/ in the sense that lim %.x/ D %.x/

!1

for every x 2 X:

A convex and left-continuous function % W X ! Œ0; 1 is called a semimodular on X if (i) %. / D 0; (ii) %.x/ D %.x/ for every x 2 X; (iii) if %.x/ D 0 for every  2 R, then x D  . A semimodular is called a modular on X if %.x/ D 0 if and only if x D  . A semimodular is called continuous if the mapping  7! %.x/ is continuous on Œ0; 1/ for every fixed x 2 X. Definition 1.5.2. Let X be a vector space and let % be a semimodular or a modular on X. Then the space ² ³ X% WD x 2 XI lim %.x/ D 0 !0

is called a semimodular space or a modular space, respectively.

8

Chapter 1 Preliminaries

Notation 1.5.3. Let X be a vector space and let % be a semimodular on X. We then denote by k  k% the functional, given for every x 2 X by ³ ²   1 x 1 : (1.5.1) kxk% WD inf  > 0I %  Remark 1.5.4. Let X be a vector space and let % be a semimodular on X. Then it can be easily shown that the functional k  k% defined by (1.5.1) is a norm on X% . It is also known as the Minkowski functional of the set ¹x 2 XI %.x/  1º. A detailed proof can be found for example in [63, Theorem 2.1.7]. Definition 1.5.5. Let X be a vector space and let % be a semimodular on X. Then the functional k  k% defined by (1.5.1) is called the Luxemburg norm on X% . Remark 1.5.6. Let X be a vector space and let % be a semimodular on X. Then, for every fixed x 2 X, the mapping  7! %.x/ is nondecreasing on Œ0; 1/. Moreover, by the convexity of %, we have ´  %.x/ for every  2 Œ0; 1; %.x/ (1.5.2)  %.x/ for every  2 Œ1; 1/: Proposition 1.5.7. Let X be a vector space, let % be a semimodular on X and let x 2 X. Then %.x/  1 if and only if kxk%  1: Proof. Assume that %.x/  1. Then the definition of the Luxemburg norm also implies that kxk%  1. Now assume that kxk%  1. Then, for every  > 1, one has %.1 x/  1. Since % is left-continuous, we obtain %.x/  1. The proof is complete. Proposition 1.5.8. Let X be a vector space, let % be a semimodular on X and let x 2 X. (i) If kxk%  1, then %.x/  kxk% . (ii) If kxk% > 1, then %.x/  kxk% . (iii) For every x 2 X, kXk%  %.x/ C 1. Proof.

(i) When x D 0, there is nothing to prove. Assume that 0 < kxk%  1. By Proposition 1.5.7 and by the fact that kxkxk1 % k% D 1, we obtain   x %  1: kxk% Because kxk%  1, the claim follows from (1.5.2).

9

Section 1.5 Modular spaces

(ii) If kxk% > 1, then for every  2 .1; kxk% /, we have %.1 x/ > 1. Thus, by (1.5.2), we obtain 1 %.x/ > 1. Since this is so for an arbitrary  < kxk% , the claim follows. (iii) This is an immediate consequence of (ii). Proposition 1.5.9. Let X be a vector space, let % be a semimodular on X and let ¹xn º1 nD1 be a sequence in X. Then limn!1 kxn k% D 0 if and only if limn!1 %.xn / D 0 for every  > 0. Proof. Assume first that limn!1 kxn k% D 0 and let  > 0. Then, for every K > 1, there exists an n0 2 N such that for every n 2 N, n  n0 , one has kKxn k% < 1: Therefore also %.Kxn /  1 and, by (1.5.2),   1 1 1 Kxn  %.Kxn /  ; %.xn / D % K K K establishing limn!1 %.xn / D 0. Now assume that limn!1 %.xn / D 0. Then there exists an n0 2 N such that for every n 2 N, n  n0 , one has %.xn /  1. Especially, for such n, we have kxn k% 

1 : 

Since  was arbitrary, we obtain limn!1 kxn k% D 0, as desired. The proof is complete. Definition 1.5.10. Let X be a vector space, let % be a modular on X and let ¹xn º1 nD1 be a sequence in X. We say that ¹xn º1 nD1 is modular convergent to some x 2 X if lim %.xn  x/ D 0:

n!1

Corollary 1.5.11. It follows from Proposition 1.5.8 that if X is a modular space, then the norm convergence always implies the modular convergence. Proposition 1.5.12. Let X be a modular space. Then the modular convergence on X is equivalent to the norm convergence if and only if %.xn / ! 0 implies %.2xn / ! 0. Proof. ) Assume that the modular convergence on X is equivalent to the norm. Let ¹xn º1 nD1 be a sequence in X such that %.xn / ! 0. Also, kxn k% ! 0, and it follows immediately from Proposition 1.5.8 applied to  D 2 that %.2xn / ! 0. ( Assume conversely that the condition holds. Let ¹xn º1 nD1 be a sequence in X satisfying %.xn / ! 0. Given  > 0, we find m 2 N such that   2m . Then,

10

Chapter 1 Preliminaries

iterating the condition m times, we obtain %.2m xn / ! 0 as n ! 1. Thus, by (1.5.2), we obtain 0  lim %.xn /  2m lim %.2m xn / D 0: n!1

n!1

Owing to Proposition 1.5.9, this establishes xn ! 0, as desired. The proof is complete. Theorem 1.5.13. Let .X; %X / and .Y; %Y / be two modular spaces, X  Y . Assume that there exists a function h W .0; 1/ ! .0; 1/, which is bounded on some right neighborhood of zero. Suppose that %Y .x/  h.%X .x//

for every x 2 X:

(1.5.3)

Then there exists a positive constant C such that, for every x 2 X, kxk%Y  C kxk%X :

(1.5.4)

Proof. Suppose that the assertion is not true. Then for every n 2 N there exists some xn 2 X such that kxn k%Y > nkxn k%X . Set xQ n WD

xn ; kxn kpn%X

n 2 N:

Then kxQ n k%X ! 0 and kxQ n k%Y ! 1. By Proposition 1.5.8, we get %X .xQ n / ! 0 and %Y .xQ n / ! 1. That, however, is a contradiction with (1.5.3), since h is bounded on some neighborhood of zero. The proof is complete. For more details on modular spaces, see, e.g., [63, 160, 163]. Most of the material in this section can be found in [63]. Theorem 1.5.13 is a special case of a more general result in [125], see also [178].

1.6 Inner product, inner product space Definition 1.6.1. Let X be a vector space. A real-valued function on X  X, whose value at the ordered pair .u; v/, u; v 2 X, is denoted by hu; vi, is called an inner product on X if it satisfies the following axioms: (i) hu; ui > 0 for every u 2 X, u 6D  ; (ii) hu; vi D hv; ui for every u; v 2 X; (iii) hu C v; wi D hu; wi C hv; wi for every u; v; w 2 X; (iv) hu; vi D hu; vi for every u; v 2 X and all scalars . A vector space X endowed with an inner product is called an inner product space (or a unitary space).

11

Section 1.7 Convergence, Cauchy sequences

Definition 1.6.2. Let X be a unitary space. For u 2 X, we define 1

kuk D hu; ui 2 : The inequalities jhu; vij 5 kuk kvk;

u; v 2 X;

(the so-called Cauchy–Schwarz inequality) and ku C vk 5 kuk C kvk;

u; v 2 X;

holds. In particular, every unitary space is a normed linear space with respect to the norm kuk generated by the inner product hu; vi.

1.7 Convergence, Cauchy sequences Definition 1.7.1. Let X be a metric space with respect to the metric % and let ¹un º1 nD1 be a sequence in X. We say that un converges to u 2 X (and write un ! u in X) if limn!1 %.un ; u/ D 0, i.e. if for every " > 0 there exists an n0 D n0 ."/ 2 N such that %.un ; u/ < " for all n > n0 . We shall also say that un converges to u strongly in X or in the norm of X. If un ! u in X, then the sequence ¹un º1 nD1 is said to be convergent in X and u is called the limit of ¹un º1 . nD1 Definition 1.7.2. Let X be a normed linear space, let ¹un º1 nD1 be a sequence in X and let u 2 X. If   n   X   lim u  uk  D 0; n!1   we say that the series

P1

kD1

nD1 un

X

converges to u in X and write uD

1 X

un :

nD1

Definition 1.7.3. A sequence ¹un º1 nD1 in a metric space .X; %/ is called a Cauchy sequence if lim %.um ; un / D 0; m;n!1

i.e. if for every " > 0 there exists an n0 D n0 ."/ 2 N such that %.um ; un / < " for all m; n 2 N, m; n > n0 . Remark 1.7.4. In every metric space, each convergent sequence is a Cauchy sequence. The converse is not true in general.

12

Chapter 1 Preliminaries

The following simple auxiliary assertion will be useful later. Lemma 1.7.5. Let ¹un º1 nD1 be a Cauchy sequence in a metric space X and let ¹unk º1 be its subsequence. If unk ! u in X then un ! u in X. kD1 Proof. Let " > 0. Then, by the Cauchy property, there exists an index n0 2 N such that for every m; n 2 N, m; n  n0 , one has %.xm ; xn / < ". Next, there is an index k0 2 N such that nk0  n0 and %.xnk0 ; x/ < ". Thus, for every n 2 N, n  nk0 , one has %.xn ; x/  %.xn ; xnk0 / C %.xnk0 ; x/ < " C " D 2": The proof is complete. Definition 1.7.6. Let M be a subset of a metric space X. The closure of M in X, X denoted by M or M , is defined as the set of all elements u 2 X such that there exists a sequence ¹un º1 nD1 in M for which un ! u in X. Remark 1.7.7. Clearly, if M is a subset of a metric space X, then M  M . A set M is closed in X if and only if M D M .

1.8 Density, separability Definition 1.8.1. A subset M of a metric space X is said to be dense in X if M D X. Definition 1.8.2. A metric space X is called separable if it contains a countable dense subset. Remark 1.8.3. Let .X; %/ be a metric space. Assume that there exists an uncountable subset M of X and a ı > 0 such that %.u; v/ > ı

for every u; v 2 M; u 6D v:

Then X is not separable.

1.9 Completeness Definition 1.9.1. A metric space X is said to be complete if every Cauchy sequence in X is convergent in X. Definition 1.9.2. A complete normed linear space is called a Banach space. Definition 1.9.3. Let X be a unitary space endowed with an inner product hu; vi. 1 If the normed linear space X is complete with respect to the norm kuk D hu; ui 2 , then X is called a Hilbert space.

13

Section 1.10 Subspaces

Definition 1.9.4. We say that a normed linear space .X; k  kX / has the Riesz–Fischer property if for each sequence ¹un º1 nD1 such that 1 X

kun kX < 1;

(1.9.1)

nD1

P there exists an element u 2 X such that 1 nD1 un D u in X, that is,  n  X    lim  uk  u D 0: n!1   kD1

X

Theorem 1.9.5. A normed linear space is complete if and only if it has the Riesz– Fischer property. Proof. Assume first that X is a Banach space and let ¹un º1 nD1 be a sequence in X such that (1.9.1) holds. Then ´ n μ1 X uk kD1

nD1

is a Cauchy sequence in X, hence it converges in X to some element u 2 X. Thus, uD

1 X

un :

nD1

Conversely, assume X is a normed linear space with the Riesz–Fischer property and let ¹un º1 nD1 be a Cauchy sequence in X. Then a subsequence ¹unk ºk2N can be chosen so that 1 X kunk  unkC1 kX < 1: kD1

  P Then, necessarily, the series 1 kD1 unk  unkC1 converges in X. In particular, there exists an element u 2 X such that unk ! u in X. But then since ¹un º1 nD1 is a Cauchy sequence, we also have un ! u in X, as desired.

1.10 Subspaces Definition 1.10.1. A subset of a normed linear space X is called a subspace of X if it is a linear set which is closed in X. Remark 1.10.2. We shall distinguish between linear subsets and subspaces: a linear subset need not be closed in X.

14

Chapter 1 Preliminaries

Remarks 1.10.3. (i) A subspace M of a normed linear space .X; k  kX / is again a normed linear space with the norm k  kM defined by kukM WD kukX for u 2 M: (ii) A subspace of a separable normed linear space X is itself a separable normed linear space. (iii) A subspace of a Banach space is also a Banach space.

1.11 Products of spaces Remarks 1.11.1. (i) Let n 2 N, and let X1 ; X2 ; : : : ; Xn be normed linear spaces, X D X1  X2      Xn the Cartesian product of X1 ; : : : ; Xn , i.e. the set of all (ordered) n-tuples u D .u1 ; : : : ; un / such that ui 2 Xi , i D 1; : : : ; n. Then X is also a normed linear space; the norm in X can be defined in various ways, for example kukX WD

n X

p

! p1

kui kXi

for some p 2 Œ1; 1/

iD1

or kukX WD max kui kXi : iD1;:::;n

(ii) Let X1 ; : : : ; Xn be separable normed linear spaces. Then the product space X D X1      Xn is also a separable normed linear space. (iii) Let X1 ; : : : ; Xn be Banach spaces. Then the product space X D X1      Xn is also a Banach space.

1.12 Schauder bases Definition 1.12.1. Let X be a Banach space. A sequence ¹un º1 nD1 in X is called a Schauder basis of X if for every u 2 X there exists a unique sequence ¹an º1 nD1 of scalars such that 1 X an un : uD nD1

The (uniquely determined) numbers an D an .u/ are called the coefficients of u with respect to the Schauder basis ¹un º1 nD1 .

15

Section 1.13 Compactness

Remark 1.12.2. Every Banach space with a Schauder basis is separable. The converse implication does not hold, i.e. there exists a separable Banach space without Schauder basis. Definition 1.12.3. A Schauder basis ¹un º1 nD1 in a Banach space X is called unconditional if the convergence of a series of the form 1 X

an un

nD1

implies the convergence of the series 1 X

a.n/ u.n/

nD1

for every permutation of the set N. Remark 1.12.4. A basis ¹un º1 nD1 is unconditional if and only if at least one of the following conditions holds: P (i) P The convergence of a series 1 nD1 an un implies the convergence of the series 1 nD1 "n an un for any choice of "n equal to C1 or 1. P (ii) P The convergence of a series 1 nD1 an un implies the convergence of the series 1 a u for any subsequence ¹nk º1 of N. kD1 nk nk kD1

1.13 Compactness Definition 1.13.1. Let .X; %/ be a metric space. A set M  X is said to be relatively compact if every sequence in M contains a convergent subsequence (i.e. a subsequence with a limit in X). If M is closed and relatively compact then M is said to be compact. Definition 1.13.2. Let .X; %/ be a metric space. Let " > 0 and let M be a subset of X. A set E in X is called an "-net of M if for every u 2 M there exists a u" 2 E such that %.u; u" / < ": A set M  X is called totally bounded (or precompact) in X if for every " > 0 there exists a finite "-net of M in X. Remarks 1.13.3. Let .X; %/ be a metric space. Then (i) every relatively compact set in X is bounded;

16

Chapter 1 Preliminaries

(ii) every closed subset of a compact set is itself compact; (iii) if X is complete, then each its subset is relatively compact if and only if it is totally bounded. Remark 1.13.4. For i D 1; : : : ; n, let Mi be a relatively compact subset of a Banach space Xi . Then M1      Mn is a relatively compact subset of X1      Xn .

1.14 Operators (mappings) Definition 1.14.1. Let X; Y be normed linear spaces, and M a set in X. Suppose that a rule is given by which to every u 2 M there corresponds a uniquely determined element in Y . We denote this element by Au or

A.u/

and say that the rule defines an operator A on M . The set M is called the domain of the operator A and denoted by Dom.A/; the set Rng.A/ D ¹uQ 2 Y I uQ D Au for some

u 2 Mº

is called the range of the operator A. If for all u; v 2 M we have Au 6D Av provided u 6D v, then to every uQ 2 Rng.A/ is assigned a uniquely determined element u 2 M by the rule Au D u. Q We write this as u D A1 uQ and call A1 the inverse operator to A. We have that Dom.A1/ D Rng.A/, Rng.A1/ D Dom.A/ D M . We say that the operator A is an operator from X into Y which maps M into Y . If Rng.A/ D Y , we say that A maps M onto Y . The expressions function, abstract function or mapping are frequently used instead of operator. Definition 1.14.2. Let X; Y be normed linear spaces. An operator A from X into Y is called linear if Dom.A/ is a linear set and if the following two conditions are satisfied: (i) A.u C v/ D Au C Av (ii) A.u/ D Au for all u; v 2 Dom.A/ and for every scalar . Definition 1.14.3. Let X; Y be normed linear spaces. An operator A from X into Y is said to be continuous if un ! u in X

implies

Aun ! Au in Y

for any sequence ¹un º1 nD1 provided un 2 Dom.A/, u 2 Dom.A/.

17

Section 1.14 Operators (mappings)

Definition 1.14.4. Let X; Y be normed linear spaces. A linear operator A from X into Y is said to be bounded if sup kAukY < 1; where the supremum is taken over all u 2 Dom.A/ such that kukX 5 1. Remark 1.14.5. A linear operator between two normed linear spaces is bounded if and only if it is continuous. We shall now introduce the notion of a norm of a continuous linear operator (the so-called operator norm). Definition 1.14.6. Let X; Y be normed linear spaces. Let A be a continuous linear operator from X into Y . Then kAkX!Y WD sup ¹kAukY I u 2 Dom.A/; kukX 5 1º :

Remark 1.14.7. Let X; Y be normed linear spaces and let A be a continuous linear operator from X into Y . Then the following formulas for the norm kAkX!Y of A will often be useful: kAukY : kAkX!Y D sup kAukY D sup u2Dom.A/ u2Dom.A/ kukX kukX D1

u6D

Theorem 1.14.8 (Banach). Let X; Y be Banach spaces, and A a linear operator from X onto Y with Dom.A/ D X (and Rng.A/ D Y ). Suppose that A is continuous and that A1 exists. Then A1 is continuous. Definition 1.14.9. Let A be a linear operator from a normed linear space X into a normed linear space Y with Dom.A/ D X. The operator A is said to be compact (or completely continuous) if it maps every bounded set in X onto a relatively compact set in Y . Remark 1.14.10. Every compact linear operator is continuous. Theorem 1.14.11 (Banach–Steinhaus). Let X be a Banach space, Y a normed linear space. The sequence ¹An º1 nD1 of bounded linear operators from X into Y with Dom.An / D X satisfies An u ! Au in Y for every u 2 X if and only if the following two conditions are satisfied: (i) the sequence ¹kAn kº1 nD1 is bounded; (ii) An u ! Au in Y for every u 2 M , where M is a dense subset of X.

18

Chapter 1 Preliminaries

1.15 Isomorphism, embeddings Definition 1.15.1. Two normed linear spaces X; Y are said to be isomorphic if there exists a continuous linear operator A such that Dom.A/ D X, Rng.A/ D Y , and A1 exists and is continuous. This operator A is called an isomorphism mapping or briefly an isomorphism between X and Y . Definition 1.15.2. (i) Two metric spaces .X; %/ and .Y; / are said to be isometric if there exists a mapping T such that Dom.T / D X, Rng.T / D Y and %.x; y/ D .T x; T y/ for every pair x; y 2 X. (ii) Two normed linear spaces X; Y are said to be isometrically isomorphic if there exists a linear operator A such that Dom.A/ D X, Rng.A/ D Y and ku  vkX D kAu  AvkY for every pair u; v 2 X. Remark 1.15.3. In particular, two isometrically isomorphic spaces are isomorphic. Remark 1.15.4. Let X; Y be two isomorphic normed linear spaces. Then (i) if X is separable then Y is also separable; (ii) if X is complete then Y is also complete; (iii) if X has a Schauder basis then Y also has a Schauder basis. Definition 1.15.5. Let X; Y be two normed linear spaces and let X  Y . We define the identity operator Id from X into Y with Dom.Id/ D Rng.Id/ D X as the operator which maps every element u 2 X onto itself: Id u D u, regarded as an element of Y . If the identity operator is continuous, that is, if there exists a constant c > 0 such that kukY 5 ckukX

for every u 2 X;

then we say that the space X is embedded into the space Y and we shall call the operator Id the embedding operator from X to Y . Alternatively, we may sometimes say that there exists a continuous (or bounded) embedding of X to Y . By the operator norm of Id we call the number kf kY : f 60 kf kX

k Id kX,!Y WD sup

We shall also call this quantity an embedding constant.

Section 1.16 Continuous linear functionals

Remarks 1.15.6.

19

(i) The operator Id from Definition 1.15.5 is obviously linear.

(ii) While the elements in X and in Rng.Id/  Y coincide, their respective norms in X and in Y may be different. Notation 1.15.7. If X and Y are two normed linear spaces and there exists a continuous embedding from X into Y , we write X ,! Y: If simultaneously X ,! Y and Y ,! X; then we shall write X  Y: If the embedding operator is compact (see Definition 1.14.9), we write X ,!,! Y: Definition 1.15.8. Let X be a vector space and suppose that k  k1 and k  k2 are two norms on X. These norms are said to be equivalent if there exist constants c > 0, d > 0 such that ckuk1 5 kuk2 5 d kuk1 for all u 2 X: In other words, the norms k  k1 and k  k2 are equivalent if and only if .X; k  k1 /  .X; k  k2 /. In particular, the embedding from .X; k  k1 / into .X; k  k2 / is an isomorphism.

1.16 Continuous linear functionals Definition 1.16.1. Let X be a normed linear space. A linear operator from X into R is then called a linear functional. Remark 1.16.2. In this section we denote linear functionals by Greek letters: '; ˆ; : : : . The value of the functional ' at u 2 X will usually be denoted by '.u/I the notation h'; ui is also very frequently used in the literature. Since a functional is an operator, some concepts introduced in connection with operators (see Section 1.14) can be transferred to functionals. We shall deal here only with continuous linear functionals, that is, continuous linear operators from X into R.

20

Chapter 1 Preliminaries

Definition 1.16.3. A norm of a linear functional ' is defined by k'k D sup j'.u/j where the supremum is taken over all u 2 Dom.'/ such that kukX 5 1. Continuous linear functionals defined on linear subsets of X can be extended to the entire space X as stated in the following theorem. Theorem 1.16.4 (Hahn–Banach). Let ' be a continuous linear functional defined on a linear subset M of a normed linear space X. Then there exists a continuous linear functional ˆ defined on X such that ˆ.u/ D '.u/ for u 2 M and kˆk D k'k:

1.17 Dual space, weak convergence Definition 1.17.1. Let us denote by X  the set of all continuous linear functionals defined on X. Then X  is a vector space if we define the addition of functionals and the multiplication of a functional by a scalar in the natural way, namely .' C where ';

/.u/ WD '.u/ C

.u/; .'/.u/ WD '.u/

2 X  and  is a scalar.

Remarks 1.17.2. If X is a normed linear space, then the space X  from Definition 1.17.1, endowed with the norm from Definition 1.16.3, is itself also a normed linear space. Moreover, the space X  is a Banach space, that is, it is always complete. Definition 1.17.3. Let X be a normed linear space, ¹un º1 nD1 a sequence in X. We say w

that un converges weakly to u 2 X (notation un * u or un ! u) if limn!1 '.un / D '.u/ for every ' 2 X  . Remark 1.17.4. Every weakly convergent sequence is bounded. Theorem 1.17.5 (Banach–Steinhaus theorem for weak convergence). Let X be a Banach space. The sequence ¹un º1 nD1 converges weakly to u 2 X if and only if the following two conditions are satisfied: (i) the sequence ¹kun kX º1 nD1 is bounded; (ii) limn!1 '.un / D '.u/ for all ' 2 , where  is a dense subset of the dual space X  . Theorem 1.17.6 (Banach–Alaoglu). Let X be a Banach space and X  its dual. Then the unit ball ¯ ® ƒ 2 X  I kƒkX   1 is weakly* compact in X  .

Section 1.18 The principle of uniform boundedness

21

1.18 The principle of uniform boundedness Theorem 1.18.1 (Uniform boundedness principle). Let X be a Banach space and Y a normed linear space. Let ¹T˛ º˛2I be a set of linear operators from X to Y , where I is an arbitrary index set (without any restriction on its cardinality). Assume that sup kT˛ xkY < 1 for every x 2 X: ˛2I

Then sup kT˛ kX!Y < 1: ˛2I

1.19 Reflexivity Definition 1.19.1. Let X be a Banach space, X  its dual space. Then we can define the dual of X  by setting X  WD .X  / : Let us denote the elements of X  by u ; v  ; : : : . The operator J from X into with Dom.J / D X, defined by the formula

X 

.J u/.'/ D '.u/ for ' 2 X  ; u 2 X; is called the canonical mapping from X into X  . (J u is the element u 2 X  which satisfies u .'/ D '.u/.) Remark 1.19.2. Let X be a Banach space. Denote by J.X/ the image of X in the canonical mapping J . Then J is an isometric isomorphism between X and J.X/. Definition 1.19.3. A Banach space X is said to be reflexive if J.X/ D X  : Remarks 1.19.4.

(i) Every subspace of a reflexive Banach space is reflexive.

(ii) A Cartesian product of a finite number of reflexive Banach spaces is a reflexive Banach space. (iii) A Banach space isomorphic to a reflexive Banach space is reflexive. (iv) If a Banach space has separable dual space, then it is itself separable. (v) The dual space of a separable reflexive Banach space is separable.

22

Chapter 1 Preliminaries

1.20 Measure spaces: general extension theory Definition 1.20.1. Let X be a nonempty set and let S  exp X be a collection of subsets of X. Then S is called an algebra if the following three conditions are satisfied: (i) ;; X 2 S; (ii) A \ B 2 S for every A; B 2 S; (iii) Ac 2 S for every A 2 S. Remark 1.20.2. For every collection F of subsets of a set X, there exists a unique algebra S of subsets of X such that F  S and if S1 is another algebra containing F , then S  S1 . Definition 1.20.3. Let X be a nonempty set and let S  exp X be a collection of subsets of X. Then every function  W S ! Œ0; 1 is called a set function on S. We say that a set function  is monotone on S if .A/  .B/ whenever A; B 2 S; A  B: We say that  is finitely additive on S if ! n n [ X Ai D .Ai /;  iD1

iD1

for every n 2 N and pairwise disjoint sets Ai 2 S, i D 1; : : : ; n, such that S. We say that  is countably additive on S if ! 1 1 [ X An D .An /  nD1

iD1 Ai

2

nD1

for all pairwise disjoint sets An 2 S, n 2 N, such that is countably subadditive on S if .A/ 

Sn

1 X

S1

nD1 An

2 S. We say that 

.An /

nD1

for every A 2 S such that A 

S1

nD1 An ,

where An 2 S for each n 2 N.

Definition 1.20.4. Let X be a nonempty set and let S  exp X be a collection of subsets of X satisfying ; 2 S. A countably additive set function  W S ! Œ0; 1 is called a measure on S if .;/ D 0.

23

Section 1.20 Measure spaces: general extension theory

We shall now extend a measure to a set function on the entire exp X. We pay for this extension by a possible loss of nice properties of the original measure. In particular, the extended set function need not be countably additive any more. Definition 1.20.5. Let X be a nonempty set and let S  exp X be an algebra of subsets of X satisfying ; 2 S. Let  W S ! Œ0; 1 be a measure on S. For every A  X we define ´1 μ 1 X [  .A/ WD inf .Ai /I Ai 2 S; A  Ai : iD1

iD1

The function  is called the outer measure induced by . Remark 1.20.6. An outer measure is well-defined since S1 for every A 2 exp X there  exists at least one sequence ¹An º1 such that A  nD1 An . The set function  nD1 can attain infinite value. In the next proposition we shall collect some properties of an outer measure. Proposition 1.20.7. Let X be a nonempty set and let  W exp X ! Œ0; 1 be an outer measure on X. Assume that  is induced by a measure  defined on an algebra S  exp X. Then (i)  .A/  0 for every A 2 exp X; (ii)  .;/ D 0; (iii)  is monotone; (iv)  is countably subadditive; (v)  is an extension of  on S in the sense that  .A/ D .A/ for every A 2 S. We shall now extend the notion of an outer measure to set functions about which we do not a priori know that they were induced by a measure. Definition 1.20.8. Let X be a nonempty set and let  W exp X ! Œ0; 1 be a set function such that the properties (i)–(iv) of Proposition 1.20.7 are satisfied. Then  is called an outer measure on X. We have seen how an outer measure is defined on exp X by an extension of a given measure on a subalgebra of exp X. Now we shall take the converse path. We start with an outer measure and our aim will be to build a measure from it. This will be done by choosing an appropriate subclass of exp X on which the given outer measure behaves like a measure.

24

Chapter 1 Preliminaries

Definition 1.20.9. Let X be a nonempty set and let  W exp X ! Œ0; 1 be an outer measure on X. We say that a set A 2 exp X is -measurable if .T / D .T \ A/ C .T \ Ac / for every “test set” T 2 exp X:

(1.20.1)

We shall denote by M the set of all -measurable subsets of X. Remark 1.20.10. If A 2 M, then also Ac 2 M due to the symmetry in (1.20.1). Definition 1.20.11. Let X be a nonempty set and let S  exp X be a collection of subsets of X. Then S is called a -algebra if the following three conditions are satisfied: (i) ;; X 2 S; S 1 (ii) 1 nD1 An 2 S for every countable sequence ¹An ºnD1  S; (iii) Ac 2 S for every A 2 S. In the next proposition we shall collect some properties of the class of measurable sets. Proposition 1.20.12. Let X be a nonempty set, let  W exp X ! Œ0; 1 be an outer measure on X, and let M be the set of all -measurable subsets of X. Then (i) M is a -algebra of subsets of X; (ii)  is countably additive when restricted to M; (iii) if  was induced by some measure defined on an algebra S, then S  M; (iv) the set N WD ¹A 2 exp XI .E/ D 0º satisfies N  M. Remark 1.20.13. For every collection A of subsets of a set X, there exists a unique algebra S of subsets of X such that A  S and if S1 is another algebra containing A, then S  S1 . In such cases, we can say that the -algebra S is generated by A and write S D S.A/. Definition 1.20.14. Let X be a topological space. Let G and F denote the set of all open and closed subsets of X, respectively. Then S.G / D S.F /: We call the -algebra generated by open (or closed) sets the -algebra of Borel subsets of X and denote it by B.X/.

Section 1.20 Measure spaces: general extension theory

25

Definition 1.20.15. Let X be a nonempty set, let S  exp X be a collection of subsets of X and let  W S ! Œ0; 1 be a set function on S. We say that  is finite on S if .A/ < 1 for every A 2 S. We say that  is -finite on S if there exists a sequence ¹Xn º1 nD1 of pairwise disjoint S subsets of sets Xn 2 S, n 2 N, such that .Xn / < 1 for every n 2 N and X D 1 nD1 Xn . Definition 1.20.16. Let X be a nonempty set, let S  exp X be a -algebra of subsets of X and let  W S ! Œ0; 1 be a measure on S. The pair .X; S/ is then called a measurable space and the triple .X; S; / is called a measure space. The elements of S are called measurable sets. Definition 1.20.17. Let .X; S; / be a measure space. Define N WD ¹A 2 exp XI there exists N 2 S such that A  N and .N / D 0º : Then the elements of N are called the -null subsets of X. We say that .X; S; / is a complete measure space if N  S. Remark 1.20.18. It is not difficult to realize that every measure can be completed. Therefore it is not such a restriction to assume that the measure space in question is complete. Definition 1.20.19. Let .X; S; / be a measure space. A set A 2 S is called an atom if .A/ > 0 and for every set B 2 S, B  A, either .B/ D 0 or .A n B/ D 0. The measure space .X; S; / is called completely atomic (or just atomic or discrete) if there exists a set M  X such that .X n M / D 0 and .¹xº/ ¤ 0 for every x 2 M . The measure space .X; S; / is called nonatomic if there do not exist any atoms in S. We often for short say that the measure  is nonatomic. Such measure is also called continuous. Example 1.20.20. Let X be a nonempty set and let S WD exp X. For A 2 S, define ´ the number of elements of A if A is finite; .A/ WD 1 if A is infinite: Then  is a measure. This measure is called the counting measure on X. In the particular case when X D N, the counting measure is denoted by m and is called the arithmetic measure on N. The triple .N; exp N; m/ is a typical example of a completely atomic space with all atoms having the same measure. Definition 1.20.21. Let .X; %/ be a metric space and let  be a measure defined on B.X/. We say that  is outer regular if, for every A 2 B.X/, one has .A/ D inf¹.G/I G open; A  Gº D sup¹.F /I F closed; F  Aº:

26

Chapter 1 Preliminaries

We say that  is inner regular if, for every A 2 B.X/ satisfying 0 < .A/ < 1 and every " > 0, there exists a compact set K  A such that .A n K/ < ". If a measure is both inner and outer regular, then we say that it is regular. Definition 1.20.22. Let X be a nonempty set and E  X. The function E W X ! Œ0; 1/, defined by ´ 1 if x 2 E; E .x/ WD 0 if x 62 E; is called the characteristic function of E. Definition 1.20.23. Let .X; S; / be a complete -finite measure space. The function s W X ! Œ0; 1/ is called a -simple function (or just simple function) if it is a finite linear combination of characteristic functions of -measurable sets of finite measure, i.e. if there exist an m 2 N, real numbers ¹a1 ; : : : ; am º and disjoint measurable subsets of X of finite measure ¹E1 ; : : : ; Em º such that ´ aj ; x 2 Ej ; j D 1; : : : ; m; s.x/ D S 0; x 2 X n jmD1 Ej : Definition 1.20.24. Let .X; %/ be a metric space and let u W X ! Œ0; 1. The set ¹x 2 XI u.x/ 6D 0º; where the bar denotes the closure in the space .X; %/, is called the support of the function u and is denoted by supp u. Remark 1.20.25. Let .X; S; / be a complete -finite measure space endowed further with a metric. Then a function s W X ! R is simple if and only if its range is a finite set and its support is of finite measure. Definition 1.20.26. Let .X; S; / be a complete -finite measure space and let A 2 S. We say that a certain statement, say V .x/, holds almost everywhere on A (or a.e. on A for short) with respect to  (we will also say for almost all x 2 A) if the set E WD ¹x 2 AI V .x/ does not holdº satisfies E 2 S and .E/ D 0. Definition 1.20.27. Let .X; S; / be a complete -finite measure space and let s be a simple function on X with representation s.x/ D

m X iD1

ai Ei ; x 2 X;

27

Section 1.20 Measure spaces: general extension theory

where ai 2 R and Ei 2 S, i D 1; : : : ; m. We then define the integral of s by Z X

s.x/ d.x/ WD

m X

ai .Ei /:

iD1

Definition 1.20.28. Let .X; S; / be a complete -finite measure space and let f W X ! Œ0; 1. We say that f is S-measurable (or just measurable) if there exists a nondecreasing sequence of nonnegative simple functions ¹sn º1 nD1 satisfying f .x/ D lim sn .x/; n!1

x 2 X:

We shall denote by SC .X/ the set of all nonnegative measurable functions on X. For a nonnegative measurable function f W X ! Œ0; 1, we define its integral by Z Z f .x/ d.x/ WD lim sn .x/ d.x/: n!1 X

X

The first important property of nonnegative measurable functions is the following theorem on monotone convergence. Theorem 1.20.29 (monotone convergence theorem). Let .X; S; / be a complete finite measure space and let ¹fn º1 nD1 be a sequence of nonnegative integrable functions defined on X such that fn .x/  fnC1 .x/ for every n 2 N and x 2 X. Let f .x/ WD lim fn .x/; n!1

x 2 X:

Then f 2 SC .X/ and Z Z Z lim fn .x/ d.x/ D f .x/ d.x/ D lim fn .x/ d.x/:  n!1



n!1 

Our aim is now to extend the notion of measurability to functions that are not necessarily nonnegative. Definition 1.20.30. Let .X; S; / be a complete -finite measure space and let f W X ! Œ1; 1. We define the nonnegative part f C of f by ´ f .x/ if f .x/  0; C f .x/ WD 0 if f .x/ < 0: Similarly, we define the nonpositive part f  of f by ´ 0 if f .x/  0; f  .x/ WD f .x/ if f .x/ < 0:

28

Chapter 1 Preliminaries

Remark 1.20.31. The functions f C and f  are nonnegative and f D f C  f ;

jf j D f C C f  :

Definition 1.20.32. Let .X; S; / be a complete -finite measure space and let f W X ! Œ1; 1. We say that f is S-measurable (or just measurable) if both f C and f  are measurable in the sense of Definition 1.20.28. We denote by M.X/ the class of all measurable functions on X. Remark 1.20.33. Let .X; S; / be a complete -finite measure space. Then the characteristic function E of a subset E of X is measurable if and only if the set E is measurable. Theorem 1.20.34 (Fatou lemma). Let .X; S; / be a complete -finite measure space and let ¹fn º1 nD1 be a sequence of nonnegative measurable functions on X. Then Z Z lim inf fn .x/ d.x/  lim inf fn .x/ d.x/: X n!1

n!1

X

Definition 1.20.35. Let .X; S; / be a complete -finite measure space and R letCf W d X ! Œ1; 1. We say that f is -integrable (or just integrable) if both Xf R and X f  d are finite. We then define the integral of f by Z Z Z f .x/ d.x/ WD f C .x/ d.x/  f  .x/ d.x/: (1.20.2) X

X

X

In such cases, we say that the integral of f over X converges. We denote by L1 .X; S; / (or just L1 .X/ or L1 ./) the set of all -integrable functions on X. R R In the case when exactly one of the values X f C .x/ d.x/, X f  .x/ d.x/ is infinite, we again define the integral of f by the formula (1.20.2), that is, as 1 or 1, and we say that R X exists (although it does not converge). When R the integral of f over both values X f C .x/ d.x/ and X f  .x/ d.x/ are infinite, then we say that the integral of f over X does not exist. We shall now collect basic properties of integrable functions in the following proposition. Proposition 1.20.36. Let .X; S; / be a complete -finite measure space and let f; g W X ! Œ1; 1 and a; b 2 R. Then (i) f 2 L1 .X/ holds if and only if jf j 2 L1 .X/ and ˇZ ˇ Z ˇ ˇ ˇ f .x/ d.x/ˇ  jf .x/j d.x/I ˇ ˇ X

X

(ii) if jf .x/j  g.x/ for a.e. x 2 X and g 2 L1 .X/, then f 2 L1 .X/;

29

Section 1.21 The Lebesgue measure and integral

(iii) if f .x/ D g.x/ for a.e. x 2 X and f 2 L1 .X/, then g 2 L1 .X/ and Z Z f .x/ d.x/ D g.x/ d.x/I X

X

(iv) if f; g 2 L1 .X/, then .f C g/ 2 L1 .X/ and Z Z Z .af .x/ C bg.x// d.x/ D a f .x/ d.x/ C b g.x/ d.x/I X

X

X

(v) if E 2 S and f 2 L1 .X/, then f E 2 L1 .X/ and Z Z E .x/f .x/ d.x/ D f .x/ d.x/: X

E

Another important result concerning convergence is the following assertion. Theorem 1.20.37 (Lebesgue dominated convergence theorem). Let .X; S; / be a complete -finite measure space and let ¹fn º1 nD1 be a sequence of measurable functions 1 on X and let g 2 L .X/ be such that jfn .x/j  g.x/ for all n 2 N and almost all x 2 X. Assume that lim fn .x/ D f .x/

n!1

for a.e. x 2 X:

Then the following statements hold: (i) f 2 L1 .X/; R R (ii) X f .s/ d.x/ D limn!1 X fn .x/ d.x/; R (iii) limn!1 X jfn .x/  f .x/j d.x/ D 0.

1.21 The Lebesgue measure and integral In this section we apply the abstract general theory developed in Section 1.20 to the particular case when X D , where  is an open set in RN , N 2 N, S is the algebra generated by all N -dimensional intervals and  is the length function defined by N Y .bi  ai /: ..a1 ; b1 /      .aN ; bN // WD iD1

We denote the set of all N -dimensional intervals in RN by IN . The restriction to an open set  is temporary and will be abandoned in Chapter 6 and subsequent chapters, where more general measure spaces will be considered.

30

Chapter 1 Preliminaries

Definition 1.21.1. The Lebesgue outer measure  is defined for every A 2 exp./ by ´1 μ 1 X [   .A/ WD inf .In \ /I In 2 IN ; Im \ In D ; for m ¤ n; A  In : nD1

nD1

The corresponding -algebra of  -measurable sets is called the -algebra of Lebesgue measurable sets. The resulting measure is denoted by  and is called the Lebesgue measure. The measure space .; B./; / is called the Lebesgue measure space. Proposition 1.21.2. The Lebesgue measure on RN , N 2 N, is a translation-invariant (that is, if A 2 M and x 2 RN , then the set A C x WD ¹y 2 RN ; y  x 2 Aº satisfies A C x 2 M and .A C x/ D .A/) -finite regular measure on B.RN /. In fact, it is unique, up to multiplication by a positive constant. R Convention 1.21.3. Throughout the text, we shall often write  jf .x/j dx instead R of  jf .x/j d.x/ when no confusion can arise. In the rest of this section we shall collect the most important assertions concerning the Lebesgue integral. Some of them will be the “Lebesgue” versions of their appropriate abstract counterparts from Section 1.20. Theorem 1.21.4 (Levi monotone convergence theorem). Let ¹fn º1 nD1 be a sequence of integrable functions on a measurable set   RN such that fn .x/  fnC1 .x/ for every n 2 N and almost all x 2  and that Z f1 .x/ dx > 1: 

Then, for almost all x 2 , the limit f .x/ D lim fn .x/ n!1

exists, the function f is integrable and Z Z Z lim fn .x/ dx D f .x/ dx D lim fn .x/ dx:  n!1



n!1 

Theorem 1.21.5 (Lebesgue dominated convergence theorem). Let ¹fn º1 nD1 be a sequence of measurable functions on a measurable set   RN which converges for almost all x 2  to f .x/. Suppose that there exists a function g with the finite Lebesgue integral over  such that jfn .x/j  g.x/ for all n 2 N and almost all x 2 .

31

Section 1.21 The Lebesgue measure and integral

Then fn , n 2 N, and f have finite integrals and Z Z f .x/ dx D lim fn .x/ dx: n!1 



Theorem 1.21.6 (Fatou lemma). Let ¹fn º1 nD1 be a sequence of measurable functions which are nonnegative almost everywhere on . Then lim inf fn .x/ n!1

is integrable and Z

Z lim inf fn .x/ dx  lim inf

 n!1

n!1

fn .x/ dx: 

A generalization of Theorem 1.21.5 is given by the following result. Theorem 1.21.7 (Vitali). Let ¹fn º1 nD1 be a sequence of functions with finite integrals over a measurable set   RN : Suppose that lim fn .x/ D f .x/

n!1

for almost all x 2  and let f be an almost everywhere finite function. Suppose that the following condition is satisfied: (P) for every " > 0 there exists a ı > 0 with the property: if B  ; .B/ < ı; then Z jfn .x/j dx < " B

for all n 2 N. Then the function f has a finite integral over  and Z Z fn .x/ dx D f .x/ dx: lim n!1 



Very close to the preceding result is the following theorem. Theorem 1.21.8 (Vitali–Hahn–Saks). Let ¹fn º1 nD1 be a sequence of functions with finite integrals over a measurable set   RN : Suppose that, for an arbitrary measurable set E  , the limit Z fn .x/ dx lim n!1 E

exists and is finite. Then the condition (P) in Theorem 1.21.7 is satisfied.

32

Chapter 1 Preliminaries

The following assertion is a direct consequence of Theorem 1.21.5. Theorem 1.21.9 (continuous dependence of the integral on a parameter). Let   RN be measurable and let .X; %/ be a metric space. Let f .x; ˛/ be defined on   X. Suppose that (i) for almost all x 2 ; f .x; / is continuous on XI (ii) for every ˛ 2 X; f .; ˛/ is measurable on I (iii) there exists a g with a finite integral over  such that jf .x; ˛/j  g.x/ for all ˛ 2 X and almost all x 2 : Then the function F , defined by Z F .˛/ WD

f .x; ˛/ dx; 

˛ 2 X;

is continuous on X. The following assertion is of great importance. Theorem 1.21.10 (derivative of the integral with respect to a parameter). Let   RN be measurable, 1  a < b  1 and let f .x; ˛/ be defined on   .a; b/. Define the function Z F .˛/ WD f .x; ˛/ dx; ˛ 2 .a; b/; 

and suppose that (i) F .˛/ is finite for at least one ˛ 2 .a; b/I (ii) for every ˛ 2 .a; b/, f .; ˛/ is measurable on ; (iii) the partial derivative @f .x; ˛/ @˛ exists and is finite for every ˛ 2 .a; b/ and almost every x 2 ; (iv) there exists a function g with finite integral over  such that ˇ ˇ ˇ @f .x; ˛/ ˇ ˇ ˇ ˇ @˛ ˇ  g.x/ for almost every x 2  and all ˛ 2 .a; b/.

33

Section 1.21 The Lebesgue measure and integral

Then, for all ˛ 2 .a; b/, the integral

Z

F .˛/ D is finite and F 0 .˛/ D

f .x; ˛/ dx 

Z 

@f .x; ˛/ dx: @˛

To handle integrals over subsets of R we shall use the Fubini theorem (sometimes in literature called the Fubini–Tonelli theorem). N

Theorem 1.21.11 (Fubini). Let i  RNi , i D 1; 2, be measurable and set  D 1  2 : Let f .x; y/ be integrable over : Then for almost all x 2 1 and y 2 2 the integrals Z Z f .x; y/ dx

f .x; y/ dy

and

1

exist. Moreover, Z Z f .x; y/ dx dy D 

2



Z f .x; y/ dy

1

2

dx D



Z

Z

f .x; y/ dx dy: 2

1

A characterization of measurable functions is given by the Luzin theorem. Theorem 1.21.12 (Luzin). Let   RN be measurable; let f be defined almost everywhere on : Then f is measurable on  if and only if, for every " > 0, there exists an open set M  , .M / < "; such that the restriction of f to  n M is continuous on  n M . We shall now recall a useful result on the absolute continuous dependence of an integral on the integration domain. Theorem 1.21.13 (absolute continuity of integral). Let f be a function with a finite Lebesgue integral over   RN : Then, for every " > 0, there exists a ı D ı."/ > 0 such that for every measurable subset E of  with .E/ < ı; we have ˇ ˇZ ˇ ˇ ˇ f .x/ dx ˇ < ": ˇ ˇ E

Definition 1.21.14. Let  be a -additive set function defined on the family of all Lebesgue measurable subsets of . Let .;/ D 0: We say that  is absolutely continuous with respect to Lebesgue measure  and write  2 AC Œ, if .E/ D 0 implies for every -measurable sets E  :

.E/ D 0

34

Chapter 1 Preliminaries

Theorem 1.21.15 (Radon–Nikodým). Let  2 AC Œ be a finite set function. Then there exists exactly one function f with a finite Lebesgue integral over  such that Z f .x/ dx .E/ D E

for every Lebesgue measurable subset E  :

1.22 Modes of convergence Throughout this section we shall assume that .X; S; / is a -finite complete measure space. As already noticed, the requirement of completeness is not too restrictive because every measure space can be easily completed. The main reason why it is reasonable to assume completeness is that in a complete space, the following implication holds: if f and g are functions on X, f is -measurable and  .¹x 2 XI f .x/ ¤ g.x/º/ D 0; then g is also -measurable. We shall study several types of convergence of a sequence of functions. Definition 1.22.1. Let fn , n 2 N, and f be -measurable functions defined on a finite complete measure space .X; S; /. (i) We say that the sequence ¹fn º1 nD1 converges to f pointwise on X if lim fn .x/ D f .x/

n!1

for every x 2 X. We write fn ! f . (ii) We say that the sequence ¹fn º1 nD1 converges to f uniformly on X if for every " > 0 there exists an n0 2 N such that for every n  n0 and every x 2 X we have jfn .x/  f .x/j < ": We write fn  f . (iii) If X is further endowed with a metric %, then we say that the sequence ¹fn º1 nD1 converges to f locally uniformly on X if ¹fn º1 converges to f uniformly on nD1 loc

each compact subset K of X. We write fn  f . (iv) We say that the sequence ¹fn º1 nD1 converges to f uniformly up to small sets on X if for every " > 0 there exists an M  X, .M / < ", such that ¹fn º1 nD1 converges uniformly to f on X n M .

35

Section 1.22 Modes of convergence

(v) We say that the sequence ¹fn º1 nD1 converges to f almost everywhere on X if there exists a set M  X of measure zero such that ¹fn º1 nD1 converges a.e.

pointwise to f on X n M . We write fn ! f . (vi) We say that the sequence ¹fn º1 nD1 converges to f in measure on X if for every " > 0, lim .¹x 2 XI jfn .x/  f .x/j  "º/ D 0: n!1



We write fn ! f . Remark 1.22.2. Let fn , n 2 N, and f be -measurable functions defined on a finite complete measure space .X; S; /. Then the following implications hold: fn  f

)

loc

fn  f

)

fn ! f

)

a.e.

fn ! f:

(1.22.1)

Moreover, it is easy to construct examples showing that none of the implications can be reversed. However, there is a certain substitute result in this direction as the following theorem shows. Theorem 1.22.3 (Egorov). Let .X; S; / be a complete -finite measure space, let ¹fn º1 nD1 be a sequence of measurable functions on X and let f be a measurable function on X such that lim fn .x/ D f .x/ a.e. on X. Let E  X be such that .E/ < 1. Then, for every " > 0, there exists a measurable set M  E such that .M / < " and ¹fn º1 nD1 converges uniformly to f on E n M . Remark 1.22.4. The assertion of the Egorov theorem can also be restated as follows: a.e. If fn ! f on a set of finite measure, then fn ! f uniformly up to small sets. As a corollary, we get the following assertion. Proposition 1.22.5. Let .X; S; / be a complete -finite measure space such that .X/ < 1, let ¹fn º1 nD1 be a sequence of measurable functions on X and let f be a.e.

a measurable function on X. Then fn ! f on X if and only if fn ! f on X uniformly up to small sets. Now we shall study the relations between convergence in measure and other modes of convergence. Proposition 1.22.6. Let .X; S; / be a complete -finite measure space such that .X/ < 1, let ¹fn º1 nD1 be a sequence of measurable functions on X and let f be a.e.

a measurable function on X. Assume that fn ! f on X. Then the sequence ¹fn º1 nD1 converges to f in measure on X. The converse implication in Proposition 1.22.6 is not true. Moreover, the assumption .X/ < 1 is indispensable. We shall demonstrate these facts with examples.

36

Chapter 1 Preliminaries

Examples 1.22.7. (i) Let X D R, let  be the one-dimensional Lebesgue measure and S the -algebra of Lebesgue measurable sets. Define fn WD Œn;nC1 and f .x/ WD 0 for every x 2 R. Then fn ! f on R but lim .¹x 2 RI jfn .x/  f .x/j  1º/ D 1;

n!1

so ¹fn º1 nD1 does not converge to f in measure. (ii) Let X D .0; 1/, let  be the one-dimensional Lebesgue measure restricted to .0; 1/ and S the -algebra of sets of type A \ .0; 1/, where A is a Lebesgue measurable set on R. For every n 2 N we find the unique integer m 2 N such that 2m  n < 2mC1 . Then n D 2m C k for some k D 0; 1; : : : ; 2m . Define   k kC1 In WD m ; m 2 2 (note that the correspondence n 7! In is unique) and fn WD In . Define further f .x/ WD 0 for every x 2 R. Then ¹fn º1 nD1 converges to f in measure but not a.e. Again, there is at least a “partial” result in the converse direction, as the following theorem shows. Theorem 1.22.8 (Riesz). Let ¹fn º1 nD1 be a sequence of measurable functions which which converges in measure to f on . Then there exists a subsequence ¹fnk º1 kD1 converges to f almost everywhere on . We shall return to the study of modes of convergence in Section 3.13.

1.23 Systems of seminorms, Hahn–Saks theorem Definition 1.23.1. Let X be a Banach space and let P be a system of seminorms on X. We say that P is separating if for every x 2 X, x ¤  , there exist some p 2 P such that p.x/ ¤ 0. Definition 1.23.2. Let X be a vector space. We say that a subset B of X is balanced if for every x 2 B and every ˛ 2 Œ1; 1, we have ˛x 2 B. Theorem 1.23.3. Let X be a vector space and let P be a separating system of seminorms on X. Define ² ³ 1 V .p; n/ WD x 2 XI p.x/ < ; p 2 P; n 2 N; n and set B 0 WD ¹V .p1 ; n1 / \ : : : V .pk ; nk /; k 2 N; pi 2 P; ni 2 N; i D 1; : : : ; kº :

Section 1.23 Systems of seminorms, Hahn–Saks theorem

37

Then B 0 is a convex balanced basis of topology such that every p is continuous and for every bounded E  X and every p 2 P, p.E) is bounded in R. We shall now quote a classical result from measure theory (see, e.g., [105, Exercise 19.68, p. 339]). Theorem 1.23.4 (Hahn–Saks). Let .X; S; / be a measure space and let ¹n º1 nD1 be a sequence of finite measures on S. Assume that for every n 2 N, the measure n is absolutely continuous with respect to . Let  be a functional on S. Assume further that for every E 2 S such that .E/ < 1, one has lim n .E/ D .E/:

n!1

Then (i) The measures ¹n º are uniformly absolutely continuous with respect to , that is, 8" > 0 9ı > 0 8n 2 N; 8E  SW .E/ < ı ) n .E/ < "I (ii)  is a measure, which is absolutely continuous with respect to .

Chapter 2

Spaces of smooth functions

In this chapter we shall focus on spaces of continuous and smooth functions. Since however the role of such spaces for our purpose is in principal auxiliary, we will just briefly survey some of their very basic properties.

2.1 Multiindices and derivatives Definition 2.1.1. Let N be a positive integer (i.e. N 2 N). A vector ˛ D .˛1 ; : : : ; ˛N / with components ˛i 2 N0 .i D 1; : : : ; N /, where N0 D N [ ¹0º, is said to be a multiindex of dimension N . The number j˛j D

N X

˛i

iD1

is called the length of the multiindex ˛. We introduce the following notation (“calculus of multiindices”): given multiindices ˛; ˇ, ˛ C ˇ WD .˛1 C ˇ1 ; : : : ; ˛N C ˇN /; ˛Š WD ˛1 Š    ˛N Š; ! ˛Š ˛ WD ; ˇ ˇŠ.˛  ˇ/Š for ˛ D .˛1 ; : : : ; ˛N / a multiindex and x D .x1 ; : : : ; xN / 2 RN , ˛N : x ˛ WD x1˛1 x2˛2    xN

Notation 2.1.2. Let k 2 N0 . We denote by MN;k the set of all multiindices of dimension N whose length does not exceed k. For the sake of convenience, denote by  the multiindex with all components zero (i.e.  is the multiindex with zero length). Further, we denote by D .k/ D N .k/ the number of all elements of the set MN;k , i.e. the number of all different multiindices ˛ such that j˛j 5 k, and by

Q D .k/ Q D Q N .k/ the number of all elements of the set MN;k  MN;k1 , i.e. the total number of different multiindices ˛ such that j˛j D k.

Section 2.2 Classes of continuous and smooth functions

39

Remark 2.1.3. One can easily prove the following relations: .N C k  1/Š ; kŠ.N  1/Š .N C k/Š

.k/ D : N ŠkŠ

.k/ Q D

Notation 2.1.4. The concept of the “classical” partial derivative of a function of N real variables u D u.x/; x D .x1 ; : : : ; xN /; is well known. Here, we shall use the following notation: for ˛ a multiindex D ˛ u WD

@j˛j u ˛N : : : : @xN

@x1˛1 @x2˛2

In the following we consider functions all of whose derivatives up to a fixed order k are continuous. Thus the fact that for example, the same symbol D ˛ u for N D 2, k D 2, ˛ D .1; 1/ means both     @ @ @u @2 u @u @2 u D D and @x1 @x2 @x1 @x2 @x2 @x1 @x2 @x1 causes no trouble since

@2 u @x1 @x2

D

@2 u @x2 @x1

due to the continuity of the derivatives.

2.2 Classes of continuous and smooth functions Definition 2.2.1. Let  be a domain (i.e. an open connected set) in RN . We introduce: (i) C./ or C 0 ./ – the set of all functions defined and continuous on ; (ii) C k ./ with k 2 N – the set of all functions defined on  which have continuous derivatives up to the order k on ; (iii) C 1 ./ – the set of all functions defined on  which have derivatives of any order on , i.e. 1 \ 1 C k ./: C ./ WD kD0

(iv) Let k 2 N0 . Then C0k ./ denotes the set of all functions u 2 C k ./ whose supports are compact subsets of . (v) We denote C01 ./

WD

1 \ kD1

C0k ./:

40

Chapter 2 Spaces of smooth functions

Definition 2.2.2. Let  be a bounded domain in RN . (i) We denote by C./ or C 0 ./ the set of all functions in C./ which are bounded and uniformly continuous on . (ii) For k 2 N we denote by C k ./ the set of all functions u 2 C k ./ such that D ˛ u 2 C./ for all ˛ 2 MN;k . (iii) We denote C 1 ./ WD

1 \

C k ./:

kD0

Remark 2.2.3. Let  be a bounded domain in RN . Let k 2 N0 [ ¹1º. Then u 2 C k ./ means that for an arbitrary multiindex ˛, j˛j 5 k, there exists a uniquely determined function u˛ 2 C./ and such that the restriction of u˛ to  coincides with D ˛ u. In what follows, we shall extend D ˛ u on  as u˛ for u 2 C k ./ and j˛j 5 k. Notation 2.2.4. Let  be a bounded domain in RN , and let  2 .0; 1. (i) For u 2 C k ./ and ˛ a multiindex, j˛j 5 k, let us denote H˛; .u/ D sup

x;y2 x6Dy

jD ˛ u.x/  D ˛ u.y/j : jx  yj

(2.2.1)

(ii) By C k; ./ we denote the subset of all functions u 2 C k ./ such that H˛; .u/ < 1 for all ˛ with j˛j D k: Definition 2.2.5. A function u defined on  is said to satisfy the Hölder condition with exponent , 0 <  5 1, if there exists a nonnegative constant c D c.u/ such that ju.x/  u.y/j 5 cjx  yj

(2.2.2)

for all x; y 2 . If  D 1, we say that u satisfies the Lipschitz condition. Accordingly, the functions in C 0; ./ are said to be -Hölder continuous or, when  D 1, Lipschitz continuous or Lipschitz functions for short). Exercise 2.2.6.

(i) Show that H˛; .u/ 5 H˛; .u/  .diam / ; H˛; .u/ 5 .2 max jD ˛ u.x/j/ x2

for 0 <  5 .

 

.H˛; .u//

(2.2.3)  

(2.2.4)

Section 2.2 Classes of continuous and smooth functions

41

(ii) Show that a function u satisfying condition (2.2.2) with the exponent  > 1 is necessarily constant on the domain . Definition 2.2.7. Let  be a domain in RN , k 2 N0 and  2 .0; 1. Then C k;;0 ./ denotes the subset of all functions u 2 C k; ./ satisfying the following condition: For every " > 0 there exists a ı D ı."; u/ such that for all x; y 2  with 0 < jx  yj 0 there Since ¹un º1 nD1 is a fundamental sequence in C exists an n0 D n0 ."/ 2 N such that if n > n0 and m > n0 then

sup x;y2 x6Dy

jD ˛ .un  um /.x/  D ˛ .un  um /.y/j 0 there exists an n 2 N and real numbers a0 ; : : : ; an such that ju.t /  a0  a1 t      an t n j < " for all t 2 Œa; b. There are various proofs of this important assertion. We shall give here a proof based on a lemma by Korovkin from [120]. Definition 2.4.3. Let L W C.Œ0; 1/ ! C.Œ0; 1/ be a linear operator. We say that L is monotone if for every u; v 2 C.Œ0; 1/ such that u.t /  v.t / for all t 2 Œ0; 1, we have .Lu/.t /  .Lv/.t / for all t 2 Œ0; 1: Lemma 2.4.4 (Korovkin). Let ¹Hn º1 nD1 be a sequence of monotone linear operators from C 0 .Œ0; 1/ into C 0 .Œ0; 1/. Let lim kHn ei  ei k0 D 0

n!1

(2.4.1)

where ei .t / WD t i1 , i D 1; 2; 3. Then for every function u 2 C 0 .Œ0; 1/, lim kHn u  uk0 D 0:

n!1

Proof. Let u 2 C 0 .Œ0; 1/ be arbitrary. Since u is uniformly continuous, for every " > 0 there exists a ı > 0 such that ju.t1 /  u.t2 /j < 12 "

(2.4.2)

provided t1 ; t1 2 Œ0; 1, jt1  t2 j < ı. For s; t 2 Œ0; 1 we have ju.t /  u.s/j  12 " C 2kuk0 ı 2 .t  s/2 ; since if jt  sj < ı, inequality (2.4.2) implies (2.4.3), and if jt  sj  ı, then ju.t /  u.s/j  2kuk0  2kuk0 ı 2 .t  s/2 : Let s 2 Œ0; 1 be fixed for the moment. If we denote v.t / WD u.t /  u.s/ D u.t /  u.s/e1 .t /

(2.4.3)

46

Chapter 2 Spaces of smooth functions

and w.t / D 12 " C 2kuk0 ı 2 .t  s/2 D 12 "e1 .t / C 2kuk0 ı 2 ¹e3 .t /  2se2 .t / C s 2 e1 .t /º; we can rewrite (2.4.3) as w.t /  v.t /  w.t /: From the monotonicity and linearity of Hn it follows that .Hn w/.t /  .Hn v/.t /  .Hn w/.t / with .Hn v/.t / D .Hn u/.t /  u.s/.Hn e1 /.t / and .Hn w/.t / D 12 ".Hn e1 /.t / C 2kuk0 ı 2 ¹.Hn e3 /.t /  2s.Hn e2 /.t / C s 2 .Hn e1 /.t /º; which implies for arbitrary n 2 N and for arbitrary s; t 2 Œ0; 1 that j.Hn u/.t /  u.s/.Hn e1 /.t /j 

1 2 ".Hn e1 /.t /

C 2kuk0 ı

2

(2.4.4) 2

¹.Hn e3 /.t /  2s.Hn e2 /.t / C s .Hn e1 /.t /º:

By making s D t the right-hand side in (2.4.4) becomes 1 2"

C 12 "Œ.Hn e1 /.t /  e1 .t / C 2kuk0 ı 2 ¹Œ.Hn e3 /.t /  e3 .t / 2t Œ.Hn e2 /.t /  e2 .t / C t 2 Œ.Hn e1 /.t /  e1 .t /º:

Thus, j.Hn u/.t /  u.t /j  j.Hn u/.t /  u.t /.Hn e1 /.t /j C ju.t /.Hn e1 /.t /  u.t /e1 .t /j  12 " C ¹ 12 " C ju.t /j C e3 .t /  2kuk0 ı 2 ºj.Hn e1 /.t /  e1 .t /j C 4kuk0 ı 2 je2 .t /j  j.Hn e2 /.t /  e2 .t /j C 2kuk0 ı 2 j.Hn e3 /.t /  e3 .t /j and kHn u  uk0  12 " C ¹ 12 " C kuk0 C ke3 k0  2kuk0 ı 2 ºkHn e1  e1 k0 C 4kuk0 ı 2 ke2 k0 kHn e2  e2 k0 C 2kuk0 ı 2 kHn e3  e3 k0 : Now assumption (2.4.1) implies the assertion. Proof of Theorem 2.4.2. Let us define the operators Bn from C 0 .Œ0; 1/ into C 0 .Œ0; 1/ by !   n X n i u t i .1  t /ni ; n 2 N; t 2 Œ0; 1: .Bn u/.t / D i n iD0

47

Section 2.4 Separability, bases

Obviously, .Bn e1 /.t / D 1 D e1 .t /; .Bn e2 /.t / D t D e2 .t /; .Bn e3 /.t / D t 2 C

t  t2 t  t2 D e3 .t / C ; n n

and thus conditions (2.4.1) are satisfied. The operators Bn are linear and monotone, so Lemma 2.4.4 implies that lim kBn u  uk0 D 0:

n!1

However, Bn u is a polynomial of degree at most n (the so-called Bernstein polynomial), so that we have proved the assertion of Theorem 2.4.2 for a D 0, b D 1. Using a suitable transformation we obtain the assertion of Theorem 2.4.2 also for C 0 .Œa; b/. Theorem 2.4.2 asserts that every function in C 0 .Œa; b/ can be approximated in the norm of the space C 0 .Œa; b/ by a sequence of polynomials. The following generalization (for the proof see, e.g., [161]) to the case of more than one variable holds (let us recall that, according to the above convention, only bounded domains are admitted). Theorem 2.4.5. Let u 2 C 0 ./ and let " > 0. Then there exists a polynomial X Pn .x/ D a˛ x ˛ ; x 2 ; (2.4.5) j˛jn

where n 2 N and a˛ 2 R, j˛j  n, such that ku  Pn k0 < ":

(2.4.6)

Now we can easily prove the following result. Theorem 2.4.6. The space C 0 ./ is separable. Proof. Let " > 0. Let Pn be the polynomial from (2.4.5) satisfying (2.4.6) and let

D N .n/ be the number of its coefficients a˛ (cf. Notation 2.1.2). Further, let d be a positive number such that  is contained in the N -dimensional cube Cd D Œd; d N : For any coefficient a˛ of Pn there exists a rational number r˛ such that ja˛  r˛ j
1 is based on the same idea. Remark 2.4.8. We note that the boundedness of  is indispensable in Theorem 2.4.2. For example, when N D 1 and  D .0; 1/, then the space C 0 ./ is not separable. Indeed, for " 2 .0; 12 / and A  N, define 8 ˆ if k 2 A ˆ ˆuA .k/ D 1 ˆ 1 is known only for some special domains  (see, e.g., [51, 52, 191, 196, 197]). For 1 instance, let  D .0; 1/  .0; 1/ and let ¹en º1 nD1 be a Schauder basis in C .Œ0; 1/. We enumerate the set N  N in the following way: .1; 1/; .1; 2/; .2; 1/; .2; 2/; : : : ; .n; n/; .1; n C 1/; .2; n C 1/; : : : ; : : : ; .n; n C 1/; .n C 1; 1/; .n C 1; 2/; : : : ; .n C 1; n C 1/; : : :

(2.4.10)

and set hp .x; y/ D ei .x/  ej .y/ where .i; j / is the pth element in the sequence (2.4.10). The functions hp , p D 1; 2; : : : , form a basis in C 1 ./ (see [196]). An idea how to construct Schauder bases in the spaces C k ./ for domains  with “sufficiently smooth” boundary @ is contained in [192].

2.5 Compactness It is not our intention to investigate more deeply compactness of subsets of the spaces considered in this chapter, and, therefore, we give only the fundamental Arzelà–Ascoli theorem on the characterization of compact sets in C 0 ./ and some results on compact sets in C k ./ and C k; ./. Let us note that Convention 2.4.1 takes place. Definition 2.5.1. Let K be a subset of C 0 ./. Then K is said to be equicontinuous if to every " > 0 there exists a ı D ı."/ > 0 such that ju.x/  u.y/j < " holds for all u 2 K and all x; y 2  for which jx  yj < ı. Exercise 2.5.2. Prove the following assertion: A set K  C 0 ./ is equicontinuous if and only if for every " > 0 and every x 2  there exists a neighborhood V .x/ of x such that sup sup ju.x/  u.y/j  ": u2K y2V .x/

(Hint: Use the fact that  is bounded and consequently  is compact in RN .) In what follows we shall use this equivalent definition of equicontinuity. Theorem 2.5.3 (Arzelà–Ascoli). A subset K of C 0 ./ is relatively compact if and only if it is bounded and equicontinuous. Proof.

(I) Let K be relatively compact. Then for " > 0 there exists a finite "0 -net u1 ; : : : ; ur ."0 D 13 "/, i.e. for every u 2 K there exists an index i D i.u/, 1  i  r, such that (2.5.1) ku  ui k0 < 13 ":

52

Chapter 2 Spaces of smooth functions

For x 2  and i 2 ¹1; 2; : : : ; rº there exists a neighborhood Vi .x/ of x such that jui .x/  ui .y/j < 13 " T for all y 2 Vi .x/. The set V .x/ D riD1 Vi .x/ is a neighborhood of x and jui .x/  ui .y/j < 13 "

(2.5.2)

for all i  r and and all y 2 V .x/. Now, it follows from (2.5.1) and (2.5.2) that for any u 2 K, ju.x/  u.y/j  ju.x/  ui .x/j C jui .x/  ui .y/j C jui .y/  u.y/j < " for every y 2 V .x/. Consequently, K is equicontinuous. The boundedness of K follows from (2.5.1): kuk0 D ku  ui C ui k0  ku  ui k0 C kui k  13 " C max kui k0 : 1ir

(II) Let K be bounded and equicontinuous. The equicontinuity of K implies that for x 2  there exists a neighborhood V .x/ of x such that ju.x/  u.y/j < 14 " S for y 2 V .x/ and u 2 K. Since x2 V .x/  , a finite set of neighborhoods Vi D V .xi / .i D 1; : : : ; r/ can be chosen so that r [

V .xi /  :

iD1

As K is bounded, the set K.x/ D ¹u.x/I u 2 Kº is also a bounded subset b D of in R. Therefore K SrR and, consequently, K.x/ is relatively compact 1 0 iD1 K.xi / is also relatively compact in R. Let " D 4 ". Then there exists some s 2 N and a finite "0 -net ˛1 ; ˛2 ; : : : ; ˛s , i.e. jˇ  ˛i j < 14 " b and some ˛i (with i depending on ˇ). for every ˇ 2 K Let us denote by ˆ the set of mappings ' W ¹1; 2; : : : ; rº ! ¹1; 2; : : : ; sº and for ' 2 ˆ set K' D ¹u 2 KI ju.xi /  ˛'.i/ j  14 "; i D 1; : : : ; rº:

53

Section 2.5 Compactness

S We have that K  '2ˆ K' (the set ˆ is obviously finite). Indeed, for u 2 K we can construct ' 2 ˆ such that u 2 K' in the following way: For u.x1 / 2 K.x1 / there exists an ˛ t such that ju.x1 /  ˛ t j  14 ": We set '.1/ D t and define '.2/; : : : ; '.r/ analogously. Further diam K'  ": Indeed, for u; v 2 K' and x 2  we have that x 2 V .xi / for some i  r and ju.x/  v.x/j ju.x/  u.xi /j C ju.xi /  ˛'.i/ j C j˛'.i/  v.xi /j C jv.xi /  v.x/j < " which follows from the equicontinuity of K and from the definition of the set K' . Therefore, diam K' D sup sup ju.x/  v.x/j  ": u;v2K' x2

We now take one function u' from each set K' which yields a finite set of functions ¹u' º'2ˆ  K. If u 2 K then u 2 K' for some ' and, consequently, ku  u' k0  diam K'  ": Hence, K is relatively compact and the theorem is proved. By virtue of the Arzelà–Ascoli theorem (Theorem 2.5.3) we can easily prove the following theorem. Theorem 2.5.4. A set K  C k ./ is relatively compact if and only if the following conditions are satisfied: (i) K is bounded in C k ./; (ii) the sets K˛ D ¹D ˛ uI u 2 Kº are equicontinuous for all ˛ such that j˛j  k. Proof.

(i) Let conditions (i) and (ii) be satisfied and let ¹un º1 nD1 be a sequence 0 in K. Because K is also bounded in C ./ and K0 is equicontinuous, it follows from Theorem 2.5.3 that there exists a subsequence ¹un1 º of ¹un º which converges in C 0 ./. Now we use the assertion of Theorem 2.5.3 for the set K D ¹D ˛ un1 º with a fixed ˛, j˛j D 1. The set K is bounded in C 0 ./ and equicontinuous, thus there exists a subsequence ¹un2 º of ¹un1 º such that ¹D ˛ un2 º converges in C 0 ./. Repeating this process with the other derivatives D ˇ , we obtain after steps (for see Section 1.1) a subsequence ¹un º of ¹un º such that ¹D un º converges in C 0 ./ for all multiindices  which j j  k, i.e. the sequence ¹un º converges in C k ./.

54

Chapter 2 Spaces of smooth functions

(ii) Let K be a relatively compact subset of C k ./. Then every set K˛ .j˛j  k/ is relatively compact in C 0 ./ and thus K˛ is bounded and equicontinuous in C 0 ./. Hence, it follows that conditions (i) and (ii) of the theorem are satisfied. Definition 2.5.5. Let K be a subset of C k; ./. Then K is said to be .k; /-equicontinuous if for every " > 0 there exists a ı D ı."/ > 0 such that jD ˛ u.x/  D ˛ u.y/j 0, a sequence of functions un 2 K .n 2 N/, a multiindex ˛ .j˛j D k/ and two sequences of points xn ; yn .n 2 N/ belonging to  such that jD ˛ un .xn /  D ˛ un .yn /j 1 and  ": (2.5.3) 0 < jxn  yn j < n jxn  yn j Since K is relatively compact we can suppose un ! u in C k; ./ with u 2 C k;;0 ./ (see Exercise 2.3.4). For arbitrary n 2 N we have jD ˛ un .xn /  D ˛ un .yn /j jxn  yn j ˛ jD un .xn /  D ˛ un .yn /  D ˛ u.xn / C D ˛ u.yn /j jD ˛ u.xn /  D ˛ u.yn /j  C : jxn  yn j jxn  yn j

"

For n ! 1, the right-hand side in the last relation tends to zero. Indeed, the first term tends to zero since un ! u in C k; ./ and the other term tends to zero since u 2 C k;;0 ./. This is a contradiction. Remark 2.5.8. If the domain  has some special properties, e.g., if it satisfies condition (S) introduced in Definition 2.2.10, it can be shown that boundedness and .k; /equicontinuity of the set K  C k;;0 ./ is also a sufficient condition for K to be relatively compact in C k; ./.

55

Section 2.6 Continuous linear functionals

Exercise 2.5.9. Prove the following assertion: A .k; /-equicontinuous set K  C k;;0 ./ is relatively compact in C k; ./ if and only if it is relatively compact in C k ./. (Hint: First prove that if a .k; /-equicontinuous sequence ¹un º1 nD1 of functions in k;;0 ./ converges in C k ./ to a function u 2 C k ./ then u belongs to C k;;0 ./ C and un ! u in C k; ./). We shall now formulate several inclusions between the spaces of smooth functions. We will use the symbols ,! and ,!,! introduced in Notation 1.15.7. Theorem 2.5.10. Let k 2 N0 ,  2 .0; 1/. Then C k; ./ ,! C k ./:

(2.5.4)

C k; ./ ,!,! C k ./:

(2.5.5)

Moreover, Proof.

(i) Embedding (2.5.4) follows from the definition of the spaces under consideration.

(ii) To prove (2.5.5), let K be a bounded set in C k; ./. Then K is obviously bounded in C k ./ and further, there exists a constant c > 0 such that jD ˛ u.x/  D ˛ u.y/j  H˛; .u/  c jx  yj for all u 2 K (where j˛j D k). This implies that the sets K˛ D ¹D ˛ uI u 2 Kº are equicontinuous for j˛j D k. It can be shown by integration that the sets K˛ are also equicontinuous for j˛j < k and the compactness of K in C k ./ follows from Theorem 2.5.4. Remark 2.5.11. The assertion of Theorem 2.5.10 can be strengthened: For k 2 N0 and 0 <  <   1, C k; . ,!,! C k; ./ (see, e.g., [242]; cf. also Exercise 2.2.6 (i)).

2.6 Continuous linear functionals Definition 2.6.1. Let K be a compact subset of RN . We define the space C.K/ of all real-valued functions defined and continuous on K. This space is endowed with the norm kukC.K/ WD sup ju.x/j: x2K

56

Chapter 2 Spaces of smooth functions

Our next aim is to characterize the dual space ŒC.K/ of the space C.K/. We recall (cf. Definition 1.20.14) that, for a compact subset K of RN , the smallest -algebra containing all closed subsets of K is denoted by B.K/ and every M 2 B.K/ is called a Borel subset of K. Any measure defined on B.K/ is called a Borel measure. We also introduce the symbol var.; K/ WD sup

m X

j.Mi /j

iD1

where the supremum is taken over all finite systems ¹Mi ºm iD1 , Mi 2 B.K/ such that Mi \ Mj D ; for i 6D j . As usual, the norm of a functional ˆ in ŒC.K/ is given by kˆk WD sup jˆ.u/j: kuk0 1

Now we shall formulate the Riesz representation theorem. For the proof see, e.g., [66] or [222]. Theorem 2.6.2 (Riesz). Let K be a compact subset of RN and let ˆ be a continuous linear functional on C.K/. Then there exists a uniquely determined real Borel measure  on K such that Z u.x/ d.x/ ˆ.u/ D K

for every u 2 C.K/. Moreover, kˆk D kˆkŒC.K/ D sup

m X

j.Mi /j;

iD1

where the supremum is extended over all finite pairwise disjoint systems ¹Mi ºm iD1 , Mi 2 B.K/. Later we shall use the following important result: Theorem 2.6.3. The space C.K/ is not reflexive. Proof. For ˆ 2 ŒC.K/ let ˆ be the measure from Theorem 2.6.2, i.e. ˆ is a measure such that Z u.x/ dˆ .x/ ˆ.u/ D K

for all u 2 C.K/. Let x0 be a fixed point in K and let us define a functional F0 on ŒC.K/ by F0 .ˆ/ D uˆ .¹x0 º/

57

Section 2.6 Continuous linear functionals

for every ˆ 2 ŒC.K/ . The functional F0 is obviously linear and sup jF0 .ˆ/j D

kˆk1

j.¹x0 º/j D 1;

sup var.;K/1

so that F0 is also bounded and, consequently, F0 2 ŒC.K/ . Let us suppose that C.K/ is reflexive. Thus there exists a function u0 2 C.K/ such that Z u0 .x/ dˆ .x/ F0 .ˆ/ D ˆ.u0 /; i.e. ˆ .¹x0 º/ D K

for any ˆ 2

ŒC.K/ .

If we define the Borel measure 0 by Z u0 .x/ dx 0 .M / D M

for M 2 B.K/, then for every u 2 C.K/, Z Z u.x/ d0 .x/ D u.x/u0 .x/ dx: K

K

Let us denote by ˆ0 the functional in the space ŒC.K/ which is defined by the measure 0 . Then Z F0 .ˆ0 / D 0 .¹x0 º/ D u0 .x/ dx D 0: (2.6.1) ¹x0 º

On the other hand, ˆ0 is not the null-functional and, consequently, u0 is not everywhere zero. Therefore, Z Z u0 .x/ d0 .x/ D Œu0 .x/2 dx F0 .ˆ0 / D ˆ0 .u0 / D K

K

is a positive number which contradicts (2.6.1). Consequently, C.K/ cannot be reflexive. Let us now consider the space C k;;0 ./ with k 2 N0 and  2 .0; 1/. For u 2 the function

C k;;0 ./,

Fˇ .x; y/ D

D ˇ u.x/  D ˇ u.y/ ; jˇj D k jx  yj

is defined for x; y 2  such that x 6D y. If we set Fˇ .x; x/ D 0 we obtain a function which is defined and continuous on   .

(2.6.2)

58

Chapter 2 Spaces of smooth functions

Let us introduce the product space …./ D C./  C./      C./  C.  /      C.  / ƒ‚ … „ ƒ‚ … „ Q -times

-times

(for the definition of and Q see Notation 2.1.2). Then we can identify the space C k;;0 ./ with a closed subspace of …./, namely, with the subspace of elements ¹.u˛ .x/; vˇ .x; y//I j˛j  k; jˇj D kº 2 …./ corresponding to an element u 2 C k;;0 ./ by the formulas u˛ .x/ D D ˛ u.x/; vˇ .x; y/ D Fˇ .x; y/: On the other hand, …./ can be identified with the space C.K/ where K is a compact subset of RM with M D N. .k/ C 2Q .k//, and the problem of characterizing the continuous linear functionals on C k;;0 ./ can by virtue of the Hahn–Banach theorem (Theorem 1.16.4) be reduced to the problem considered in Theorem 2.6.2. Hence we have the following theorem. Theorem 2.6.4. Let ˆ be a continuous linear functional defined on C k;;0 ./. Then there exist two uniquely determined families of Borel measures ˛ on  .j˛j  k/; vˇ on    .jˇj D k/ such that ˆ.u/ D

X Z j˛jk

X Z

˛



D u.x/ d˛ .x/ C



jˇ jDk

D ˇ u.y/  D ˇ u.z/ dvˇ .y; z/ jy  zj (2.6.3)

for every u 2 C k;;0 ./. Moreover, X X var.˛ ; / C var.vˇ ;   /: kˆk D j˛jk

jˇ jDk

Remark 2.6.5. (i) For functionals on C k ./ we can prove analogously the same assertion (with vˇ D 0 for jˇj D k) as in Theorem 2.6.4, i.e. if ˆ 2 ŒC k ./ then X Z D ˛ u.x/ d˛ .x/ ˆ.u/ D j˛jk



for every u 2 C k ./ and kˆk D

X j˛jk

var.˛ ; /:

59

Section 2.7 Extension of functions

(ii) For functionals on C k; ./ we cannot use the above argument because the functions Fˇ .x; y/ from (2.6.2) cannot in general be extended in this case to all of    so as to be continuous on   . Nonetheless, these functions are bounded on .  /  S , where S has measure zero, and instead of …./ from (1.7.4.2) we can introduce another product space …1 ./ D C./      C./  L1 .  /      L1 .  / ƒ‚ … „ ƒ‚ … „

(2.6.4)

Q -times

-times 1

(for L .  / see Section 2.11). Then we obtain that every continuous linear functional ˆ on C k; ./ can be expressed in the form (2.6.3) but with more restrictive conditions on the measure vˇ , jˇj D k. For details see, e.g., [236]. Remark 2.6.6. To complete the assertion of Theorem 2.6.3 it can be shown that the spaces C k ./, C k;1 ./ and C k; ./, C k;;0 ./ for 0 <  < 1 are not reflexive. The nonreflexivity of the last two spaces follows from the following characterization of the second dual space to C k;;0 ./ (for the details see [87] or [61]): Let 0 <  < 1. Then the spaces ŒC k;;0 ./ and C k; ./ are isometrically isomorphic. On the other hand, C k;1;0 ./ (as a finite dimensional space, see Exercise 2.2.8 (i)) is reflexive.

2.7 Extension of functions Let u be a real-valued function defined on a set M  RN . A function U defined on RN is said to be an extension of u if U.x/ D u.x/

for

x 2 M:

We shall now study the problem of extending u in such a way that its extension U has – roughly speaking – the same properties as u itself. First, let us remark that a necessary condition for the existence of continuous extensions of continuous functions is that u is defined on a closed set. Indeed, let M n M 6D ;. We construct an example of a continuous function u on M which has no continuous extension to M and, consequently, cannot be extended continuously to all of RN . Suppose that x0 2 M n M . We choose a sequence xn 2 M , n 2 N, such that x0 D lim xn and jxnC1  x0 j < jxn  x0 j n!1

for n 2 N. There exists a continuous real-valued function g defined on R such that 1 for n 2 N; n g.t / D 0 if and only if t  0:

g.jxn  x0 j/ D

60

Chapter 2 Spaces of smooth functions

(For instance, we can set g.t / D 0 for t  0, g.t / D 1 for t  intervals ŒjxnC1  x0 j; jxn  x0 j, n 2 N.) The function u.x/ D sin

1 ,

g linear on the

1 g.jx  x0 j/

is obviously continuous and bounded on M . Nevertheless, a continuous extension of u to M does not exist. Before we state the fundamental results due to Whitney concerning extensions of functions in C k ./ and C k; ./ we prove one easy assertion. Let us introduce the necessary notation. Notation 2.7.1. For u defined and continuous on a closed set M  RN denote kuk0;M WD sup ju.x/j; x2M

H;M .u/ WD sup

x;y2M x6Dy

ju.x/  u.y/j jx  yj

.0 <   1/:

Further, C 0; .M / denotes the space of continuous functions u on M such that kuk0;M C H;M .u/ < 1: The following result from [16] is an extension of the classical Tietze extension theorem (see [223] or [130]). Theorem 2.7.2. Let M be a closed set in RN , let u 2 C 0; .M / and let 0 <   1. Then there exists an extension U 2 C 0; .RN / of u such that kU k0;RN D kuk0;M

(2.7.1)

H;RN .U / D H;M .u/:

(2.7.2)

v.x/ D sup ¹u.y/  H;M .u/jx  yj º

(2.7.3)

and Proof. For x 2 RN set y2M

and define U by

´ v.x/ U.x/ D kuk0;M

for x such that for x such that

jv.x/j  kuk0;M ; v.x/ < kuk0;M :

(2.7.4)

Let x 2 M . It follows from (2.7.3) that v.x/  u.x/. If we suppose that v.x/ > u.x/ then there exists a point y 2 M such that u.y/  H;M .u/jx  yj > u.x/;

61

Section 2.7 Extension of functions

which leads to a contradiction with the definition of H;M .u/. Therefore, u.x/ D v.x/

for

x2M

U.x/ D u.x/

for

x 2 M;

(2.7.5)

and (2.7.4) implies i.e. U is an extension of u. From (2.7.4) it also follows that kU k0;RN D kuk0;M and we have (2.7.1). Further, let x; y 2 RN . The case U.x/ D U.y/ is devoid of interest. If U.x/ 6D U.y/ we can suppose that U.x/ > U.y/. Then we have from (2.7.4) that 0 < U.x/  U.y/  v.x/  v.y/ and in view of v.x/  v.y/ D sup ¹u.z/  H;M .u/jx  zj º  sup ¹u.z/  H;M jy  zj º z2M 

z2M 

 H;M .u/ sup ¹jy  zj  jx  zj º  H;M .u/jx  yj z2M

we have H;RN .U /  H;M .u/: We have that H;RN .U /  H;M .u/ since U is an extension of u. Thus we have (2.7.2) and the proof is complete. Remark 2.7.3. Formula (2.7.5) in the foregoing proof shows that the function v from (2.7.3) is also an extension of u; v is continuous and H;RN .v/ < 1, but v is not necessarily bounded.

Chapter 3

Lebesgue spaces

3.1 Lp -classes and some integral inequalities Notation 3.1.1. Let p 2 .0; 1/ (the case p D 1 will be considered in Section 3.10). Let  be a (Lebesgue) measurable subset of the Euclidean space RN . Denote by Lp ./ the set of all real-valued measurable functions f defined almost everywhere on  and such that the Lebesgue integral Z jf .x/jp dx (3.1.1) 

is finite. For  2 R and f; g 2 Lp ./, we define f and f C g by the relations .f /.x/ WD f .x/;

.f C g/.x/ WD f .x/ C g.x/

for x 2 : For f 2 Lp ./ we set Z Np .f / WD

p



jf .x/j dx

 p1

:

(3.1.2)

Remark 3.1.2. It follows immediately from (3.1.2) that, given p; q 2 Œ1; 1/, we have   p Nq jf jp D Npq .f / : (3.1.3) Definition 3.1.3. For every p 2 .1; 1/, we define its conjugate Lebesgue index p 0 by the relation 1 1 C 0 D 1: p p In other words, we have p0 D

p : p1

(3.1.4)

In what follows, we shall first prove some inequalities which will be useful. Theorem 3.1.4 (Young inequality). Let ' be a continuous real-valued strictly increasing function defined on Œ0; 1/ such that lim '.u/ D 1

u!1

Section 3.1 Lp -classes

63 φ

b

φ b

ψ (b)

ψ (b)

ϕ (a) 0

b > φ (a)

ϕ (a) a

0

b < φ (a)

a

Figure 3.1. Young inequality.

and '.0/ D 0: Let

D ' 1 : For all x 2 Œ0; 1/, define Z ˆ.x/ WD

x

'.u/ du

0

Z

and ‰.x/ WD Then, for all a; b 2 Œ0; 1/,

x

.v/ dv: 0

ab  ˆ.a/ C ‰.b/I

the equality occurs if and only if b D '.a/: Proof. The above inequality is seen to be highly plausible by observing Figure 3.1. A formal proof can be given by virtue of the fact that Z c Z '.c/ ˆ.c/ C ‰.'.c// D '.u/ du C .v/ dv D c'.c/ 0

0

for all c 2 Œ0; 1/. Corollary 3.1.5. Given p 2 .1; 1/ and nonnegative real numbers a and b; we have 0

ab 

bp ap C 0; p p

(3.1.5) 0

where p 0 is defined by (3.1.4). Equality holds in (3.1.5) if and only if ap D b p : Proof. For u 2 Œ0; 1/, define '.u/ D up1 ; ' is continuous and strictly increasing, 1 lim '.u/ D 1 and '.0/ D 0: The inverse of ' is given by .v/ D v p1 : The u!1 corollary follows immediately from the Young inequality (Theorem 3.1.4).

64

Chapter 3 Lebesgue spaces 0

Theorem 3.1.6 (Hölder inequality). Let p 2 .1; 1/, f 2 Lp ./ and g 2 Lp ./. Then fg 2 L1 ./ and ˇZ ˇ Z ˇ ˇ ˇ f .x/g.x/ dx ˇ  (3.1.6) jf .x/g.x/j dx  Np .f /Np 0 .g/: ˇ ˇ 



If there exists a positive constant c such that jf .x/jp D cjg.x/jp

0

(3.1.7)

for almost every x 2 , then the equality Z jf .x/g.x/j dx D Np .f /Np 0 .g/

(3.1.8)



holds. Proof. If either of the functions f or g is zero almost everywhere, then (3.1.6) is trivial. Otherwise, applying inequality (3.1.5) to aD we have

jf .x/j ; Np .f /

bD

jg.x/j ; Np 0 .g/ 0

1 jf .x/jp 1 jg.x/jp jf .x/j jg.x/j    p C 0  p0 Np .f / Np 0 .g/ p Np .f / p N 0 .g/ p

(3.1.9)

for all x 2 . Integrating (3.1.9) we obtain (3.1.6) since 1 1 C 0 D 1: p p If (3.1.7) is satisfied for some c 2 .0; 1/ and almost every x 2 , then (3.1.8) follows from Corollary 3.1.5. In Section 3.11, we shall need a version of the Hölder inequality for three functions. We shall omit the proof. Theorem 3.1.7 (Hölder inequality for three functions). Let p; q; r 2 .1; 1/ be such that 1 1 1 C C D 1: p q r Furthermore, let f 2 Lp ./, g 2 Lq ./ and h 2 Lr ./. Then fgh 2 L1 ./ and ˇZ ˇ Z ˇ ˇ ˇ f .x/g.x/h.x/ dx ˇ  jf .x/g.x/h.x/j dx  Np .f /Nq .g/Nr .h/: (3.1.10) ˇ ˇ 



Section 3.1 Lp -classes

65

Theorem 3.1.8 (Minkowski inequality). Let p 2 Œ1; 1/ and let f; g 2 Lp ./. Then f C g 2 Lp ./ and Np .f C g/  Np .f / C Np .g/:

(3.1.11)

Proof. For p D 1, the assertion follows from the triangle inequality. Suppose that p 2 .1; 1/. We first note that f C g 2 Lp ./. Indeed, that follows from integrating over  the estimate jf .x/ C g.x/jp  .jf .x/j C jg.x/j/p  Œ2 max¹jf .x/j; jg.x/jºp  2p .jf .x/jp C jg.x/jp /; which holds for almost all x 2 . In particular, Np .f C g/ < 1. We can also assume that Np .f C g/ > 0 as otherwise there is nothing to prove. By the Hölder inequality (3.1.6), by (3.1.3) applied to q D p 0 and by using the relation p 0 .p  1/ D p, we get Z Z p p jf .x/ C g.x/j dx D jf .x/ C g.x/jjf .x/ C g.x/jp1 dx Np .f C g/ D   Z Z p1 jf .x/jjf .x/ C g.x/j dx C jg.x/jjf .x/ C g.x/jp1 dx  



 Np .f /Np 0 .jf C gjp1 / C Np .g/Np 0 .jf C gjp1 /   D Np .f / C Np .g/ Np 0 .jf C gjp1 /   p1 D Np .f / C Np .g/ Np .f C g/ ; p1  . and our claim follows on dividing this inequality by Np .f C g/ Corollary 3.1.9. The set Lp ./ is a vector space. The Minkowski inequality implies that for p 2 Œ1; 1/ the functional f 7! Np .f / satisfies the triangle inequality. Furthermore, it obviously satisfies the homogeneity axiom (ii) of Definition 1.4.1. Remark 3.1.10 (Clarkson inequalities). Let f; g 2 Lp ./: If p 2 Œ2; 1/, then Npp .f C g/ C Npp .f  g/  2p1 .Npp .f / C Npp .g//

(3.1.12)

while if p 2 Œ1; 2/, then Npp .f C g/ C Npp .f  g/  2.Npp .f / C Npp .g//: The proof can be obtained by considering the function '.t / D

.1 C t /p C .1  t /p ; 1 C tp

t 2 Œ0; 1:

(3.1.13)

66

Chapter 3 Lebesgue spaces

Remark 3.1.11. Let p 2 .0; 1/ and let f; g 2 Lp ./, f  0 and g  0. Then, again, f C g 2 Lp ./, but instead of the Minkowski inequality we have the reverse inequality, namely Np .f C g/  Np .f / C Np .g/: (3.1.14) Remark 3.1.12 (generalized Hölder inequality). Let n 2 N and let pi 2 .1; 1/, i D 1; : : : ; n, be such that n X 1 D 1: pj j D1

Let fi 2 L ./, i D 1; : : : ; n. Then pi

f1 f2   fn 2 L1 ./ and

Z 

jf1 .x/f2 .x/    fn .x/j dx  Np1 .f1 /Np2 .f2 /    Npn .fn /:

In particular, if f1 ; f2 ; : : : ; fn 2 Ln ./, then f1 f2   fn 2 L1 ./ and

Z 

jf1 .x/f2 .x/    fn .x/j dx  Nn .f1 /Nn .f2 /    Nn .fn /:

3.2 Lebesgue spaces Convention 3.2.1. From now on, two measurable functions f; g on  will be considered to be equal if f .x/ D g.x/ for almost all x 2 . Notation 3.2.2. Let p 2 .0; 1/. Then we denote by Lp ./ the set of (equivalence classes of) functions from Lp ./. The elements of Lp ./ will be called “functions.” In the case that  is an open interval .a; b/ in R, we shall simply write Lp .a; b/ instead of Lp ..a; b//. Lemma 3.2.3. Let p 2 Œ1; 1/ and set kf kp D Np .f /:

(3.2.1)

Then Lp ./ endowed with kf kp is a normed linear space. Proof. By Corollary 3.1.9, Lp ./ is a linear set. The homogeneity axiom (ii) of Definition 1.4.1 for the norm follows immediately from the properties of the Lebesgue integral. The fact that kf kp > 0 if f ¤ 0 is guaranteed by the equality relation on the set Lp ./. The triangle inequality is a consequence of the Minkowski inequality (3.1.11).

67

Section 3.3 Mean continuity

Exercise 3.2.4. Let   RN and let 1  p2 < p1 < 1. (i) If ./ < 1 and f 2 Lp1 ./, then also f 2 Lp2 ./ and kf kp2  ..//

p1 p2 p1 p2

kf kp1 :

(3.2.2)

(Hint: The assertion follows immediately from the Hölder inequality 3.1.6 applied to g.x/ 1.) (ii) If ./ D 1, then there is no inclusion between the sets Lp1 ./ and Lp2 ./.

3.3 Mean continuity Convention 3.3.1. Here we will, when convenient, consider a function f defined almost everywhere on  to be extended outside of  by zero. We thus obtain a function F defined for almost all x 2 RN by ´ f .x/ if x 2 ; F .x/ D 0 if x … : Instead of F .x/ we shall often simply write f .x/ also for x … : Definition 3.3.2. Let p 2 .0; 1/ and f 2 Lp ./. The function f is said to be p-mean continuous if for every " > 0 there exists a ı D ı."/ > 0 such that Z p



jf .x C h/  f .x/j dx

 p1

0: According to the absolute continuity of the Lebesgue integral (Theorem 1.21.13), there exists an  > 0 such that for each E   satisfying .E/ < 4 we have  p1 Z " p (3.3.1) jf .x/j dx < : 4 E For this , there exists a % > 0 such that .H% / < ; where H% D ¹x 2 I dist.x; @/  %º

68

Chapter 3 Lebesgue spaces

and @ is the boundary of  defined by @ WD  \ R n . Set % WD  n H% . Clearly, f is measurable on % and thus the Luzin theorem (Theorem 1.21.12) implies the existence of a closed set F1  % such that the restriction of the function f to F1 is continuous, .% n F1 / <  and thus . n F1 / < 2: The function f is uniformly continuous on F1 since F1 is compact. Hence, there exists a ı 2 .0; %/ such that jf .x C h/  f .x/j
0 and let " be an arbitrary number from .0; 12 kf k1 /. The obvious inequality 1 kf kp  kf k1 ..// p implies lim sup kf kp  kf k1 : p!1

Then there exists a set B   of positive measure such that for every x 2 B; jf .x/j  kf k1  ": Moreover, Z kf kp 

B

jf .x/jp dx

 p1

1

 ..B// p .kf k1  "/:

Hence, lim inf kf kp  kf k1  "; p!1

and thus lim inf kf kp  kf k1  lim sup kf kp : p!1

p!1

Theorem 3.10.8. Let ./ < 1: Let 1  p1  p2  : : : and suppose lim pk D 1:

k!1

Let

1 \

f 2

Lpk ./

kD1

and a D sup kf kpk < 1:

(3.10.3)

k2N

Then f 2 L1 ./: Proof. Set M D ¹x 2 I jf .x/j  a C 1º: If .M / D 0, then the assertion is obvious. Suppose .M / > 0: Then Z kf kpk 

M

jf .x/j

pk

 p1 dx

k

1

 .a C 1/ ..M // pk :

Hence, we obtain 1

a D sup kf kpk  lim .a C 1/ ..M // pk D a C 1; k2N

which is a contradiction.

k!1

82

Chapter 3 Lebesgue spaces

Example 3.10.9. Obviously, log x 2 Lp .0; 1/ for every p 2 Œ1; 1/ but log x … L1 .0; 1/: This example shows that assumption (3.10.3) is essential for the assertion of Theorem 3.10.8. The following result follows immediately from the properties of the uniform convergence of functions. Theorem 3.10.10. L1 ./ is a Banach space. The most important feature of the space L1 ./ consists in the fact that its elements represent all bounded linear functionals on L1 ./. We shall now study this topic in detail. Theorem 3.10.11. Let  be a nonempty bounded open subset of RN : (i) Let g 2 L1 ./ and define the functional ˆg on L1 ./ by Z ˆg .f / D f .x/g.x/ dx 

for f 2 L1 ./: Then,

  ˆg 2 L1 ./

and

  ˆg  D kgk : 1

(ii) Let ˆ be a bounded linear functional on L1 ./: Then there exists one and only one g 2 L1 ./ such that Z ˆ.f / D g.x/f .x/ dx 

for f 2 L1 ./. Proof. The assertion can be verified in an analogous way as those of Theorems 3.8.1 and 3.8.3. The proof of (ii) uses the following result, analogous to Lemma 3.8.2. Lemma 3.10.12. Let g be a measurable function on : Suppose that there exists an M > 0 such that, for arbitrary f 2 L1 ./; fg 2 L1 ./I ˇZ ˇ ˇ ˇ ˇ f .x/g.x/ dx ˇ  M kf k : 1 ˇ ˇ 

Then g 2 L1 ./ and kgk1  M:

(3.10.4) (3.10.5)

83

Section 3.11 Hardy inequalities

Proof. In the same way as in Lemma 3.8.2 we verify that jgjn sign g 2 L1 ./ and

Z 

jg.x/j

n

 n1

1

 M..// n

for arbitrary n 2 N: Thus, g2

1 \

Ln ./;

nD1

sup kgkn < 1

n2N

and Theorems 3.10.7 and 3.10.8 yield the assertion. Theorem 3.10.13. Let  be a nonempty bounded open subset of RN : Then L1 ./ is not reflexive. Proof. Let us suppose that L1 ./ is reflexive. Since it is separable (see Theorem   1 3.6.1), L ./ is also separable. On the other hand, L1 ./ is isomorphic with the nonseparable Banach space L1 ./, which is a contradiction. Theorem 3.10.14. Let  be a nonempty bounded open subset of RN : Then L1 ./ is not reflexive. Proof. The assertion follows from the fact that L1 ./ contains the nonreflexive Banach space C./ as a subspace (see Theorem 2.6.3).

3.11 Hardy inequalities In this section we shall collect some very important inequalities involving integrals and suprema under a common title of Hardy inequalities. Such inequalities are indispensable for many applications. There is a vast amount of literature available for this topic, thus let us name just a few, for instance [128, 127, 148, 149, 171]. Here we follow the explanation of [149, Section 1.3]. We start with recalling the Minkowski integral inequality. Theorem 3.11.1 (Minkowski integral inequality). Let p 2 Œ1; 1 and let F .x; y/ be a measurable function of two variables defined on .0; 1/  .0; 1/. Assume that F .; y/ 2 Lp .0; 1/ for a.e. fixed y 2 .0; 1/ and that Z 1 kF .; y/kp dy < 1: 0

84 Then

Chapter 3 Lebesgue spaces

R1 0

F .x; y/ dy converges for a.e. x 2 .0; 1/ and  Z 1 Z 1      F .x; y/ dy kF .; y/kp dy:   0

(3.11.1)

0

p

Proof. When p D 1, the assertion is obvious. Let p < 1 and set Z 1 f .x/ WD F .x; y/ dy; x 2 .0; 1/: 0

0

Let g 2 Lp .0; 1/ be such that kgkp 0  1. Then, by the Fubini theorem (1.21.11) and the Hölder inequality (Theorem 3.1.6), we get ˇZ 1 ˇ Z 1Z 1 ˇ ˇ ˇ f .x/g.x/ dx ˇˇ  jF .x; y/jjg.y/j dx dy ˇ 0 Z0 1 0  kF .; y/kp kgkp 0 dy 0 Z 1 kF .; y/kp dy:  0

Thus, by Lemma 3.8.2, we get f 2 Lp .0; 1/ and the estimate (3.11.1). The proof is complete. Lemma 3.11.2. Let f be a positive measurable function on .0; 1/ and p 2 .1; 1/. Then p

1 Z t

Z

f .s/ ds 0

0

dt tp

! p1 p

0

Z

1

p

f .x/ dx

 p1

:

(3.11.2)

0

Proof. Using the Hölder inequality, we get p

1 Z t

Z

f .s/ ds 0

0

dt D tp

Z

1 Z t

1

1

f .s/s pp0 s  pp0 ds

0 0 1Z t

p

dt tp p1 ds dt

Z t 1 1 f .s/p s p0 ds s p 0 0 Z 1 Z0 1 1 p. 1 1/ 0 p1 p p0 f .s/ s t .p0 /2 dt ds D .p / s Z 10 f .s/p ds; D .p 0 /p 

Z

0

and the assertion follows on taking pth roots.

85

Section 3.11 Hardy inequalities

Remark 3.11.3. The preceding lemma can be trivially extended to the case when p D 1. More precisely, for a positive measurable function f on .0; 1/, one has Z 1 t f .s/ ds  ess sup f .t /: (3.11.3) ess sup t2.0;1/ t 0 t2.0;1/ Indeed, this follows immediately from setting K WD ess sup t2.0;1/ f .t / as then 1 t2.0;1/ t

Z

t

ess sup

0

1 t2.0;1/ t

Z

f .s/ ds  ess sup

t 0

K ds D K:

On the other hand, the assertion of Lemma 3.11.2 does not hold for p D 1. This can be observed for example by taking f D .0;1/ , since then Z 1 f .x/ dx D 1 0

but

Z

1

1 t

0

Z

Z

t 0

f .s/ ds dt 

1

1

1 t

Z

Z

1

f .s/ ds dt D

0

1 1

dt D 1: t

For p D 1, (3.11.2) is not true even when .0; 1/ is replaced with a finite interval. For instance, there is no positive constant C that would render the inequality Z 1 Z 1 Z t 1 f .s/ ds dt  C f .x/ dx 0 t 0 0 true for all positive functions on .0; 1/. To see this, take for example 1 f .t / WD  2 ; t log 2t Then, again,

R1 0

t 2 .0; 1/:

f .x/ dx < 1 but, with appropriate c, Z

1 0

1 t

Z

Z

t 0

f .s/ ds dt D c

1 0



1

t log 2t

 dt D 1:

Definition 3.11.4. Let w be a nonnegative Lebesgue measurable function defined on the interval .0; 1/. We then say that w is a weight on .0; 1/. Lemma 3.11.5. Assume that f and g are nonnegative functions on Œ0; 1/ and let p 2 Œ1; 1/. Then Z

Z

1

g.x/ 0

p

x

f .t / dt 0

 p1 dx

Z 

Z

1

f .t / 0

 p1

1

g.x/ dx t

dt:

(3.11.4)

86

Chapter 3 Lebesgue spaces

Proof. We have Z

1

Z

g.x/ 0

p

x

f .t / dt

 p1 dx

0

1 Z 1

Z D

p

1 p

f .t /g.x/ Œt;1/ .x/ dt 0

 p1 dx

:

0

By the Minkowski integral inequality (Theorem 3.11.1), we obtain 1 Z 1

Z 0



f .t /g.x/ Œt;1/ .x/ dt

0

1 Z 1

Z

p

1 p

1 p

 p1 dx

p

Œf .t /g.x/ Œt;1/ .x/ dx 0

0

Z D

Z

1

dt

 p1

1

f .t /

g.x/ dx

0

 p1

dt;

t

as desired. The proof is complete. Lemma 3.11.6. Assume that 1  q < p < 1, let b 2 .0; 1 and let w be a positive measurable function on .0; b/. Then the following two statements are equivalent. (i) There exists a positive constant C such that Z

Z

b

w.t /

! q1

q

t

g.s/ ds

0

Z C

dt

0

b

! p1 g.t /p dt

(3.11.5)

0

for every positive measurable function g. (ii) The estimate 0 A WD @

Z

Z

b

w.s/ ds 0

1 pq pq

p ! pq

b

t

.q1/p pq

dt A

t

1. Integrating by parts on the left-hand side p p , q1 , of (3.11.5) and using the Hölder inequality (3.1.10) for the parameters pq p (observe that Z

pq p

q1 p

C Z

b

1 p

D 1), we get ! q1

q

t

w.t /

g.s/ ds

0

Dq

C

dt

0 1 q

Z

b

Z

Z

b

w.s/ ds g.t / 0

t

g.y/ dy 0

! q1

q1

t

dt

87

Section 3.11 Hardy inequalities

2

Z

1 q

q 4 0 Z @ 

! p1

b

Z

Z

b

p

g.y/ dy

g.s/ ds

0

b

t

0

Z

.q1/p pq

p ! pq

b

t

p

! q1 p dt

0

w.s/ ds

0

p

t

t

3 q1 1 pq q 7 dt A 5 :

By (ii) and the power-weight inequality (3.11.2), we see that the last expression is majorized by ! p1 Z b 1 0 q0 q1 p A.p / q g.t / dt ; 0

as desired. (i) ) (ii) We shall restrict ourselves to the case when b D 1, the proof is similar when b < 1. We first note that when (3.11.5) holds for a weight w, then it also holds for the weight wN WD w .0;N / , where N 2 N, with unchanged constant. Let 1 Z 1  pq q1 wN .t / dt x pq ; x 2 .0; 1/; gN .x/ WD x

and

1 Z 1

Z AN WD

wN .s/ ds 0

p  pq

t

q1 pq

! pq p :

dt

t

By (i), we get q pq

CAN

Z DC

p

gN .x/ dx

 p1

0

Z 

1

Z

1

wN .t /

gN .s/ ds

0

! q1

q

t

dt

:

0

By integrating by parts we verify that the last expression is equal to Z q1 ! q1 Z Z 1

q

1

gN .t / 0

t

wN .s/ ds t

gN .s/ ds

dt

:

(3.11.7)

0

Calculation shows that q1 Z Z t Z t q1 pq gN .s/ ds D x 0

0

 

p1 pq

1

wN .s/ ds

!q1

1  pq

dx

x

1q Z

 .p1/.q1/ pq

1

wN .s/ ds t

q1

t pq :

88

Chapter 3 Lebesgue spaces

Hence, the expression in (3.11.7) is no less than ! q1 p  pq  1q Z 1 Z 1 p.q1/ p1 q q wN .s/ ds t pq dt pq 0 t  1q  p 1 p1 q pq D qq AN : pq 1 q



We thus get

 1q pq q AN  q C; p1 and letting N ! 1 we get the same estimate with AN replaced by A.  q1



In the case when q D 1 the argument is similar and the expressions to work with are even simpler. We omit the details. Remark 3.11.7. We note that the assertion of Lemma 3.11.6 can be extended in a suitable way to the case p D 1. The argument of the proof is then similar, using (3.11.3) instead of (3.11.2). We omit the details. Theorem 3.11.8 (weighted Hardy inequalities). Let p; q 2 Œ1; 1/ and let w; v be two a.e. positive weights on .0; 1/. Then there exists a positive constant C such that the inequality ! q1 q Z 1 Z Z 1

1

t

f .s/ ds 0

C

w.t / dt

0

f .t /p v.t / dt

p

(3.11.8)

0

holds for every nonnegative measurable function f on .0; 1/ if and only if either 1 < p  q < 1 and  q1 Z t  p10 Z 1 1p 0 w.s/ ds v.s/ ds 0

t

0

or 1 D p  q < 1 and Z B1 WD sup t>0

 q1

1

w.s/ ds

ess sup

t

s2.0;t/

1 0, run the same argument and then let " ! 0C ; we thereby obtain (3.11.13) again. In each case, this proves (3.11.9), hence the necessity. Sufficiency. We denote Z g.t / WD

t

v

1p 0

 qp1 0 .s/ ds

t 2 .0; 1/:

;

0

Then, by the Hölder inequality, applied to the inner integral, we get

f .s/ ds 0

! q1

q

1 Z t

Z

w.t / dt

0 1 Z t

Z 

0

f .s/p g.s/p v.s/ ds

 pq Z

t

p 0

g.s/

0

1p 0

v.s/

! q1

 pq0 ds

w.t / dt

0

(3.11.14) By Lemma 3.11.5, the last expression is no bigger than Z

1 Z x

Z

1

p

p

f .t / g.t / v.t / 0

p 0

g.s/ t

1p 0

v.s/

! pq

 q0 ds

p

w.x/ dx

dt:

0

Inserting the formula that defines the function g now yields that the inner integral can be rewritten as Z

1

Z

x

Z

s

v t

0

0

1p 0

.y/ dy

 q1

1p 0

v.s/

! pq0 ds

w.x/ dx:

(3.11.15)

90

Chapter 3 Lebesgue spaces

Calculation now gives Z

x

Z

s

v 0

1p 0

.y/ dy

 q1

1p 0

v.s/

0

ds D q

0

Z

x

v

 10

1p 0

.s/ ds

q

;

0

whence (3.11.15) becomes 0

.q /

q p0

1 Z x

Z

1p 0

v.s/ t

 ds

q p0 q0

w.x/ dx:

0

By (3.11.9), the last expression is no bigger than q q0

0

B .q /

q p0

 10

1 Z 1

Z

w.s/ ds t

q

x

w.x/ dx D B

q1

q p0

Z

1

.q / q

0

q

0

q p0

w.s/ ds t

Z

 B .q / q

t

1p 0

v.s/

 q1  1 p0

ds

0

q

D B q .q 0 / p0 qg.t /q : Altogether, the right-hand side of (3.11.14) is majorized by the expression Z 1 Z 1 q p p p q 0 p0 q q 0 p1 q f .t / g.t / v.t /B .q / qg.t / dt D B .q / q f .t /p v.t / dt; 0

0

establishing the desired Hardy inequality (3.11.8) for the case 1 < p  q < 1. When p D 1, then we get from Lemma 3.11.5 that 1 Z t

Z

f .s/ ds 0

0

! q1

q w.t / dt

Z 

Z

1

f .t / 0

 B1

 q1

1

w.x/ dx t

Z

1 v.t / dt v.t /

1

f .t /v.t / dt: 0

The necessity follows by setting f D v1 . The proof is complete. Remark 3.11.9. The assertion of Theorem 3.11.8 can be, again, extended to the case when either p D 1 or q D 1, or both. We will omit the details and just note how the proof goes in these situations. In the case when p D q D 1, then the appropriate modification of (3.11.8) follows from the estimate # Z x Z x" dt : f .t / dt  ess sup w.x/ ess sup v.t /f .t / ess sup w.x/ v.t / 0 0 x2.0;1/ x2.0;1/ t2.0;x/

91

Section 3.11 Hardy inequalities

If p D 1 and q D 1, we have Z x f .t / dt  ess sup ess sup w.x/ 0

x2.0;1/

x2.0;1/ Z 1

 B1

!

x

f .t /v.t / dt 0

f .t /v.t / dt: 0

Finally, when p > 1 and q D 1, one has Z x f .t / dt ess sup w.x/ 0 x2.0;1/ Z x  10 Z p 1p 0 v.t / dt  ess sup w.x/ 0

x2.0;1/ Z 1

B

Z

1 w.x/ ess sup t2.0;x/ w.t /

x

p

f .t / v.t / dt 0

f .t /v.t / dt: 0

In the case when 1  q < p  1, we set Z x 0 v.y/1p  .x/ WD 0

and define w, Q vQ by w.x/ D w. Q .x//;

v.x/ WD v. Q .x//;

x 2 .0; 1/:

Then (3.11.8) becomes Z

b

Z

q

t 0

Z

where g. .x// D

! q1

w.t Q / v.t Q / dt

g.s/ ds 0

p0

q

Z C Z

x

f .y/ dy 0

and b D

1

! p1

b

p

g.t / dt 0

0

v.y/1p dy:

0

Thus, for p < 1, the sufficiency follows from Lemma 3.11.6. Finally, when p D 1, one has M D

! q1  Z dx dy q1 1 w.y/ dy v.x/ 0 v.y/ x  Z x  q1 Z 1 dy q w.x/ dx : 0 0 v.y/

1 Z x

Z 0 1

D q q Therefore, Z

f .s/ ds 0

as desired.

0

! q1

q

1 Z t

w.t / dt

1

 M q q ess sup f .x/v.x/; x2.0;1/

;

 p1

92

Chapter 3 Lebesgue spaces

3.12 Sequence spaces Let N be the set of all positive integers endowed with the arithmetic measure. As we already know from Example 1.20.20, this is a typical completely atomic measure space. We can then define Lebesgue spaces of functions acting on this measure space in the same way as we have built the spaces Lp ./ for p 2 .0; 1 on a nonatomic measure space. The “functions” on such a measure space however turn out to be infinite sequences of real numbers. The resulting spaces are traditionally denoted as `p , 0 < p  1. Definition 3.12.1. Let 0 < p < 1. We then define the sequence space `p by ´ μ 1 X p 1 p ` WD ¹an ºnD1 ; an 2 R for every n 2 N; jan j < 1 : nD1

We equip the set `p with the functional k  k`p , defined for every sequence ¹an º1 nD1 of real numbers as ! p1 1 X p jan j : k¹an ºk`p WD nD1

Definition 3.12.2. The sequence space `1 is defined by ² ³ ; a 2 R for every n 2 N; sup ja j < 1 : `1 WD ¹an º1 n n nD1 n2N

We equip the set `1 with the functional k  k`1 , defined for every sequence ¹an º1 nD1 of real numbers as k¹an ºk`1 WD sup jan j: n2N

Definition 3.12.3. The sequence spaces c and c0 are defined by ³ ² ; a 2 R for every n 2 N; lim a < 1 exists c WD ¹an º1 n n nD1 n2N

and

² ³ ; a 2 R for every n 2 N; lim a D 0 : c0 WD ¹an º1 n n nD1 n2N

We equip both the sets c and c0 with the functional k  k`1 . Remark 3.12.4. The space .`p ; k  k`p / is a Banach space for every p 2 Œ1; 1 and is a quasi-normed space when p 2 .0; 1/. Moreover, for every 1  p  q  1, one has the relations `1 ,! `p ,! `q ,! c0 ,! c ,! `1 :

Section 3.13 Modes of convergence

93

3.13 Relations between various types of convergence We have seen relations between various types of convergence of a sequence of measurable functions ¹fn º1 nD1 on a general measure space in Section 1.22. Definition 3.13.1. Let 1  p < 1. We say that a sequence of measurable function p p p ¹fn º1 nD1 on  converges in L ./ (or in the L -mean) to a function f 2 L  if lim kfn  f kLp ./ D 0:

n!1

Theorem 3.13.2. Let 1  p < 1. (i) Let a sequence of functions ¹fn º in Lp ./ converge in Lp ./ to a function f 2 Lp ./. Then kfn kp ! kf kp : (ii) The convergence in the Lp ./-mean does not in general imply convergence almost everywhere. (iii) The convergence in the Lp ./-mean always implies convergence in measure. (iv) In the special case ./ < 1 the uniform convergence implies convergence in the Lp -mean but none of uniform convergence up to small sets, convergence in measure or convergence a.e. in general need not imply. (v) If ./ D 1, then even the uniform convergence in general need not imply the convergence in the Lp -mean. The diagram in Figure 3.2 describes the mutual relations between the types of convergence for the case when  is of finite measure. Remarks 3.13.3. (i) The implications .˛/; .ˇ/; . /; .ı/ are immediate consequences of the definitions, see also Remark 1.22.2, Propositions 1.22.5 and 1.22.6 and Theorem 3.13.2. (ii) The implication ."/ holds in general, but only under the assumption that fn 2 Lp ./ for every n 2 N: It can be proved by means of the Lebesgue dominated convergence theorem (Theorem 1.21.5). (iii) The implication ( ) follows from the Lebesgue dominated convergence theorem (Theorem 1.21.5). (iv) The implication (!) is the Egorov theorem (Theorem 1.22.3). (v) The implication () is the Riesz theorem (Theorem 1.22.8).

94

Chapter 3 Lebesgue spaces Uniform convergence

(ε)

(α) Locally uniform convergence

(β) Convergence in the space Lp (Ω)

Pointwise convergence

(γ) Convergence almost everywhere

(ψ)

(ω)

Uniform convergence up to small sets

(υ)

Existence of a subsequence which converges almost everywhere

(δ) Convergence in measure

Figure 3.2. Modes of convergence.

3.14 Relatively compact subsets of Lp ./ In this section we shall use the convention stated at the beginning of Section 3.3. Definition 3.14.1. Let p 2 Œ1; 1/. We say that a set K  Lp ./ is p-mean equicontinuous if for every " > 0 there exists a ı > 0 such that Z jf .x C h/  f .x/jp dx < "p 

for each f 2 K and h 2 RN with jhj < ı: Theorem 3.14.2 (Riesz compactness theorem). Let p 2 Œ1; 1/ and let K  Lp ./. Then K is relatively compact in Lp ./ if and only if the following two conditions are satisfied: (i) the set K is bounded, i.e. there exists a c > 0 such that kf kp  c for every f 2 K; (ii) the set K is p-mean equicontinuous. Proof. Let K be a relatively compact subset of Lp ./: Thus K is bounded, whence (i) follows. Let " > 0 and let f1 ; : : : ; fn be an 3" net of the set K. According to Theorem 3.3.3, f1 ; : : : ; fn are p-mean continuous. Thus there exists a ı > 0 such that Z " p jfi .x C h/  fi .x/jp dx < 3  if i D 1; : : : ; n and h 2 RN ; jhj < ı:

95

Section 3.15 Weak convergence

Let f 2 K; h 2 RN ; jhj < ı; kfi  f kp < 3" . The triangle inequality now implies (ii). Conversely, let us assume that (i) and (ii) are satisfied. Let  > 0: It follows from (3.4.2), (3.4.5) and the Arzelà–Ascoli theorem (Theorem 2.5.3) that for a sufficiently small " > 0, the set R" .K/ D ¹R" uI u 2 Kº is relatively compact in the space 1 C./. Thus, there exists a finite 0 -net (0 D 12  ..// p ) of R" .K/ in C./, which is an -net of K in Lp ./: By virtue of Remark 1.13.3 (iii), the proof is complete. Remarks 3.14.3. (i) The theorem just proved is due to Riesz [187] in the case of  D .0; 1/. Essentially the same proof as that given here was given by Kolmogorov [119]. (ii) Further criteria for relative compactness of subsets of the spaces Lp ./ can be found, e.g., in the books by Natanson [164] and Krasnosel’skiˇi et al. [124], where the following theorem is also proved. Theorem 3.14.4. Let M be a relatively compact subset of the space L2 ./ and let p > 2: Suppose that for every " > 0 there exists a ı > 0 such that Z ju.x/jp dx < " F

for every u 2 M and F  ; .F / < ı: Then M is a subset of Lp ./ and is relatively compact in Lp ./: Exercise 3.14.5. Let 1  q < p. Then the embedding Lp ./ ,! Lq ./ is not compact. (Hint: It suffices to show that the embedding L1 ./ ,! L1 ./ is not compact. If  D .0; / then the sequence ¹sin nxº1 nD1 does not contain a 1 subsequence which converges in L .0; /. For a general nonempty bounded open set   RN , an analogous example may be constructed.)

3.15 Weak convergence p Theorem 3.15.1. Let p 2 .1; 1/ and let ¹fn º1 nD1  L .0; 1/ be a sequence of 1 functions. Then ¹fn ºnD1 converges weakly to a function f 2 Lp .0; 1/ if and only if the following two conditions are satisfied:

(i) the sequence ¹kfn kp º1 nD1 is bounded; (ii) for every  2 .0; 1/, we have Z lim

n!1 0

Z



fn .t / dt D



f .t / dt: 0

96

Chapter 3 Lebesgue spaces

Proof. The proof is based on the Banach–Steinhaus theorem (Theorem 1.17.5). Set ´ 1 for 0  t  ; ˛ .t / D 0 for  < t  1: It follows from Exercise 3.5.2 (iv) that the linear hull of all such functions forms a 0 dense subset of Lp .0; 1/: Thus, according to the second condition in Theorem 1.17.5 and Section 3.8 it is necessary and sufficient to show that Z lim

n!1 0

Z

1

fn .t /˛ .t / dt D

1

f .t /˛ .t / dt 0

for arbitrary ˛ ; which is nothing else than (ii). In the same way it is possible to prove the following assertion dealing with the space Lp ./: Theorem 3.15.2. Let  be a nonempty bounded open subset of RN : Let p 2 .1; 1/ P p and let fn 2 L ./, n D 0; 1; : : : . Let 0 be the collection of measurable subsets of  such that the linear hull of their characteristic functions forms a dense sub0 set in Lp ./: Then the sequence ¹fn º1 nD1 converges weakly to f if and only if the following two conditions are satisfied: (i) the sequence ¹kfn kp º1 nD1 is bounded; Z Z P fn .x/ dx D f .x/ dx for every E 2 0 . (ii) lim n!1 E

E

3.16 Isomorphism of Lp ./ and Lp .0; .// Theorem 3.16.1. Let  be a nonempty bounded open subset of RN : Then there exists a one-to-one bounded linear mapping from Lp ./ onto Lp .0; .//; i.e. the spaces Lp ./ and Lp .0; .// are isomorphic. Moreover, it is possible to choose this mapping to be an isometry. Sketch of the proof. We explain the idea of the proof in the case N D 2 only. Since  is a bounded set, there exists a square B0  R2 such that   B0 . Consider the sequence of equidistant divisions ¹Tn º1 nD1 of the set B0 by the lines parallel to the coordinate axes such that every Tn contains 4n equal squares Bnk , k D 1; 2; : : : ; 4n . Passing from n to n C 1 each square of Tn is divided into 4 equal squares of TnC1 . A mapping of the set of all squares obtained onto the set of subsegments of the interval I0 D Œ0; ./ in which the length of the segment Ink corresponding to the square Bnk equals .\Bnk / is constructed as follows: Let Œ˛; ˇ  I0 correspond by

97

Section 3.17 Schauder bases

the mapping in question to a square Bnk of Tn : Denote the four squares obtained from Bnk when passing to the division TnC1 by B I ; B II ; B III ; B IV in the same manner as the quadrants in the plane are numbered. Now divide the segment Œ˛; ˇ by the points 1 ; 2 ; 3 such that ˛  1  2  3  ˇ; and 1  ˛ D . \ B I /;

3  2 D . \ B III /;

2  1 D . \ B II /;

ˇ  3 D . \ B IV /:

Define the segments Œ˛; 1 ; Œ1 ; 2 , Œ2 ; 3 ; Œ3 ; ˇ as the values of our mapping corresponding successively to B I ; B II ; B III ; B IV : For every P  ; let .P /  I0 be the intersection of the intervals corresponding to all squares from the cover of P in the mapping defined above. We have .P / D . .P //: Setting now u.t Q / D u. 1 .t // for every u 2 Lp  we obtain the desired isomorphism. (Be careful about the sets of measure zero.) Remark 3.16.2. For unbounded domains the space Lp ./ is isomorphic with Lp .0; 1/, too (see Triebel [227]).

3.17 Schauder bases in Lp ./ First we shall consider the space Lp .0; 1/; p = 1: Since we proved in Theorem 3.6.1 that these spaces are separable, the investigation of the existence of Schauder bases makes sense. Definition 3.17.1. On the interval Œ0; 1 we define the following family of functions: for t 2 Œ0; 1; f00 .t / D 1 8 n i h 2k2 2k1 ˆ 2 ; if t 2 ; 2 ˆ nC1 nC1 ˆ 2 2 < i k n fn .t / D ; 2k ; 2 2 if t 2 2k1 ˆ 2nC1 2nC1 ˆ ˆ :0 otherwise; n D 0; 1; : : : ; k D 1; 2; : : : ; 2n . We order this family into groups so that every group contains the functions with the same index nI ordering in the groups is done according to increasing k; the groups are ordered with increasing n. The sequence ¹hi º1 iD1 obtained in this way is called the Haar system. Remark 3.17.2. Using elementary computation one can verify that the Haar system is orthogonal, i.e. Z 1 hi .t /hj .t / dt D 0 for i ¤ j:

0

98

Chapter 3 Lebesgue spaces

Theorem 3.17.3. Let p 2 Œ0; 1/, f 2 Lp .0; 1/ and set Z ci D Then

Z

1

f .x/hi .x/ dx: 0

ˇ ˇp n ˇ X ˇ ˇ ci hi .x/ˇ dx D 0: ˇf .x/  ˇ ˇ

1ˇ

lim

n!1 0

iD1

Proof. Denote Z snk .f .x//

D

f00 .x/

1 0

Z f00 .t /f .t /dt

C  C

fnk .x/

1 0

fnk .t /f .t / dt

and Knk .s; t / D f00 .s/f00 .t / C    C fnk .s/fnk .t / for n D 0; 1; : : : ; and k D 1; 2; : : : ; 2n : Clearly, Z snk .f .x// D

0

1

Knk .x; t /f .t / dt:

(3.17.1)

The function Knk .s; t / is equal almost everywhere on Œ0; 1  Œ0; 1 to the function K.s; t / defined by 8 i h i h m m ˆ  2nC1 ; 2mC1 ; 2mC1 ˆ 2nC1 for .t; s/ 2 2nC1 nC1 nC1 ; ˆ ˆ ˆ ˆ ˆ m D 0; 1; : : : ; 2k  1I <  m mC1  K.s; t / D 2n  2n ; 2n ; for .t; s/ 2 2mn ; mC1 2n ˆ ˆ ˆ n ˆ m D 2k; : : : ; 2  1I ˆ ˆ ˆ :0 otherwise: Let t 2 Œ0; 1; n 2 N and k 2 ¹1; 2; : : : ; 2n º be fixed. Then there exists an integer m; 0  m  2n  1; such that t 2 Œ 2mn ; mC1 2n : The following possibilities may occur: (a) m  k; (b) m > k: Assume (a): Then either (˛): 

2m 2m C 1 t 2 nC1 ; nC1 2 2 or (ˇ):





 2m C 1 2m C 2 ; nC1 : t2 2nC1 2

99

Section 3.17 Schauder bases

In the case (˛)

Z snk .f .t //

2mC1 2nC1

nC1

D2

2m 2nC1

f . / d;

and for the case (ˇ) we have Z snk .f .t //

D2

2mC1 2nC1

Assume (b):

Z snk .f .t //

2mC2 2nC1

nC1

n

D2

m C1 m 2n

f . / d:

2n f . / d:

In all cases the function snk .f .t // is independent of t; thus there exist intervals Œa0 ; a1 ; : : : ; Œar 1 ; ar  covering the interval Œ0; 1 and on the .i C 1/st interval snk .f .t // is constant and equal to Z .aiC1  ai /1

ai C1

f . / d:

(3.17.2)

ai

By virtue of the Hölder inequality (Theorem 3.1.6) it follows that Z

Z ˇp ˇ k ˇ ˇsn .f .t //ˇ dt 

1ˇ 0

1 0

jf .t /jp dt:

(3.17.3)

Further, lim s k .f .t // n!1 n

D f .t /

(3.17.4)

for almost all t 2 Œ0; 1: Suppose first that f is bounded, i.e. there exists an M > 0 such that jf .t /j  M for all t 2 Œ0; 1: From (3.17.3) we obtain ˇ ˇ ˇ ˇ sup ˇsnk .f .x//ˇ  sup jf .x/j  M (3.17.5) x2Œ0;1

and thus

x2Œ0;1

ˇ ˇ ˇ ˇ sup ˇsnk .f .x//  f .x/ˇ  .2M /p :

x2Œ0;1

ˇ ˇp ˇ ˇ lim ˇsnk .f .x//  f .x/ˇ D 0

So

n!1

and the Lebesgue dominated convergence theorem (Theorem 1.21.5) yields Z lim

n!1 0

1ˇ

ˇp ˇ k ˇ ˇsn .f .x//  f .x/ˇ dx D 0:

100

Chapter 3 Lebesgue spaces

In this way the assertion is proved under the additional assumption that the function f is bounded. To prove the claim in a general case, let " > 0: According to Theorem 3.5.1, there exists a bounded function f1 and a function f2 2 Lp .0; 1/ such that f D f1  f2 and Z 1

0

From (3.17.3) we have

Z

jf2 .x/jp dx < ":

1ˇ

0

ˇp ˇ k ˇ ˇsn .f2 .x//ˇ dx < "

and, further, Z

ˇp  p1 ˇ ˇ k ˇf .x/  sn .f .x//ˇ dx

1ˇ 0

Z



0

Z 

ˇp  p1 Z ˇ ˇ k C ˇf1 .x/  sn .f1 .x//ˇ dx

1ˇ

1 0

ˇp  p1 1 ˇ ˇ k C 2" p : ˇf1 .x/  sn .f1 .x//ˇ dx

1ˇ 0

p

 p1

jf2 .x/j dx Z 1 ˇ ˇp  p1 ˇ k ˇ C ˇsn .f2 .x//ˇ dx 0

From the first part of the proof we have Z lim sup n!1

ˇ p  p1 1 ˇ ˇ k  2" p ˇf .x/  sn .f .x//ˇ dx

1ˇ 0

which immediately implies the assertion in the general case, since " > 0 was arbitrary. Theorem 3.17.4. Let p 2 Œ1; 1/. Then the Haar orthogonal system forms a Schauder basis in the space Lp .0; 1/. Proof. Theorem 3.17.3 shows that every f 2 Lp .0; 1/ may be written as f D

1 X

ci hi

iD1

(the convergence being considered in the space Lp .0; 1/). Now it suffices to show the uniqueness of ci : Let ¹ci º1 iD1 be a sequence of real numbers such that lim

n!1

n X iD1

ci hi D 0

in Lp .0; 1/:

101

Section 3.18 Weak Lebesgue spaces

Thus

Z

1

lim

n!1 0

hj .t /

n X

! dt D 0

ci hi .t /

iD1

for arbitrary j D 1; 2; : : : ; which implies Z 0 D lim

n!1 0

1

hj .t /

n X

! ci hi .t /

dt D cj :

iD1

So cj D 0 for j D 1; 2; : : : , and the uniqueness is proved. Remarks 3.17.5. (i) The Haar system is an unconditional basis in Lp .0; 1/ for 1 < p < 1 (see [174] and [85]), but not in L1 .0; 1/: (ii) The trigonometric system 1; cos x, sin x, cos.2x/, sin.2x/, . . . forms a Schauder basis in Lp .0; 1/I 1 < p < 1 (see [247]). (iii) The trigonometric system from (ii) is not an unconditional basis in Lp .0; 1/, p ¤ 2 (see, e.g., [133]). Theorem 3.16.1 together with Theorem 3.17.4 and Remark 1.15.4 enables us to state a general theorem on the existence of Schauder bases in Lp ./. Theorem 3.17.6. Let p 2 Œ1; 1/ and let  be a nonempty open subset of RN : Then there exists a Schauder basis in Lp ./.

3.18 Weak Lebesgue spaces Notation 3.18.1. Let f be a (Lebesgue) measurable function on a nonempty bounded open set   RN : For > 0; define S.f; / D ¹x 2 I jf .x/j > º: For every > 0 the set S.f; / is measurable. Lemma 3.18.2. Let f 2 Lp ./, 1  p < 1. Then Z 1 Z p

p1 .S.f; // d : jf .x/j dx D p 

0

Proof. Define F W .0; 1/   ! R by ´ 1 if < jf .x/j ; F . ; x/ D 0 if  jf .x/j :

(3.18.1)

102

Chapter 3 Lebesgue spaces

In other words, the function F . ; / is the characteristic function of the set S.f; /. The assertion follows from the chain ! Z Z Z jf .x/j p p1 p

d dx jf .x/j dx D 



Z Z D



0



p1

Z

0



1 Z

p

F . ; x/ d dx D 0  Z Z

p1 F . ; x/ dx d D p

p

0 1

Z

Dp

1



1

p1

F . ; x/ dx

d



p1 .S.f; // d ;

0

by virtue of the Fubini theorem (Theorem 1.21.11). Definition 3.18.3. Let p 2 Œ1; 1/. We shall then say that the collection of all measurable functions f on  such that there exists a c D c.f / > 0 satisfying .S.f; t // 

c tp

for every t > 0, is the weak Lebesgue space or the Marcinkiewicz space denoted by Lp;1 ./. Lemma 3.18.4. Let 1  p < 1. Then Lp ./  Lp;1 ./: Proof. Let t > 0: Then f 2 Lp ./ satisfies Z t Z t p p1

d  p

p1 .S.f; // d

t .S.f; t // D .S.f; t //p 0 0 Z 1 p1

.S.f; // d D kf kpp ; p 0

according to Lemma 3.18.2. Example 3.18.5. The inclusion in Lemma 3.18.4 is sharp: Let p D 1,  D .0; 1/ and f .x/ D x1 : Then f 2 Lp;1 ./ but f … Lp ./: Definition 3.18.6. Let p 2 Œ1; 1/. For u 2 Lp;1 ./, set 1

kukp;1 WD sup t ..S.u; t /// p : t>0

(3.18.2)

103

Section 3.18 Weak Lebesgue spaces

Remarks 3.18.7. Let p 2 Œ1; 1/. (i) For u; v 2 Lp;1 ./ and  2 R, the following relations hold: .˛/ kukp;1 D 0 if and only if u D 0 almost everywhere on ; .ˇ/ ku C vkp;1  2.kukp;1 C kvkp;1 /I . / kukp;1 D jj kukp;1 : It turns out that k  kp;1 is not a norm. For p 2 .1; 1/, there is a partial remedy for this, because it is at least equivalent to a norm. However, when p D 1, then a norm on L1;1 ./ that would be equivalent to k  k1;1 does not exist. We shall return to these questions in detail in Chapter 8, see in particular Theorem 8.2.2 and Corollary 8.2.4. In particular, the proof of (ˇ) will be given in Corollary 8.6.4. (ii) Lp;1 ./ is a vector space. We consider equality in Lp;1 ./ to be in the sense of almost everywhere. For p D 1, expression (3.18.2) does not define a norm on L1;1 ./ since the triangle inequality is not satisfied (for example, let  D .0; 1/, u.t / D t , v.t / D 1  t ; kuk1;1 D 14 , kvk1;1 D 14 and ku C vk1;1 D 1). Theorem 3.18.8. Let p > 1 and let 0 < "  p  1: Then Lp ./  Lp;1 ./  Lp" ./:

(3.18.3)

Proof. The first inclusion follows from Lemma 3.18.4. In order to verify the second one, let f 2 Lp;1 ./ and let " > 0. Then ´ ./ if t 2 .0; 1/; p t p1" .S.f; t //  kukp;1 if t 2 Œ1; 1/: t 1C" From this and Lemma 3.18.2, we have Z 1 Z p" dx D .p  "/ t p"1 .S.f; t // dt jf .x/j 0  Z p  .p  "/./ C .p  "/ kukp;1

1 1

1 t 1C"

dt < 1;

establishing f 2 Lp" ./, as desired. The proof is complete. We shall prove in Chapter 8, where we will have the operation of a nonincreasing rearrangement of a function at our disposal, that although L1;1 is not equivalently normable, the functional k  k1;1 satisfies certain weaker conditions.

104

Chapter 3 Lebesgue spaces

Remark 3.18.9. Weak Lebesgue spaces constitute a particular instant in the scale of two-parameter Lorentz spaces which will be studied in Chapter 8, where also more details about these spaces will be provided, framed in a more general context. They are also a special case of the endpoint Marcinkiewicz spaces, studied in Section 7.10. Weak Lebesgue spaces will also appear in Chapter 5.

3.19 Miscellaneous remarks and related results Remark 3.19.1 (The space L2 ./). The most important space in applications is L2 ./ because it is a Hilbert space with respect to the inner product Z f .x/g.x/ dxI f; g 2 L2 ./: hf; gi D 

For p ¤ 2, the space Lp ./ is not Hilbert. Remark 3.19.2 (The dual space of L1 ./). Let M be the family of all measurable subsets of . The space ŒL1 ./ is isometrically isomorphic to the space of all finitely additive measures on M which are absolutely continuous with respect to the Lebesgue measure . The norm in this space is the total variation of a measure. Remark 3.19.3 (Weak convergence in L1 ./). (i) The sequence ¹fn º1 nD1 is weakly 1 convergent to f0 in L ./ if and only if Z Z fn d ! f0 d 



for every  2 L1 ./ (see [66]). 1 (ii) If the sequence ¹fn º1 nD1 weakly converges to f0 in L ./ then fn ! f0 in p every L ./; 1  p < 1. For the proof see [246].

Remark 3.19.4 (Weak completeness of Lp ./). (i) A sequence ¹xn º1 nD1 of elements of a Banach space X is said to be weakly Cauchy if for every x  2 X  the sequence ¹x  .xn /º1 nD1 is a Cauchy sequence of scalars. We say that a Banach space is weakly complete if every weakly Cauchy sequence is weakly convergent. (ii) Every reflexive Banach space is weakly complete (see, e.g., [66]). (iii) Thus the spaces Lp ./, 1 < p < 1, are weakly complete. Moreover, the space L1 ./, which is not reflexive, still is weakly complete (see again [66]).

105

Section 3.19 Remarks

Remark 3.19.5 (The case ./ D 1). If the set  has infinite measure, some of the properties of Lp ./ with ./ < 1 are preserved. For instance, the assertions concerning separability or reflexivity of these spaces still hold. However, for example, the inclusions L1 ./  Lp1 ./  Lp2 ./  L1 ./ for 1 > p1 > p2 = 1 are no longer true when ./ D 1 (notice that any constant nonzero function serves as a counterexample for the first inclusion). In fact, once ./ D 1, then there is no inclusion between Lebesgue spaces Lp ./ and Lr ./ unless p D r. Sufficient conditions for f 2 Lp .1; 1/ to be an element of Lq .1; 1/, q > p; are given in [84, 224]. In the latter, the following result is proved: Let 1  p < q: Let f 2 Lp .1; 1/ be an even function. Suppose 1 X

n

q p2

 !pq

nD1

where !p .ıI f / D sup



1 If n

 < 1;

jf .t C h/  f .t /jp dt

 p1

:

0 p; are given in [231]. A complete characterization of embeddings between Lebesgue spaces with respect to different measures was given by Kabaila [112]. Remark 3.19.6. The spaces Lp ./ for 0 < p < 1. Let p 2 .0; 1/. Then we can construct the space Lp ./ analogously as in Section 3.2 starting with Lp ./ (see Remark 3.1.10). The space obtained is a linear topological space which is not locally convex (see [60, 121]). Moreover, the only continuous linear functional on the topological vector space Lp ./; 0 < p < 1 is the zero functional. Remark 3.19.7. Uniformly convex Banach spaces. We say that a Banach space is uniformly convex if for every " > 0 there exists a ı D ı."/ > 0 such that     1 .x C y/ inf 1  = ı."/:  2 kxkDkykD1 kxyk="

It follows immediately from Clarkson inequalities (see Remark 3.1.10 (ii)) that the spaces Lp ./ .1 < p < 1/ are uniformly convex. We recall that every uniformly convex Banach space is reflexive (see [4, p. 286], [152, 177]; for a short proof using the so-called James characterization of reflexivity see, for example, [75, Theorem 9.12]).

106

Chapter 3 Lebesgue spaces

In every uniformly convex Banach space X; the following assertion holds: 1 Let ¹xn º1 nD1 ; ¹yn ºnD1 be sequences in X such that kxn k D kyn k D 1, lim kxn C yn k D 2. Then

n!1

lim kxn  yn k D 0:

n!1

Spaces in which the above assertion holds are called locally uniformly convex Banach spaces. For such spaces, the implication xn * x;

kxn k ! kxk ) xn ! x

(3.19.1)

holds (see [76]). Thus in Lp ./, 1 < p < 1, the implication (3.19.1) is true. Remark 3.19.8. Weighted Lebesgue spaces. In approximation theory and also in the theory of partial differential equations, the spaces with weights are of interest. Let % be a weight, that is, a measurable almost everywhere positive function on . Let p = 1. Then Lp .; %/ denotes the set of all measurable functions f defined almost everywhere on  and such that Z  p1 Np;% .f / D %.x/ jf .x/jp dx < 1: 

Defining equality in Lp .; %/ as equality almost everywhere we obtain analogously as in Section 3.2 that Lp .; %/ is a vector space. Np;% .f / is then a norm on Lp .; %/. The space Lp .; %/ is complete for p 2 Œ1; 1 and separable for p 2 Œ1; 1/. The main tool for working with weighted spaces is the Hardy inequality (see, e.g., Theorem 3.11.8). Remark 3.19.9. Spaces with a mixed norm. Consider a set of functions defined on  D 1      n ; where the i are nonempty open subsets of RNi (i D 1; : : : ; n), which are elements of Lpi .i / “with respect to the variable xi ”. More precisely, let p D .p1 ; : : : ; pn /; 1  pi  1, i D 1; : : : ; n. We denote by Lp ./ the set (with equality in the sense almost everywhere) of all measurable functions f on   RN1 CCNn such that kf kp D Npn .Npn1 .: : : .Np1 .f .x1 ; : : : ; xn // : : : /// < 1: Formula (3.19.2) should be understood as follows: For fixed .x2 ; : : : ; xn / 2 2      n ; set Np1 .f .; x2 ; : : : ; xn // D 1 .x2 ; : : : ; xn /: Thus we obtain a function 1 on 2      n and set Np2 .1 .; x3 ; : : : ; xn // D 2 .x3 ; : : : ; xn /;

(3.19.2)

107

Section 3.19 Remarks

etc. In particular, if pi 2 Œ1; 1/, i D 1; : : : ; n, then Z

Z kf kp D

::: n

1

jf .x1 ; : : : ; xn /jp1 dx1

 pp2 1

! p1n : : : dxn

:

Given the vector p D .p1 ; : : : ; pn / let us define the conjugate vector p0 D .p10 ; : : : ; pn0 /. We immediately obtain an analogue of the Hölder inequality: ˇZ ˇ ˇ ˇ ˇ f .x/g.x/ dx ˇ  kf k  kgk 0 p p ˇ ˇ  0

for all f 2 Lp ./; g 2 Lp ./. It is easy to see that Lp ./ is a vector space and kf kp forms the so-called mixed norm on Lp ./. Moreover, this norm makes Lp ./ a Banach space which is separable if the coordinates of the vector p are finite. The Banach space Lp ./ is reflexive if and only if pi 2 .1; 1/ for every i D 1; : : : ; n: The above assertions and, in fact, a discussion of the whole theory of the spaces Lp ./ can be found in [11].

Chapter 4

Orlicz spaces

4.1 Introduction Let us return for a moment to the Lebesgue spaces Lp ./, p  1, which were studied in Chapter 3. Again,  is supposed throughout this chapter to be an open set in RN . A function u defined on an open set   RN belongs to Lp ./ if Z (4.1.1) ju.x/jp dx < 1 

and the corresponding Lebesgue space Lp ./ is normed by Z kukp WD



ju.x/jp dx

 p1

:

(4.1.2)

If we define the function ˆ for t 2 Œ0; 1/ by ˆ.t / WD t p , then we can rewrite (4.1.1) and (4.1.2) in the form Z 

ˆ.ju.x/j/ dx < 1;

and 1

kukp D ˆ

(4.1.3) 

Z

ˆ.ju.x/j/ dx ;

(4.1.4)

 1

respectively, where ˆ1 is the inverse function of ˆ, that is, ˆ1 .t / D t p . Now, we can ask whether we can replace the power function ˆ.t / D t p by a more general function. Definition 4.1.1. Let ˆ be a nonnegative measurable function defined on Œ0; 1/. We shall denote by (4.1.5) Lˆ ./ the set of all Lebesgue measurable functions u defined almost everywhere on  such that Z ˆ.ju.x/j/ dx < 1: (4.1.6) 

The set L ./ will be called an Orlicz class and we shall use the notation Z ˆ.ju.x/j/ dx: (4.1.7) %.u; ˆ/ WD ˆ



109

Section 4.2 Young function, Jensen inequality

Convention 4.1.2. In a way similar to that in the case of Lp ./, the equality relation in Lˆ ./ is introduced to be equality almost everywhere (see Notation 3.1.1) and the elements of Lˆ ./ are called “functions” in this text. Examples 4.1.3. (i) The Lebesgue spaces Lp ./ with p  1 are special cases of Orlicz classes Lˆ ./. Indeed, setting ˆ.t / D ct p with an arbitrary positive constant c, we obtain Lp ./ D Lˆ ./. (ii) A case that occurs very frequently is that of the function ´ t logC t; for t 2 .0; 1/; ˆ.t / D 0 for t D 0; where logC t D max¹0; log t º. The corresponding Orlicz class Lˆ ./ is in some sense “close” to the Lebesgue space L1 ./ which, as we know from Chapter 3, is a rather exceptional case among Lebesgue spaces. In many tasks of analysis in which for some reason L1 ./ “does not work,” the class Lˆ ./ with this special ˆ serves as its certain appropriate “replacement”. (iii) If we set ˆ.t / D jsin t j ;

t 2 Œ0; 1/;

then obviously every measurable function on  belongs to the class Lˆ ./ if ./ < 1. (iv) Let us consider N D 1,  D .0; 1/ and ˆ.t / D et . Then the function u.x/ D  12 log x belongs to Lˆ ./ while the function v.x/ D  log x D 2u.x/ does not. Remark 4.1.4. Example 4.1.3 (iv) shows that Lˆ ./ need not be a linear set. And, even in the cases when it happens to be a linear set, it is not clear whether and how it is possible to introduce a norm on it. The analogy with Lebesgue spaces might suggest a possibility of using formula (4.1.4), but our assumptions on ˆ in Definition 4.1.1 do not imply the existence of the inverse function ˆ1 . Therefore it is necessary first of all to restrict the set of functions ˆ in Definition 4.1.1.

4.2 Young function, Jensen inequality Definition 4.2.1. We shall say that ˆ is a Young function if there exists a function ' W Œ0; 1/ ! Œ0; 1/ such that Z t '.s/ ds; t  0; (4.2.1) ˆ.t / WD 0

110

Chapter 4 Orlicz spaces

and ' has the following properties: (i) '.0/ D 0; (ii) '.s/ > 0 for s > 0; (iii) ' is right-continuous at any point s  0; (iv) ' is nondecreasing on Œ0; 1/; (v) lims!1 '.s/ D 1. The properties of Young functions are given in the following lemma. Lemma 4.2.2. A Young function ˆ is continuous, nonnegative, strictly increasing and convex on Œ0; 1/. Moreover, ˆ.0/ D 0;

ˆ.1/ WD lim ˆ.t / D 1;

(4.2.2)

t!1

ˆ.t / D 0; t ˆ.t / D 1; lim t!1 t ˆ.˛t /  ˛ˆ.t / for every ˛ 2 Œ0; 1 and t  0; ˆ.ˇt /  ˇˆ.t / for every ˇ > 1 and t  0: lim

t!0C

(4.2.3) (4.2.4) (4.2.5) (4.2.6)

Proof. The continuity, nonnegativity, monotonicity of ˆ and the properties in (4.2.2) are obvious. Regarding convexity, let us consider  2 .0; 1/ and 0  s  t . We shall prove that ˆ.s C .1  /t /  ˆ.s/ C .1  /ˆ.t /: (4.2.7) We have Z ˆ.s C .1  /t / D Z D

0

'.r/ dr C

Z

D

'.r/ dr 0 sC.1/t

Z

s

sC.1/t

'.r/ dr s

Z

s 0

'.r/ dr C .1  /

Z

s 0

'.r/ dr C

sC.1/t

'.r/ dr: s

Since ' is nondecreasing and right-continuous, we have Z sC.1/t '.r/ dr  .1  /.t  s/'.s C .1  /t / s

and

Z

t

sC.1/t

'.r/ dr  .t  s/'.s C .1  /t /:

(4.2.8)

111

Section 4.2 Young function, Jensen inequality

The comparison of the foregoing two inequalities yields Z

'.r/ dr  .1  /

 s

that is Z sC.1/t s

Z

sC.1/t

Z '.r/ dr D 

t

'.r/ dr; sC.1/t

Z

sC.1/t

'.r/ dr C .1  /

s

Z

 .1  /

sC.1/t

'.r/ dr s

t

'.r/ dr: s

Substituting into (4.2.8) we obtain Z t Z s Z s '.r/ dr C .1  / '.r/ dr C .1  / '.r/ dr ˆ.s C .1  /t /   0 s 0 Z s Z t D '.r/ dr C .1  / '.r/ dr 0

0

which is inequality (4.2.7). Further, Z ˆ.t / 1 t D lim '.s/ ds D '.0/ D 0 lim t!0C t t!0C t 0 by Definition 4.2.1 (i) and (iii). Thus we have shown (4.2.3). From 4.2.1 (iv) we have for t > 0     Z Z 1 t t 1 t 1 t 1t ˆ.t / D ' D ' '.s/ ds  '.s/ ds  t t 0 t 2t t2 2 2 2 and hence property (4.2.4) follows by Definition 4.2.1 (iv). Rewriting (4.2.7) in the form ˆ..1  ˛/s C ˛t /  .1  ˛/ˆ.s/ C ˛ˆ.t /;

0  ˛  1;

0  s  t;

and setting s D 0 we obtain property (4.2.5) from (4.2.2). Let ˇ > 1. Inequality (4.2.5) for ˛ D ˇ1 and t D ˇs implies 

1 ˆ ˇs ˇ

 

1 ˆ.ˇs/; ˇ

which is nothing else than (4.2.6). From Lemma 4.2.2 we can derive some properties of the Orlicz class Lˆ ./ provided ˆ is a Young function.

112

Chapter 4 Orlicz spaces

Theorem 4.2.3. Let ˆ be a Young function. Then Lˆ ./ is a convex set and Lˆ ./  L1 ./

(4.2.9)

for ./ < 1. Proof. If we set s D ju.x/j and t D jv.x/j in (4.2.7) and integrate over , we obtain the convexity of Lˆ ./ from the monotonicity and convexity of ˆ. To prove (4.2.9) let u 2 Lˆ ./. According to (4.2.4), there exists a k > 0 such that for ju.x/j > k ˆ.ju.x/j/ > 1; ju.x/j

i.e.

ju.x/j < ˆ.ju.x/j/:

Denoting k D ¹x 2 I

ju.x/j > kº;

Z

Z

we conclude that Z 

ju.x/j dx D Z 

k

k

ju.x/j dx C

nk

ju.x/j dx

ˆ.ju.x/j/ dx C k. n k /

 %.u; ˆ/ C k./ < 1; hence u 2 L1 ./. Remark 4.2.4. The inclusion (4.2.9) is strict because there exists a function u defined on  such that (4.2.10) u 2 L1 ./ but u … Lˆ ./: Such a function u can be constructed as follows: From (4.2.4) we have tn > 1, n 2 N, such that n 2 N: tn1 ˆ.tn /  2n ; We choose a sequence of disjoint subsets n  , n 2 N, such that .n / D

1 ./; tn 2n

n 2 N:

Then we have that 1 X

.n / D ./

nD1

Now let us define

´ tn u.x/ D 0

1 X nD1

1 < ./: tn 2n

for x 2 n ; n 2 N; otherwise in :

113

Section 4.2 Young function, Jensen inequality

Then Z 

but

1 X

ju.x/j dx D

nD1

Z 

tn .n / D

ˆ.ju.x/j/ dx D

1 X

tn

nD1

1 X

ˆ.tn /.n / 

nD1

1 ./ D ./ < 1; tn 2n 1 X

tn 2n

nD1

1 ./ D 1; tn 2n

so that u satisfies (4.2.10). The Lebesgue space L1 ./ can be considered as the union of all Orlicz classes where ˆ varies over all Young functions.

Lˆ ./

Theorem 4.2.5. Let ./ < 1 and let u 2 L1 ./. Then there exists a Young function ˆ such that u 2 Lˆ ./. Proof. We denote n D ¹x 2 I n  1  ju.x/j < nº; Z

Then



ju.x/j dx D  D

1 Z X nD1 n 1 X

n 2 N:

ju.x/j dx

.n  1/ .n /

nD1 1 X

n.n /  ./:

nD1

P 1 The series 1 nD1 n.n / converges, because u 2 L ./ and ./ < 1. Moreover, there exists a nondecreasing sequence of numbers ˛n such that ˛n > 1; If we define

lim ˛n D 1 and

n!1

1 X

˛n n.n / < 1:

nD1

´ t '.t / D ˛n

for t 2 Œ0; 1/; for t 2 Œn; n C 1/; n 2 N;

then

Z ˆ.t / D

t

'.s/ ds 0

114

Chapter 4 Orlicz spaces

is a Young function such that ˆ.n/  n˛n for n 2 N. Hence, we have Z 1 Z 1 X X ˆ.ju.x/j/ dx D ˆ.ju.x/j/ dx  ˆ.n/.n / 



nD1 n 1 X

nD1

n˛n .n / < 1;

nD1

i.e. u 2 Lˆ ./. Examples 4.2.6. (i) For '.t / D t p1 with p > 1, we obtain the Young function 1 p ˆ.t / D p t . (ii) For '.t / D et  1, we obtain the Young function ˆ.t / D et  t  1. ˛

˛

(iii) For ˛ > 0 and '.t / D ˛t ˛1 et , we obtain the Young function ˆ.t / D et 1. (iv) The function ˆ.t / D t , corresponding to the Lebesgue space L1 ./, is not a Young function as it satisfies neither (4.2.3) nor (4.2.4). (v) The functions ˆ in Examples 4.1.3 (ii) and (iii) are not Young functions. Remark 4.2.7. It is shown in [123] that if ˆ is continuous, increasing and convex in Œ0; 1/ and if (4.2.3) and (4.2.4) hold, then ˆ is a Young function. Remark 4.2.8. In some literature, the definition of a Young function is relaxed in the sense that assumptions (4.2.3) and (4.2.4) are not required. Sometimes the notion of an N -function for such functions is used. Remark 4.2.9. We can relax the definition of Young function by replacing conditions (ii), (iii) in Definition 4.2.1 by the following ones: (ii)* '.s/  0 for s > 0; (iii)* '.s/ is left-continuous at any point s > 0. See, for example [141, 241]. In general, after this modification all results concerning Orlicz classes remain valid. Remark 4.2.10. Let us mention that the function in Example 4.1.3 (ii) is not a Young function, but it is a relaxed Young function in the sense of the preceding remark. Theorem 4.2.11 (Jensen inequality). Let ˆ be a convex function on R. (i) Let t1 ; : : : ; tn 2 R and let ˛1 ; : : : ; ˛n be positive numbers. Then   ˛1 t1 C    C ˛n tn ˛1 ˆ.t1 / C    C ˛n ˆ.tn / : ˆ  ˛1 C    C ˛n ˛1 C ˛2 C    C ˛n (ii) Let ˛ D ˛.x/ be defined and positive almost everywhere on .

(4.2.11)

115

Section 4.3 Complementary functions

R

Then, ˆ

Ru.x/˛.x/ dx

R



 ˛.x/dx



dx  ˆ.u.x//˛.x/ R

(4.2.12)

 ˛.x/ dx

for every nonnegative function u provided all the integrals in (4.2.12) are meaningful. Proof. We shall give only a sketch. (i) For n D 2 inequality (4.2.11) is just the definition of the convexity of ˆ (see also (4.2.7)). For n > 2 we obtain inequality (4.2.11) by induction. (ii) Let  > 0 be fixed. From the convexity of ˆ it follows that there exists a k 2 R such that ˆ.t /  ˆ. /  k.t   / for all t  0: Setting t D u.x/ and multiplying the resultant inequality by ˛.x/ we obtain after integration over  that Z

Z 

ˆ.u.x//˛.x/ dx  ˆ. /



˛.x/ dx ²Z ³ Z k u.x/˛.x/ dx   ˛.x/ dx : 

Inequality (4.2.12) now follows by setting Z u.x/˛.x/ dx Z D : ˛.x/ dx



(4.2.13)



4.3 Complementary functions Definition 4.3.1. Let ˆ be a Young function generated by a function ', that is, Z t ˆ.t / D '.s/ ds; t 2 Œ0; 1/: (4.3.1) 0

We set .t / D sup s

(4.3.2)

'.s/t

and

Z ‰.t / D

t

.s/ ds; 0

t 2 Œ0; 1/:

The function ‰ is called the complementary function to ˆ.

(4.3.3)

116

Chapter 4 Orlicz spaces

Exercises 4.3.2. (i) Prove that ‰ is also a Young function (i.e. prove that fies the corresponding conditions of Definition 4.2.1). (ii) Prove that if

satis-

is given by (4.3.2), then '.t / D sup s;

t  0:

(4.3.4)

.s/t

Remark 4.3.3. If ' is continuous and strictly increasing in Œ0; 1/ then is the inverse function ' 1 and vice versa. In the general case, ' and are mutually “inverse” in a generalized sense. Consequently, if ‰ is complementary to ˆ then ˆ is complementary to ‰. We can also call ˆ; ‰ a pair of complementary Young functions. We shall now introduce the Young inequality. Note that a special form of it appeared in Theorem 3.1.4. Theorem 4.3.4 (Young inequality). Let ˆ; ‰ be a pair of complementary Young functions. Then for all a; b 2 Œ0; 1/ we have ab  ˆ.a/ C ‰.b/:

(4.3.5)

Equality holds in (4.3.5) if and only if b D '.a/

or a D

.b/:

(4.3.6)

For the idea how to prove (4.3.5) see Theorem 3.1.4. A detailed proof of a more general assertion is given in [241]. Examples 4.3.5.

(i) Let p 2 .1; 1/ and let ˆ.t / WD

tp ; p

t 2 Œ0; 1/:

Then the complementary function ‰ of ˆ satisfies 0

‰.t / D

tp ; p0

t 2 Œ0; 1/;

where p 0 is the Lebesgue conjugate index (see Definition 3.1.3). (ii) Let ˛ > 1. Then the function ˛

ˆ.t / WD et  1;

t 2 Œ0; 1/;

is a Young function generated by ˛

'.s/ D ˛s ˛1 es ;

s 2 Œ0; 1/:

117

Section 4.3 Complementary functions

We are unable to give an analytical expression for either its complementary function ‰ or the function (see (4.3.2)). It can be however shown that ‰.t / is asymptotically (for t ! 1/ equivalent to p e / D t log t ‰.t (for details see Section 4.5). (iii) Let ˆ.t / WD et  t  1;

t 2 Œ0; 1/:

Then ˆ is a Young function and its complementary function ‰ satisfies ‰.t / D .1 C t / log.1 C t /  t;

t 2 Œ0; 1/

(see Example 4.2.6 (ii)). Example 4.3.6. Let

´ t logC t ˆ.t / WD 0

for t 2 .0; 1/; for t D 0:

(4.3.7)

Then ˆ is a relaxed Young function and ' (see (4.3.1)) is defined by ´ 0 if 0 < t < 1; '.t / D log t C 1 if t  1: Set

8 ˆ n0 , m > n0 , Z 

jum .x/  un .x/j jv.x/j dx < ":

(4.9.1)

Let us decompose  into aSsequence of its subsets ¹n º1 nD1 , n 2 N, with 0 < 1 .n / < 1 such that  D nD1 n , i \ j D ; for i ¤ j . Let us choose k > 0 such that 1 ; ‰.k/  .1 / and define a function v by ´ k for x 2 1 ; v.x/ D 0 for x 2  n 1 : Then

Z %.v; ‰/ D

Z 

‰.jv.x/j/ dx D

1

and using this function v in (4.9.1) we obtain Z " jum .x/  un .x/j dx < k 1

‰.k/ dx D ‰.k/.1 /  1;

for

n > n0 ;

m > n0 :

1 This means that ¹un º1 nD1 is a Cauchy sequence in L .1 /. This space is complete by Theorem 3.7.1 and it follows from the proof of this theorem that there exists a 1 subsequence ¹un;1 º of ¹un º1 nD1 which converges to a function u 2 L .1 / almost everywhere on 1 . Now we use the same procedure for 2 and ¹un;1 º to obtain a subsequence ¹un;2 º of ¹un;1 º which converges almost everywhere to a function (let us denote it by u again) in L1 .2 /. Repeating this procedure we obtain a sequence of subsequences

¹un º  ¹un;1 º  ¹un;2 º  : : : and each subsequence ¹un;k º converges to u on k almost everywhere. Replacing um by um;m in (4.9.1) we obtain from the convergence of um;m .x/ to u.x/ for almost all x 2  and from the Fatou lemma (Theorem 1.21.6) that Z ju.x/  un .x/j jv.x/j dx  " 

provided v 2

L‰ ./,

%.v; ‰/  1 and n  n0 . This means that

138

Chapter 4 Orlicz spaces

(i) un  u 2 Lˆ ./ and, consequently, also u D un  .un  u/ 2 Lˆ ./I moreover, (ii) limn!1 kun  ukˆ D 0.

4.10 Convergence in Orlicz spaces The usual (norm) convergence in the Orlicz space Lˆ ./ can be introduced in terms of the Orlicz norm kkˆ as follows: un ! u in Lˆ ./ means

limn!1 kun  ukˆ D 0.

We shall now introduce another important type of convergence in Orlicz spaces. ˆ Definition 4.10.1. Let ˆ be a Young function. A sequence ¹un º1 nD1 in L ./ is said ˆ to converge in ˆ-mean to u 2 L ./ if Z lim %.un  u; ˆ/ D lim ˆ.jun .x/  u.x/j/ dx D 0: (4.10.1) n!1

n!1 

Remark 4.10.2. Let ˆ be a Young function. The ˆ-mean convergence is also sometimes called the modular convergence, since it is the convergence with respect to the (convex) modular %.; ˆ/. By Lemma 4.8.4 (i) and by Theorem 4.8.5, one has, for w 2 Lˆ ./, (4.10.2) %.w; ˆ/  kwkˆ ˆ once kwkˆ  1. Hence, for a sequence ¹un º1 nD1 and a function u 2 L ./, we always have ) %.un  u; ˆ/ ! 0: kun  ukˆ ! 0

In other words, in an Orlicz space, the norm topology is stronger than the modular one and, therefore, also the norm convergence is stronger than the modular one. This is a very important fact which is worth formulating as a proposition. Proposition 4.10.3. Let ˆ be a Young function. Then convergence in Lˆ ./ always implies ˆ-mean convergence. The assertion of Proposition 4.10.3 cannot be in general reversed as can be demonstrated with the following counterexample. Example 4.10.4. Assume that ˆ is a Young function which does not satisfy the condition 2 . Then there exists a strictly increasing sequence of positive numbers ¹tn º1 nD1 such that ˆ.2tn / > 2n ˆ.tn /; n 2 N:

139

Section 4.10 Convergence in Orlicz spaces

In  one can find a sequence of disjoint sets ¹En º1 nD1 such that their union E WD

1 [

En

nD1

satisfies .E/ < 1. Next, for each fixed n 2 N and for all k 2 N, 1  k  n, there is a finite sequence ¹En;k ºnkD1 of disjoint subsets of En such that .En;k / D

.E/ ; n2k ˆ.tk / ´

We then set un .x/ WD Then we have Z n Z X ˆ.un .x// dx D E

kD1

if x 2 En;k ; k D 1; : : : ; n otherwise:

tk 0

En;k

In other words, we get

n 2 N; k D 1; : : : ; n:

n X

ˆ.un .x// dx D

ˆ.tk /.En;k / 

kD1

.E/ : n

Z lim

n!1 E

ˆ.un .x// dx D 0;

so the sequence ¹un º converges to zero in ˆ-mean. However, it does not converge to zero in norm, since, if it was so, then, by (4.10.2), we would have Z lim ˆ.2un .x// dx  lim k2un kˆ D 0; n!1 E

n!1

which is impossible as Z E

ˆ.2un .x// dx D

n Z X kD1

En;k

ˆ.2un .x// dx D

n X

ˆ.2tk /.E/ > .E/;

kD1

which is a contradiction. The fact that the Young function ˆ in Example 4.10.4 does not satisfy the 2 condition is of crucial importance. In fact, a certain converse of (4.10.2) is available when ˆ 2 2 . Proposition 4.10.5. Let ˆ be a Young function which satisfies the 2 -condition (with T D 0 if ./ D 1). Let w 2 Lˆ ./. If there exists an m 2 N such that %.w; ˆ/ 

1 km

(4.10.3)

140

Chapter 4 Orlicz spaces

where k is the constant in (4.4.1), then kwkˆ 

c ; 2m

(4.10.4)

where c D 2 if ./ D 1 and c D ˆ.T /./ C 2 if ./ < 1. Proof. Let m 2 N be fixed. If ./ < 1, we denote 1 D ¹x 2 I 2m jw.x/j  T º: Then, ˆ.2m jw.x/j/  ˆ.T / if x 2 1 ; ˆ.2m jw.x/j/  k m ˆ.jw.x/j/ if x 2  n 1 ;

(4.10.5) (4.10.6)

(to obtain the latter inequality the 2 -condition was used m-times). So we have Z Z Z m m ˆ.2 jw.x/j/ dx D ˆ.2 jw.x/j/ dx C ˆ.2m jw.x/j/ dx  1 n1 Z Z m ˆ.T / dx C k ˆ.jw.x/j/ dx  1 Z 1 ˆ.jw.x/j/ dx;  ˆ.T /.1 / C k m 

i.e. %.2m w; ˆ/  ˆ.T /./ C k m %.w; ˆ/:

(4.10.7)

If ./ D 1, we set 1 D ; and by virtue of inequality (4.10.6) we obtain immediately that %.2m w; ˆ/  k m %.w; ˆ/: If now (4.10.3) holds then we have in both cases (./ D 1 as well as ./ < 1) that %.2m w; ˆ/  c  1; where c D 2 for ./ D 1 and c D ˆ.T /./ C 2 for ./ < 1. Thus (4.3.8) yields Z 

j2m w.x/j jv.x/j dx  %.2m w; ˆ/ C %.v; ‰/  c

where v 2 L‰ ./ is such that %.v; ‰/  1 and, consequently, k2m wkˆ  c, which is (4.10.4). Now we are able to prove the equivalence of convergence in Lˆ ./ and ˆ-mean convergence.

141

Section 4.10 Convergence in Orlicz spaces

Theorem 4.10.6. Let ˆ be a Young function which satisfies the 2 -condition (with T D 0 if ./ D 1). Then un converges to u in Lˆ ./ if and only if un converges to u in ˆ-mean. Proof. In view of Proposition 4.10.3 it suffices to show that ˆ-mean convergence ˆ ˆ implies convergence in Lˆ ./. So, let ¹un º1 nD1  L ./, u 2 L ./ and let lim %.un  u; ˆ/ D 0:

n!1

Given " > 0, we choose an m 2 N such that " > c2m , where c D 2 if ./ D 1 and c D 2 C ˆ.T /./ if ./ < 1. Then we choose an n0 D n0 ."/ so that %.un  u; ˆ/  k1m for n  n0 . By (4.10.4) (for w D un  u), we have kun  ukˆ  c2m < "

for n  n0 ;

and the proof is complete. We shall now present an analogue of the Lebesgue dominated convergence theorem (Theorem 1.21.5) for Orlicz spaces. The proof is left to the reader as an exercise. ˆ Theorem 4.10.7. Let ˆ be a Young function. Let ¹un º1 nD1 be a sequence in L ./. Suppose that there exists c > 0 such that jun .x/j  c for all n 2 N and almost all x 2 . Let limn!1 un .x/ D u.x/ almost everywhere in . Then un ! u in ˆ-mean.

With regard to Theorem 4.10.7, let us mention that ˆ-mean convergence does not imply pointwise convergence. This is shown by the following example. Example 4.10.8. Let ˆ be an arbitrary Young function. Let N D 1,  D .0; 1/, 1 D .0; 12 /, 2 D . 12 ; 1/, 3 D .0; 13 /, 4 D . 13 ; 23 /, 5 D . 23 ; 1/, 6 D .0; 14 /; : : : and define, for n 2 N, ´ 1 for x 2 n ; un .x/ D 0 elsewhere in : Then

Z lim

n!1 0

1

ˆ.jun .x/j/ dx D 0;

but un does not converge pointwise to zero. Remark 4.10.9. Let u 2 Lˆ ./. Then u.x/ is a finite number for almost all x 2 . If we denote n D ¹x 2 I ju.x/j > nº

142

Chapter 4 Orlicz spaces

then n  nC1 for n 2 N and limn!1 .n / D 0. If we introduce further the ˆ sequence ¹un º1 nD1 in L ./ by ´ 0 for x 2 n ; un .x/ D (4.10.8) u.x/ elsewhere in ; Z

then lim %.un  u; ˆ/ D lim

n!1

n!1  n

ˆ.ju.x/j/ dx D 0:

(4.10.9)

We thus get the following assertion. Corollary 4.10.10. Let ˆ be a Young function. Then (i) every function u 2 Lˆ ./ can be approximated in the sense of the ˆ-mean convergence by a sequence of bounded measurable functions; (ii) if moreover ./ < 1, then the set of all bounded measurable functions on  is a dense subset of Lˆ ./ in the topology of ˆ-mean convergence. Proof. The assertion (i) follows from the observations made in Remark 4.10.9. If ./ < 1, then every bounded measurable function belongs to Lˆ ./, and (ii) follows. Exercise 4.10.11. With the help of Proposition 4.10.3, formulate assertions analogous to those in Corollary 4.10.10 (i) and (ii) for convergence in Lˆ ./. We shall now introduce the notion of a ˆ-mean boundedness. Definition 4.10.12. Let ˆ be a Young function. A subset K of Lˆ ./ is said to be ˆ-mean bounded if there exists a c > 0 such that Z ˆ.ju.x/j/ dx  c for all u 2 K: 

Remark 4.10.13. Let ˆ be a Young function. Then it follows from (4.6.3) that each ˆ-mean bounded set K in Lˆ ./ is automatically bounded (we shall call it norm bounded), as we have kukˆ  c C 1

for all u 2 K:

Evidently, a norm bounded set in Lˆ ./ need not be ˆ-mean bounded. As expected in view of Theorem 4.10.6, we have the following assertion. Proposition 4.10.14. If ˆ satisfies the 2 -condition (with T D 0 if ./ D 1) then norm boundedness and ˆ-mean boundedness are equivalent.

143

Section 4.11 Separability

Proof. Let K  Lˆ ./ be a norm bounded set, that is, there exists a c1 > 0 such that kukˆ  c1 for all u 2 K. We choose an m 2 N such that c1  2m , take w D 2um in (4.10.7) and use the fact that  

u u ; ˆ  1I % m;ˆ  % 2 kukˆ see Lemma 4.7.2. Thus, ´ ˆ.T /./ C k m %.u; ˆ/  km

if ./ < 1; if ./ D 1;

for all u 2 K, i.e. K is ˆ-mean bounded.

4.11 Separability In this section we shall characterize when an Orlicz space is separable. Theorem 4.11.1. Let ˆ be a Young function which satisfies the 2 -condition (with T D 0 if ./ D 1). Then the Orlicz space Lˆ ./ is separable. Proof. Let S be the set of all open cubes in RN with vertices at points with rational coordinates and with edges parallel to the coordinate axes. The set S is evidently countable. Further, let F denote the set of all functions on RN of the type

X ri Ci .x/; x 2 RN ; f .x/ D iD1

where D .f / 2 N, ri are rational numbers, C1 ; C2 ; : : : ; C are pairwise disjoint cubes from S and Ci is the characteristic function of Ci . Evidently F is also countable. We shall prove that a subset of F forms a dense subset in the Orlicz space Lˆ ./. Step 1. Given u 2 Lˆ ./ and " > 0, let c D 2 if ./ D 1 and c D 2Cˆ.T /./ if ./ < 1. We choose m 2 N such that 2cm < 3" and  > 0 such that   k1m (k; T are the constants in the 2 -condition). Step 2. By Theorem 4.7.3, Lˆ ./ D Lˆ ./, and consequently, u 2 Lˆ ./. Therefore there exists a function u0 on  such that (i) u0 .x/ D u.x/ for x 2 0 with some 0  , .0 / < 1; (ii) u0 .x/ D 0 for x 2  n 0 ; (iii) %.u0  u; ˆ/ < .

144

Chapter 4 Orlicz spaces

The term 0 can be taken to be intersection of  with a sufficiently large ball in RN . Then %.u0  u; ˆ/ < k1m , and from Proposition 4.10.3 it follows that ku0  ukˆ 

c 1 < ": m 2 3

(4.11.1)

Step 3. Now we can write u0 in the form u0 D u1  u2 ; where u1 ; u2 are nonnegative functions in Lˆ ./ which vanish outside of 0 . Let us consider, e.g., the function u1 . Then there exists a nondecreasing sequence of simple functions ¹fn º1 nD1 , such that every fn is nonnegative, vanishes outside of 0 , the range of fn contains only rational points, and u1 .x/ D lim fn .x/ n!1

for almost all x 2 0 . Since ˆ.ju1 .x/  fn .x/j/  ˆ.ju1 .x/j/ and lim ˆ.ju1 .x/  fn .x/j/ D 0 almost everywhere in ;

n!1

we have by Theorem 4.10.7 that Z ˆ.ju1 .x/  fn .x/j/ dx D 0: lim n!1 

Using Proposition 4.10.3 again with w D u1  fn , we obtain that for sufficiently large n 1 ": ku1  fn kˆ < 12 Proceeding analogously with u2 , we can eventually construct a simple function v0 with rational values, which vanishes outside of 0 and is such that 1 kv0  u0 kˆ < ": 3

(4.11.2)

So the function v0 has the form v0 .x/ D

m X

ri i .x/;

iD1

where the i are the characteristic functions S of the subsets i of 0 (i D 1; 2; : : : ; m) such that i \ j D ; for i ¤ j and m iD1 i D 0 ; ri 2 Q, ri ¤ 0.

Section 4.12 The space E ˆ ./

145

Step 4. For the characteristic function M of the set M   we have Z ˆ.1/ dx D ˆ.1/.M /: %. M ; ˆ/ D M

When .M / ! 0 then also %. M ; ˆ/ ! 0, and consequently, by Proposition 4.10.5, k M kˆ ! 0. This implies that for each set i in Step 3 there exists a set Si which is the union of a finite number of cubes from S and   "  i  S   ; i D 1; 2; : : : ; m: i 3m jri j If we set w0 .x/ D

m X

ri Si .x/

iD1

then w0 2 F and kw0  v0 kˆ D

m X iD1

  1 jri j  Si  i ˆ  ": 3

(4.11.3)

Step 5. From (4.11.1), (4.11.2), (4.11.3) we conclude that ku  w0 kˆ  ku  u0 kˆ C ku0  v0 kˆ C kv0  w0 kˆ < "; and since F is countable, the proof is complete. Remark 4.11.2. The fact that ˆ satisfies the 2 -condition is not only sufficient but also necessary for Lˆ ./ to be separable. This will be shown in Section 6.5.

4.12 The space E ˆ ./ In this section we shall introduce an important new function space related to a given Young function ˆ. Definition 4.12.1. Let ˆ be a Young function. The space E ˆ ./ is defined by ° ± E ˆ ./ WD u 2 Lˆ ./I ku 2 Lˆ ./ for every k > 0 : Remark 4.12.2. Let ˆ be a Young function. It is obvious that the space E ˆ ./ is a vector space and that the inclusions hold.

E ˆ ./  Lˆ ./  Lˆ ./

(4.12.1)

Converse inclusions to (4.12.1) are not valid in general, but we have the following result.

146

Chapter 4 Orlicz spaces

Proposition 4.12.3. If ˆ is a Young function which satisfies the 2 -condition (with T D 0 if ./ D 1), then E ˆ ./ D Lˆ ./ D Lˆ ./:

(4.12.2)

Proof. Let u 2 Lˆ ./. Then (see Remark 4.7.1 and Lemma 4.7.2) there exists some  > 0 such that u 2 Lˆ ./. Let k > 0. Then it follows from ˆ 2 2 that there exists a C > 0 such that ˆ.k t /  C ˆ.t /

for every t 2 ŒT; 1/:

Thus, in particular, Z

Z 

ˆ.kju.x/j/ dx  C



ˆ.ju.x/j/ dx < 1;

hence u 2 E ˆ ./. This shows that Lˆ ./  E ˆ ./, and (4.12.2) follows from (4.12.1). Remark 4.12.4. If ˆ is a Young function such that ˆ … 2 then E ˆ ./ ¦ Lˆ ./ ¦ Lˆ ./; since E ˆ ./ and Lˆ ./ are vector spaces and Lˆ ./ is not a linear set according to Theorem 4.5.3. Exercise 4.12.5. Show that E ˆ ./ is the maximal subspace (i.e. maximal closed linear subset) of Lˆ ./ which is contained in Lˆ ./. It is of interest to study the distance of a given function from the set E ˆ ./. Remark 4.12.6. For u 2 Lˆ ./ let us denote d.uI E ˆ .// WD

inf

w2E ˆ ./

ku  wkˆ :

(4.12.3)

For r > 0, let If ˆ … 2 then

Br;ˆ WD ¹u 2 Lˆ ./ W d.uI E ˆ .// < rº:

(4.12.4)

B1;ˆ ¦ Lˆ ./ ¦ B1;ˆ ;

(4.12.5)

where the closure B1;ˆ is taken with respect to the Orlicz norm. Formula (4.12.5) has various interesting consequences. For example, since equality is excluded in (4.12.5), the Orlicz class is neither open nor closed subset of the corresponding Orlicz space. For the proof of (4.12.5), see [123]. We now aim for yet another important characterization of the set E ˆ ./.

Section 4.12 The space E ˆ ./

147

Notation 4.12.7. Let us denote by S./ the set of all simple functions defined on . We recall that simple functions were introduced in Definition 1.20.23. Theorem 4.12.8. Let ˆ be a Young function and let ./ < 1. Then E ˆ ./ coincides with the closure S./ of S./ in the norm k  kˆ . Proof. Let u 2 E ˆ ./. With no loss of generality, assume that u  0. Consequently, there exists a sequence of simple functions ¹un º1 nD1 such that 0  un " u   a.e. on : Then, for every  > 0 and n 2 N, one has un 2 E ˆ ./ and ˆ.u/  ˆ..u  un //: Therefore, by the Lebesgue dominated convergence theorem (Theorem 1.21.5), Z ˆ..u  un // dx ! 0; n ! 1: 

Fix " > 0 and take  > 2" . Then there is a n0 2 N such that for every n 2 N, n  n0 , one has Z ˆ..u  un // dx  1: 

Let ‰ be the complementary Young function of ˆ. Then, by (4.6.3), ku  un kˆ D 1 k.u  un /kˆ  Z 1  ˆ..u  un // dx C 1 



2 

< ":

In other words, u 2 S./. Conversely, assume that u 2 S./ and fix  > 0. By (4.8.3), the Orlicz and the Luxemburg norms are equivalent on Lˆ ./, hence, there exists a simple function u0 such that 1 jjju  u0 jjjˆ  2 : By Lemma 4.8.4 (i), we obtain %.2.u  u0 /; ˆ/  1: Thus, in particular, we have 2.u  u0 / 2 Lˆ ./:

148

Chapter 4 Orlicz spaces

The set of simple functions S./ is obviously contained in the Orlicz class Lˆ ./, hence also 2u0 2 Lˆ ./: Finally, by the convexity of the Orlicz class Lˆ ./, we get u D

2.u  u0 / C 2u0 2 Lˆ ./: 2

Because  was an arbitrary positive number, this implies that u 2 E ˆ ./, as desired. The proof is complete. Based on the characterization of the space E ˆ ./ given in Theorem 4.12.8, we can now establish certain approximative property of this space. Lemma 4.12.9. Let ˆ be a Young function and let ./ < 1. Let u 2 Lˆ ./: Then d.uI E ˆ .// D lim ku  un kˆ ;

(4.12.6)

n!1

´

where un .x/ D

u.x/ 0

if ju.x/j  n; if ju.x/j > n:

Proof. The sequence ¹ju.x/  un .x/jº1 nD1 is nonincreasing for almost every x 2 . Consequently, the sequence ¹ku  un kˆ º1 nD1 is nonincreasing as well (see Remark 4.6.8), therefore limn!1 ku  un kˆ exists and we have (4.12.7) lim ku  un kˆ  d.uI E ˆ .// n!1

for un bounded. Let " 2 .0; 2/ and 1 1 0 such that   " 1 1 ; D ı‰ ı 2M since the function t ! t‰

1

  1 t

is nondecreasing. If now E   is such that .E/ < ı, then, by (4.6.10),   " 1 1 ku E kˆ  ku  v E kˆ C M k E kˆ  C M.E/‰ 2 .E/   " 1 D ";  C M ı‰ 1 2 ı hence u 2 Lˆ a ./. Conversely, let u 2 Lˆ a ./. Denote En WD ¹x 2 I ju.x/j  nº ;

n 2 N:

Since u 2 L1 ./ and ./ < 1, we have lim . n En / D 0:

n!1

Thus, by the absolute continuity of the norm, lim ku  u En kˆ D 0:

n!1

In other words, u is a limit in Lˆ ./ of a sequence of bounded functions, whence, by Theorem 4.12.8, u 2 E ˆ ./. The proof is complete. Definition 4.12.14. Let X be a Banach space containing functions defined on  and let K  X. We say that functions in K have uniformly absolutely continuous norms in X if for every " > 0 there exists a ı > 0 such that for every measurable subset E   with .E/ < ı we have ku E kX < "

for all u 2 K:

(4.12.9)

Section 4.13 Continuous linear functionals

151

Theorem 4.12.15 (de la Vallée-Poussin). Let ˆ be a Young function and let K be a ˆmean bounded subset of Lˆ ./. Then the functions u 2 K have uniformly absolutely continuous norms in L1 ./. Proof. Assume that K is a ˆ-mean bounded set in Lˆ ./. Then there exists a positive constant C such that %.u; ˆ/  C for every u 2 K. By (4.6.3), this implies that kukˆ  C C 1; u 2 K: Let E  . Then, by the complementary version of (4.6.10), we have   1 1 : k E k‰ D .E/ˆ .E/ However, in view of (4.2.3), one has lim t ˆ. 1t / D 0:

t!0C

Thus, given " > 0, there exists a ı > 0 such that k E k‰ < "

for every E   satisfying .E/ < ı:

Altogether, then, for every such E   and every u 2 K, we have by (4.7.4) Z ku E kL1 D ju.x/j dx  kukˆ k E k‰  .C C 1/"; E

and the assertion follows. The proof is complete.

4.13 Continuous linear functionals In Section 3.8 we characterized bounded linear functionals on the spaces Lp ./. If Lp ./ is replaced by Lˆ ./, then only a partial analogue of the Riesz representation theorem (see Theorems 3.8.3 and 3.8.1) holds. Theorem 4.13.1. Let ˆ; ‰ be a pair of complementary Young functions, let v be a fixed function in L‰ ./. Then the formula Z u.x/v.x/ dx; u 2 Lˆ ./; (4.13.1) F .u/ D 

defines a continuous linear functional F on Lˆ ./ and 1 kvk‰  kF k  kvk‰ ; 2   where kF k is the norm of F in L‰ ./ .

(4.13.2)

152

Chapter 4 Orlicz spaces

Proof. From formula (4.7.4) we have that jF .u/j  kukˆ kvk‰ :

(4.13.3)

Thus the second inequality in (4.13.2) clearly holds. The first inequality in (4.13.2) follows from (4.6.3) and from the definition of the Orlicz norm kvk‰ . Indeed, for %.u; ˆ/  1 we have kukˆ  %.u; ˆ/ C 1  2 and ˇZ ˇ ˇ ˇ ˇ kvk‰ D sup ˇ u.x/v.x/ dx ˇˇ  sup jF .u/j D sup jF .2u/j  2 kF k : %.u;ˆ/1

kukˆ 2



kukˆ 1

We have also used Exercise 4.6.6. Remark 4.13.2. The analogue of Theorem 3.8.3 does not hold. Firstly, formula (4.13.1) need not be the general form of a bounded linear functional (see Theorem 4.13.5) and, secondly, we do not generally have kF k D kvk‰ as would be reasonable to require. Setting k.v/ D

kvk‰ ; kF k

v 2 L‰ ./;

kvk‰ ¤ 0

(4.13.4)

we have from (4.13.2) only the estimate 1  k.v/  2: Example 4.13.3. Let p 2 .1; 1/ and set ˆ.t / WD 1

tp p.

Then we have

1

k.v/ D .p/ p .p 0 / p0 for every v 2 L‰ ./, where k is the function from (4.13.4). In case of the Lebesgue space we had 1 1 Q k.v/ D C 0 D 1: p p Remark 4.13.4. The function k in (4.13.4) defines a mapping of the set L‰ ./ n ¹0º into R, more precisely into the interval Œ1; 2. In the foregoing example the function k.v/ was constant. Salekhov [193] investigated the functions k.v/ in detail and has shown that k.v/ is constant only if ˆ.t / D ct p with c > 0, p > 1. One can show (see, e.g., [123]) that the range of the function k contains all numbers of the form 1 1 ˆ .˛/‰ 1 .˛/; ˛

˛>

1 : ./

153

Section 4.13 Continuous linear functionals

‰  According  to Theorem 4.13.1 we can consider the space L ./ to be a subset of ˆ L ./ . But, in general h i L‰ ./ ¤ Lˆ ./ :

Theorem 4.13.5. Assume that ˆ does not satisfy the 2 -condition. Then there exists a continuous linear functional on Lˆ ./ which cannot be expressed in the form (4.13.1). Proof. Under the assumptions of the theorem, E ˆ ./ is a proper subspace Lˆ ./ (see Remark 4.12.4), and thus there exists a function u0 2 Lˆ ./ such that u0 … E ˆ ./. Let F be a bounded linear functional on Lˆ ./ with F .u0 / D 1;

F .u/ D 0 for u 2 E ˆ ./:

(4.13.5)

The existence of at least one functional F with property (4.13.5) follows immediately from the Hahn–Banach theorem (Theorem 1.16.4). Let us suppose that F can be expressed in the form (4.13.1) with a function v 2 L‰ ./ and let us define the sequence of functions ¹vn º1 nD1 by ´ v.x/ for jv.x/j  n; vn .x/ D 0 for jv.x/j > n: Obviously vn 2 E ˆ ./, and hence Z vn .x/v.x/ dx D 0 F .vn / D 

for n 2 N:

This means that v.x/ D 0 almost everywhere on  and thus also F .u0 / D 0, which contradicts the first condition in (4.13.5). The following theorem is an “almost-complete” analogue of Theorem 3.8.3. Theorem 4.13.6. Let F be a bounded linear functional on E ˆ ./. Then there exists a uniquely determined v 2 L‰ ./ such that Z F .u/ D u.x/v.x/ dx; u 2 E ˆ ./: (4.13.6) 

Proof. Since the uniqueness of v is obvious, it is enough to show its existence. For a measurable subset M of , set .M / D F . M /: Using Example 4.6.9 we have j.M /j D jF . M /j  kF k k M kˆ D kF k .M /‰

1



 1 ; .M /

154

Chapter 4 Orlicz spaces

and hence lim

.M /!0

.M / D 0;

1

(lim t!1 ‰ t .t/ D 0 because lim t!1 ‰.t/ D 1). Hence  is a measure which is t absolutely continuous with respect to the Lebesgue measure . The Radon–Nikodým Theorem (Theorem 1.21.15) implies the existence of a function v 2 L1 ./ such that Z v.x/ dx: (4.13.7) .M / D If u is a simple function, i.e. u.x/ D Mi  ;

Pm

M

iD1 ˛i Mi .x/

Mi \ Mj D ;

with

for i ¤ j;

we have, according to (4.13.7), F .u/ D D

m X iD1 m X iD1

˛i F . Mi / D

m X

˛i . Mi / D

iD1

Z ˛i 

v.x/ Mi .x/ dx D

m X

Z ˛i

iD1

Z

v.x/ dx Mi

(4.13.8)

u.x/v.x/ dx: 

Let u 2 E ˆ ./. Then there exists a sequence of simple functions ¹un º1 nD1 such that u.x/ D lim un .x/; n!1

jun .x/j  ju.x/j

for almost all x 2  (see the proof of Theorem 4.11.1). Hence (see Remark 4.6.8) kun kˆ  kukˆ and, moreover, lim jun .x/v.x/j D ju.x/v.x/j

n!1

almost everywhere on . The Fatou lemma 1.21.6 implies that ˇZ ˇ Z ˇ ˇ ˇ u.x/v.x/ dx ˇ  sup jun .x/v.x/j dx ˇ ˇ 

n2N



D sup F .jun j sign v/  kF k sup kun kˆ  kF k kukˆ : n2N

n2N

Thus v 2 L‰ ./. Now let F1 be the functional which corresponds to this function v by formula (4.13.1), that is, Z F1 .u/ WD u.x/v.x/ dx; u 2 E ˆ ./: 

In view of (4.13.8) we obtain F1 .u/ D F .u/ for simple functions u. However, the set of simple functions is dense in E ˆ ./ due to Theorem 4.12.8, and consequently F1 .u/ D F .u/ for all u 2 E ˆ ./. The proof is complete.

155

Section 4.14 Compact subsets of Orlicz spaces

Remark 4.13.7. Let F be a bounded linear functional on Lˆ ./ with norm ± ° kF k D sup jF .u/j I u 2 Lˆ ./; kukˆ  1 : Since the restriction of F to E ˆ ./ (denote it again by F ) is a bounded linear functional on E ˆ ./ with norm ° ± kF kE D sup jF .u/j I u 2 E ˆ ./; kukˆ  1 ; there exists (according to Theorem 4.13.6) a function v 2 L‰ ./ such that Z u.x/v.x/ dx; u 2 E ˆ ./: (4.13.9) F .u/ D 

Obviously, kF kE  kF k :

(4.13.10)

Moreover, if we suppose that (4.13.9) holds not only for u 2 E ./ but also for all u 2 Lˆ ./ then kF k D kF kE . (The proof proceeds by contradiction.) ˆ

Remark 4.13.8. Using the isomorphism given by (4.13.6), it is possible to rewrite the assertions of Theorems 4.13.6 and 4.13.1 as h i (4.13.11) L‰ ./ D E ˆ ./ : If ˆ 2 2 , then, according to Proposition 4.12.3, h i L‰ ./ D Lˆ ./ and if ‰ 2 2 , then

h i Lˆ ./ D L‰ ./ :

The last two equalities are of the same type as in the case of Lebesgue spaces and we can prove the following theorem in the same way as Theorem 3.9.1. Theorem 4.13.9. Let ˆ; ‰ be a pair of complementary Young functions. Then the Orlicz space Lˆ ./ is reflexive if and only if both ˆ and ‰ satisfy the 2 -condition.

4.14 Compact subsets of Orlicz spaces In this section we shall give necessary and sufficient conditions for K to be a relatively compact subset of Lˆ ./. First we introduce some notions which are analogous to those introduced in Definition 3.14.1.

156

Chapter 4 Orlicz spaces

Definition 4.14.1. A subset K of Lˆ ./ is said to be ˆ-equicontinuous for every " > 0 there exists a ı D ı."/ > 0 such that for h 2 RN with jhj < ı and for every u 2 K we have (4.14.1) ku  uh kˆ < "; where uh is the function defined by ´ u.x C h/ uh .x/ WD 0

for x 2  with x C h 2 ; otherwise in RN :

(4.14.2)

Notation 4.14.2. For r > 0 and x 2 RN let B.x; r/ be the ball ¹y 2 RN I jx  yj < rº and mr the measure of B.x; r/. Further, for u 2 Lˆ ./ let us set u.y/ D 0 if y … . The so-called Steklov function Sr .u/ corresponding to u is defined as follows: Z Z 1 1 u.y/ dy D u.x C y/ dy: (4.14.3) Sr .u/.x/ WD mr B.x;r / mr jyj 0 such that .Bx;y / < ı for jx  yj < . With respect to (4.12.9) we have for such x; y Z 1 " ju.z/j dz  jSr .u/.x/  Sr .u/.y/j  mr Bx;y mr provided Sr .u/ 2 Kr , i.e. the set Kr is equicontinuous (see Definition 1.5.1). The assertion now follows from the Arzelà–Ascoli theorem (Theorem 2.5.3).

158

Chapter 4 Orlicz spaces

Now we are able to prove a criterion for relative compactness in E ˆ ./. Theorem 4.14.6. A subset K of E ˆ ./ is relatively compact if and only if the following conditions are satisfied: (i) the set K is bounded in E ˆ ./; (ii) for every " > 0 there exists a ı D ı."/ > 0 such that ku  Sr .u/kˆ < "

(4.14.7)

for 0 < r < ı and every u 2 K. Proof. Conditions (i), (ii) are sufficient for K to be relatively compact as follows from Proposition 4.14.5. Indeed, the corresponding "-net in E ˆ ./ can be taken to be the "-net in C./, the existence of which is guaranteed by Proposition 4.14.5, and then we use (4.14.7). Conversely, let K be relatively compact in E ˆ ./. Then there exists a finite 13 "net ¹w 1 ; w 2 ; : : : ; w s º. Because K  E ˆ ./, we can take the functions ¹w i º, i D 1; : : : ; s, to be continuous on . Let c D k  kˆ and let us choose a number r > 0 such that ˇ ˇ " ˇ i ˇ i .i D 1; : : : ; s/ (4.14.8) .x/  w .y/ ˇw ˇ< 3c for x; y 2  with jx  yj < r. Then in view of (4.14.8) the corresponding Steklov functions Sr .w i / satisfy ˇ ˇ Z ˇ ˇ ˇ 1 Z ˇ 1 ˇ ˇ ˇ ˇ i i i w.y/ dy  w .x/ dy ˇ ˇSr .w /.x/  w .x/ˇ  ˇ ˇ mr Bx;y ˇ mr Bx;y ˇZ ˇ ˇ 1 ˇˇ " ˇ .w i .y/  w i .x// dy ˇ < ; D ˇ ˇ 3c mr ˇ Bx;y and the monotonicity of the Orlicz norm (see Remark 4.6.8) yields    " "     Sr .w i /  w i  <    D : ˆ 3c 3 ˆ

(4.14.9)

Let u be an arbitrary function in K. Then there exists an index , 1   s such that " ku  w kˆ < : 3

(4.14.10)

On the other hand, it follows from (4.14.4) that also " kSr .u  w /kˆ D kSr .u/  Sr .w /kˆ < : 3

(4.14.11)

159

Section 4.14 Compact subsets of Orlicz spaces

Using (4.14.9), (4.14.10) and (4.14.11), we have ku  Sr ukˆ  ku  w kˆ C kw  Sr .w /kˆ C kSr .w /  Sr .u/kˆ < "; so that condition (ii) is necessary for K to be relatively compact. As the necessity of (i) is obvious, the theorem is proved. Remark 4.14.7. If ˆ satisfies the 2 -condition then E ˆ ./ D Lˆ ./ so that Theorem 4.14.6 gives necessary and sufficient conditions for K to be relatively compact in Lˆ ./. For ˆ 2 2 norm convergence and norm boundedness are equivalent to ˆ-mean convergence and ˆ-mean boundedness, respectively (see Theorem 4.10.6, Remark 4.10.13) so that we can reformulate Theorem 4.14.6 as follows. Theorem 4.14.8. Let ˆ satisfy the 2 -condition. Then a subset K of Lˆ ./ is relatively compact in Lˆ ./ if and only if the following conditions are satisfied: (i) there exists a C > 0 such that Z ˆ.ju.x/j/ dx  C 

for all u 2 KI

(ii) for every " > 0 there exists a ı D ı."/ > 0 such that Z ˆ.ju.x/  Sr .u/.x/j/ dx < " 

for 0 < r < ı and for all u 2 K. Another criterion for relative compactness, more similar to that from the Riesz theorem (Theorem 3.14.2) and using the concept of ˆ-equicontinuity introduced in Definition 4.14.1, is given by the following theorem. Theorem 4.14.9. A subset K of E ˆ ./ is relatively compact if and only if the following conditions are satisfied: (i) the set K is bounded in E ˆ ./; (ii) the set K is ˆ-equicontinuous. Proof. Let u 2 K and let Sr .u/ be the corresponding Steklov function. We have ˇ ˇ Z ˇ 1 ˇ .u.x/  u.x C y// dy ˇˇ ju.x/  Sr .u/.x/j D ˇˇ mr jyj n0 ;

so that uQ n converges to u in Lˆ ./.

4.15 Further properties of Orlicz spaces Definition 4.15.1. Let ˆ be a Young function. A function u 2 Lˆ ./ is said to be ˆ-mean continuous if for every " > 0 there exists a ı D ı."/ > 0 such that kuh  ukˆ < "

(4.15.1)

for h 2 RN with jhj < ı, where ´ u.x C h/ if x 2  and x C h 2 ; uh .x/ D 0 otherwise in RN : p

Remark 4.15.2. Let p 2 .1; 1/. For ˆ.t / D tp , inequality (4.15.1) means the same (possible up to a multiplicative constant) as Z ˆ.ju.x C h/  u.x/j/ dx < "; 

i.e. %.uh  u; ˆ/ < ": Moreover, every function u 2 Lp ./ D Lˆ ./ is ˆ-mean continuous (i.e. in the terminology introduced in Definition 3.3.2, u 2 Lp ./ is p-mean continuous – see Theorem 3.3.3).

162

Chapter 4 Orlicz spaces

These assertions do not hold in general for Orlicz spaces. Nonetheless, the following analogous assertion, whose proof is left to the reader, holds for E ˆ ./. Theorem 4.15.3. Let u 2 E ˆ ./. Then for every " > 0 there exists a ı D ı."/ > 0 such that for h 2 RN with jhj < ı we have %.uh  u; ˆ/ < ": If ˆ satisfies the 2 -condition then Lˆ ./ D E ˆ ./. Using Theorem 4.10.6, we obtain immediately from Theorem 4.15.3 the following result. Theorem 4.15.4. If ˆ satisfies the 2 -condition, then every function u 2 Lˆ ./ is ˆ-mean continuous. For u 2 E ˆ ./, the same assertion as above holds without any restriction on ˆ: Theorem 4.15.5. If u 2 E ˆ ./, then u is ˆ-mean continuous. Proof. The assertion follows from Theorem 4.15.3. Let u 2 E ˆ ./ and let " > 0. Let k > 0 be arbitrary but fixed, and let us denote v D uk . Then vh D ukh , v 2 E ˆ ./ and Theorem 4.15.3 used for v yields that there exists a ı D ı."/ > 0 such that for h 2 RN with jhj < ı we have %.vh  v; ˆ/ D %.

uh  u ; ˆ/  1: k

If we choose k D 12 " we conclude from the definition of the Luxemburg norm that 1 jjjuh  ujjjˆ < "; 2 and from (4.8.3) we immediately obtain kuh  ukˆ  2jjjuh  ujjjˆ < ": As the general representation of functionals on general Orlicz spaces is not known, we shall now introduce the notion of a modified weak convergence: ˆ ‰ Definition 4.15.6. A sequence ¹un º1 nD1 in L ./ is said to converge E -weakly to ˆ u 2 L ./ if Z

lim

n!1 

.un .x/  u.x//v.x/ dx D 0

for all v 2 E ‰ ./. Remark 4.15.7. If both complementary Young functions ˆ, ‰ satisfy the 2 -condition then the concept of E ‰ -weak convergence coincide with that of the usual weak convergence.

163

Section 4.16 Isomorphism properties, Schauder bases

Standard theorems concerning weak convergence enable us to derive various assertions, see, e.g., [123], where also further properties connected with the concept of E ˆ -weak convergence can be found. We shall give only the following example. Example 4.15.8. For u 2 Lˆ ./ we define a sequence ¹un º1 nD1 of bounded functions as follows: ´ u.x/ for x 2 n D ¹x 2 I ju.x/j  nº; un .x/ D 0 for x 2  n n : Then

Z lim

n!1 

Z .u.x/  un .x//v.x/ dx D lim

n!1 n n

u.x/v.x/ dx D 0

for every v 2 L‰ ./ and thus also for every v 2 E ‰ ./ since lim . n n / D 0:

n!1

‰ This means that ¹un º1 nD1 converges E -weakly to u. 1 Note that the sequence ¹un ºnD1 constructed above does not in general converge to u in Lˆ ./.

Remark 4.15.9. In Example 4.15.8 we have constructed for every u 2 Lˆ ./ a sequence of bounded functions un , i.e. un 2 E ˆ ./, such that ¹un º1 nD1 converges E ‰ -weakly to u. This shows that Lˆ ./ is the closure of E ˆ ./ with respect to E ‰ -weak convergence.

4.16 Isomorphism properties, Schauder bases Let  be bounded and let us denote by J the open interval .0; t0 / where t0 D ./. The connection between the space Lˆ ./ of functions of N variables and the space Lˆ .J / of functions of one variable is described by the following theorem. Theorem 4.16.1. The spaces Lˆ ./ and Lˆ .J / are isometrically isomorphic. The proof follows literally the ideas given in the proof of Theorem 3.16.1 and is omitted. Remark 4.16.2. The mapping which realizes the isometric isomorphism between Lˆ ./ and Lˆ .J / simultaneously maps the space E ˆ ./ onto E ˆ .J /. Theorem 4.16.1 enables us to formulate and prove various results for the simpler cases of the space Lˆ .J /. These results can then be transferred to the general space Lˆ ./.

164

Chapter 4 Orlicz spaces

Remark 4.16.3. We know from Remark 4.11.2 that the space Lˆ ./ need not be separable, so that in general it makes no sense to speak of Schauder bases in Orlicz spaces. However, the space E ˆ ./ is separable (see Theorem 4.12.11) so that the problem of the existence of a Schauder basis in E ˆ ./ is meaningful. First we shall consider the case N D 1,  D .0; 1/, denoting the space E ˆ ./ by E ˆ .0; 1/ instead of by E ˆ ..0; 1//. Recall that the Haar system ¹hi º1 iD1 was introduced in Definition 3.17.1. Theorem 4.16.4. The Haar system ¹hi º1 iD1 forms a Schauder basis in the space ˆ E .0; 1/, i.e.   n   X   lim u  ci hi  D 0 (4.16.1) n!1   iD1

for u 2

E ˆ .0; 1/,

ˆ

where the numbers ci , i D 1; : : : ; n, are uniquely determined by u: Z 1 ci D u.x/hi .x/ dx; i 2 N: (4.16.2) 0

Proof. Let the ci , i 2 N, be defined by the relations (4.16.2). Denote sn .u; x/ D

n X

ci hi .x/

iD1

for x 2 .0; 1/ and n 2 N. Let the numbers a1 ; a2 ; : : : ; ar .r D r.n// be defined as follows: 0 D a1 < a2 <    < ar 1 < ar D 1 and a1 ; a2 ; : : : ; ar 1 are precisely all the points at which the Haar functions h1 ; : : : ; hn have jumps. Then, for x 2 .a ; a C1 /, one has Z a C1 1 u.y/ dy: (4.16.3) sn .u; x/ D a C1  a a (We note that the proof is the same as that of Theorem 3.17.3.) Now using the Jensen integral inequality (4.2.12) with juj instead of u and ˛ the characteristic function of the interval Œa ; a C1 , we obtain   Z a C1 1 ˆ.jsn .u; x/j/  ˆ ju.y/j dy a C1  a a Z a C1 1 ˆ.ju.y/j/ dy  a C1  a a for every x 2 .a ; a C1 / and hence Z Z a C1 ˆ.jsn .u; x/j/ dx  a

a C1

ˆ.ju.x/j/ dx a

165

Section 4.16 Isomorphism properties, Schauder bases

and

Z 0

If kukˆ  1 then also

Z

1

ˆ.jsn .u; x/j/ dx  Z

1 0

1

ˆ.ju.x/j/ dx: 0

ˆ.ju.x/j/ dx  1

(see (4.10.2)) and from inequality (4.6.3) for such a function u we conclude that ksn .u; x/kˆ  %.sn .u; x/; ˆ/ C 1  %.u; ˆ/ C 1  2: This means that the operator norm (in E ˆ .0; 1/) of sn satisfies ksn k D

sup ksn .u; x/kˆ  2

(4.16.4)

kukˆ 1

for every n 2 N. Denote by H the set of all jump points of all Haar functions hi . Then H is obviously a countable set, and because (4.16.3) implies lim sn .u; x/ D u.x/

n!1

uniformly on Œ0; 1 n H

for u 2 C.Œ0; 1/, we have limn!1 sn .u; x/ D u.x/ in Lˆ .0; 1/ for every u 2 C.Œ0; 1/, i.e. the operators sn converge to the identity operator I on the set C.Œ0; 1/ which is dense in E ˆ .0; 1/. Since, in view of (4.16.4), all operators are uniformly bounded, we have by the Banach–Steinhaus theorem (Theorem 1.17.5) that sn .u; x/ ! u.x/

in Lˆ .0; 1/

for all u 2 E ˆ .0; 1/, which is the same as (4.16.1). The uniqueness of the decomposition of u into a series 1 X

ci hi

iD1

follows in the same way as in Theorem 3.17.4, which completes the proof of our theorem. In the general case we have the following theorem. Theorem 4.16.5. Let ./ < 1. Then there exists a Schauder basis in the space E ˆ ./. Proof. The assertion follows from Theorems 4.16.4 and 4.16.1. Indeed, the functions t gi .t / D hi . / t0 with t0 D ./ form a Schauder basis in E ˆ .0; t0 / and the mapping from Theorem ˆ 4.16.1 maps ¹gi º1 iD1 onto a basis in E ./. If ˆ 2 2 then it suffices to use Lemma 4.12.3.

166

Chapter 4 Orlicz spaces

Corollary 4.16.6. Let ./ < 1 and let ˆ be a Young function satisfying the 2 condition. Then a Schauder basis exists in the space Lˆ ./. Remark 4.16.7. Let us suppose ˆ … 2 . Formula (4.16.2) which defines the coefficients ci is meaningful for all u 2 Lˆ .0; 1/, not only for u 2 E ˆ .0; 1/. However if we suppose for u 2 Lˆ .0; 1/ that uD

1 X

ci hi ;

(4.16.5)

iD1

then necessarily u 2 E ˆ .0; 1/. Consequently, where the series converges in Lˆ .0; 1/, P ˆ for u 2 Lˆ .0; 1/ n E ˆ .0; 1/ the series 1 iD1 ci hi cannot converge in L .0; 1/.

4.17 Comparison of Orlicz spaces In Exercise 3.2.4 (i) we have shown that if ./ < 1 and 1 < p < q then Lq ./ ,! Lp ./: We can obtain a similar result for the Orlicz spaces Lˆ1 ./ and Lˆ2 ./ making use of the ordering introduced in Definition 4.5.6. Theorem 4.17.1. Let ˆ1 ; ˆ2 be two Young functions. Then the inclusion Lˆ1 ./  Lˆ2 ./

(4.17.1)

ˆ2 ˆ1 :

(4.17.2)

holds if and only if Proof. Let (4.17.2) hold, i.e. let c > 0 and T  0 (T D 0 if ./ D 1) be such that ˆ2 .t /  ˆ1 .ct / for

t  T:

Let u 2 Lˆ1 ./. Then there exists a  > 0 such that  u 2 Lˆ1 ./; i.e. %. u; ˆ1 / < 1 (see Lemma 4.7.2). Let us denote ³ ² T : 1 D x 2 I ju.x/j < c  It follows from (4.17.3) that for x 2  n 1 we have

 ju.x/j  ˆ1 . ju.x/j/ ˆ2 c

(4.17.3)

167

Section 4.17 Comparison of Orlicz spaces

and consequently, Z Z Z



   ju.x/j dx D ju.x/j dx C ju.x/j dx ˆ2 ˆ2 ˆ2 c c c  1 n1 Z  ˆ2 .T /./ C ˆ1 . ju.x/j/ dx  ˆ2 .T /./ C %. u; ˆ1 / < 1; n1

i.e.

 u 2 Lˆ2 ./; c

which means that u 2 Lˆ2 ./ (see, e.g., the generalization mentioned in Remark 4.7.1). So we have proved inclusion (4.17.1). Conversely, let us suppose that condition (4.17.2) is not satisfied. Then there exists a sequence ¹tn º1 nD1 such that 0 < t1 < t2 < : : : ;

lim tn D 1;

n!1

and ˆ2 .tn / > ˆ1 .2n ntn /;

n 2 N:

(4.17.4)

From the formula (4.2.5) used for the Young function ˆ1 with t D 2n ntn ; ˛ D 2n we obtain ˆ1 .ntn /  2n ˆ1 .2n ntn / which together with (4.17.4) yields ˆ2 .tn / > 2n ˆ1 .ntn /: Let us choose a sequence of disjoint subsets n in ; n 2 N such that .n / D

ˆ1 .t1 /./ : 2n ˆ1 .tn /

This is possible since 1 X

.n /
of (4.17.5), that Z 1 X ˆ2 . ju.x/j/ dx D ˆ2 . ntn /.n / 

 D

nD1 1 X nDm 1 X

ˆ2 .tn /.n / >

1 X

1 m

and we have, in virtue

2n ˆ1 .ntn /.n /

nDm

ˆ1 .t1 /./ D 1:

nDm

Consequently, condition (4.17.2) is necessary for the inclusion (4.17.1) and the theorem is proved. The following theorem follows directly from Theorem 4.17.1. Theorem 4.17.2. Let ˆ1 , ˆ2 be two Young functions. Then Lˆ1 ./ D Lˆ2 ./

(4.17.6)

if and only if ˆ1 and ˆ2 are equivalent (in the sense of Definition 4.5.6). Remark 4.17.3. Equality (4.17.6) is a set identity. Nevertheless, we can also show that the norms of the two spaces considered are equivalent. This fact will follow from the next theorem which states that the inclusion Lˆ1 ./  Lˆ2 ./ implies the embedding Lˆ1 ./ ,! Lˆ2 ./: Theorem 4.17.4. Let Lˆ1 ./  Lˆ2 ./. Then there exists a k > 0 such that kukˆ2  k kukˆ1 for all u 2 Lˆ1 ./.

(4.17.7)

169

Section 4.17 Comparison of Orlicz spaces

Proof. Suppose that (4.17.7) fails. Then there exists a sequence ¹un º1 nD1 of functions ˆ 1 un  0 in L ./ such that kun kˆ1 5 1

but

kun kˆ2 > n3 for all n 2 N:

We know Orlicz spaces are complete, hence they have the Riesz–Fischer property. P uthat n Thus, converges in Lˆ1 ./ to some function u 2 Lˆ1 ./. By the hypothesis n2 Lˆ1 ./  Lˆ2 ./ we get also u 2 Lˆ2 ./. However, since 0 5 n2 un 5 u and so kukˆ2 = n2 kun kˆ2 > n; for all n, we get a contradiction. Example 4.17.5. If ˆ1 , ˆ2 are equivalent Young functions then the norms kkˆ1 and kkˆ2 are equivalent. In example 4.5.7 (iii) it was shown that the functions ˆ.t /, ˆk .t / D ˆ.k t / with k > 0 are equivalent provided ˆ is a Young function. In this case we even have that kukˆk D k kukˆ ; which is a consequence of the fact that the corresponding complementary functions ‰, ‰k satisfy ‰k .t / D ‰. kt / (see Exercise 4.3.7). If we use the ordering ˆ2

ˆ1 introduced in Definition 4.5.10 we obtain sometimes more than the inclusion Lˆ1 ./  Lˆ2 ./: Theorem 4.17.6. If ˆ2

ˆ1 then Lˆ1 ./ ,! Lˆ2 ./:

(4.17.8)

Proof. Let u 2 Lˆ1 ./. Then there exists a  > 0 such that  u 2 Lˆ1 ./. Since ˆ2

ˆ1 we have that lim

t!1

ˆ2 . t / D0 ˆ1 . t /

for every > 0:

(4.17.9)

Let us fix ; then (4.17.9) means that there exists a T  0 such that ˆ2 . t / < ˆ1 . t /

for every t  T :

However, Theorem 4.5.2 implies that u 2 Lˆ2 ./. This result is true for all

> 0 so that we have

u 2 Lˆ2 ./

for all > 0;

which means that u 2 E ˆ2 ./. So we have Lˆ1 ./  E ˆ2 ./: In view of E ˆ2 ./  Lˆ2 ./, the embedding in (4.17.8) follows from Theorem 4.17.4.

170

Chapter 4 Orlicz spaces

Theorem 4.17.7. Let ˆ2

ˆ1 and let K be a bounded subset of Lˆ1 ./. Then the functions u 2 K have uniformly absolutely continuous norms in the space Lˆ2 ./. Proof. From ˆ2

ˆ1 we conclude by Exercise 4.5.13 that the complementary functions ‰1 ; ‰2 satisfy ‰1

‰2 : (4.17.10) Let us denote by V the set of all v 2 L‰2 ./ such that %.v; ‰2 /  1. Moreover, let W be the set of all functions w of the form w.x/ D ‰1 .jv.x/j/;

v 2 V;

 > 0 fixed:

Then it follows from (4.17.10) and the de la Vallée-Poussin theorem (Theorem 4.12.15) that all w 2 W have uniformly absolutely continuous L1 -norms, i.e. for every " > 0 there exists a ı D ı."/ > 0 such that for M   with .M / < ı we have Z Z " (4.17.11) jw.x/jdx D ‰1 .jv.x/j/ dx < 2 M M for all v 2 V . Let K be a bounded subset of Lˆ1 ./, i.e. let c > 0 be such that kukˆ1  c for all u 2 K. Let 0 < " < 1 and let us take  D 2c " in (4.17.11). For u 2 K and v 2 V we have by the Young inequality and (4.10.2) that   Z Z Z ju.x/j ju.x/v.x/j dx  ˆ1 ‰1 .jv.x/j/ dx dx C  M M M Z u     C ‰1 .jv.x/j/ dx:  ˆ1 M " kukˆ1  2" ; if we choose M   so that .M / < ı we can We have k u kˆ1 D 2c use (4.17.11) and thus we obtain for every u 2 K Z " " ju.x/v.x/j dx < C D ": ku M kˆ2 D sup 2 2 v2V M

Remark 4.17.8. In view of Theorem 4.14.10 the preceding theorem shows that for ˆ2

ˆ1 the embedding Lˆ1 ./ ,! E ˆ2 ./ is “almost-compact”. Nonetheless, this embedding still fails to be compact (see Exercise 3.14.5 and Exercise 4.5.11 (i)). An “almost-compact embedding” turns out to be an important relation between function spaces. In the literature it is also often called an absolutely continuous embedding. We shall study this type of relation in a far more general context in Section 7.11. It was shown in Remark 4.5.12 that the relation ˆ2 ˆ1 does not generally imply the relation ˆ2

ˆ1 . Some additional conditions guaranteeing this implication are given in the following theorem which is in a sense the converse of Theorem 4.17.7.

171

Section 4.17 Comparison of Orlicz spaces

Theorem 4.17.9. Let ˆ2 ˆ1 and let for every bounded subset K of Lˆ1 ./ all u 2 K have uniformly absolutely continuous Lˆ2 -norms. Then ˆ2

ˆ1 :

(4.17.12)

Proof. Assume that (4.17.12) does not hold. Then Exercise 4.5.13 implies that ‰1

‰2 is impossible, i.e. there exists a constant  > 0 and a sequence ¹tn º1 nD1 with limn!1 tn D 1 so that ‰1 .tn / > ˆ2 .tn / for all n 2 N:

(4.17.13)

If we denote sn D ‰1 .tn / then (4.17.13) implies that the inverse functions to ‰1 and ‰2 satisfy ‰21 .sn / > tn D ‰11 .sn / for all n 2 N: (4.17.14) Let n .n 2 N/ be subsets of  with .n / D s1n and let us define a sequence of functions un : ´ s n for x 2 n ; 1 ‰ .sn / 1 un .x/ D 0 otherwise in : From Example 4.6.9 we have kun kˆ1 D

sn 1 ‰1 .sn /

k n kˆ1 D

sn 1 1 ‰ .sn / 1 s ‰1 .sn / n 1

D 1;

ˆ1 i.e. the sequence ¹un º1 nD1 is bounded in L ./. By the assumptions of our theorem, all un have uniformly absolutely continuous Lˆ2 -norms, i.e.

lim

.M /!0

kun M kˆ2 D 0

uniformly with respect to n 2 N. We can set M D n ; as limn!1 .n / D 0 we obtain lim kun kˆ2 D 0: n!1

However, this leads to a contradiction, because (4.17.14) yields kun kˆ2 D

sn 1 ‰1 .sn /

k n kˆ2 D

sn 1 ‰1 .sn /



1 1 ‰ .sn / >  > 0 sn 2

for all n 2 N. Because of this contradiction, (4.17.12) must hold. In conclusion we shall prove that, for ˆ2

ˆ1 , ˆ1 -mean convergence implies norm convergence in Lˆ2 ./ :

172

Chapter 4 Orlicz spaces

ˆ1 Theorem 4.17.10. Let ˆ2

ˆ1 and let ¹un º1 nD1 be a sequence in L ./ such that Z ˆ1 .jun .x/j/ dx D 0: (4.17.15) lim n!1 

Then lim kun kˆ2 D 0:

n!1

(4.17.16)

Proof. By virtue of kun kˆ1  %.un ; ˆ1 / C 1 it follows from (4.17.15) that the seˆ1 quence ¹un º1 nD1 is bounded in L ./. According to Theorem 4.17.7 all the un have uniformly absolutely continuous Lˆ2 -norms. Simultaneously, it follows from (4.17.15) that un converges in measure to zero: If we denote n D ¹x 2 I jun .x/j  º and if we had .n /   > 0 for some fixed  > 0 then we should have Z Z ˆ1 .jun .x/j/ dx  ˆ1 .jun .x/j/ dx  ˆ1 ./.n /  ˆ1 ./ > 0; 

n

which contradicts (4.17.15). Now relation (4.17.16) follows from Theorem 4.14.10.

Chapter 5

Morrey and Campanato spaces

5.1 Introduction In this chapter we shall study systematically the subspaces of real Lebesgue spaces characterized by the property that the “mean oscillation” of their elements are bounded in some sense. These spaces were studied in close connection with the theory of partial differential equations and helped to obtain many interesting results. Some special cases of these spaces were introduced by Morrey in 1938 in [157]. The early 1960s may be considered the beginning of an intense development of the general theory; we mention here the papers by John and Nirenberg [110], Meyers [151], Campanato [25], [26], Stampacchia [211], and a survey paper by Peetre [176]. In our exposition of the theory we follow the lecture notes of Campanato [28]. Our notation and terminology p; instead differ slightly from that commonly used (we denote Morrey spaces by LM p; p; p; of L -spaces, Campanato spaces by LC instead of L -spaces, etc.).

5.2 Marcinkiewicz spaces and their connection with the spaces Lp;1./ Definition 5.2.1. We suppose in this section that   RN is a domain with ./ < 1. Let ˇ 2 .0; 1/. Denote by Mˇ ./ the set of all measurable functions on  for which Z 1 kukMˇ D sup ju.x/j dx < 1 (5.2.1) ..E//ˇ E where the supremum is taken over all measurable sets E   with positive measure. The set Mˇ ./ is called the Marcinkiewicz space (see also Section 3.18). Theorem 5.2.2. The quantity k  kMˇ defined by the relation (5.2.1) is a norm on the vector space Mˇ ./. Mˇ ./ with this norm is a Banach space. Proof. The first part of the theorem is obvious. We shall prove the completeness. Let ¹un º1 nD1 be a Cauchy sequence in the space Mˇ ./. From the relation Z jf .x/j dx  ..//ˇ kf kMˇ ; kf k1 D 

valid for every f 2 Mˇ , we conclude that ¹un º1 nD1 is a Cauchy sequence in the space L1 ./. The completeness of the space L1 ./ (see Theorem 3.7.1) guarantees the

174

Chapter 5 Morrey and Campanato spaces

existence of a function u 2 L1 ./ such that lim kun  uk1 D 0:

n!1

It suffices now to prove that u 2 Mˇ ./ and lim kun  ukMˇ D 0:

n!1

For each measurable set E  , .E/ > 0 we have Z 1 ju.x/j dx ..E//ˇ E Z Z 1 1 ju .x/  u.x/j dx C jun .x/j dx  n ..E//ˇ E ..E//ˇ E Z 1  jun .x/  u.x/j dx C kun kMˇ : ..E//ˇ E

(5.2.2)

The sequence ¹un º1 nD1 in Mˇ ./ is Cauchy and hence bounded, i.e. there exists a constant c > 0 such that kun kMˇ  c for all n 2 N. This together with the convergence un ! u in L1 ./ and relation (5.2.2) implies Z 1 ju.x/j dx  c; ..E//ˇ E hence kukMˇ  c. Consequently, u 2 Mˇ ./. Further, we obtain for E as above that Z 1 ju.x/  un .x/j dx ..E//ˇ E Z Z 1 1  jun .x/  um .x/j dx C ju.x/  um .x/j dx ..E//ˇ E ..E//ˇ E Z 1  kun  um kMˇ C ju.x/  um .x/j dx: (5.2.3) ..E//ˇ E For every " > 0 there exists an n0 2 N such that kun  um kMˇ < " for all n; m  n0 . From this and inequality (5.2.3) it follows that Z Z 1 1 ju.x/  u .x/j dx  " C ju.x/  um .x/j dx n ..E//ˇ E ..E//ˇ E for all n; m  n0 . Passing to the limit as m ! 1 and using relation (5.2.1) we obtain ku  un kMˇ  " for all n  n0 . Thus the sequence ¹un º1 nD1 converges to u in the space Mˇ ./.

175

Section 5.2 Marcinkiewicz spaces

In Section 3.18 we introduced the spaces Lp;1 ./. Let us recall here the notation. For a measurable function f on  and for any positive number we define S.f; / D ¹x 2 I jf .x/j > º: For 1  p < 1 we denote 1

kf kp;1 D sup.t Œ.S.f; t // p /: t>0

p;1

./ is the set of all functions u which are meaThen the weak Lebesgue space L surable on  and for which kukp;1 < 1. We shall now establish the following theorem. Theorem 5.2.3. Let p 2 .1; 1/ and let ˇ D 1  Lp;1 ./ coincide.

1 p.

Then the sets Mˇ ./ and

Proof. (I) Suppose u 2 Lp;1 ./. Then for measurable E with .E/ > 0 and for ˛ 2 R we have (see Lemma 2.18.2) Z 1 Z ju.x/j dx D .¹x 2 EI ju.x/j > t º/ dt (5.2.4) E

Z 

0 ..E //˛

Z .¹x 2 EI ju.x/j > t º/ dt C

0

Z ˛C1

 ..E//

p

C .kukp;1 /

1



..E //˛

1 t

1

..E //˛ p

.S.u; t // dt

dt

D ..E//˛C1 C ..E//˛.1p/ .p  1/1 .kukp;1 /p : Setting ˛ D  p1 in (5.2.4) we arrive at Z

which implies



.kukp;1 /p ju.x/j dx  1 C p1 E

 ..E//ˇ

kukMˇ  1 C .kukp;1 /p .p  1/1 :

Consequently, u 2 Mˇ ./. (II) Suppose now that u 2 Mˇ ./, t > 0. Then Z Z dx  ju.x/j dx  kukMˇ ..S.u; t ///ˇ ; t.S.u; t // D t S.u;t/

S.u;t/

so that kukp;1  kukMˇ :

176

Chapter 5 Morrey and Campanato spaces

Remark 5.2.4. In the subsequent sections we shall investigate certain generalizations of the spaces Mˇ ./. The method of generalization consists in taking the supremum in relation (5.2.1) not over the class of all measurable sets E   with positive measure but over some narrower class of more special subsets of . Moreover, the integrand ju.x/j will be replaced by another expression depending on the function u.x/.

5.3 Morrey and Campanato spaces: Definitions and basic properties Definition 5.3.1. For a bounded domain   RN denote ı D diam  D sup ¹jx  yjI x; y 2 º: The bounded domain   RN is said to be of type A if there exists a constant A > 0 such that for every x 2  and all % 2 .0; ı/ ..x; %//  A%N ; where .x; %/ D ¹y 2 I jx  yj < %º: Example 5.3.2. A square in the plane is a set of type A with A D hand, the domain

1 2.

On the other

 D ¹.x; y/ 2 R2 I 0 < x < 1; 0 < y < x 2 º is not of type A for any A > 0. (It should be noted that the origin is a cuspidal point of the boundary of .) In the rest of this chapter we shall consider only domains of type A. Definition 5.3.3. Denote by ı the Cartesian product   .0; ı/ (with ı from Definition 5.3.1). For   0 and p 2 .1; 1/, set ² ³ Z 1 p; ju.y/jp dy < 1 LM ./ WD u 2 Lp ./I sup  r .x;r / where the supremum is taken over all ordered pairs .x; r/ 2 ı . Define ´ M

kukp; WD

sup .x;r /2ı

1 r

μ p1

Z p

.x;r /

ju.y/j d

:

p; ./ equipped with the above norm is called the Morrey space. The set LM

177

Section 5.3 Morrey and Campanato spaces

Remarks 5.3.4. (i) The reader will easily verify that M k  kp; is a norm on the p; linear space LM ./. Using the assumption that  is of type A, we obtain that the expression μ p1 ´ Z 

..x; r// N

sup

.x;r /

.x;r /2ı

ju.y/jp dy

p;

defines another norm in the space LM ./ which is equivalent to M k  kp; . (ii) Let u 2 L1 ./. Denote by G the set of all points x 2  for which Z 1 u.y/ dy: u.x/ D lim r !0C .B.x; r// B.x;r /

(5.3.1)

(The set B.x; r/ is a ball with its center at the point x and radius r.) The set G is called the Lebesgue set of u and its elements are called the Lebesgue points of u. Recall that . n G / D 0: (5.3.2) It is proved in [212] that the equality (5.3.2) remains valid if we replace the limit in formula (5.3.1) by the expression Z 1 u.x  y/ dy; lim E2F .E/ E .E/!0 where F is the so-called regular family of subsets in RN (as a simple example of F we can take the family of all N -dimensional cubes with their centers at the origin). We shall now introduce yet another class of function spaces. Definition 5.3.5. Let   0 and p 2 .1; 1/. Define ´ μ Z 1 p sup ju.y/  ux;r jp dy < 1 ; Lp; C ./ WD u 2 L ./I  .x;r / .x;r /2ı r where ux;r Denote

1 D ..x; r//

´ Œup; WD

sup .x;r /2ı

1 r

Z u.y/ dy: .x;r /

μ p1

Z .x;r /

ju.y/  ux;r jp dy

:

178

Chapter 5 Morrey and Campanato spaces

The expression Œup; is a seminorm in the vector space Lp; C ./ and Œup; D 0 if and only if u is constant almost everywhere on . Define the norm in the space Lp; C ./ by C kukp; WD kukp C Œup;: The set Lp; C ./ with the above norm will be called the Campanato space.

5.4 Completeness p;

p;

Theorem 5.4.1. The spaces LM ./ and LC ./ are Banach spaces. The proof is essentially the same as that of Theorem 5.2.2.

5.5 Relations to Lebesgue spaces We shall now study in detail connections between Morrey, Campanato and Lebesgue spaces as well as their embedding properties. Theorem 5.5.1.

(i) Let p 2 .1; 1/. Then, p;0 LM ./  Lp ./:

(ii) Let p 2 .1; 1/. Then,

LM ./  L1 ./: p;N

(iii) Let 1  p  q < 1, let  and  be nonnegative numbers. If N N  ; p q then Proof.

q; p; LM ./ ,! LM ./:

Assertion (i) follows immediately from the equality M

kukp;0 D kukp :

p;N ./ is complete (see Theorem 5.4.1), and so is the space (ii) The space LM 1 L ./. We shall prove that the identity operator maps L1 ./ continuously p;N onto LM ./. Then we complete the proof by making use of Theorem 1.14.8.

Thus, let u 2 L1 ./. For every ordered pair .x; r/ 2 ı the inequality Z ..x; r// 1 p ju.y/jp dy  kukp 1  cN kuk1 N r rN .x;r /

179

Section 5.5 Relations to Lebesgue spaces

holds, where cN is the volume of the unit ball in the space RN and where k  k1 is a norm in L1 ./. Thus, M

1

kukp;N  cNp kuk1 p;N

and the continuity of the identity operator Id W L1 ./ ! LM ./ follows from the last inequality. p;N

Suppose now that the operator Id does not act onto LM ./. Let p;N u 2 LM ./ n L1 ./:

Inasmuch as u 2 Lp ./, almost all points of  are Lebesgue points of ju.x/jp (see Remark 5.3.4 (ii)). Denote the set of all Lebesgue points of the function ju.x/jp by G . It follows from the fact that u … L1 ./, i.e. kuk1 D 1, that for every K > 0 the set S.u; K/ D ¹x 2 I ju.x/j > Kº has positive measure. So G \ S.u; K/ 6D ;. If x 2 G \ S.u; K/, we have 1 r !0C ..x; r//

Z

lim

.x;r /

ju.y/jp dy D ju.x/jp > K p :

From this fact it follows easily that for every C > 0 there exists .x; r/ 2 ı such that Z 1 ju.x/jp dx > C; ..x; r// .x;r / and so M kukp;N D 1 (see Remark 5.3.4 (i)). This contradicts the assumption p;N that u 2 LM ./. (iii) By the Hölder inequality for Lebesgue norms (3.1.6) we obtain for p  q and .x; r/ 2 ı Z

1 p q

p

.x;r /

Z

ju.y/j dy  ...x; r/// 

p p 1 p cN q r N.1 q /C q

q

.x;r /



1 r

ju.y/j dy

Z

q

.x;r /

According to the inequality 

p N 1 q we have

r N.1 pq /C pq ı

 C



 pq

p q

r  ı

;

ju.y/j dy

 pq

: (5.5.1)

180

Chapter 5 Morrey and Campanato spaces

so that

p

p

p

p

r N.1 q /C q  r  ı N.1 q /C q  :

Using the last relation we transform the formula (5.5.1) into 

1 r

Z .x;r /

ju.y/jp dy

 p1

 c

1 r

Z .x;r /

ju.y/jq dy

 q1

and taking the supremum first on the right-hand side and then on the left-hand side of the previous inequality we obtain M

kukp;  c M kukq;v ;

which is the assertion occurring in (iii). Theorem 5.5.2.

(i) Let p 2 .1; 1/. Then p Lp;0 C ./  L ./:

(ii) Let 1  p  q < 1 and let  and  be nonnegative numbers. If N N  p q then

q;

p;

LC ./ ,! LC ./: Proof. Both of these assertions can be proved in a way similar to that of the corresponding assertions (i) and (iii) of Theorem 5.5.1. p;

Remarks 5.5.3. (i) It is easy to see that LM ./ D ¹0º for  > N . Further, p; it follows from Theorem 5.5.1 that the collection ¹LM ./º2Œ0;N  for fixed p 2 Œ1; 1/ generates a certain “scale of spaces” between Lp ./ and L1 ./. 1 (ii) As will be proved later (Lemma 5.8.1) the spaces Lp;N C ./ and L ./ do not coincide, so that the analogue to assertion (ii) of Theorem 5.5.1 for Campanato spaces is not valid.

181

Section 5.6 Some lemmas

5.6 Some lemmas Now we prove several lemmas to be used in the next section. Unless otherwise stated we shall suppose that  is a set of type A and, moreover, 1  p < 1 and   0. p;

Lemma 5.6.1. A function u belongs to the space LC ./ if and only if u 2 Lp ./ and   Z 1 p p D sup ju.y/  cj dy < 1: (5.6.1) jjjujjjp; inf  c2R .x;r / .x;r /2ı r Proof. Evidently

p p  Œup; : jjjujjjp;

Let u 2 Lp ./ and assume (5.6.1). Then the inequalities Z ju.y/  ux;r jp dy .x;r / Z p1 ju.y/  cjp dy 2 .x;r / ˇZ ˇp  Z ˇ ˇ ...x; r///p ˇˇ .c  u.y// dy ˇˇ dz C .x;r / .x;r / Z p p ju.y/  cj dy 2 .x;r /

hold for every c 2 R. Hence we conclude that Œup;  2jjjujjjp;; which proves the lemma. Remark 5.6.2. In the course of the proof of Lemma 5.6.1 we have established the p; equivalence of the norms C kukp; and jjjujjjp; C kukp on the space LC ./. Lemma 5.6.3. Let A be the constant from Definition 5.3.1. There exists a constant C D C.p; A/ such that the implication % C  0 < < % < ı ) jux;%  ux; j  C.p; A/

N p;

holds for all u 2 LC ./ and all x 2 .

! p1 Œup;

(5.6.2)

182

Chapter 5 Morrey and Campanato spaces

Proof. For almost all y 2 .x; /  .x; %/ the inequality jux;%  ux; jp  2p1 .jux;%  u.y/jp C jux;  u.y/jp /

(5.6.3)

holds. Integrating (5.6.3) with respect to the variable y over the set .x; / we obtain Z jux;%  ux; jp dy .x; / Z  Z  2p1 jux;  u.y/jp dy C jux;%  u.y/jp dy .x; /

.x;%/

p  2p1 .% C  /Œup; :

Since  is of type A (see Definition 5.3.1) we have Z jux;%  ux; jp dy D jux;%  ux; jp ..x; //  jux;%  ux; jp A N : .x; /

Thus, jux;%  ux; jp 

2p1 % C  p Œup;; A

N

and the inequality in (5.6.2) follows. Lemma 5.6.4. There exists a constant C D C.p; ; A/ such that jux;%  ux;

% 2n

j  C.p; ; A/Œup; %

N p

n1 X

2m

N  p

(5.6.4)

mD0

whenever u 2 Lp; C ./, .x; %/ 2 ı and n 2 N. p;

Proof. Fix u 2 LC ./ and .x; %/ 2 ı . Lemma 5.6.3 implies for m D 0; 1; 2; : : : that 1

jux;

% 2mC1

 ux;

% 2m

j  C Œup;

% .. 2%m / C . 2mC1 / / p

D C 0 Œup;%

N

% . 2mC1 /p N p

 2m

N  p

;

(5.6.5)

where C 0 is a constant which is independent of m. Hence we obtain (5.6.4) by summing (5.6.5) over m D 0; 1; : : : ; n  1. p;

Lemma 5.6.5. Let  > N . Then for every u 2 LC ./ there exists a function uQ defined on  such that u equals uQ almost everywhere on  and Q lim ux;% D u.x/

%!0C

for all x 2 , the convergence being uniform on .

183

Section 5.6 Some lemmas

Proof. According to Remark 5.3.4 (ii) we have lim ux;% D u.x/

%!0C

almost everywhere in . It remains prove that the convergence of ux;% is uniform with respect to x. p; Let u 2 LC ./. Fix % 2 .0; ı/. By Lemma 5.6.4 we have jux;

% 2n

 ux;

% 2nCq

j  C 00 Œup;

% N p ; n 2

where the constant C 00 is independent of x and q. We see that the sequence ¹ux; %n º1 nD1 2 is Cauchy uniformly with respect to x. Let u.x/ Q D lim ux; n!1

% 2n

; x 2 :

We shall prove that uQ does not depend on the choice of %. Let 0 < < ı. Since Q  jux; %n  u.x/j Q C jux; %n  ux; n j jux; n  u.x/j 2 2 2 2 % N p C jux; %n  ux; 2n j  C 00 Œup; n 2 2 ! p1 % N  C % N 

p  C 00 Œup; n C C Œup;  2n p N 2 Œmin. ; %/  C.u; %; /  2n

N  p

! 0 (as n ! 1/;

we conclude that uQ is the uniform limit of any sequence of the type ¹ux; 2n º1 nD1 , where is an arbitrary real number in the interval .0; ı/. Using Lemma 5.6.4 once more we obtain jux;  ux; 2n j  C 0 Œup;

N p

n1 X

2

m.n/ p

:

mD0

Letting n ! 1, we obtain Q  C Œup;

jux;  u.x/j and consequently Q lim ux; D u.x/

!0C

uniformly on .

N p

;

(5.6.6)

184

Chapter 5 Morrey and Campanato spaces

Lemma 5.6.6. Let 0   < N . Then there exists a constant C.A; p; ; N / > 0 such p; that for any u 2 LC ./ and all .x; %/ 2 ı the inequality jux;% j  ju j C C.A; p; ; N /Œup; % holds, where

1 u D ./

N p

Z u.x/ dx: 

Proof. Fix u 2 Lp; C ./ and % 2 .0; ı/. Then jux;% j  ju j C ju  ux; Choose n 2 N with

jux;

ı 2n

ı 2n

j C jux;

 ux;% j  C Œup;

ı 2n

(5.6.7)

%
N , the function u can be replaced in our considerations by the uniform limit uQ of the mean values ux;% . The inequality in Lemma 5.6.6 describes the situation in the case 0   < N . It yields a local estimate ux;% D O.%

N p

/ .% ! 0C/

and a global estimate for ux;% on the interval 0 < % < ı.

185

Section 5.7 Embeddings

Remark 5.6.8. The case  D N is not dealt with in this section, because the methods of proofs of Lemmas 5.6.5 and 5.6.6 cannot be used. This case will be studied in Section 5.8. Now we prove another lemma. Lemma 5.6.9. There exists a constant C D C .A; N; / such that the implication % D 2jx  yj ) jux;%  uy;% j  C .A; N; /Œu1; jx  yjN holds for every u 2 L1; C ./ and all x; y 2 . Proof. Fix x; y 2 , u 2 L1; C ./ and set J% D .x; %/ \ .y; %/: We have jux;%  uy;% j  jux;%  u.t /j C ju.t /  uy;% j for almost all t 2 J% . Since .x; 12 %/  J% and .J% /  A. 12 %/N ; integration with respect to the variable t over J% yields jux;%  uy;% j  C%N Œu1; D C Œu1; jx  yjN : The proof is complete. p;

p;

5.7 Relations between the spaces LC ./, LM ./ and C 0;˛ ./ Theorem 5.7.1. Let 1  p < 1. We have that p; (i) Lp; C ./  LM ./ provided  2 Œ0; N /, p;

(ii) LC ./  C 0;˛ ./ with ˛ D

N p

provided  2 .N; N C p.

p;

Proof. Let  2 Œ0; N / and u 2 LM ./. According to Remark 5.6.2, there exists a constant Q > 0 such that C

p

kukp;  Q.kukpp C jjjujjjpp/ DQ

kukpp

C

 Q.kukpp C

sup

.x;r /2ı p M kukp; /

1 r



!

Z p

inf

c2R .x;r / p

 Q1 M kukp; :

ju.y/  cj dy

186

Chapter 5 Morrey and Campanato spaces

Hence u 2 Lp; C ./ and the identity operator p;

p;

Id W LM ./ ! LC ./ is continuous. Now let u 2 Lp; C ./. We have (using Lemma 5.6.6) that Z Z Z ju.y/jp dy  2p1 ju.y/  ux;r jp dy C .x;r /

.x;r / p

 .x;r /

jux;r jp dy

p

 2p1 .r  Œup; C C r N .r n Œup; C ju jp // p C r N kukpp /:  C1 .r  Œup;

Now the inequality M

p

p

kukp;  C2 C kukp;

p; easily follows. Thus u 2 LM ./ and the identity operator p;

p;

Id W LC ./ ! LM ./ is continuous, which proves assertion (i). 0;˛ ./. We have Let  > N and ˛ D N p ,u2C Z ju.y/  ux;r jp dy .x;r /

Z

D

p

ˇZ ˇ ˇ ˇ

ˇp  ˇ .u.y/  u.t // dt ˇˇ dy .x;r /  ju.y/  u.t /jp dt dy:

...x; r/// Z Z 1  .x;r / ..x; r// .x;r / .x;r /

(5.7.1)

Recall the notation introduced in Notation 2.2.4 and Lemma 2.3.1: H0;˛ .u/ D sup

x;y2 x6Dy

ju.x/  u.y/j jx  yj˛

and kukC 0;˛ ./ D kukC./ C H0;˛ .u/:

(5.7.2) From formula (5.7.1) we now conclude that Z ju.y/  ux;r jp dy .x;r / Z  Z 1 ju.y/  u.t /jp ˛p  jy  t j dt dy jy  t j˛p .x;r / ..x; r// .x;r / Z  Z 1 p ˛p H0;˛ .u/r dt dy  .x;r / ..x; r// .x;r / p

p

D ..x; r//H0;˛ .u/r p˛  CN H0;˛ .u/r 

187

Section 5.8 The John–Nirenberg space

and consequently Œup; C kukp  C.H0;˛ .u/ C kukC./ /: Thus u 2 Lp; C ./ and the identity operator p;

Id W C 0;˛ ./ ! LC ./

(5.7.3)

is continuous. p; Let u 2 LC ./. Since, by Theorem 5.5.2 (ii), we have 1;˛CN Lp; ./; C ./ ,! LC

./. By virtue it suffices to prove that u 2 C 0;˛ ./ for ˛ 2 .0; 1/ and u 2 L1;˛CN C of (5.7.3) the continuity of the mapping p;

Id W LC ./ ! C 0;˛ ./ follows from the Banach theorem (Theorem 1.14.8). ./. We prove that Let u 2 L1;˛CN C lim ux;% D uQ 2 C 0;˛ ./

%!0C

and use Remark 5.6.7. Let x; y 2 , % D 2jx  yj. From (5.6.6) and Lemma 5.6.9 we have ju.x/ Q  u.y/j Q  ju.x/ Q  ux; j C jux;%  uy;% j C juy;%  u.y/j Q  C Œu1;N C˛ jx  yj˛ : Thus H0;˛ .u/  C Œu1;N C˛ and the proof of the theorem is complete. p;

Remark 5.7.2. For  > N C p the spaces LC ./ and C 0; constant functions on .

N p

p;N

5.8 LC ./ and the John–Nirenberg space In Section 5.3 we postponed the proof of the assertion LC ./ 6D L1 ./ p;N

(see Remark 5.5.3 (ii)). We shall finish this proof in this section.

./ contain only

188

Chapter 5 Morrey and Campanato spaces

Lemma 5.8.1. Let p 2 Œ1; 1/. Then L1 ./ ¤ LC ./: p;N

Proof. Let u 2 L1 ./ and .x; %/ 2 ı . Then Z  Z p p1 p p ju.y/  ux;% j dy  2 ju.y/j dy C ..x; %//jux;% j .x;%/

.x;%/

p

 2 ..x; %//k jujp k1  c%N k jujp k1 ; whence Œup;N  ckuk1 : In order to prove that 1 Lp;N C ./ n L ./ 6D ;; p;1

note that, for N D 1 and  D .0; 1/, one has log x 2 LC ./ for any p  1, but log x … L1 ./. It is easy to construct a similar example for N > 1 and general   RN . Remark 5.8.2. Lemma 5.8.1 and Theorem 5.7.1 suggest that the spaces Lp;N C ./ play an important part in the family of Campanato spaces. If  is an N -dimensional p;N cube then the space LC ./ will be shown to coincide with a certain vector space JN./, which was introduced by John and Nirenberg [110]. The space JN./ is more familiar under the notation BMO./ (the space of functions with bounded mean oscillation). There is a vast literature available on this space, see, e.g., [14, 79]. Definition 5.8.3. Denote by Q the N -dimensional cube whose edges are parallel with the coordinate axes. We shall denote by Q0 the cubes contained in Q and homothetic with Q. It is easy to see that the norm C

kukp;N D Œup;N C kukp

which was used until now is equivalent for  D Q to the norm kukp C huip;N ;

(5.8.1)

where huip;N uQ 0

! p1 Z 1 D sup ju.y/  uQ0 jp dy ; 0 Q0 Q . / 0 Z 1 D u.x/ dx: .Q0 / Q0

(5.8.2)

From now on we shall denote by C kukp;N the norm defined by relation (5.8.1), since it is more suitable for our purpose.

189

Section 5.8 The John–Nirenberg space

Definition 5.8.4. For any measurable function u on the cube Q and for b > 0 denote S.u; b; Q0 / D ¹x 2 Q0 I ju.x/  uQ0 j > bº:

(5.8.3)

Let u 2 L1 .Q/ be a function for which there exist two positive constants ˇ > 0 and H > 0 such that for every b > 0 and every cube Q0  Q we have .S.u; b; Q0 //  H eˇ b u.Q0 /: We shall denote the set of all functions with this property by JN.Q/. p;N

Lemma 5.8.5. JN.Q/  LC .Q/ for every p  1. Moreover, the inequality huip;N 

1 1 .H  .p C 1// p ˇ

holds for all u 2 JN.Q/ where  is the Gamma function. Proof. Let u 2 JN.Q/. Using Lemma 3.18.2, we obtain Z Z 1 ju.x/  uQ0 jp dx D p t p1 .S.u; t; Q0 // dt Q0 0 Z 1 t p1 eˇ t dt D Hˇ p .Q0 /.p C 1/  pH.Q0 / 0

and this implies the assertion. p;N

Our next aim will be to prove that LC .Q/ D JN.Q/ for all p  1. We start with some auxiliary material. Lemma 5.8.6. Let u 2 L1 .Q/ and K 2 .0; 1/. Suppose Z 1 ju.x/j dx  K: .Q/ Q Then there exists a countable collection ¹Qi0 º of pairwise disjoint open cubes Qi0  Q such that S (i) ju.x/j  K almost everywhere on Q n Qi0 ; (ii) jujQi0  2N K for each cube Qi0 in the collection ¹Qi0 º; R P 1 (iii) .Qi0 /  K Q ju.x/j dx.

190

Chapter 5 Morrey and Campanato spaces

Proof. Halving the edges of Q we decompose it into 2N equal cubes Q0 . Because the measures of all these cubes are the same the following holds: Z Z 1 X 1 1 ju.x/j dx D N ju.x/j dx (5.8.4) .Q/ Q .Q0 / Q0 2 0 ; Q0 ; : : : (we sum over all cubes obtained by the decomposition). Denote by Q1;1 1;2 0 those of the cubes Q for which Z 1 ju.x/j dx  K: .Q0 / Q0

This inequality together with (5.8.4) implies Z 0 0 ju.x/j dx  K2N .Q1;i /; K.Q1;i /  0 Q1;i

i 2 N:

Now we decompose each of the remaining cubes Q0 for which the mean value jujQ0 is less than K into 2N equal cubes and then repeat the process described above. Proceeding this way we obtain a countable collection of cubes. Let us arrange this collection in a sequence ¹Qj0 º. The interiors of the members of ¹Qj0 º form a collection satisfying (i)–(iii). Indeed, these sets Int Qj0 , j D 1; 2; : : : are pairwise disjoint open cubes and K.Qj0 /

Z 

Qj0

ju.x/j dx  2N K.Qj0 /; j D 1; 2; : : : :

Hence assertion (ii) is valid. Summing the inequalities Z ju.x/j dx K.Qj0 /  Qj0

we obtain inequality (iii). It remains to prove (i). Inasmuch as u 2 L1 .Q/ we have that almost all the points of Q are Lebesgue points of the function u (see Remark 5.3.4 (ii)). Let x S be a Lebesgue point of u belonging to Q n Qj0 . It follows from the construction of ¹Qj0 º that for every " > 0 there exists a cube Q" with its edge shorter than " containing the point x 2 Q" and such that Z 1 ju.x/j dx < K: .Q" / Q" Thus

1 ju.x/j D lim "!0C .Q" /

Z Q"

ju.y/j dy  K:

191

Section 5.8 The John–Nirenberg space

In the following lemma, we shall use the symbols hui1;N from (5.8.2) and S.u; b; Q0 / from (5.8.3). Lemma 5.8.7. Let u 2 L1;N C .Q/. Then u 2 JN.Q/ and there exist positive constants A; ˛ such that   ˛b .S.u; b; Q0 //  A exp  (5.8.5) .Q0 / hui1;N for any Q0  Q. Proof. Let u be a nonconstant function in L1;N C .Q/. Suppose that uQ D 0

and hui1;N D 1:

(5.8.6)

Define the function F W .0; 1/ ! R by .S.u; b; Q// F .b/ D sup R Q ju.x/j dx (here the supremum is taken over the set of all functions satisfying (5.8.6)). Notice that the definition of F does not depend on the length of the edge of Q. In fact, if Q0 is a cube, Q0  Q and if we define 0

b; Q // e .b/ D sup .S.u; R F ju.x/j dx Q0 0 where the supremum is taken over the set of all functions in L1;N C .Q / for which 0 e uQ0 D 0 and hui1;N D 1 (calculated on Q ), then F .b/ D F .b/. The function F has the following property: For every b  2N e and for every K 2 Œ1; 2N b,

F .b/  K 1 F .b  2N K/:

(5.8.7)

This inequality will be proved later. We show first of all how assertion of Lemma 5.8.7 follows from it. Fix a positive number ". Let A be a positive number such that for all b 2 Œ"; 2N e C ", 1 F .b/  Ae˛b ; ˛ D N : (5.8.8) 2 e Such an A exists. (Its existence follows from the definition of F : We have .S.u; "; Q// .S.u; "; Q// F ."/ D sup R D sup R 1 ju.x/j dx Q 0 .S.u; t; Q// dt 1 .S.u; "; Q//  " 0 .S.u; t; Q// dt

 sup R "

192

Chapter 5 Morrey and Campanato spaces

and the function F is nonincreasing.) Then (5.8.8) is valid for all b 2 ."; 1/. Indeed, set K D e in (5.8.7). Hence F .b/  e1 F .b  2N e/ provided b 2 Œ2N e; 1/. For b 2 Œ2N e C "; 2N C1 e C "/, it follows from here and from (5.8.8) that   1 N F .b/  A exp 1  N .b  2 e/ D Ae˛b : 2 e We have thus proved that (5.8.8) is satisfied in the interval Œ"; 2N C1 e C ". Using the induction on k we can easily prove the validity of (5.8.8) in each interval Œ"; 2N Ck e C ". So the formula (5.8.8) is valid in Œ"; 1/. It follows from the estimate (5.8.8) and the definition of F that .S.u; b; Q//  A.Q/e˛b

(5.8.9)

for all b 2 Œ"; 1/. In the general case, for a function u 2 L1;N C .Q/ we deduce (substituting u.x/  uQ U.x/ D hui1;N into (5.8.9)) that



˛b .S.u; b; Q//  A.Q/ exp hui1;N

 (5.8.10)

for all b 2 Œ"; 1/. Regarding .S.u; b; Q//  .Q/ it is possible, increasing A if necessary, to achieve estimate (5.8.10) on the entire interval Œ0; 1/. Finally, by virtue of the fact that F is independent of the choice of Q0 , it is easy to see that (5.8.10) is valid (with the same values of A and ˛) for any cube Q0  Q. It remains to prove assertion (5.8.7). Let u satisfy (5.8.6) and let b  2N e, K 2 Œ1; 2N b. According to Lemma 5.8.6 there exists an at most countable collection ¹Qj0 º of cubes from Q with the properties: S (i) ju.x/j  K almost everywhere on Q  j Qj0 ; (ii) jujQj0  2N K, j D 1; 2; : : : ; (iii)

P

j

.Qj0 / 

1 K

R

Q

ju.x/j dx.

We can suppose S.u; b; Q/ 6D ;. (Otherwise everything is obvious.) Let x 2 S.u; b; Q/. Thus ju.x/j > b > K. According to (i) either there exists a Qj0 such that S x 2 Qj0 or x belongs to a subset of Q  j Qj0 which has measure zero. Hence it  [ follows that ® ¯ 0 x 2 Qj I ju.x/j > b : (5.8.11) .S.u; b; Q// D  j

193

Section 5.8 The John–Nirenberg space

The sets in the union on the right-hand side of (5.8.11) are pairwise disjoint and .¹x 2 Qj0 I ju.x/j > bº/  .¹x 2 Qj0 I ju.x/  uQj0 j > b  jujQj0 º/  .¹x 2 Qj0 I ju.x/  uQj0 j > b  2N Kº/ D .S.u; b  2N K; Qj0 //: This estimate together with (5.8.11) yields X .S.u; b; Q//  .S.u; b  2N K; Qj0 //:

(5.8.12)

j

On Qj0 the functions u  uQj0 satisfy the conditions .u  uQj0 /Qj0 D 0; hu  uQj0 i1;N D 1: From this and the definition of the function F we conclude that X .S.u; b  2N K; Qj0 //  .Q/F .b  2N K/: j

Substituting this inequality into (5.8.12) and dividing the resulting inequality by Z ju.x/j dx Q

we obtain (also using Lemma 5.8.6 (iii)) 1 .S.u; b; Q// R  F .b  2N K/: K Q ju.x/j dx Finally, taking the supremum over the set of all functions u satisfying (5.8.6) we obtain (5.8.7). Theorem 5.8.8. Let p 2 Œ1; 1/. Then p;N

LC .Q/ D JN.Q/: p;N

q;N

Proof. It follows from the definition of LC .Q/ that if 1  p < q, then LC .Q/  Lp;N C .Q/ and huip;N  huiq;N : By virtue of Lemmas 5.8.5 and 5.8.7 we obtain immediately the assertion of Theorem 5.8.8.

194

Chapter 5 Morrey and Campanato spaces

5.9 Another definition of the space JN.Q/

e

Theorem 5.9.1. Let J N .Q/ be the set of all functions u 2 L1 .Q/ with the following property: There exist k D k.u/ > 0 and M D M.u/ > 0 such that for each Q0  Q Z exp.kju.x/  uQ0 j/ dx  M.Q0 / (5.9.1) Q0

e

holds. Then

e

JN.Q/ D J N .Q/:

Proof. Let u 2 J N .Q/. Then we have that Z exp.kju.x/  uQ0 j/ dx  M.Q0 /: ekb .S.u; b; Q0 //  0

Hence .S.u; b; Q0 //  ekb M.Q0 / and so u 2 JN.Q/. Now let u 2 JN.Q/. Then Z 1 Z exp.kju.x/  uQ0 j/ dx D k ekt .S.u; t; Q0 // dt: 0

(5.9.2)

0

(This assertion can be proved in the same manner as Lemma 3.18.2.) Inasmuch as .S.u; t; Q0 //  Ae˛t .Q0 /, we conclude from (5.9.2) (choosing k D 12 ˛) that Z 1  Z ˛

˛  ˛t 2 ju.x/  uQ0 j dx  exp e dt A.Q0 /: 2 2 0 Q0

e

Consequently, u 2 J N .Q/. Definition 5.9.2. Let X1 be the set of all functions u which are defined almost everywhere on Q and satisfy the conditions T (i) u 2 p2N Lp .Q/; (ii) to every u there exists a constant Cu such that kukp  Cu p for all p 2 N. Theorem 5.9.3. JN.Q/ ¤ X1 . Proof. Let u 2 JN.Q/. According to Lemma 3.18.2, Z Z 1 ju.x/  uQ jp dx D p t p1 .S.u; t; Q// dt Q 0 Z 1 A.Q/ t p1 e˛t dt D p.p/:  Ap.Q/ ˛p 0

195

Section 5.9 Another definition of the space JN.Q/ y 2

1–logx

1

0

1 1 1 4 3 2

x

1

Figure 5.1. A logarithmic function.

Using the well-known formula 1

p.p/

p pC 2 .p ! 1/ ep

and the estimate  Z Z Z p p1 p p ju.x/j dx  2 ju.x/  uQ j dx C juQ j dx ; Q

Q

Q

we obtain that u 2 X1 . Now we shall construct a function belonging to X1 n JN.Q/. In the case N D 1, Q D .0; 1/ set   1 ; n D 2; 3; : : : Qn D n1 ; n1 and define g.x/ D n.1 C log n/Œ.n  1/x  1 for x 2 Qn , n D 2; 3; : : : (Figure 5.1). It is evident that for p 2 N 0  g p .x/  .1  log x/p  2p1 .j log xjp C 1/ and so

Z

1

p

g .x/ dx 0

 p1

1

From the last estimate we obtain, by virtue of 1

.p C 1/ that g 2 X1 .

1

 .2p1 ..p C 1/ C 1// p  2..p C 1// p :

p pC 2 ep

.p ! 1/;

196

Chapter 5 Morrey and Campanato spaces

In order to prove g … JN..0; 1// it suffices to show that g … L1;1 C ..0; 1//. However, gQn D and thus 1 .Qn /

1 1 n1 

Z 1 n

1 n1 1 n

g.x/ dx D 12 .1 C log n/

Z Qn

jg.x/  gQn j dx D 14 .1 C log n/ ! 1

(as n ! 1). Theorem 5.9.4. Let u be a measurable function on Q. Suppose there exists a C > 0 such that for all p 2 N and all Q0  Q Z Z 1 ju.x/  u.y/jp dy dx  C p p p : (5.9.3) ..Q0 //2 Q0 Q0

e

Then u 2 J N .Q/. Proof. We have 1 .Q0 /

Z Q0

exp.kju.x/  uQ0 j/ dx D

Z 1 X kp 1 ju.x/  uQ0 jp dx: (5.9.4) pŠ .Q0 / Q0

pD0

If there exists a C > 0 such that for all p 2 N and all Q0  Q Z 1 ju.x/  uQ0 jp dx < C p p p ; .Q0 / Q0

(5.9.5)

we can choose k such that the series in (5.9.4) converges and so u 2 JN.Q/. However, from condition (5.9.5) we easily obtain condition (5.9.3) by expressing uQ0 explicitly and using the Hölder inequality. Exercises 5.9.5.

(i) Denote kukp

 D

1 .Q/

Z p

Q

ju.x/j dx

 p1

and define for ˛ 2 Œ0; 1 the set X˛ .Q/ as the set of all u 2 L1 .Q/ for which there exists a C > 0 such that kukp  Cp ˛ for every p 2 N. Prove that X˛ .Q/ is a Banach space with the norm kukX˛ D sup

p2N

kukp p˛

:

197

Section 5.10 Spaces Np; .Q/

(ii) We have

X0 .Q/ D L1 .Q/:

For ˛ < ˇ prove that X˛ .Q/ ,! Xˇ .Q/: (iii) Show that X˛ .Q/ n JN.Q/ 6D ; and JN.Q/ n X˛ .Q/ 6D ; for ˛ 2 .0; 1/: (In the case of ˛ D 0, ˛ D 1 the situation is described in Lemma 5.8.1 and Theorem 5.8.8.) (Hint: f .x/ D  log x belongs to JN..0; 1// but not to X˛ ..0; 1//. To prove X˛ ..0; 1// n JN..0; 1// 6D ; we construct a function g which is analogous to the function g in the proof of Theorem 5.8.8, but taking the majorant . log x/˛ C 1 instead of the majorant . log x/ C 1.)

5.10 Spaces Np; .Q/ and their relation p; to the spaces LC .Q/ p;

Let us introduce another type of spaces which are closely related to LC .Q/. As in the previous sections we denote by Q an N -dimensional cube, Q0 is an arbitrary cube contained in Q with its edges parallel to the edges of Q. By ¹º we denote the collection of all finite decompositions  of the cube Q into subcubes Q0 . The symbol P 0  denotes the sum over all cubes Qi of . With the exception of the last assertion of this section we omit the proofs. Theorem 5.10.1. Let p 2 .1; 1/. The function u 2 L1 .Q/ belongs to Lp .Q/ if and only if p Z X ..Qi0 //1p ju.x/j dx < 1: sup Qi0

2¹º 

Proof. For the proof, see, e.g., [28, 187]. Definition 5.10.2. We say that u 2 L1 .Q/ belongs to the class Np; .Q/ for p 2 .1; 1/ and  2 R if and only if  Kp; .u/ WD

sup ..Qi0 //1p

2¹º

Z Qi0

ju.x/  uQi0 j dx

The following result is stated and proved in [110].

p  p1

< 1:

(5.10.1)

198

Chapter 5 Morrey and Campanato spaces

Theorem 5.10.3 (John–Nirenberg). Let u 2 L1 .Q/ and suppose that for some p 2 .1; 1/ the expression Kp;0 .u/ is finite. Then there exists a constant A > 0 such that for all > 0, p Kp;0 .u/ .¹x 2 QI ju.x/  uQ j > º/  A :

p Remark 5.10.4. It is easy to prove that for every p 2 .1; 1/ the inclusion u  uQ 2 Lp;1 .Q/, where Lp;1 .Q/ is the weak Lebesgue space, introduced in Definition 3.18.3, implies u 2 Lp;1 .Q/. Thus Theorem 5.10.3 gives a sufficient condition for u to be in Lp;1 .Q/. Theorem 5.10.5. Let u 2 L1 .Q/. Then 

lim Kp;pC p .u/ D sup ..Q0 // N

p!1

and so

N

Q0 Q

u 2 L1; C .Q/ if and only if

Proof. Set 0

 N

M D sup ..Q //

Q0

ju.x/  uQ0 j dx;

(5.10.2)

lim Kp;pC p .u/ < 1:

p!1

N

Z Q0

Q0 Q

Z

ju.x/  uQ0 j dx:

If M D 0, then u is constant and Kp;pC p .u/ is also zero. If M > 0, choose M 0 so N that 0 < M 0 < M . There exists at least one cube Q0  Q such that Z  ju.x/  uQ0 j dx  M 0 ..Q0 // N Q0

and so

Z Kp;pC p .u/  N



Q0



1

ju.x/  uQ0 j dx ..Q0 // N ..Q0 // p 1

0

 M ..Q0 // p : Hence we obtain lim inf Kp;pC p .u/  M: p!1

N

On the other hand

(5.10.3)

1

Kp;pC p .u/  M..Q// p N

and so lim sup Kp;pC p .u/  M: p!1

N

(5.10.4)

Relations (5.10.3) and (5.10.4) imply (5.10.2) for M > 0. This completes the proof of the theorem.

199

Section 5.11 Miscellaneous remarks

5.11 Miscellaneous remarks Remark 5.11.1 (the spaces Lp;;m ./). Let   RN be a domain of type A (see C Definition 5.3.1). In Remark 5.6.2 we have seen that Lp; C ./ can be renormed by the equivalent norm k  kp C jjj  jjjp;, where the seminorm jjj  jjjp; is defined by  jjjujjjp; D

sup .x;%/2ı

1 %



Z p

inf

c2R .x;%/

ju.y/  cj dy

  p1

:

(5.11.1)

p So Lp; C ./ is a subspace of L ./ which consists of those functions u for which jjjujjjp; < 1. This definition can be generalized in the following way: We replace the space of all constant functions over which we take the infimum in formula (5.11.1) by another subspace Y of Lp ./. The case of Y D Pm (the set of all polynomials in N variables and of degree  m) is especially important. Therefore, let us define

 jjjujjjp;;m WD

sup .x;/2ı

1 %



Z p

inf

P 2Pm

.x;%/

ju.y/  P .y/j dy

Lp;;m ./ WD ¹u 2 Lp ./I jjjujjjp;;m < 1º C

  p1

;

(5.11.2) (5.11.3)

./ be given by the expression k  kp C jjj  jjjp;;m . Using and let the norm in Lp;;m C p;;m ./ instead of Lp; ./ for this notation we can write Lp;;0 C C ./. The spaces LC 1  p < 1,   0, m 2 N0 are Banach spaces. p;;m

./). As was proved in Theorem Remark 5.11.2 (properties of the spaces LC p;;0 p; p;;0 5.7.1, LC ./  LM for 0   < N and LC ./  C 0;˛ ./ for N <   N C p with ˛ D N p (this result was established independently by Campanato .Q/ D JN.Q/ [26] and Meyers [151]). In Theorem 5.8.8 we proved that Lp;N;0 C p;;0 provided Q is a cube. LC ./ is the set of constant functions if  > N C p. For m > 0, the following result is due to Campanato [27]: Let m  1, p 2 .1; 1/. Then p;;m

./  LC

p;;m

./ is isomorphic to a certain limit space Em ./ for  D N C mp;

LC

LC

p;;m1

./ for 0   < N C mp;

./  C m;˛ ./ with ˛ D

Lp;;m C

N p

 m for N C mp <   N C .m C 1/p;

p;;m

for  > N C .m C 1/p, the space LC only.

./ consists of the polynomials P 2 Pm

200

Chapter 5 Morrey and Campanato spaces p, λ,0

p, λ,1

p, λ,2

LC

LC

λ

p, λ,3

LC

LC

0 p, λ

LM N

JN

N+p

N+2p

N+3p

p, λ,0

LC

p, λ,1

LC

α = λp–N

constant functions (=P0)

C

0, α

E1

p, λ,2

LC

C 1, α

α=

λ –(N+p) p

E2 C

P1

2, α

α=

λ –(N+2p) p

P2

C 3, α

E3

Figure 5.2. Campanato spaces.

p; Consequently, fixing m 2 N0 we obtain all the spaces LM ./, C 0;˛ ./, . . . , C m;˛ ./ by changing only one parameter, . Figure 5.2 is an attempt to describe the situation graphically.

Remark 5.11.3 (the spaces Lp; C .; /). We obtain another generalization of the p; spaces LC ./ if we replace .x; %/ in (5.11.1) by some other sets; the following choice is of great importance: Let be a metric in RN with the following properties (i) B .0; %/ WD ¹y 2 RN I .0; y/  %º is a convex set for all  > 0; (ii) there exist two constants M1 > 0, M2 > 0 and a number m  N such that for every % > 0 M1 %m  .B .0; %//  M2 %m : Set again ı. / D sup¹ .x; y/I x; y 2 º and define the seminorm  sup x2;0 0 the inequality Z

ı 0

'.t / dt < 1 t

'

holds then every function u 2 LC .Q/ is continuous and its modulus of continuity satisfies the estimate Z r '.t / dt !.u; r/  c t 0 (c depends on u; r is sufficiently small). (ii) If

'.t/ t

is nonincreasing and the integral Z

ı 0

'.t / dt t

is not finite, then there exists a function u 2 L'C .Q/ which is neither continuous nor essentially bounded.

Chapter 6

Banach function spaces

In the previous chapters we have studied in detail several types of function spaces, emphasizing their distinctive properties. In this chapter we shall abandon this approach and focus instead on properties that are common for many classes of function spaces. More precisely, we shall develop an abstract and rather general theory of the so-called Banach function spaces of measurable functions defined on a measure space on which only very mild assumptions will be adopted. A Banach function space is an abstract structure that covers many important examples of scales of function spaces including Lebesgue, Orlicz and Morrey spaces and their various modifications. It also includes a very important class of Lorentz spaces and their likes and, more generally, rearrangement-invariant (r.i.) spaces; these classes of function spaces will be studied in subsequent chapters. We will basically follow the exposition in [14, Chapter 1].

6.1 Banach function spaces Convention 6.1.1. We shall work in this chapter with more general measure spaces than in the preceding chapters. We shall assume once and for all that .R; / is a -finite measure space and that there exists a sequence of sets ¹Rn º1 nD1 such that .Rn / < 1 for every n 2 N and RD

1 [

Rn :

nD1

Notation 6.1.2. Let .R; / be a –finite nonatomic measure space. Let M.R; / be the set of all –measurable real functions on R. Let M0 .R; / denote the class of functions in M.R; / that are finite -a.e. By MC .R; / we denote the subset of M0 .R; / consisting of nonnegative functions. When R is an interval .a; b/, 1  a < b  1, and  is the one-dimensional Lebesgue measure, then we just write MC .a; b/. Remark 6.1.3. As usual, any two functions coinciding -a.e. will be identified. The natural vector space operations are well-defined on M0 (although not on all of M), and when M0 is given the topology of convergence in measure on sets of finite measure it becomes a metrizable topological vector space. One of the possible corresponding

204

Chapter 6 Banach function spaces

metrics on this space, is given by d.f; g/ WD

1 X nD1

2n

Z Sn

jf .x/  g.x/j d.x/; 1 C jf .x/  g.x/j

where Sn are disjoint sets such that .Sn / < 1 for every n 2 N and Moreover, the metric space .M; d / is complete.

S n2N

Sn D R.

Convention 6.1.4. By A . B and A & B we mean that A  CB and B  CA, respectively, where C is a positive constant independent of appropriate quantities involved in A and B. We write A  B when both of the estimates A . B and B . A are satisfied. We shall use throughout the convention 0  1 D 0, 00 D 0 and 1 1 D 0. Definition 6.1.5. We say that a function % W MC ! Œ0; 1 is a Banach function norm if, for all f , g and ¹fn º1 nD1 in MC , for every  = 0 and for all -measurable subsets E of R, the following five properties are satisfied: (P1) %.f / D 0 , f D 0 -a.e.; %.f C g/ 5 %.f / C %.g/ (P2) 0 5 g 5 f -a.e. (P3) 0 5 fn " f -a.e. (P4) .E/ < 1 (P5) .E/ < 1

%.f / D %.f /;

) %.g/ 5 %.f / (the lattice property); ) %.fn / " %.f /

(the Fatou property);

) %. E / < 1; R ) E f d  CE %.f /.

for some constant CE 2 .0; 1/ and all f . Definition 6.1.6. Let % be a Banach function norm. We then say that the set X D X.%/ of those functions in M for which %.jf j/ < 1, is a Banach function space. For each f 2 X, we then define (6.1.1) kf kX WD %.jf j/: Remark 6.1.7. We note that the functional kf kX is defined for every f 2 M0 but it can be infinite. A function f then belongs to X D X.%/ if and only if kf kX < 1. The notion of a simple function was introduced in Definition 1.20.23. We shall now for the sake of completeness recall this definition in the context suitable for this chapter. Definition 6.1.8. A real-valued function s on the measure space .R; / is called a simple function if it is a finite linear combination of characteristic functions of measurable sets of finite measure, i.e. if there exist an m 2 N, a finite sequence of real

205

Section 6.1 Banach function spaces

numbers ¹a1 ; : : : ; am º and a finite sequence ¹E1 ; : : : ; Em º of disjoint -measurable subsets of R of finite measure such that ´ aj ; x 2 Ej ; j D 1; : : : ; m; s.x/ D S 0; x 2 R n jmD1 Ej : The set of all -simple functions will be denoted by S . If no confusion can arise, we will say shortly simple instead of -simple. Theorem 6.1.9. Let X be a Banach function space generated by a Banach function norm %. Then .X; kkX / is a normed linear space. Moreover, the inclusions S  X ,! M0

(6.1.2)

hold, where S is the set of simple functions on R. Proof. Since  is a -finite measure, locally integrable functions are -a.e. finite. Hence, X  M0 . Thanks to the property (P5) from Definition 6.1.5 of the norm k  kX , all functions in X are locally integrable on R. Thus, since M0 is a vector space, so is X. It follows immediately from (P1) that X is a normed space. By (P4), E 2 X for every set E such that .E/ < 1. Consequently, by the linearity of X, we get S  X. It remains to show that the embedding X ,! M0 is continuous. Assume that a sequence ¹fn º1 nD1 satisfies fn ! f in X. Then, by (6.1.1), %.jfn  f j/ ! 0 as n ! 1: Given " > 0 and a set E  R such that .E/ < 1, we get from (P5) that Z 1 CE ¹x 2 E W jf .x/  fn .x/j > "º  %.jf  fn j/; jf  fn j d  " E " which converges to 0 as n ! 1 since CE is independent of n. Therefore, fn ! f in measure on every set of finite measure, in other words, fn ! f in M0 . Remark 6.1.10. It follows in particular from Theorem 6.1.9 that if fn ! f in X, then fn ! f in measure on sets of finite measure. Hence, it follows for example from [188, p. 92] that there exists a subsequence of ¹fn º1 nD1 that converges pointwise -a.e. to f . Remark 6.1.11. Let X be a Banach function space and let fn 2 X, n 2 N. Assume that 0 5 fn " f -a.e. for some function f 2 M. Then we have the following two possibilities:

either f 2 X, hence kf kX < 1, and kfn kX " kf kX ;

or kfn kX " 1 and f … X. Indeed, this fact is an immediate consequence of Definition 6.1.6 and (P3).

206

Chapter 6 Banach function spaces

We shall now show that the Fatou lemma, familiar from the theory of the Lebesgue integral (cf. Theorem 1.21.6), holds for every Banach function space. The key ingredient of the proof is the Fatou property (P3). Lemma 6.1.12 (Fatou lemma for Banach function spaces). Let X be a Banach function space and assume that fn 2 X, n 2 N, and fn ! f -a.e. for some f 2 M. Assume further that lim inf kfn kX < 1: n!1

Then f 2 X and

kf kX 5 lim inf kfn kX : n!1

Proof. Denote gn .x/ WD infm=n jfm .x/j. Then 0 5 gn " jf j -a.e, whence, by (P2) and (P3), kf kX D lim kgn kX 5 lim inf kfm kX D lim inf kfn kX < 1: n!1

n!1 m=n

n!1

Hence f 2 X and kf kX 5 lim infn!1 kfn kX : Remark 6.1.13. As a consequence of the Fatou lemma, every Banach function space is complete. We shall now turn our attention to the Riesz–Fischer property (see Definition 1.9.4). Theorem 6.1.14. Every Banach function space has the Riesz–Fischer property. Proof. Let X be a Banach function space, let ¹fn º1 nD1  X and suppose that 1 X

kfn kX < 1:

(6.1.3)

nD1

P P We denote, for every n 2 N, gn WD nkD1 jfk j, and g D 1 nD1 jfn j, so that 0 5 gn " g. Since n 1 X X n 2 N; kgn kX 5 kfk kX 5 kfn kX ; nD1

kD1

it follows from (6.1.3) and Lemma 6.1.12 that g belongs P1 to X. By the embedding 6.1.3, the series X ,! M0 in (6.1.2) and Remark nD1 jfn .x/j converges pointP f .x/: We set wise -a.e. and hence so does 1 nD1 n f WD

1 X nD1

fn

207

Section 6.1 Banach function spaces

and hn D

n X

n 2 N;

fk ;

kD1

then hn ! f -a.e. Hence, for any m 2 N, we have hn  hm ! f  hm

-a.e. as n ! 1:

Furthermore, lim inf khn  hm kX 5 lim inf n!1

n!1

n X

kfk kX D

kDmC1

1 X

kfk kX ;

kDmC1

which tends to 0 as m ! 1 because of (6.1.3). Thus, by Lemma 6.1.12, we get f  hm 2 X, therefore also f 2 X, and kf  hm kX ! 0 as m ! 1. This implies that, for every m 2 N, kf kX 5 kf  hm kX C khm kX 5 kf  hm kX C

m X

kfk kX :

kD1

By letting m ! 1, we get kf kX 5

1 X

kfn kX :

(6.1.4)

nD1

Corollary 6.1.15. Every Banach function space is complete. Proof. This is an immediate consequence of Theorems 6.1.14 and 1.9.5. Remark 6.1.16. Let us summarize the basic properties of Banach function spaces. Let X be a Banach function space generated by a Banach function norm %. Assume that kf kX D %.jf j/ for every f 2 X. Then .X; kkX / is a Banach space and the following properties hold for all f; g; fn , n 2 N, in M and for all measurable subsets E  R: (i) (The lattice property) If jgj 5 jf j -a.e. and f 2 X; then g 2 X and kgkX 5 kf kX . (ii) In particular, a function f 2 M satisfies f 2 X if and only if jf j 2 X, and kf kX D k jf j kX . (iii) (The Fatou property) Suppose that fn 2 X, fn = 0, n 2 N, and fn " f -a.e. If f 2 X, then kfn kX " kf kX , while if f … X, then kfn k " 1.

208

Chapter 6 Banach function spaces

(iv) (Fatou lemma) If fn 2 X, n 2 N, fn ! f -a.e., and lim inf kfn kX < 1; n!1

then f 2 X and

kf kX 5 lim inf kfn kX : n!1

(v) The space X contains the set S of all simple functions. (vi) To each set E of finite measure there corresponds a positive constant CE depending only on E such that Z jf j d 5 CE kf kX E

for all f 2 X: (vii) If fn ! f in X; then fn ! f in measure on every set of finite measure; in particular, some subsequence of ¹fn º converges to f pointwise -a.e. We conclude this section with a simple, but useful, observation that a set-theoretic inclusion between two Banach function spaces in fact already implies the continuous embedding. Theorem 6.1.17. Let X and Y be Banach function spaces over the same measure space and assume that X  Y . Then X ,! Y . Proof. Suppose X ,! Y fails. Then there exists a sequence ¹fn º1 nD1 of functions fn  0 in X such that kfn kY > n3 ; n 2 N: kfn kX 5 1; P fn converges in X to some function f 2 X. By the hypothesis By Theorem 6.1.14, n2 X  Y we get also f 2 Y . However, since 0 5 n2 fn 5 f and so kf kY = n2 kfn kY > n for all n 2 N, we get a contradiction with f 2 Y . Exercise 6.1.18. Let   RN be a suitable domain. (i) Let 1  p  1. Then the Lebesgue space Lp ./ is a Banach function space. (ii) Let 1  p  1 and let % be a weight on . Then the weighted Lebesgue space Lp .; %/ (cf. Remark 3.19.8) is a Banach function space. (iii) Let ˆ be a Young function. Then the Orlicz space Lˆ ./ is a Banach function space. p;

(iv) Let   0 and p 2 .1; 1/. Then the Morrey space LM ./ is a Banach function space.

209

Section 6.2 Associate space

(v) Let   0 and p 2 .1; 1/. Then the Campanato space Lp; C ./ is not a Banach function space. We shall see more examples of Banach function spaces in subsequent chapters.

6.2 Associate space In this section we shall study the important concept of the associate space to a given Banach function space, modeled upon the example of the duality in Lebesgue spaces 0 between Lp and Lp , where 1  p  1. Definition 6.2.1. Let % be a Banach function norm. Then the functional %0 , defined on M C by ²Z ³ fg d W f 2 M C ; %.f / 5 1 ; g 2 MC; (6.2.1) %0 .g/ D sup R

is called the associate norm of %. Theorem 6.2.2. Let % be a Banach function norm. Then %0 is also a Banach function norm. Proof. Suppose that %.f / 5 1. Then (6.1.2) implies that f .x/ < 1 -a.e. If moreover g D 0 -a.e., then Z fg d D 0; R

%0 .g/

hence, by (6.2.1), R D 0. If %0 .g/ D 0, then R fg d D 0 for all f 2 M C with %.f / 5 1. If E  R is a -measurable set with 0 < .E/ < 1, then 0 < %. E / < 1 by the properties (P1) and (P4) of %. The function f WD E =%. E / satisfies %.f / D 1, therefore Z Z g d D fg d D 0: %. E /1 E

R

Thus, g D 0 -a.e. on E. Because E was chosen arbitrarily, we get that g D 0 a.e. The positive homogeneity and the triangle inequality for %0 can be easily verified. This shows (P1). Next, again, (P2) trivially follows from the definition of %0 . We shall show (P3). Let ¹gn º1 nD1  M and assume that 0 5 gn " g -a.e. for some g 2 M. We already know that %0 has the lattice property (P2). Thus, for every m; n 2 N, m  n, %0 .gm /  %0 .gn /  %0 .g/. We can with no loss of generality " < %0 .gn /. assume that %.gn / < 1 for every n 2 N. Let " be any number satisfying R By (6.2.1), there is a function f in M C with %.f / 5 1 and such that fg d > ": Now 0 5 fgRn " fg -a.e. R so the monotone convergence theoremR (Theorem 1.21.4) implies that fgn " fg: Hence there is n0 2 N such that fgn > " for all

210

Chapter 6 Banach function spaces

n = n0 . Thus, by (6.2.1), we obtain %0 .gn / > " for all n = n0 . Consequently, we get %0 .gn / " %0 .g/, in other words, %0 enjoys the property (P3). In order to verify (P4) for %0 , we use (P5) for %, and vice versa. Let E  R satisfy .E/ < 1, then, by (P5) for %, there is constant CE < 1 for which Z E f d 5 CE %.f /; .f 2 M C /: R

Together with (6.2.1) this gives %0 . E / 5 CE < 1, proving (P4) for %R0 : Finally, let E  R be such that .E/ < 1. If .E/ D 0, then E f d D 0, hence (P5) holds automatically. When .E/ > 0, we have by (P4) for % that %. E / < E . Then 1 and by (P1) for % that %. E / > 0. We set CE0 WD %. E / and f WD %. E/ C %.f / D 1, whence, for any g 2 M , we obtain from (6.2.1) Z Z 0 g d D CE fg d 5 CE0 %0 .g/; R

E

proving (P5) for %0 . The proof is complete. Exercise 6.2.3. Prove that the functional  , defined on Lebesgue measurable functions g over the interval .0; 1/ by Z

1

 .g/ WD sup

g.t /˛t ˛1 dt;

0 n3 ; n 2 N: kfn kX 5 1; R

It follows from Theorem 6.1.14 that the function f D However, Z Z 2 jfgj d > n jfn gj d > n; R

R

P1

nD1 n

2

fn belongs to X:

n 2 N;

which is a contradiction. Notation 6.2.8. Let X be a Banach function space and let X 0 be its associate space. Then the associate space .X 0 /0 of X is called the second associate space of X and is denoted by X 00 .

212

Chapter 6 Banach function spaces

Theorem 6.2.9 (Lorentz–Luxemburg theorem). Let X be a Banach function space. Then X coincides with its second associate space X 00 in the sense that f 2X

f 2 X 00

,

and kf kX D kf kX 00 ;

f 2 M:

(6.2.4)

Proof. If f 2 X, then by the Hölder inequality (6.2.3) fg 2 L1 for every g 2 X 0 . It follows therefore from Theorem 6.2.7 (applied to X 0 instead of X) that f 2 X 00 . We thus get X  X 00 : We also obtain from (6.2.2) and (6.2.3) that ²Z ³ jfgj d W kgkX 0 5 1 5 kf kX : kf kX 00 D sup R

Hence, in order to complete the proof we need only show X 00  X and kf kX 5 kf kX 00 ;

f 2 X 00 :

(6.2.5)

Conversely, let f 2 X 00 and n 2 N. Set fn .x/ WD min.jf .x/j ; n/ Rn .x/;

n 2 N;

(6.2.6)

where ¹Rn º1 nD1 is the sequence of sets from Convention 6.1.1. Now, 0 5 fn 5 n Rn ; hence, by Remark 6.1.16 (i) and (v), fn 2 X and likewise fn 2 X 00 . We further know that both X and X 00 have the Fatou property and that 0 5 fn " jf j : Hence, in order to verify (6.2.5), we just need show that kfn kX 5 kfn kX 00 ;

n 2 N:

(6.2.7)

From now on we shall suppose that f and n are fixed. With no loss of generality we may assume that kfn kX > 0. We note that the space Mn WD ¹f 2 M; supp f  Rn º ; endowed with the norm

Z kgkMn WD

Rn

jg.x/j d.x/;

213

Section 6.2 Associate space

is a Banach space. Let SX denote the closed unit ball in X. Then the set U WD SX \ Mn is a convex subset of Mn . If ¹hk º1  U and hk ! h in Mn , then there exists kD1 a subsequence ¹hkj ºj1D1 such that hkj ! h -a.e. on R. Since hkj 2 SX for every j 2 N, it follows from the Fatou lemma (Remark 6.1.16 (iv)) that h 2 SX . Hence also h 2 U , which means that U is a closed subset of Mn . Let now  > 1. Then g WD 

fn 2 .Mn n U / : kfn kX

Therefore, by the Hahn–Banach theorem, there is a closed hyperplane that separates g and U . In other words, there exists a nonzero function ' 2 L1 on R with supp '  Rn and some  2 R such that Z Z h d <  < g d (6.2.8) Rn

Rn

for every h in U . Writing ' D j'j in polar form, where j j D 1, and since jhj 2 U if and only if h 2 U , we get Z Z Z g d  sup (6.2.9) jhj d   < jgj d: h2U

Rn

Rn

Rn

For every h 2 S we have h.x/ D lim hk .x/ WD min.h.x/; k/ Rn .x/; k!1

x 2 Rn :

Obviously, hk 2 Mn , so hk 2 U for every k 2 N. A function h 2 S can thus be approximated by the sequence ¹hk º  U . Since U  S , we get from the monotone convergence theorem (Theorem 1.21.4) that Z Z j'hj d D sup j'hj d: sup h2U

Rn

h2S

Rn

Thus, by the definition of X 0 and (6.2.8), one obtains (recall that supp '  Rn ) Z Z  k'kX 0 D sup j'hj d 5  < j'fn j d: kfn kX Rn h2S Rn This together with the Hölder inequality (6.2.3), applied to X 0 and X 00 , implies ˇ Z ˇ ˇ ˇ ˇfn  ˇ d 5  kfn k 00 : kfn kX <  X ˇ kk 0 ˇ R X On letting  ! 1, we obtain (6.2.7). The proof is complete.

214

Chapter 6 Banach function spaces

Given a Banach function space X, it is of interest to compare its associate space X 0 with its dual space X  . It is clear from the definitions that one always has X 0  X  . On the other hand, it is equally clear from Theorem 6.2.9 that the converse inclusion is not always true (take X D L1 for instance). The following lemma is the first step in our study of relations between X 0 and X  . Lemma 6.2.10. Let X be a Banach function space and X 0 its associate space. Then ˇ ²ˇZ ³ ˇ ˇ (6.2.10) kgkX 0 D sup ˇˇ fg dˇˇ I f 2 X; kf kX 5 1 ; g 2 M: R

Proof. By Theorem 6.2.6, we have for every f 2 X, kf kX  1, and every g 2 X 0 , ˇZ ˇ Z ˇ ˇ ˇ fg dˇ 5 jfgj d  kf kX kgkX 0  kgkX 0 ; ˇ ˇ R

R

which yields immediately ˇ ³ ²ˇZ ˇ ˇ sup ˇˇ fg dˇˇ I f 2 X; kf kX 5 1  kgkX 0 : R

In order to prove the reverse inequality, denote by SX the unit ball in X. Given g 2 M, denote E WD ¹x 2 R; g.x/ ¤ 0º and write g in the polar form: g.x/ D jg.x/j .x/;

x 2 E;

where j j D 1. Let further f 2 SX . Denote x 2 R:

h.x/ WD jf .x/j .x/ E .x/; Then jh.x/j  jf .x/j

for x 2 R;

hence, by the lattice property of X, also h 2 SX . Altogether, Z Z Z jfgj d D jfgj d D jf j g d: R

Thus,

E

Z

Z R

jfgj d D

R

E

ˇZ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ hg d 5 ˇ hg dˇ 5 sup ˇ fg dˇˇ : R

f 2SX

R

Passing on the left-hand side to the supremum over all f in SX and using (6.2.2), we obtain the inequality ˇ ²ˇZ ³ ˇ ˇ ˇ ˇ kgkX 0  sup ˇ fg dˇ I f 2 X; kf kX 5 1 ; g 2 M; R

finishing the proof.

215

Section 6.2 Associate space

Remark 6.2.11. Let X be a Banach function space and X 0 its associate space. It might be useful to note that there exists an alternative way of expressing kgkX 0 for g 2 M, namely, ˇ ˇR ˇ fg dˇ R : (6.2.11) kgkX 0 D sup kf kX f 60 We shall now recall the notion of a norm-fundamental subspace of the dual space. Definition 6.2.12. Let B  X  be a closed linear subspace. We say that B is normfundamental in X, if kf kX D sup ¹jƒ.f /jI ƒ 2 B; kƒkX   1º ;

f 2 X:

In other words, a space is norm-fundamental if it is rich enough so that it can reproduce the norm of each f 2 X. Theorem 6.2.13. Let X be a Banach function space and let X 0 be its associate space. Then X 0 is isometrically isomorphic to a norm-fundamental subspace of X  . Proof. Given g 2 X 0 , we define the functional ƒg on X by Z ƒg .f / WD fg d; f 2 X: R

Then ƒg is obviously linear and by the Hölder inequality it is bounded on X. Next, ƒg .f / D 0 if and only if f 0 on R. Therefore, the mapping g 7! ƒg

(6.2.12)

is an isomorphism between X 0 and some subspace Y of X  . Moreover, by the definition of the norm in X  , by the definition of ƒg and by Lemma 6.2.10, we subsequently get ® ¯ kƒg kX  D sup jƒg .f /jI kf kX  1 ˇ ³ ²ˇZ ˇ ˇ ˇ ˇ D sup ˇ fg dˇ I kf kX  1 R

D kgkX 0 ; hence the mapping in (6.2.12) is an isometry. Since X 0 is a Banach space, this mapping has closed range in X  . It remains to verify the norm-fundamental property of X 0 . By Theorem 6.2.9 and Lemma 6.2.10, we have ˇ ²ˇZ ³ ˇ ˇ ˇ ®ˇ ¯ kf kX D kf kX 00 D sup ˇˇ fg dˇˇ I kgkX 0  1 D sup ˇƒg .f /ˇ I kƒg kX   1 ; R

in other words,

X0

is norm-fundamental. The proof is complete.

216

Chapter 6 Banach function spaces

Proposition 6.2.14. Let X and Y be Banach function spaces. Then X ,! Y

,

Y 0 ,! X 0 ;

and the embedding constants coincide. Proof. By (6.2.11), we have k Id kX,!Y

ˇR ˇR ˇ ˇ ˇ fg dˇ ˇ fg dˇ kf kY R R D sup D sup sup D sup sup f 60 kf kX f 60 g60 kf kX kgkY 0 g60 f 60 kf kX kgkY 0 kgkX 0 D k Id kY 0 ,!X 0 ; kf kY 0 g60

D sup

and the assertion follows.

6.3 Absolute continuity of the norm Absolute continuity of the norm, which will be now studied in detail, is one of the basic concepts in the theory of Banach function spaces and it is indispensable for example for the investigation of basic properties such as reflexivity or separability of the spaces. Definition 6.3.1. Let ¹En º1 nD1 be a sequence of -measurable subsets of R. We say that En tends to the empty set and write En ! ; if En ! 0 -a.e. on R. We write En # ; when En ! ; and moreover the sequence ¹En º1 nD1 is monotonically decreasing, that is, En  EnC1 . Remarks 6.3.2. (i) Let ¹En º1 nD1 be a sequence of -measurable subsets of R. Then En ! ; -a.e. if and only if the set lim sup En WD n!1

1 1 [ \

En

mD1 nDm

has measure zero. (ii) It follows from (i) that En ! ; -a.e. if and only if En can be modified on a set of measure zero in order that the resulting sequence converges to ; everywhere. (iii) It is important to note that the sets En can have infinite measure. Definition 6.3.3. Let X be a Banach function space and let f 2 X. We say that f has an absolutely continuous norm in X if kf En kX ! 0

217

Section 6.3 Absolute continuity of the norm

for every sequence ¹En º1 nD1 satisfying En ! ; -a.e. We shall denote Xa WD ¹f 2 XI f has an absolute continuous norm in Xº : If Xa D X, we say that the space X has an absolutely continuous norm. We first note that we can restrict ourselves in Definition 6.3.3 to nonincreasing sequences. Proposition 6.3.4. Let X be a Banach function space and let f 2 X. Then f has an absolutely continuous norm in X if and only if kf En kX # 0

(6.3.1)

for every sequence ¹En º1 nD1 satisfying En # ; -a.e. Proof. The “only if” part is clear. As for the “if” part, assume that f 2 X satisfies (6.3.1) and let ¹Fn º1 nD1  R be a sequence for which Fn ! ; -a.e. For n 2 N, denote 1 [ Fm : En WD mDn

Then the sequence ¹En º is decreasing and satisfies lim sup Fn D lim sup En : n!1

n!1

However, we know that lim supn!1 Fn is a set of measure zero, hence so is lim supn!1 En . Moreover, En  EnC1 . By (6.3.1), we get kf En kX # 0: Since Fn  En for all n we also have kf Fn kX ! 0. In other words, f has an absolutely continuous norm in X, as desired. Examples 6.3.5. (i) Let X D Lp , p 2 Œ1; 1/. Then X D Xa , that is, every p function in L has an absolutely continuous norm. (ii) Let X D L1 and let .R; / be a nonatomic measure space. Then .L1 .R; //a D ¹0º; that is, in this space, only the function which is identically equal to zero, has an absolutely continuous norm. (iii) Let .R; / D .N; m/ where m is the arithmetic measure on N, and let X D `1 . Then Xa D c0 .

218

Chapter 6 Banach function spaces

(iv) Let ˆ be a Young function and let Lˆ be the corresponding Orlicz space. Then ˆ Lˆ a DE :

Our next aim is to search for a reasonable characterizations of the space of functions with absolute continuous norms. Lemma 6.3.6. Let X be a Banach function space and let f 2 X. Then f 2 Xa if and only if for every " > 0 there exists a ı > 0 such that whenever E  R satisfies .E/ < ı, then, necessarily, kf E kX < ". Proof. Assume that there exists an " > 0 and a sequence ¹En º1 nD1 of measurable sets in R such that .En / < 21n and kf En kX  ": Then, for every m 2 N, 

 [ 1 nDm

 X 1 En  .En / < 21m ; nDm

whence En & ; -a.e. on R. Consequently, kf En kX  ", which is a contradiction, since f 2 Xa . Now we are in a position to characterize the set Xa in terms of assertions analogous to the monotone convergence theorem and the dominated convergence theorem which we know from the theory of Lebesgue spaces (cf. Theorems 1.21.4 and 1.21.5). Theorem 6.3.7. Let X be a Banach function space and let f 2 X. Then f 2 Xa if and only if, for every sequence ¹fn º1 nD1 satisfying fn  jfn j and fn # 0 -a.e., we have kfn k # 0. Proof. ( This implication is obvious (it suffices to take fn D f En ). ) Assume that f 2 Xa and let a sequence ¹fn º1 nD1 satisfy jf j  fn and fn & 1 º is the fixed sequence of sets in R 0 -a.e. on R. Let " > 0. Recall that ¹R k kD1 S satisfying 0 < .R/ < 1 and 1 R D R. Set Q k WD R n Rk , k 2 N. Since kD1 k f 2 Xa , there exists a k0 2 N such that for every k 2 N, k  k0 , one has " kf Qk kX < : 2 Now fix k  k0 and let

" ˛ WD k Rk kX 4

and En WD ¹x 2 Rk ; fn .x/  ˛º:

Section 6.3 Absolute continuity of the norm

219

Then lim .Ek / D 0;

k!1

since fn & 0 -a.e. on R and Rk is of finite measure. Therefore, by Lemma 6.3.6, there exists an n0 2 N such that for every n 2 N, n  n0 , we have " kf En kX < : 4 Finally, for such n, one gets kfn kX  kfn Qk kX C kfn Rk kX  kfn Qk kX C kfn En kX C kfn Rk nEn kX  kfn Qk kX C kf En kX C ˛k Rk nEn kX " " " < C C 2 4 4 D ": The proof is complete. Remark 6.3.8. In the theory of Lebesgue spaces, we have monotone convergence theorem and the dominated convergence theorem. The former result holds also in the theory of Banach function spaces because the Fatou axiom is postulated. As for the latter theorem, we have the following theorem which characterizes the space Xa as the biggest space contained in X on which the dominated convergence theorem holds. Theorem 6.3.9. Let X be a Banach function space and let f 2 X. Then f 2 Xa if and only if, for every sequence ¹fn º1 nD1 and every g 2 X, satisfying fn ! g -a.e., we have lim kfn  gk D 0: n!1

Proof. ( Again, this implication is clear (just take fn D f En and g 0). ) Suppose that f 2 Xa and let a sequence ¹fn º1 nD1 satisfy jf j  jfn j and fn ! g a.e. on R. Set hn .x/ WD sup jfm .x/  g.x/j; n 2 N: m n

Then 2jf j  hn & 0 -a.e. By Theorem 6.3.7, we get khn kX & 0. Thus, since kfn  gkX  khn kX , also lim kfn  gkX D 0;

n!1

as desired. The proof is complete.

220

Chapter 6 Banach function spaces

Definition 6.3.10. Let X be a Banach function space and let Y  X be a closed linear subspace of X. We say that Y is an ideal in X if for every f 2 Y and for every g 2 M, jgj  jf j -a.e. we have g 2 Y . Example 6.3.11. Let X be a Banach function space. Then both Y D X and Y D ¹0º are ideals in X. Theorem 6.3.12. Let X be a Banach function space. Then the space Xa is an ideal in X. Proof. We clearly have Xa  X. Moreover, it is also obvious that if f 2 Xa and g 2 M, jgj  jf j -a.e., then, necessarily, g 2 Xa . The only part of the assertion which is not entirely clear is the closedness of Xa in X. Suppose that ¹fn º1 nD1 is a sequence of functions in Xa and let lim kfn  f kX D 0:

n!1

Let " > 0. Then there exists n0 2 N such that for every n 2 N, n  n0 , one has " kf  fn kX < : 2 be a sequence of -measurable subsets of R such that Ek & ; -a.e. as Let ¹Ek º1 kD1 k ! 1. Let n  n0 . Since fn 2 Xa , hence there is a k0 2 N such that for every k 2 N, k  k0 , " kfn Ek kX < : 2 Thus, altogether, we have for every k  k0 kf Ek kX  k.f  fn / Ek kX C kfn Ek kX  kf  fn kX C kfn Ek kX " "  C D ": 2 2 It follows that limk!1 kf Ek kX D 0, and therefore f 2 Xa . In other words, Xa is a closed subspace of X. Theorems 6.3.7 and 6.3.9 obviously yield the following result. Corollary 6.3.13. Let X be a Banach function space and let f 2 Xa . Assume that ¹fn º1 nD1  X is a sequence of functions in X satisfying 0  fn " f

  a.e.

Then kf  fn kX ! 0:

Section 6.3 Absolute continuity of the norm

221

Definition 6.3.14. Let X be a Banach function space and let S denote the space of simple functions. We will denote by Xb the closure in X of S . Theorem 6.3.15. Let X be a Banach function space. Then Xb D ¹f 2 XI f bounded; .supp f / < 1º; where the closure is understood in the norm topology of the space X. Proof. Only the inclusion “” needs proof. Let f be a bounded function on R, and let .supp f / < 1. Assume first that f  0. Let ¹fn º1 nD1 be a sequence of simple functions on R such that supp fn D supp f and fn converge uniformly to f . Then kfn  f kX D k.fn  f / supp f kX  kfn  f kL1 k supp f kX : It follows from .supp f / < 1 and the property (P4) of X that k supp f kX < 1. Thus, lim kfn  f kX D 0; n!1

as desired. The proof is complete. Theorem 6.3.16. Let X be a Banach function space. Then Xb is an ideal in X and Xa  Xb  X: Proof. The closedness of Xb in X follows immediately from the definition. Assume that f 2 Xb and let g 2 M0 .R/, jgj  jf j -a.e. on R. Suppose further that ¹fn º1 nD1 is a sequence of simple functions on R such that lim kfn  f kX D 0:

n!1

We then define, for n 2 N and x 2 R, gn .x/ WD sign.g.x// min¹jfn .x/j; jg.x/jº: Then, for every n 2 N, gn is bounded on R and .supp fn / < 1. Moreover, jg  gn j D max¹jgj  jfn j; 0º  jjf j  jfn jj  jf  fn j ; whence kg  gn kX  kf  fn kX ! 0: By Theorem 6.3.15, we get g 2 Xb . In other words, Xb is an ideal in X. be the sequence of -measurable subsets Finally, let f 2 Xa and let ¹Rk º1 kD1 of R of finite positive measure. Then, by Theorem 6.3.15, the functions fk , defined for k 2 N and x 2 R by fk .x/ WD sign.f .x// min¹jf .x/j; k Rk .x/º;

222

Chapter 6 Banach function spaces

satisfy fk 2 Xb for every k 2 N, and lim fk .x/ D f .x/

k!1

-a.e. on R:

Theorem 6.3.9 then gives lim kfk  f kX D 0:

k!1

But, Xb is a closed subspace of X, hence also f 2 Xb . This establishes the inclusion Xa  Xb ; finishing the proof. As we know from Example 6.3.5 (ii), the space Xa can be very small. That cannot happen with Xb , which is always a rather rich structure, as the following result shows. Theorem 6.3.17. Let X be a Banach function space. Then Xb is isometrically isomorphic to a norm-fundamental subspace of .X 0 / . Proof. Applying the assertion of Theorem 6.2.13 to X 0 instead of X, and using the fact that X 00 D X (cf. Theorem 6.2.9), we get that X 00 is isometrically isomorphic to a closed subspace of .X 0 / . Since Xb is also a closed linear subspace of X 00 D X, the same is true for Xb , that is, Xb is isometrically isomorphic to a closed subspace of .X 0 / . We have to show that it is norm-fundamental. Let g 2 X 0 and let " > 0. Then, by the definition of the associate space, there exists a function f 2 X, kf kX  1, such that Z jfgj d C ": kgkX 0  R

For k 2 N, define

fk WD min¹jf j; kº Rk :

(Here, ¹Rk º1 is as usual.) Let B be the unit ball in Xb , that is, kD1 B WD ¹h 2 Xb ; khkX  1º: By Theorem 6.3.15, we have fk 2 B for every k 2 N. Since 0  fk % jf j, the monotone convergence theorem yields Z Z jfk gj d C "  sup jhgj d C ": kgkX 0  sup k2N

R

h2B

We now send " ! 0C to get

R

Z kgkX 0  sup h2B

R

jhgj d;

223

Section 6.4 Reflexivity of Banach function spaces

which, in fact, together with the Hölder inequality yields Z jhgj d: kgkX 0 D sup R

h2B

Using Lemma 6.2.10, we get kgkX 0

ˇZ ˇ ˇ ˇ ˇ D sup ˇ hg dˇˇ : R

h2B

Here we have to note that the only property used in the proof of Lemma 6.2.10 is the lattice property, which however is possessed by Xb since, by Theorem 6.3.16, it is an ideal. The proof is complete. We finish this section by characterizing when Xa D Xb . The key role is played by characteristic functions of sets of finite measure and the question whether they have absolutely continuous norms. Theorem 6.3.18. Let X be a Banach function space. Then Xa D Xb if and only if E 2 Xa for every set E  R of finite measure. Proof. ( Assume that E  R and .E/ < 1. Then, by Theorem 6.3.16, E 2 Xb  Xa : ) Assume that E 2 Xa for every set E  R of finite measure. This means that S  Xa . But Xa is a closed subspace of X by Theorem 6.3.12. Therefore, Xb  Xa . Together with Theorem 6.3.16, this shows that Xb D Xa . The proof is complete.

6.4 Reflexivity of Banach function spaces Let X be a Banach function space and let Y  X be a closed linear subspace of X such that S  Y , where S is the set of simple functions. Let g 2 X 0 . Then the mapping g 7! Lg 2 X   Y  ; Z

where Lg .f / WD

R

fg d

is injective as a map from X 0 to Y  . It is also an isometry, since, by Theorem 6.3.17, Y  Xb . Consequently, through this mapping, X 0 is isometrically isomorphic to a closed linear subspace of Y  . The following theorem shows what happens when Y is moreover an ideal in X.

224

Chapter 6 Banach function spaces

Theorem 6.4.1. Let X be a Banach function space and let Y be an ideal in X such that S  Y . Then Y  D X 0 if and only if Y  Xa . Proof. ( Step 1. Assume that Y  Xa . We know that Y contains S , hence also its closure. In other words, Xb  Xa . Together with Theorem 6.3.16, this implies Y D Xa D Xb :

(6.4.1)

We already know that X 0  Y  , so we only have to prove that Y   X 0 . R Assume that ƒ 2 Y  . We shall find a function g 2 X 0 such that ƒ.f / D R fg d for every f 2 Y . S1 Let ¹SN º1 N D1 SN D N D1 be a disjoint system of measurable subsets of R such that R and .SN / < 1 for every N 2 N. For every fixed N 2 N, we denote by AN the -algebra of all -measurable subsets of SN . We then define the function N W AN ! Œ0; 1/ by N .A/ WD ƒ. A /;

A 2 AN :

The function N is well-defined on every A 2 AN since A 2 Xb  Y . Step 2. We claim that  is countably additive on AN . Let ¹Ai º1 iD1 be a sequence of disjoint sets in AN . We then define Bn WD

n [

n 2 N;

Ai ;

iD1

and A WD

1 [

An D

nD1

1 [

Bn :

(6.4.2)

nD1

Because A 2 AN , one has A 2 Xb , that is, by (6.4.1), A 2 Xa . Next, the obvious inclusion A n Bn  A implies .AnBn /  A , and (6.4.2) implies that .A n Bn / & ;. Hence, .AnBn / & 0. By Theorem 6.3.7, this implies that lim k A  Bn kX D 0:

n!1

Now, the continuity of ƒ on Y shows that N .A/ D ƒ. A / D lim ƒ. Bn / D lim n!1

proving the claim.

n!1

n X iD1

ƒ. Ai / D

1 X iD1

ƒ. Ai /;

225

Section 6.4 Reflexivity of Banach function spaces

Step 3. Now we claim that N is bounded on AN . Indeed, for any A 2 AN , we have jN .A/j D jƒ. A /j  kƒkY  k A kY D kƒkY  k A kX  kƒkY  k SN kY < 1: Step 4. Next, we shall prove that N is absolutely continuous with respect to . To this end, let A 2 AN be such that .A/ D 0. Then jN .A/j  kƒkY  k A kX ; whence also jN .A/j D 0, establishing the claim. Step 5. We have verified all the assumptions of the Radon–Nikodým theorem (Theorem 1.21.15). Thus, for each N 2 N, there exists a uniquely determined function gN on SN such that Z A gN d; A 2 AN : ƒ. A / D N . A / D R

We set

1 X

gD

gN :

N D1

Since SN are mutually disjoint, this defines a function g on R. Then Z 1 [ A g d; A  AN : ƒ. A / D R

(6.4.3)

N D1

Step S 6. We have to show that g 2 X 0 . Assume that h 2 S , h  0 and supp h  Gn WD jnD1 Sj . We then have Z Z jhgj d D h sign.g/g d D ƒ.h sign.g// R

R

 kƒkY  kh sign.g/kY D kƒkY  khkX : Now let f 2 X be arbitrary. Then we can find a sequence ¹hn º1 nD1  S such that supp hn  Gn , satisfying 0  hn % jf j. By the monotone convergence theorem and the Fatou lemma, we obtain, for every f 2 X, Z jfgj d  kƒkY  kf kX : R

Step 7. We will prove (6.4.3) for a general case (so far we have it established only for a special case of characteristic functions). To this end, let g 2 X 0 . Then the functional ƒg , defined by Z ƒg .f / WD fg d; f 2 Y; R

226

Chapter 6 Banach function spaces

is in Y  . Let f 2 Y . Again, then there exists a sequence of simple functions ¹hn º1 nD1 such that supp hn  Gn and 0  hn % jf j -a.e. on R. Define fn WD hn sign.f /

n 2 N:

Then fn 2 S for every n 2 N and supp fn  Gn . Moreover, fn ! f -a.e. on R. However, we know that Y D Xa . This implies f 2 Xa , and, via Theorem 6.3.9, also lim kfn  f kY D 0:

n!1

Next, we have ƒ.fn / D ƒg .fn /, and since both ƒ and ƒg are continuous, we finally get ƒ.f / D ƒg .f /, as desired. It thus follows that Y   X 0 , whence, finally, X 0 D Y . ) Assume now that Y  D X 0 . We shall show that then Y  Xa by contradiction. Suppose that there exists some f 2 Y n Xa . We can with no loss of generality assume that f  0, because Y is an ideal. The fact that f does not have an absolutely continuous norm in X means that there is a sequence ¹En º1 nD1 satisfying En & ; -a.e., and an " > 0 such that kf En kX  ": We define, for n 2 N, the sets Gn WD ¹g 2 X 0 ;

Z

(6.4.4)

" jf En gj d < º: 2 R

Then it follows from the dominated convergence theorem and the assumption of the theorem that 1 [ Gn D X 0 D Y  : nD1

Again, since Y is an ideal, we have f En 2 Y for every n 2 N. Therefore, for every n 2 N, the set Gn is weakly* open in Y  . By the Banach–Alaoglu theorem (Theorem 1.17.6), there is some k 2 N and n1 ; : : : ; nk such that BY  

k [

Gnj ;

j D1

where BY  is the unit ball in Y  . Hence, for every g 2 X 0 such that kgkX 0  1 there exists a j 2 N such that g 2 Gnj . Now, because f  0 and En & ; -a.e., we have, for every n 2 N, n  max¹n1 ; : : : ; nk º. Hence, by the Lorentz–Luxemburg theorem (Theorem 6.2.9) and by Lemma 6.2.10, we finally obtain, for every n 2 N, Z Z " jf En gj d  jf nj gj d < ; 2 R R which contradicts (6.4.4). The proof is complete.

Section 6.4 Reflexivity of Banach function spaces

227

Corollary 6.4.2. Let X be a Banach function space and let Y be an ideal in X such that S  Y . Then Y D Xa D Xb : Proof. Since S  Y and Y is closed, we have Xb  Y  Xa  Xb ; and the assertion follows. Corollary 6.4.3. Let X be a Banach function space such that S  Xa . Then .Xa / D X 0 : Proof. Set Y WD Xa . Then Y is an ideal in X by Corollary 6.3.13. Since S  Y , the assertion follows from Theorem 6.4.1. Example 6.4.4. Let ˆ be a Young function. and let Lˆ be the corresponding Orlicz space. Then Xa D E ˆ . Thus, .E ˆ / D L‰ ; where ‰ is the complementary function of ˆ. Corollary 6.4.5. Let X be a Banach function space. Then X  D X 0 if and only if X D Xa . Proof. This follows by applying Theorem 6.4.1 to Y D X. Corollary 6.4.6. Let X be a Banach function space. Then X is reflexive if and only if X D Xa and X 0 D .Xa /0 . Proof. ( Assume first that both X and X 0 have absolutely continuous norms. Then, by Theorem 6.4.1, we have X D X0

and also .X 0 / D .X 0 /0 D X 00 :

Therefore, combining this with the Lorentz–Luxemburg theorem (Theorem 6.2.9), we obtain X  D .X  / D .X 0 / D X 00 D X; whence X is reflexive. ) Assume now that X is reflexive. We know that X 0  X  . Suppose that this inclusion is proper, that is, X 0 ¤ X  . By the Hahn–Banach theorem, then there exists a nonzero functional F 2 X  such that F .g/ D 0 for every g 2 X 0 . Since the space

228

Chapter 6 Banach function spaces

X  is reflexive, there exists a function f 2 X such that F .ƒ/ D ƒ.f / for every ƒ 2 X  . In particular, for every g 2 X 0 , one has Z f .x/g.x/ d: F .g/ D R

This, however, means that, for every g 2 X 0 , Z f .x/g.x/ d D 0: R

By Theorem 6.2.13, X 0 is a closed norm-fundamental subspace of X  . Thus, necessarily, f 0 -a.e. on R. This contradicts the fact that F is a nonzero functional, hence X  D X 0 . By Corollary 6.4.5, X D Xa . Using the reflexivity of X and Lorentz–Luxemburg theorem (Theorem 6.2.9) once again, we get X  D X D X 00 D .X 0 /0 : Since X has absolutely continuous norm, we have X 0 D X ; hence also

.X 0 / D .X  / :

Setting all this information together, we get .X 0 / D .X  / D X  D .X 0 /0 : By Corollary 6.4.5, this implies that also X 0 D .X 0 /a , as desired.

6.5 Separability in Banach function spaces Our main aim in this section is to characterize when a given Banach function space is separable. The main tool will be the weak topology on X generated by certain subspace of X 0 . Remark 6.5.1. Let X be a Banach function space and let Z  X 0 be an ideal in X 0 satisfying S  Z. Then Z contains .X 0 /b . By Theorem 6.3.18, Xb is a normfundamental subspace of .X 0 / and applying the same theorem to X 0 , we get that .Xb /0 is a norm-fundamental subspace of .X 00 / D X  by the Lorentz–Luxemburg theorem (Theorem 6.2.9). Therefore, also Z is a norm-fundamental subspace of X  .

229

Section 6.5 Separability in Banach function spaces

Definition 6.5.2. Let X be a Banach function space and let Z  X 0 be an ideal in X 0 satisfying S  Z. Then the system of seminorms defined by ˇZ ˇ ˇ ˇ ˇ f 7! ˇ fg dˇˇ ; g 2 Z; is separating and it defines on X a structure of Hausdorff locally convex topological linear space. This topology is called the weak topology on X generated by Z and is denoted by .X; Z/. Remark 6.5.3. Let X be a Banach function space and let Z  X 0 be an ideal in X 0 satisfying S  Z. Then, a sequence ¹fn º1 nD1  X satisfies fn ! f

in .X; Z/

for some f 2 X if and only if Z Z fn g d ! fg d

for every g 2 Z:

Definition 6.5.4. We say that a set A  X is .X; Z/ bounded if, for every g 2 Z, sup ¹jfg djº < 1:

(6.5.1)

f 2A

Lemma 6.5.5. Let X be a Banach function space and let Z  X 0 be an ideal in X 0 satisfying S  Z. Let A  X. Then A is .X; Z/ bounded if and only if it is bounded in X. Proof. ( Let A  X be norm bounded in X. By the Hölder inequality (Theorem 6.2.6), we obtain for every g 2 Z (since Z  X 0 ) ˇ³ ²ˇZ ˇ ˇ ˇ sup ˇ fg dˇˇ  sup ¹kf kX kgkX 0 º  C kgkX 0 f 2A

R

f 2A

for an appropriate positive constant C . Thus, A is .X; Z/ bounded. ) Conversely, assume that A  X is .X; Z/ bounded and g 2 Z. Then ˇ³ ²ˇZ ˇ ˇ ˇ sup ˇ fg dˇˇ < 1: f 2A

R

Let f 2 A. We define the functional ƒf at a function g 2 Z by Z fg d: ƒf .g/ WD R

230

Chapter 6 Banach function spaces

Then the Hölder inequality (Theorem 6.2.6) guarantees that ƒf 2 Z  and ˇ ²ˇZ ³ ˇ ˇ  ˇ ˇ kƒf kZ  D sup ˇ fg dˇ ; g 2 Z ; kgkX   1 D kf kX ; R

because Z is norm fundamental in X  . For every g 2 Z one thus has, by (6.5.1), ˇ ˇ sup ˇƒf .g/ˇ DW Cg < 1; f 2A

where Cg is a positive constant depending on g. Hence, by the principle of uniform boundedness (Theorem 1.18.1), we finally get sup kƒf kZ  D sup kf kX < 1:

f 2A

f 2A

The proof is complete. Theorem 6.5.6. Let X be a Banach function space and let Z  X 0 be an ideal in X 0 satisfying S  Z. Then X is .X; Z/ complete. Proof. Step 1. Let ¹fn º1 nD1 be a .X; Z/-Cauchy sequence in X, that is, for every g 2 Z, the sequence ³1 ²Z fn g d R

nD1

is Cauchy in R. In other words, ¹fn º1 nD1 is a .X; Z/ bounded sequence. By Lemma 6.5.5, it is also bounded in X, that is, there exists a positive constant M such that kfn kX  M for every n 2 N: Step 2. We claim that there exists a function f0 2 X such that kf0 kX  M , and, for every g 2 Xb0 , Z Z lim

n!1 R

fn g d D

R

f0 g d:

To this end, we define, for n 2 N, the set functions Z fn d; E  R: n W E 7! E

Then each n , n 2 N, is a measure, and it follows from the axioms of a Banach function space and the Hölder inequality that n is for every n 2 N absolutely continuous with respect to . We now fix k 2 N. Since ¹fn º1 nD1 is a Cauchy sequence and is a Cauchy sequence in R for every -measurable S  Xb , the sequence ¹n .E/º1 nD1 subset E of Rk , since E  Rk means in particular that .E/ < 1 (here the sequence

231

Section 6.5 Separability in Banach function spaces

¹Rk º1 has the usual meaning). Therefore, for every such E, there exists the finite kD1 limit lim n .E/ DW .E/: n!1

It follows from the Hahn–Saks theorem (Theorem 1.23.4) that the measures ¹n º1 nD1 , restricted to Rk , form a system of uniformly absolutely continuous measures (with respect to ) and  is a measure on Rk , which is absolutely continuous with respect to . It follows on letting k ! 1 that there exists a function f0 2 L1loc .R/ such that, for every E  R, .E/ < 1, Z f0 E d: lim n .E/ D .E/ D n!1

R

Let g 2 S . Then, by linearity of S , Z Z fn g d D f0 g d: lim n!1 R

R

(6.5.2)

We have to prove f0 2 X (so far we know that f0 2 L1loc .R/). If moreover kgkX 0  1, then we have from the Hölder inequality ˇZ ˇ ˇ ˇ ˇ f0 g dˇ  lim sup kfn kX kgkX 0  M: ˇ ˇ R

n!1

Consequently, f0 2 X 00 , and, by Theorem 6.2.9, kf0 kX 00 D kf0 kX  M: Now let g be bounded and such that supp g  E, where .E/ < 1. Then we uniformly approximate g by functions from S , and (6.5.2) follows again. Step 3. We now claim that (6.5.2) is true for every g 2 Z. Thus, for any g 2 Z and n 2 N, define the set function Z fn g d: !n .E/ WD E

Then every !n , n 2 N, is a finite measure, absolutely continuous with respect to . Since for every -measurable E, g E 2 Z, the sequence ¹!n .E/º is convergent by the hypothesis. We use the Hahn–Saks theorem (Theorem 1.23.4) once again and get that ¹!n .E/º is uniformly absolutely continuous with respect to , in other words, !n .E/ ! 0 uniformly in n 2 N as .E/ ! 0. We then define the measure !0 by Z f0 g d: !0 .E/ WD E

Then, again, !0 is absolutely continuous with respect to . We define the sequence by of sets ¹Ek º1 kD1 Ek WD ¹x 2 RI jg.x/j > kº [ .R n Rk /:

232

Chapter 6 Banach function spaces

Let " > 0. Then there exists a k0 such that for every n 2 N, " j!n .Ek0 /j < : 3 Set Fk0 WD R n Ek0 : Then g Fk0 is a bounded function and .supp.g Fk0 // < 1. For such functions (6.5.2) was already established, so, there exists an n0 2 N such that, for every n 2 N, n  n0 , one has ˇ ˇZ Z ˇ " ˇ ˇ ˇ fn g d  f0 g dˇ < : ˇ ˇ 3 ˇ Fk Fk0 0 Therefore,

ˇZ ˇ Z ˇ ˇ " ˇ fn g d  f0 g dˇˇ < C j!n .Ek0 /j C j!0 .Ek0 /j < ": ˇ 3 R R

Thus (6.5.2) holds for every g 2 Z, proving the claim. We have shown that fn ! f0 in the .X; Z/-topology. Hence, X is .X; Z/ complete, as desired. The proof is complete. Corollary 6.5.7. Let X be a Banach function space. Then X is .X; X 0 / complete. Definition 6.5.8. The measure  is called separable on R if the metric space .A; d / is separable, where A WD ¹E  R; .E/ < 1º and d is the metric defined on A by d.E; F / WD

Z R

j E  F j d:

Theorem 6.5.9. Let X be a Banach function space over .R; / and let Y  X be an ideal in X satisfying S  Y . Then Y is separable if and only if Y D Ya and  is a separable measure. Proof. ) Let Y be separable and let ¹fn º1 nD1 be a dense set in Y . Suppose that f0 2 Y n Xa . Then, by Proposition 6.3.4, there exists a sequence ¹En º1 nD1 of measurable subsets of R such that En & ; -a.e. and kf0 En kX  "

for every n 2 N:

Since X 0 is a norm-fundamental subspace of X  (Theorem 6.2.13), there exists a se0 quence ¹gn º1 nD1 of functions gn 2 X satisfying kgn kX 0  1, and such that Z " (6.5.3) jf0 gn En j d  : 2 R

233

Section 6.5 Separability in Banach function spaces

We define now the functions hn WD sign.f0 /jgn j En ;

n 2 N:

Then also khn kX 0  1, whence, by the Hölder inequality, for every k; n 2 N, ˇZ ˇ ˇ ˇ ˇ fk hn dˇ  kfk kX khn kX 0  kfk kX : ˇ ˇ R

So, for each fixed k 2 N, the sequence ˇZ ˇ ˇ ˇ ˇ fk hn dˇ ˇ ˇ R

is bounded in R. By the Bolzano–Weierstrass theorem, this sequence contains a convergent subsequence. Using the diagonalization procedure, we get a sequence ¹hnj ºj1D1 such that Z fk hnj d converges in R as j ! 1 for every k 2 N: R

Now,

¹fn º1 nD1

is dense in Y , hence, by the Hölder inequality, the limit Z f hnj d lim j !1 R

exists for every f 2 Y (recall that f 2 Y  X D X 00 ). This means that the sequence ¹hnj ºj1D1 is .X 0 ; Y /-Cauchy in X 0 . We now apply Theorem 6.5.6 to X 0 in place of X and Y in place of Z. It implies that there exists a function h 2 X 0 such that, for every f 2 Y , Z Z lim

j !1 R

f hnj d D

R

f h d:

(6.5.4)

We know that, for each j 2 N, the support of hnj is contained in Enj , a nonincreasing (with respect to set inclusion) sequence of sets. Fix j 2 N. Then, for any E  R satisfying .E/ < 1 and E \ Enj D ; we get by (6.5.4) (recall E 2 Y ) that Z h d D 0: E

Keeping this j 2 N still fixed and using the statement just proved for any such E, we obtain that h D 0 -a.e. on R n Enj . Letting j ! 1, we get that, in fact, h D 0 -a.e. on the entire R, since Enj & ; as j ! 1. Finally, (6.5.4) yields Z Z ˇ ˇ ˇf0 gn ˇ E d: 0 D lim f0 hnj d D lim j nj j !1 R

j !1 R

That, however, contradicts (6.5.3). Thus, Y  Xa .

234

Chapter 6 Banach function spaces

We shall now prove that  is a separable measure. We recall that the sequence ¹Rk º1 will have its usual meaning. For k 2 N, we denote by Ak the -algebra kD1 of -measurable subsets of Rk . Let A 2 Ak . Then, in particular, A 2 Y . Since Y is separable, each its subset is separable, too. Let k 2 N. Then there exists a countable set ¹ Ek;j ºj1D1 , where Ek;j 2 Ak , dense in the set ¹ E I E 2 Ak º. We define the set F WD ¹Ek;j I k; j 2 Nº and we claim that the set F is a countable dense subset of the metric space .A; d /. Assume, thus, that F 2 A. Then .F / < 1 and .F \ Rk / % F as k ! 1. Hence, by the dominated convergence theorem, .F n Rk / ! 0;

k ! 1:

Let " > 0. Then there exists a k0 2 N such that " .F n Rk0 / < : 2 By Remark 6.1.16(v), there exists a positive constant Ck0 such that, for every f 2 X, Z jf j d  Ck0 kf kX : Rk0

Thus, the definition of F now guarantees that there is a set E 2 F , E  Rk0 , satisfying " : k E  F \Rk0 kX  2Ck0 Finally, altogether, Z Z d.E; F / D j F  E j d D Z D

R

Rk0

Rk0

Z ˇ ˇ ˇ ˇ ˇ F  E \Rk0 ˇ d C

ˇ ˇ ˇ ˇ  ˇ F E \Rk0 ˇ d C .F n Rk0 /

RnRk0

j0  F j d

" "  C D ": 2 2 Thus, F is dense in .A; d /, that is,  is a separable measure. ( Assume that  is separable and Y D Ya . Then, by Theorem 6.4.1, also Y D Ya D Yb . Let F1 be a dense countable set in .A; d /. We denote its elements by E1 ; E2 ; : : : . Next, we define F WD ¹Ej \ Rk ; j; k 2 Nº

235

Section 6.5 Separability in Banach function spaces

and

´ D WD f 2 S; f D

K X

μ rk Fk ; K 2 N; rk 2 Q; Fk 2 F

:

kD1

Then D is obviously countable. We have to show that it is dense in Y . It will suffice to prove that it is dense in Xb . We know that ´ Xb D g 2 M0 ; g D

m X

μ cn Gn ; m 2 N ;

nD1

where the closure is in the norm of X, cn 2 R and Gn  R are such that .Gn / < 1. We can approximate cn by rational numbers, hence it is enough to replace Gn by Fk . Assume now that .G/ < 1 and " > 0. We want to find a sequence ¹Fj ºj1D1 such that k Fj  G kX ! 0: We know that G 2 Xb D Xa and that .G\Rk / % G : Thus, by Theorem 6.3.9,

" k G  .G\Rk / kX < : 2 It is thus enough to approximate G\Rk / by functions Fk . Let n 2 N. Then there exists a function Fn 2 F such that lim d.G \ Rk ; Fn / D 0:

n!1

This means that the sequence ¹ Fn º1 nD1 converges to .G\Rk / in the norm of the space L1 .R; /. Thus, there is a subsequence ¹ Fnj ºj1D1 , which converges pointwise -a.e. on R as j ! 1. Now, for every j 2 N, the function Fnj is dominated by Rk 2 Xa . Thus, by Theorem 6.3.9 again, we finally get lim k Fnj  .G\Rk / kX D 0;

j !1

establishing our claim. The proof is complete. Corollary 6.5.10. Let X be a Banach function space over .R; /. Then X is separable if and only if X D Xa and  is separable. Corollary 6.5.11. Let X be a Banach function space and let Xa D Xb . Then Xa is separable if and only if  is separable.

236

Chapter 6 Banach function spaces

Corollary 6.5.12. Let X be a Banach function space such that its dual X  is separable. Then X is reflexive. Proof. It is a classical fact (see, e.g., [66]) that when X  is separable, then also X itself is separable. Since, then, every subset of X  is separable, in particular, X 0 is separable. Now, by Theorem 6.5.9, the separability of X implies X D Xa and likewise the separability of X 0 implies X 0 D .Xa /0 . Finally, by Corollary 6.4.6, X is reflexive.

Chapter 7

Rearrangement-invariant spaces

7.1 Distribution functions and nonincreasing rearrangements As in the previous chapter, .R; / denotes a -finite measure space, similarly for .S; /. Definition 7.1.1. Let f W .R; / ! R be a measurable function. Then the function f , defined at every  > 0 by f ./ D  .¹x 2 R W jf .x/j > º/ ;

(7.1.1)

is called the distribution function of f . Remark 7.1.2. The distribution function f of a function f depends only on jf j. Moreover, f is allowed to attain the value 1. Definition 7.1.3. Let f 2 M0 .R; / and g 2 M0 .S; /. We say that f and g are equimeasurable if they have the same distribution function, that is, if f ./ D g ./ for all   0. We write f g. Proposition 7.1.4. Let f 2 M0 .R; /. Then the distribution function f of f is a nonnegative, nonincreasing, and right-continuous function on Œ0; 1/. Proof. It is clear that f is nonnegative and nonincreasing. Let E WD ¹x W jf .x/j > º;

  0:

Then E2  E1 for 1 < 2 . Moreover, for every 0 2 R we have E0 D

[ >0

E D

1 [ nD1

E0 C 1 : n

Hence, by the monotone convergence theorem (Theorem 1.21.4),  

1 D  E0 C 1 " .E0 / D f .0 /; f 0 C n n whence f is right-continuous.

238

Chapter 7 Rearrangement-invariant spaces

Exercise 7.1.5. Let f; g 2 M0 .R; /, and let ¹fn º1 nD1 be a sequence of functions from M0 .R; /. Let a 2 R, a ¤ 0. Then jgj  jf j -a.e. ) g  f ;    ;   0; af ./ D f jaj f Cg .1 C 2 /  f .1 / C g .2 /; jf j  lim inf jfn j -a.e. n!1

)

(7.1.2) (7.1.3) 1 ; 2  0;

f  lim inf fn ; n!1

(7.1.4) (7.1.5)

in particular, jfn j " jf j -a.e.

fn " f :

)

Definition 7.1.6. Let f 2 M0 .R; /. Then the function f  W Œ0; 1/ ! Œ0; 1/, defined by (7.1.6) f  .t / D inf¹ W f ./  t º; t 2 Œ0; 1/; is called the nonincreasing rearrangement of f . Convention 7.1.7. We use here the convention that inf ; D 1. In particular, if f ./ > t for every   0, then f  .t / D 1. Convention 7.1.8. When .R/ < 1, then f ./  .R/ for every  2 Œ0; 1/. Consequently, f  .t / D 0 for every t  .R/. In this case, we may regard f  as a function defined on the interval Œ0; .R//: Remark 7.1.9. If f 2 M0 .R; / is such that f is continuous and strictly decreasing on .0; 1/, then f  is just the ordinary inverse of f on a suitable interval. Moreover, from (7.1.6) and the monotonicity of f we have f  .t / D sup¹ W f ./ > t º D f .t /;

t  0:

(7.1.7)

Remark 7.1.10. Let f .x/ D 1 

1 ; xC1

x 2 Œ0; 1/:

Then it is not difficult to verify that f  .t / D 1 for all t  0. Thus, some information can be lost in passing to the nonincreasing rearrangement of a function. Remark 7.1.11. Let f 2 M0 .R; /. Then the nonincreasing rearrangement and the distribution function of f are connected in the following way: f  .f .//  ; 

f .f .t //  t;

 2 .0; 1/ t 2 .0; .R//:

(7.1.8)

239

Section 7.1 Nonincreasing rearrangements

Indeed, let  2 .0; 1/ and define t WD f ./. Then (7.1.6) implies that f  .f .// D f  .t / D inf¹0 W f .0 /  t D f ./º  ; and the first inequality in (7.1.8) follows. To prove the other one, let t 2 Œ0; .R//. Then, by (7.1.6), there exists a sequence n #  with f .n /  t . Consequently, using the right-continuity of f (Proposition 7.1.4), we arrive at f .f  .t // D f ./ D lim f .n /  t; n!1

as desired. Proposition 7.1.12. Let f; g 2 M0 .R; / and let ¹fn º1 nD1 be a sequence in M0 .R; /. Let  2 R. Then f  is a nonnegative, nonincreasing, right-continuous function on Œ0; 1/ such that jgj  jf j -a.e. on R 

)

g  .t /  f  .t /;

t 2 Œ0; .R//;



.af / D jajf ; jf j  lim inf jfn j -a.e. on R n!1

(7.1.9) (7.1.10)

) f   lim inf fn : n!1

(7.1.11)

In particular, one has jfn j " jf j -a.e. on R

)

fn " f  :

Proof. All the assertions follow immediately from Proposition 7.1.4 or from their corresponding counterparts in Proposition 7.1.4 and from the definition of the nonincreasing rearrangement. The operation of taking the nonincreasing rearrangement is not subadditive. Instead, we have the following result. Proposition 7.1.13. Let f; g 2 M0 .R; / and let s; t 2 Œ0; .R// be such that sCt 2 Œ0; .R//. Then (7.1.12) .f C g/ .s C t /  f  .s/ C g  .t /: Proof. Set  WD f  .s/ C g  .t / and y WD f Cg ./. Then, by the triangle inequality and the second inequality in (7.1.8), we obtain y D ¹x 2 RI jf .x/ C g.x/j > f  .s/ C g  .t /º  ¹x 2 RI jf .x/j > f  .s/º C ¹x 2 RI jg.x/j > g  .t /º D f .f  .s// C f .g  .t //  s C t:

240

Chapter 7 Rearrangement-invariant spaces

Thus, by the first of the inequalities in (7.1.8) and the monotonicity of .f C g/ , we get .f C g/ .s C t /  .f C g/ .y/ D .f C g/ .f Cg .//   D f  .s/ C g  .t /; and the assertion follows. The proof is complete. Remark 7.1.14. It is worth pointing out that probably the most useful special case of (7.1.12) reads as .f C g/ .t /  f  . 2t / C g  . 2t /;

f; g 2 M0 .R; /; t 2 .0; .R//:

(7.1.13)

Proposition 7.1.15. Let f 2 M0 .R; /. Then the functions f and f  are equimeasurable. Proof. There exists a sequence ¹fn º1 nD1 of nonnegative simple functions fn " f . It follows from definitions that for each n the functions fn and fn are equimeasurable, that is, (7.1.14) fn ./ D fn ./;  2 .0; 1/: We however know that fn " jf j and fn " f  (see (7.1.11)). Thus, the assertion follows from (7.1.5). Exercise 7.1.16. Let n 2 N and let ¹Ej ºjnD1 be pairwise disjoint measurable subsets of R of finite -measure. Let further ¹˛j ºjnD1 be positive real numbers satisfying ˛1 > ˛2 >    > ˛n > 0. Define f .x/ WD

n X

aj Ej .x/;

x 2 R:

(7.1.15)

j D1

For j D 1; : : : ; n, we denote ˇj WD

j X

.Ei /

(7.1.16)

iD1

and ˇ0 WD 0. Then f  .t / D

n X

aj Œˇj 1 ;ˇj  .t /;

t 2 .0; 1/:

(7.1.17)

j D1

Proposition 7.1.17. Let f 2 M0 . If 0 < p < 1, then Z1

Z p

R

Z1 p1

jf j d D p

 0

f ./ d D 0

f  .t /p dt:

(7.1.18)

241

Section 7.2 Hardy–Littlewood inequality

Furthermore, in the case p D 1, ess sup jf .x/j D inf¹I f ./ D 0º D f  .0/:

(7.1.19)

x2R

Proof. Let f be an arbitrary nonnegative simple function. Then there exist n 2 N, pairwise disjoint measurable subsets ¹Ej ºjnD1 of R of finite -measure and positive real numbers ¹˛j ºjnD1 satisfying ˛1 > ˛2 >    > ˛n > 0 such that f obeys (7.1.15). By Exercise 7.1.16, f  satisfies (7.1.17). So, (7.1.16) yields Z p

jf .x/j d.x/ D

n X

p ˛j .Ej /

n X

D

j D1

p ˛j .Œˇj 1 ; ˇj //

Z1 D

j D1

f  .t /p dt:

0

Similarly, one gets by (7.1.16) and the summation by parts, Z

1

p1



p 0

n Z X

f ./ d D p

˛j

p1



j D1 ˛j C1 n X D ˛jp .Ej / j D1

f ./ d D

n X

.˛jp  ˛jpC1 /ˇj

j D1

Z

jf .x/jp d.x/:

D

This shows the first part of the claim for an arbitrary nonnegative simple function f . The extension to a general f follows from (7.1.5), (7.1.11) and the monotone convergence theorem (Theorem 1.21.4). We have thus proved (7.1.18). The proof of (7.1.19) is even easier and hence omitted.

7.2 Hardy–Littlewood inequality Proposition 7.2.1. Let g 2 MC .R; / and let E  R be -measurable. Then Z

Z E

g d 

.E /

g  .s/ ds:

(7.2.1)

0

Proof. Let ¹Ej ºjnD1 be a finite sequence of -measurable subsets of R such that E1  E2      En and let ¹˛j ºjnD1 be real numbers. Let g.x/ D

n X

˛j Ej .x/;

x 2 R:

j D1

Then it is not difficult to verify that 

g .t / D

n X j D1

˛j Œ0;.Ej // .t /;

t 2 Œ0; .R//:

242

Chapter 7 Rearrangement-invariant spaces

Thus, Z E

g d D D

n X

˛j .E \ Ej / 

j D1 n X

˛j min..E/; .Ej //

j D1

Z

Z

.E /

.0;.Ej // ds D

˛j 0

j D1

n X

.E /

g  .s/ ds;

0

establishing (7.2.1) for nonnegative simple functions. For arbitrary g 2 MC we apply the usual density argument. One of the most important theoretical tools in the theory of rearrangement-invariant spaces is the inequality of Hardy and Littlewood, which we now shall state and prove. Theorem 7.2.2 (Hardy–Littlewood). Let f; g 2 M0 .R; /. Then Z Z 1 f  .t /g  .t / dt: jfgj d  R

(7.2.2)

0

Proof. Let first f be a simple nonnegative function, that is, let ¹Ej ºjnD1 be a finite sequence of -measurable subsets of R such that E1  E2      En , let ¹˛j ºjnD1 be real numbers and let f .x/ D

n X

˛j Ej .x/;

x 2 R:

j D1

Then, again,

n X

f  .t / D

˛j Œ0;.Ej // .t /:

j D1

Let g 2 MC .R; /. By Lemma 7.2.1, we obtain Z Z Z n n X X ˛j g d  ˛j jfgj d D R

j D1

Z

D Z D

Ej

n 1X 0

j D1

.Ej /

g  .s/ ds

0

˛j Œ0;.Ej // g  .s/ ds

j D1 1

f  .s/g  .s/ ds:

0

When f 2 MC .R; /, we find a sequence ¹fj ºj1D1 such that 0  fj % f and use the monotone convergence theorem (Theorem 1.21.4). This shows the assertion for the case when f and g are nonnegative. The general case follows in turn because the right-hand side depends only of f  and g  , and hence on the absolute value of f and g. The proof is complete.

243

Section 7.3 Resonant measure spaces

Corollary 7.2.3. Let f; g 2 M0 .R; /. Let gQ 2 M0 .R; / be equimeasurable with g on R. Then Z Z 1 f  .t /g  .t / dt: (7.2.3) Q d  jf gj R

0

Proof. The assertion is an immediate consequence of Theorem 7.2.2 and the fact that gQ  D g  .

7.3 Resonant measure spaces Definition 7.3.1. A -finite measure space .R; / is said to be resonant if, for each f and g in M0 .R; /, one has Z Z 1   f .t /g .t / dt D sup Q d; (7.3.1) jf gj g g Q

0

R

and strongly resonant if for every f; g 2 M0 .R; /, there exists a function gQ on R such that gQ g and Z Z 1 f  .t /g  dt: (7.3.2) Q d D jf gj R

0

Remark 7.3.2. Of course, strong resonance implies resonance. The converse is not true in general, as the following example shows. Example 7.3.3. The measure space .Œ0; 1/; dx/, endowed with the one-dimensional Lebesgue measure d.x/ D dx, is resonant. Let f be the function described in Remark 7.1.10, that is, f .x/ D 1 

1 ; xC1

x 2 Œ0; 1/:

Then, f  .t / 1 for every t 2 Œ0; 1/. Let now g D Œ0;1/ and gQ D Œ1;2/ . Then g  D gQ  D Œ0;1/ . However, we obviously have Z

1 0

while

Z Q dx D jf .x/g.x/j Z

1 0

2 1

1

1 x



dx D 1  log 2 < 1;

f  .t / g  .t / dt D 1;

hence our measure space is not strongly resonant. Remark 7.3.4. Not every measure space is resonant. A typical example of a nonresonant measure space is an atomic measure space having at least two atoms of nonequal measure. More precisely, let a; b 2 R be such that .a/ D ˛, .b/ D ˇ with

244

Chapter 7 Rearrangement-invariant spaces

0 < ˛ < ˇ. Let f D ¹bº and g D ¹aº . Then it is clear that every function gQ satisfying gQ g must obey b 62 supp g. Q Therefore, for every such g, Q we have Z Q d D 0: jf gj R

On the other hand, f  D Œ0;ˇ / and g  D Œ0;˛/ , hence Z 1 f  .t / g  .t / D ˛: 0

Consequently, no such measure space can be resonant. Notation 7.3.5. We shall denote by Rng./ the range of the measure , that is, Rng./ WD ¹t 2 Œ0; 1/I there exists E  R satisfying .E/ D t º: We note that if  is a nonatomic measure on R, then Rng./ D Œ0; .R/:

(7.3.3)

Lemma 7.3.6. Let .R; / be a nonatomic measure space such that .R/ < 1. Let f 2 M0 .R; / and let t 2 Œ0; .R/. Then there exists a -measurable set E t  R satisfying .E t / D t and such that Z Z t f  .s/ ds: (7.3.4) jf j d D Et

0

Moreover, 0  s  t  .R/

)

Es  E t :

(7.3.5)

Proof. First, assume that t 2 Rng.f /, that is, there exists ˛ > 0 such that f .˛/ D t . Then f  .t / D inf¹I f ./ D t º: Since f is right-continuous, this implies f .f  .t // D t . We then set E t WD ¹xI jf .x/j > f  .t /º: Then .E t / D t . Moreover, one has

´

f E t ./ D ¹x 2 E t I jf .x/j > º D On the other hand,

t if  2 Œ0; f  .t /; f ./ if  2 .f  .t /; 1/:

´ 

f  Œ0;t/ ./ D ¹s 2 Œ0; t I f .s/ > º D

t f  ./

if  2 Œ0; f  .t /; if  2 .f  .t /; 1/:

245

Section 7.3 Resonant measure spaces

By Proposition 7.1.15, f f  . Hence, f D mf  . In particular, we have fE t f  Œ0;t/ , and therefore also Z

Z Et

jf j d D

Z R

jfE t j d D

1 0

f  .s/ Œ0;t .s/ ds D

Z

t

f  .s/ ds:

0

This establishes (7.3.4) and (7.3.5) in the case when t 2 Rng.f /. Now assume that t 62 Rng..f //. Since .R/ < 1 and f 2 M0 , the dominated convergence theorem (Theorem 1.21.5) and the monotonicity of the distribution function f give lim f ./ D lim ¹xI jf .x/j > nº D ¹xI jf .x/j D 1º D 0:

!1

n!1

(7.3.6)

We denote 0 WD f  .t /. Since t … Rng.f /, (7.1.8) and (7.1.6) imply that t 2 .f .0 /; f .//

for every  2 .0; 0 /:

Next, let t1 WD lims!0 f .0  s/ (the limit exists due to the monotonicity of f ), and t0 WD f .0 /. Then we have t0 < t < t1 ;

(7.3.7)

hence, also by (7.1.6), f  .s/ D 0 ;

for every s 2 .t0 ; t1 /:

(7.3.8)

We shall now prove that t1 D ¹xI jf .x/j  0 º:

(7.3.9)

Define

1 º; n 2 N: n Then F1  F2  : : : and, since .R/ is finite, the dominated convergence theorem (Theorem 1.21.5) implies   1 D f .0 /: ¹x 2 RI jf .x/j  0 º D lim .Fn / D lim f 0  n!1 n!1 n Fn D ¹xI jf .x/j < 0 

Again, the limit exists due to the monotonicity of f . We next denote G WD ¹x 2 RI jf .x/j D 0 º. Then, by (7.3.7) and (7.3.9), one has .G t / D t1  t0 : Since .R; / is nonatomic, there exists a set F t  G such that .F t / D t  t0 . We define E t WD ¹x 2 RI jf .x/j < 0 º [ F t :

(7.3.10)

246

Chapter 7 Rearrangement-invariant spaces

Then .E t / D f .0 / C .t  t0 / D t . Next, Z Z Z jf j d D jf j d C jf j>0

Et

Ft

jf j d:

(7.3.11)

It follows from (7.1.7) that t0 2 Rng.f /. Thus, by the result just proved, we have Z

Z jf j>0

jf j d D

t0

f  .s/ ds:

0

Furthermore, since jf j D 0 on F t , one has, by (7.3.8), Z t Z f  .s/ ds: jf j d D 0 .F t / D 0 .t  t0 / D t0

Ft

This and (7.3.11) now imply (7.3.4). Finally, assume that 0 D 0. Then we get, using an argument analogous to that from the proof of (7.3.7), that ¹x 2 RI jf .x/j > 0º D t0 < t: There exists a set F t satisfying F t \ supp f D ; and .F t / D t  t0 . Let E t be the set from (7.3.10). Since f  .t / D 0 for t  t0 , we obtain Z t0 Z t Z jf j d D f  .s/ ds D f  .s/ ds; Et

0

0

and (7.3.4) follows, again. It follows easily from the construction that it can be done so that (7.3.5) is satisfied.

Theorem 7.3.7. A -finite measure space .R; / is strongly resonant if and only if .R/ < 1 and .R; / is either nonatomic or completely atomic with all atoms having the same measure. Proof. Let f; g 2 M0 .R; / be nonnegative functions. It suffices to find a nonnegative function gQ such that gQ g and Z Z 1 f gQ d D f  .t /g  .t / dt: (7.3.12) R

0

This is an easy exercise when .R; / is completely atomic with all atoms having the same measure, because then there are only a finite number of atoms. Assume thus that .R; / is nonatomic. Let ¹gn º1 nD1  S be such that 0  gn " g -a.e. on R.

247

Section 7.3 Resonant measure spaces

Let n 2 N, then there exist a m 2 N, measurable sets F1  F2  :::  Fm and nonnegative numbers ˛1 ; ˛2 ; : : : ; ˛m such that gn D

m X

˛i Fi :

iD1

By Lemma 7.3.6, there exist sets E1  E2  :::  Em satisfying .Ej / D .Fj / and Z Z Ej

f d D

.Fj /

f  .t / dt;

j D 1; 2; :::; m:

0

Set gQ n WD

m X

(7.3.13)

˛j Ej :

j D1

Then, gn D

m X

˛i Œ0;.Fi // D .gQn / ;

iD1

that is, gn gQ n . Furthermore, by (7.3.13), Z R

f gQ n d D

m X iD1

Z ˛i Ei

g d D

m X

Z

iD1

Z D Z D

.Fi /

˛i

m 1X 0 1 0

f  .t / dt

0

˛i Œ0;.Fi // f  .t / dt

iD1

f  .t /gn .t / dt:

Since gn ", it follows from Lemma 7.3.6 that the sets Ej in (7.3.13) can be constructed so that En " in the sense of inclusion. Consequently, we get gQ n " and, in turn, by the monotone convergence theorem (Theorem 1.21.4), also (7.3.12). As mentioned above, this implies that .R; / is strongly resonant. As for the “only if” part of the assertion of the theorem, let .R; / be resonant. Then the considerations given in Remark 7.3.4 show that if .R; / has any atoms at all, then they must be of an identical measure and also that there cannot be a nontrivial atom and a nontrivial nonatomic part of .R; / at the same time. Moreover, if .R/ D 1, then an example of the type pointed out in Remark 7.3.2 can be constructed ruling out strong resonance. Theorem 7.3.8. A -finite measure space .R; / is resonant if and only if it is either nonatomic or completely atomic with all atoms having equal measure.

248

Chapter 7 Rearrangement-invariant spaces

Proof. The “only if” part follows immediately from Remarks 7.3.2 and 7.3.4, as it was mentioned also in the proof of Theorem 7.3.7. part, let f; g 2 MC .R; /. With no loss of generality assume that R 1For the “if”  f .t /g .t / dt > 0, since otherwise (7.3.1) is trivially satisfied. Let ¹fn º1 nD1 and 0 1 ¹gn ºnD1 be two sequences of simple functions such that supp fn [ supp gn  Rn , n 2 N, and 0  fn " f , 0  gn " g. Let, moreover, ˛ 2 R be such that Z 1 0 ˛: R

Hence gQ g and, by (7.3.17), Z R

(7.3.16)

(7.3.17)

Z f gQ d 

R

f h d > ˛:

We have shown that, for each ˛ satisfying (7.3.14) we can find a nonnegative function gQ such that gQ g and Z R

f gQ d > ˛:

(7.3.18)

In other words, .R; / is resonant. The following result is an immediate consequence of Theorems 7.3.7 and 7.3.8. Corollary 7.3.9. A -finite measure space .R; / is strongly resonant if and only if it is resonant and .R/ < 1.

Section 7.4 Maximal nonincreasing rearrangement

249

7.4 Maximal nonincreasing rearrangement Let E be a -measurable subset of R satisfying .E/ D t 2 .0; 1/. Then, taking g D E and inserting g into the Hardy–Littlewood inequality (7.2.2), we get Z Z 1 t  1 jf j d  f .s/ ds; f 2 M0 .R; /: (7.4.1) .E/ E t 0 The last expression, that is, the integral average of f  over the integral .0; t /, turns out to be of great importance. In this section we shall study its basic properties. Definition 7.4.1. For f 2 M0 .R; /, we define the function f  on .0; .R// by Z 1 t   f .s/ ds; t 2 .0; .R//: (7.4.2) f .t / WD t 0 Remark 7.4.2. Let f 2 M0 .R; /. Then f  .t /  f  .t /;

t 2 .0; .R//:

(7.4.3)

Indeed, let t 2 .0; .R//. Then, by the monotonicity of f  , we obtain Z Z 1 t  f  .t / t  f .s/ ds  ds D f  .t /: f .t / D t 0 t 0 Proposition 7.4.3. Let f 2 M0 .R; /, then f  is a nonnegative, nonincreasing and continuous function on .0; .R//. Proof. It follows from the definition of f  .t / and the monotonicity of f  that either f  .t / < 1 for every t 2 .0; .R// or f  .t / D 1 for every t 2 .0; .R//. Obviously, in any case, f  is nonnegative and continuous. It is also nonincreasing, being an integral mean of a nonincreasing function. More precisely, let 0 < s < t < .R//. Then, by (7.4.3), Z 1 t   f .y/ dy f .t / D t 0 Z s Z 1 t  1  f .y/ dy C f .y/ dy D t 0 t s Z s f  .s/ t  f  .s/ C dy t t s

s s f  .s/ D f  .s/ C 1  t t s  s f .s/  f  .s/ C 1  t t D f  .s/; as desired. The proof is complete.

250

Chapter 7 Rearrangement-invariant spaces

Proposition 7.4.4. For every pair of functions f; g 2 M0 .R; /, every sequence ¹fn º1 nD1 of functions from M0 .R; / and every a 2 R, we have f  0

”

f D 0 -a.e.

0  g  f -a.e. 

.af /

D jajf

)

(7.4.4)

g   f 

(7.4.5)



(7.4.6)

0  fn " f -a.e.

)

fn

"f



:

(7.4.7)

Proof. All the properties of f  easily follow from the corresponding ones of f  (see Proposition 7.1.12). Proposition 7.4.5. Let t 2 .0; .R// \ Rng./ and let f 2 M0 .R; /. (i) If .R; / is resonant, then f



1 .t / D sup t

²Z

³ E

jf j dI .E/ D t :

(7.4.8)

(ii) If .R; / is strongly resonant, then there is a subset E of R with .E/ D t such that Z 1  jf j d: (7.4.9) f .t / D t E Proof. Let F  R be such that .F / D t . Define g D F . Then g  D Œ0;t/ . Therefore, if a function gQ is equimeasurable with g, then there exists a set E  R such that .E/ D t and jgj Q D E . Both assertions thus follow from (7.3.1) and (7.3.2). Corollary 7.4.6. Let .R; / be a resonant measure space. Let f; g 2 M0 .R; / and let t 2 .0; .R// \ Rng./. Then .f C g/ .t /  f  .t / C g  .t /:

(7.4.10)

Proof. The assertion follows immediately from (7.4.8) and the subadditivity of the supremum. Remark 7.4.7. The subadditivity of the functional f 7! f  , claimed in Corollary 7.4.6, holds for arbitrary measure spaces, not only for the resonant ones. This can be proved by the so-called method of retracts. The details can be found in [14, p. 54]. Example 7.4.8. Unlikely the operation f 7! f  , the functional f 7! f  is not subadditive, as simple examples show (for instance, take R D Œ0; 1/, let  be the one-dimensional Lebesgue measure, and set f D Œ0;1/ and g D Œ1;2/ ). The best we can hope for in the case of f 7! f  is the considerably weaker property (7.1.12).

251

Section 7.5 Hardy lemma

7.5 Hardy lemma As we shall see in the following section, the relation f   g  between two given functions f; g 2 M0 .R; /, is of a crucial importance. Note that it is weaker than the pointwise estimate f   g  . We shall now study this relation in detail. Definition 7.5.1. We say that the functions f; g 2 M0 .R; / are in the Hardy– Littlewood–Pólya relation, and write f g, if, for every t 2 Œ0; .R//, one has Z t Z t  f .s/ ds  g  .s/ ds: (7.5.1) 0

0

We shall now formulate the Hardy lemma, an important technical background result which will be very useful in what follows. Proposition 7.5.2 (Hardy lemma). Let a 2 .0; 1, let f and g be two nonnegative measurable functions on .0; a/ and suppose that Z t Z t f .s/ ds  g.s/ ds (7.5.2) 0

0

for all t 2 .0; a/. Let h be a nonnegative nonincreasing function on .0; a/: Then Z a Z a f .t /h.t / dt  g.t /h.t / dt: (7.5.3) 0

0

Proof. Assume first that h is of the form h.t / D

n X

cj .0;tj / .t /;

j D1

where cj 2 .0; 1/ and 0 < t1 < ::: < tn < a. Then, by (7.5.2), we obtain Z

a 0

f .t /h.t / dt D

n X j D1

Z

tj

cj 0

f .t / dt 

n X j D1

Z

tj

cj 0

Z g.t / dt D

a

g.t /h.t / dt; 0

proving the claim in the special case. For a general function, the assertion follows by using the monotone convergence theorem (Theorem 1.21.4) and the property (P3) from Definition 6.1.5. The proof is complete. Theorem 7.5.3. Let .R; / be a resonant measure space, let I be a countable index set and let ¹Ei ºi2I be a sequence of pairwise disjoint subsets of R satisfying 0 < .Ei / < 1 for every i 2 I . Define [ E WD R n Ei : i2I

252

Chapter 7 Rearrangement-invariant spaces

Assume that f 2 M0 .R; / and function Af by Af WD f E C

R

f d < 1 for every i 2 I . We define the

Ei

X i2I

1 .Ei /



Z Ei

f d Ei :

(7.5.4)

Then Af f: Proof. If I has only one element E1 , then   Z 1 Af D f E C f d E1 : .E1 / E1 We want to show that Z t 0

Z



.Af / .s/ ds 

t

f  .s/ ds;

0

t 2 .0; a/:

(7.5.5)

Suppose first that t 2 .0; a/ \ Rng./. Then there is some F  R such that .F / D t . Define t0 WD .F \ E1 /: Then

Z F

ˇ ˇ Z ˇ 1 ˇ jAf j d D jf j d C t0 ˇˇ f dˇˇ : .E1 / E1 F \E Z

By (7.4.1) and the monotonicity of the function .f E1 / , we get ˇ ˇ Z t0 Z ˇ 1 ˇ f dˇˇ  t0 .fE1 / ..E1 //  .fE1 / .s/ ds: t0 ˇˇ .E1 / E1 0

(7.5.6)

(7.5.7)

Now, .R; / is resonant and moreover .E1 / < 1. Thus, denoting the restriction of  to E1 by  again, we conclude, using Corollary 7.3.9, that the measure space .E1 ; / is strongly resonant. Next, .F \ E1 / D t0 , hence, in particular, t0 2 Rng./. Proposition 7.4.5 (ii) now guarantees the existence of some G  E1 such that .G/ D t0 and Z t0 Z Z  .f E1 / .s/ ds D jf E1 j d D jf j d: 0

G

G

Combining this with (7.5.7) and (7.5.6), we obtain Z Z Z jAf j d  jf j d C jf j d: F

F \E1

G

But F \ E1 and G are disjoint and ..F \ E1 / [ G/ D .F \ E1 / C .G/ D .t  t0 / C t0 D t;

(7.5.8)

253

Section 7.6 Rearrangement-invariant spaces

so applying (7.4.1) to the right-hand side of (7.5.8) we obtain Z t Z jAf j d  f  .s/ d s: F

0

Finally, taking the supremum over all sets F of measure t and applying Proposition 7.4.5 (i), we obtain (7.5.5), at least for all t in the range of : Since .R; / is of one of the types described in Theorem 7.3.8, however, it is clear that (7.5.5) must then hold for all t > 0. Now let us consider the case when I is an arbitrary finite index set. In such cases, it suffices to iterate the result just proved. If I is countable and infinite, we find a sequence ¹fn º1 nD1 satisfying 0  fn " f a.e. and 0  Afn " Af -a.e. Then Afn D An fn fn f; where An f WD f Rn C

n  X mD1

1 .Em /

and Rn WD R n

n [

n 2 N; 

Z Em

f d Em ;

Em :

mD1

and the result follows from (7.4.7). The proof is complete.

7.6 Rearrangement-invariant spaces We throughout this section assume that .R; / is resonant. Definition 7.6.1. Let % be a Banach function norm over .R; /. We say that % is rearrangement-invariant (r.i.) if %.f / D %.g/ for every pair of equimeasurable functions f; g 2 M0C .R; /. In such cases, the corresponding Banach function space X D X.%/ is then said to be a rearrangementinvariant Banach function space. Remark 7.6.2. A Banach function space X is rearrangement-invariant if and only if, whenever f belongs to X and g is equimeasurable with f , then g also belongs to X and kgkX D kf kX . Example 7.6.3. Assume that p 2 Œ1; 1. Then the Lebesgue space Lp .R; / is a rearrangement-invariant Banach function space. Indeed, this follows from Proposition 7.1.17.

254

Chapter 7 Rearrangement-invariant spaces

Remark 7.6.4. In many situations, a natural choice of a function equimeasurable with a given function f is its nonincreasing rearrangement f  . Proposition 7.6.5. Let % be a rearrangement-invariant function norm over .R; /. Then the associate norm %0 is also rearrangement-invariant. Moreover, we have ´Z μ .R/

%0 .g/ D sup

and

0

´Z %.f / D sup

f  .t /g  .t / dt I %.f /  1 ;

(7.6.1)

f 2 M0C :

(7.6.2)

μ

.R/





0

f .t /g .t / dt I % .g/  1 ;

0

g 2 M0C;

Proof. We know from Definition 6.2.1 that ²Z ³ 0 jfgj dI %.f /  1 ; % .g/ D sup R

g 2 M0C :

(7.6.3)

First, it follows immediately from the Hardy–Littlewood inequality (7.2.2) that ´Z μ %0 .g/  sup

.R/

0

f  .t /g  .t / dt I %.f /  1 ;

g 2 M0C :

To prove the converse inequality, note that the rearrangement-invariance of % guarantees that %.fQ/ D %.f / for every fQ equimeasurable with f . Thus, thanks to the resonance of .R; /, we have, for every g 2 M0 .R; /, ²Z

.R/

sup 0

³ ²Z ³ Q Q f .t /g .t / dt I %.f /  1 D sup jf gj dI f f; %.f /  1 R ²Z ³  sup jfgj dI %.f /  1 : 



R

Altogether, this establishes (7.6.1). Finally, applying the already proved assertion to %00 in place of %0 and using Theorem 6.2.9, we get (7.6.2). The proof is complete. Theorem 7.6.6 (Hölder inequality). Let % be a rearrangement-invariant norm over a resonant measure space .R; /. If f and g belong to M0C .R; /; then Z R

Z fg d 

.R/ 0

f  .s/g  .s/ ds  %.f /%0 .g/:

(7.6.4)

Proof. The assertion follows immediately from Theorem 7.2.2 and Proposition 7.6.5.

255

Section 7.7 Hardy–Littlewood–Pólya principle

It will be useful to formulate also the “space-versions” of the results just obtained for norms. Corollary 7.6.7. Let X be a Banach function space over a resonant measure space. Then X is rearrangement-invariant if and only if the associate space X 0 is rearrangementinvariant. Furthermore, we have ´Z μ kgkX 0 D sup and

.R/

0

´Z kf kX D sup

f  .t /g  .t / dt I kf kX  1 ;

(7.6.5)

f 2 X:

(7.6.6)

μ

.R/ 0

g 2 X 0;





f .t /g .t / dt I kgkX 0  1 ;

Corollary 7.6.8 (Hölder inequality). Let X be a rearrangement-invariant space over a resonant measure space .R; /. If f 2 X and g 2 X 0 , then Z

Z R

jfgj d D

.R/ 0

f  .t /g  .t / dt  kf kX kgkX 0 :

(7.6.7)

Exercise 7.6.9. Let   RN be a suitable domain. (i) Let 1  p  1. Then the Lebesgue space Lp ./ is a rearrangement-invariant Banach function space. (ii) Let 1  p  1 and let % be a weight on . Then the weighted Lebesgue space Lp .; %/ (cf. Remark 3.19.8) is a Banach function space which in general is not rearrangement-invariant (unless the rearrangement is taken with respect to the measure %.x/ dx). (iii) Let ˆ be a Young function. Then the Orlicz space Lˆ ./ is a rearrangementinvariant Banach function space. p;

(iv) Let   0 and p 2 .1; 1/. Then the Morrey space LM ./ is a Banach function space which is not rearrangement-invariant. We shall see more examples of which is not rearrangement-invariant Banach function spaces in subsequent chapters.

7.7 Hardy–Littlewood–Pólya principle We shall now state and prove one of the key theoretical results of the theory of rearrangement-invariant Banach function spaces. It also illustrates the significance of the Hardy–Littlewood–Pólya relation, introduced in Definition 7.5.1.

256

Chapter 7 Rearrangement-invariant spaces

Theorem 7.7.1 (Hardy–Littlewood–Pólya principle). Let .R; / be a resonant measure space. Suppose that the functions f; g 2 M0C.R; / satisfy f g. Let X be any rearrangement-invariant Banach function space over .R; /. Then kf kX  kgkX : Proof. Let h 2 M0 .R; / be such that khkX 0  1. Then, since h is nonincreasing on .R; /, we obtain from the Hardy lemma (Proposition 7.5.2) Z .R/ Z .R/   f .t /h .t / dt  g  .t /h .t / dt: 0

0

The assertion now follows from (7.6.2). Theorem 7.7.1 has many important consequences. One of them asserts that the averaging operator A introduced in (7.5.4) is bounded with norm equal to 1 on every rearrangement-invariant Banach function space. Theorem 7.7.2. Let .R; / be a resonant measure space, let I be a countable index set and let ¹Ei ºi2I be a sequence of pairwise S disjoint subsets of R such that 0 < .Ei / < 1 for every i 2 I . Let E WD R n i2I Ei . Let the operator A be defined at every g 2 M0 .R; / by  X 1 Z Ag WD g E C g d Ei : .Ei / Ei j 2I

Then kAgkX  kgkX ;

g 2 X:

Proof. Assume that f 2 X. Then also f 2 L1 .Ei /, i 2 I , thanks to Remark 6.1.16 (vi). By Theorem 7.5.3, we have Af f . Thus, the assertion follows from the Hardy–Littlewood–Pólya principle (Theorem 7.7.1).

7.8 Luxemburg representation theorem In this section it will be convenient to have a special symbol for the one-dimensional Lebesgue measure. Notation 7.8.1. We shall denote by 1 the one-dimensional Lebesgue measure. Theorem 7.8.2. Assume that  is a rearrangement-invariant norm over .Œ0; 1/; 1 /. Let .R; / be an arbitrary -finite measure space. Then the functional %, defined by %.g/ WD .g  /;

g 2 M0C .R; /;

is a rearrangement-invariant Banach function norm over .R; /.

(7.8.1)

257

Section 7.8 Luxemburg representation theorem

Proof. Suppose that f; g 2 M0C .R; /. By (7.4.10) and the fact that .f  / D f  , we have .f C g/  f  C g  D .f  C g  / : Thus, .f C g/ f  C g  . Now,  is a norm on .Œ0; 1/; 1 /, which is resonant. Thus, by Theorem 7.7.1 and the triangle inequality for , we obtain ..f C g/ /  .f  C g  /  .f / C .g  /: By (7.8.1), we get the triangle inequality for %. All the other norm properties of % required by the axiom (P1) of Definition 6.1.5 are clearly satisfied. The axioms (P2), (P3), (P4) and (P5) for % follow immediately from their corresponding counterparts for . The proof is complete. The preceding theorem describes a simple procedure in which a rearrangementinvariant Banach function norm arises from a given norm over the interval Œ0; 1/ endowed with the one-dimensional Lebesgue measure 1 . We shall now state and prove the Luxemburg representation theorem – a key result which shows that, in fact, every such norm is obtained this way as long as the underlying measure space is resonant. Theorem 7.8.3 (Luxemburg representation theorem). Let X be a rearrangementinvariant Banach function space over a resonant measure space .R; /. (i) There exists a (not necessarily unique) rearrangement-invariant Banach function space X over .Œ0; .R//; 1 /, called the representation space of X, such that (7.8.2) kgkX D kg  kX ; g 2 M0C .R; /: (ii) If X is any rearrangement-invariant function space over .Œ0; .R//; 1 / which represents X in the sense of (7.8.2), then kgkX 0 D kg  kX 0 ; Proof.

g 2 M0C .R; /:

(7.8.3)

(i) We define the norm k  kX at every h 2 M0 .Œ0; .R//; 1 / by ´Z μ .R/

khkX WD sup

0

g  .t /h .t / dt I kgkX 0  1 :

(7.8.4)

Then, by (7.6.2), we get (7.8.2). Let h1 ; h2 2 M0 .Œ0; .R//; 1 /. Then, as in the proof of the preceding theorem, we obtain .h1 C h2 / h1 C h2 :

258

Chapter 7 Rearrangement-invariant spaces

Hence, using the Hardy lemma (Proposition 7.5.2) and (7.8.4), we obtain kh1 C h2 kX  kh1 kX C kh2 kX ; and the triangle inequality for k  kX follows. Other norm properties required by (P1) are easy to verify. The lattice property (P2) follows from (7.1.9) and (7.8.4). The Fatou property (P3) is guaranteed by (7.1.11), (7.8.4), and the monotone convergence theorem. The rearrangement-invariance of k  kX is obvious from the definition. Let t 2 Œ0; .R// \ Rng./. Then there exists a set E  R such that .E/ D t . Thus, using (7.8.2), we get k Œ0;t kX D k E kX < 1 by (P4) for k  kX . Using the (already proved) triangle inequality for k  kX and the mathematical induction, we get k Œ0;kt kX < 1;

k 2 N:

So, if now t 2 Œ0; .R// is arbitrary (not necessarily in Rng./), it suffices to use (P2) to show that k Œ0;t kX < 1: This establishes (P4). In a similar way we can prove (P5). Altogether, the functional k  kX is a rearrangement-invariant function norm over .Œ0; .R//; 1 /. (ii) Assume that X is any rearrangement-invariant Banach function space over .Œ0; .R//; 1 / representing % in the sense of (7.8.2). By Theorem 7.3.8, the measure space .Œ0; .R//; 1 / is resonant. Thus, in view of (7.6.1), we have ´Z μ .R/

kgkX 0 D sup

0

f  .t /g  .t / dt I kf kX  1 ;

In particular, with g replaced by g  , we have ´Z .R/



kg kX 0 D sup

0



g 2 M0C.R; /:

μ



f .t /g .t / dt I kf kX  1 ;

g 2 M0C .R; /: (7.8.5)

Hence, it follows from (7.8.2) and (7.8.3) that kgkX 0  kg  kX 0 ; Assume that f .x/ D

k X j D1

g 2 M0C .R; /:

(7.8.6)

cj .0;tj / ;

(7.8.7)

259

Section 7.9 Fundamental function

where k 2 N, cj 2 .0; 1/, j 2 N and 0 < t1 <    < tk (in other words, f is a nonincreasing simple function). It follows from the Fatou property (P3) for the norm k  kX that, for every g 2 M0C .R; /, one has ´Z μ .R/

kg  kX 0 D sup

0

f  .t /g  .t / dt I f is of the form (7.8.7); kf kX  1 :

Now, if .R; / is nonatomic and f is of the form (7.8.7), then it is the nonincreasing rearrangement of some simple function on R. We thus get kg  kX 0  kgkX 0 ;

g 2 M0C .R; /;

which together with (7.8.6) establishes (7.8.3). In the remaining case when .R; / is completely atomic measure space with all atoms having the same positive measure, say ˛ (cf. Theorem 7.3.8), the functions f of the form (7.8.7) may not necessarily be constant on the intervals of the form Ik WD Œ.k  1/˛; k˛/, where k 2 N, so they may not be representable as nonincreasing rearrangements of simple functions on .R; /. We then replace every such f by Z X 1 Ik f: fQ D .Ik / Ik k2N

Then

Z

.R/ 0

and, by Theorem 7.7.2,





f .t /g .t / dt D

Z

.R/

fQ .t /g  .t / dt

0

kfQkX  kf kX :

The rest of the proof (with f replaced by fQ) is the same as in the nonatomic case. Remark 7.8.4. The Luxemburg representation theorem reduces rearrangement-invariant spaces to their representations on an interval and thereby it enables us to study one-dimensional analogues of rearrangement-invariant spaces without having to deal with intrinsic difficulties connected with the underlying measure spaces. This representation is not always unique. The uniqueness of the representation is however guaranteed in the cases when .R; / is nonatomic and such that .R/ D 1.

7.9 Fundamental function In this section we shall study a very important characteristic of a given r.i. space, namely the fundamental function.

260

Chapter 7 Rearrangement-invariant spaces

Convention 7.9.1. All r.i. spaces in this section are supposed to be over a resonant measure space .R; /. Definition 7.9.2. Let X be a rearrangement-invariant Banach function space. Assume that t 2 Rng./. We then define the function 'X W Rng./ ! Œ0; 1/ by 'X .t / WD k E kX ;

t 2 Rng./:

(7.9.1)

We say that the function 'X is the fundamental function of the space X. Remark 7.9.3. We have to make sure that the fundamental function is well-defined by (7.9.1) for every rearrangement-invariant Banach function space X. To this end, note that the value of 'X .t / does not depend on the choice of the set E. Indeed, if EQ is another such set, then . E / D . EQ / D Œ0;.E // ; hence k E kX D k EQ kX : Example 7.9.4. Assume that .R; / is nonatomic. (i) Let X D Lp .R; / with p 2 Œ1; 1/. Then 1

'X .t / D t p ;

t 2 Œ0; .R//:

(7.9.2)

(ii) Let X D L1 .R; /. Then

´ 1 if t 2 .0; .R//; 'X .t / D 0 if t D 0:

(7.9.3)

(iii) Let ˆ be a Young function, let Lˆ .R; / be the corresponding Orlicz space endowed with the Orlicz norm and let X D Lˆ . Let ‰ denote the complementary Young function to ˆ. Then ´   if t 2 .0; .R//; t ‰ 1 1t 'X .t / D 0 if t D 0: (iv) Let ˆ be a Young function, let Lˆ .R; / be the corresponding Orlicz space endowed with the Luxemburg norm and let X D Lˆ . Then ´ 1 if t 2 .0; .R//; 1 1 'X .t / D ˆ . t / (7.9.4) 0 if t D 0:

261

Section 7.9 Fundamental function

Example 7.9.5. Assume that .R; / D .N [ ¹0º; m/, where m is the counting measure. (i) Let X D `p with p 2 Œ1; 1/. Then 1

'X .n/ D n p ; (ii) Let X D `1 . Then

n 2 N:

´ 0 if n D 0; 'X .n/ D 1 if n 2 N:

(7.9.5)

(7.9.6)

One of the most important properties of fundamental functions is how they reflect the associate spaces. We have the following general result. Theorem 7.9.6. Let X be a rearrangement-invariant Banach function space and let X 0 be the associate space of X. Then 'X .t /'X 0 .t / D t

(7.9.7)

for every t 2 Œ0; 1/ \ Rng./. Proof. Every fundamental function satisfies 'X .0/ D 0, so the assertion is obvious for t D 0. Let t 2 .0; 1/ \ Rng./. Then there exists a measurable set E  R satisfying .E/ D t . By the Hölder inequality (7.6.7), we get Z d  k E kX k E kX 0 D 'X .t /'X 0 .t /: tD E

Conversely, we have

²Z

'X .t / D k E kX D sup

E

³ g dI g 2 X 0 ; g  0; kgkX 0  1 :

(7.9.8)

Let g 2 X 0 be a nonnegative function satisfying kgkX 0  1 and set   Z 1 g d E .x/; x 2 R: h.x/ WD t E Now, using Theorem 7.8.1, we obtain  Z  1 g d 'X 0 .t / D khkX 0  kgkX 0  1: t E

(7.9.9)

Finally, we take the supremum over all such g on the left and use (7.9.8). We arrive at 'X .t /'X 0 .t /  1; t the desired converse inequality. The proof is complete.

262

Chapter 7 Rearrangement-invariant spaces

Remark 7.9.7. Let X be a rearrangement-invariant Banach function space and let 'X be its fundamental function. Then 'X is nondecreasing on Œ0; .R//, vanishing at zero and only at zero, continuous on .0; .R// (with the only possible discontinuity at the origin – see Example 7.9.4 (ii)), and t is nondecreasing on .0; .R//: (7.9.10) 'X .t / Indeed, the monotonicity of 'X on Œ0; .R// is a direct consequence of the lattice property of Banach function spaces (Remark 6.1.16 (i)). Since X 0 is also a rearrangementinvariant Banach function space, 'X 0 is nondecreasing on Œ0; .R//, too. However, by (7.9.7), we have t ; t 2 .0; .R//: 'X 0 .t / D 'X .t / As a consequence, we get (7.9.10). When .R; / is atomic, then 'X is automatically continuous. If .R; / is nonatomic, then the monotonicity of 'X implies that there are only countably many points of discontinuity of it on .0; 1/. That, however, is impossible thanks to (7.9.10). Theorem 7.9.8. Let .R; / be nonatomic and let X be a rearrangement-invariant space over .R; /. Then the following conditions on X are equivalent: (i) lim t!0C 'X .t / D 0; (ii) Xa D Xb ; (iii) .Xb / D X 0 . If, in addition,  is separable, then each of the properties (i), (ii), and (iii) is equivalent to (iv) Xb is separable. Proof. Theorem 6.4.1 shows that (ii),(iii). When  is a separable measure, then (ii),(iv) by Theorem 6.5.9. Assume that (i) holds and E  R satisfies T .E/ < 1. Then we claim that E 2 Xa . Indeed, let En # ; a.e. Then .E En / # 0 by the dominated convergence theorem, whence k E En kX D k E \En kX  'X ..E \ En // # 0: By Theorem 6.3.18, this establishes (ii). We have thus shown (i))(ii). Conversely, let (ii) hold, that is, Xa D Xb . Let E  R with 0 < .R/ < 1. Since .R; / is nonatomic, there exists a sequence ¹En º1 nD1 of subsets of E such that .En / D 2n .E/ and EnC1  En for every n 2 N. Then En # ; -a.e. and it follows from (ii) that E 2 Xa . Therefore, 'X .2n .E// D k En kX D k E En kX # 0: Because 'X is nondecreasing on .0; .R//, this shows (i). The proof is complete.

Section 7.9 Fundamental function

263

Remark 7.9.9. If lim t!0C 'X .t / > 0; then Xa D ¹0º: To see this, suppose that 0 6 f 2 Xa is a nonnegative function. Then there is " > 0 and a set E  R, .E/ < 1, such that " E  f . Hence E 2 Xa , since, by Theorem 6.3.12, Xa is an ideal. Then (cf. the argument used in the proof of Theorem 7.9.8) 'X .t / # 0 as t # 0. Should a function ', defined on an interval of the form Œ0; a/ for a 2 .0; 1, be a fundamental function of some rearrangement-invariant Banach function space, then it has to satisfy the properties mentioned in Remark 7.9.7. It will be useful to single out the class of functions obeying these requirements. Definition 7.9.10. Let a 2 .0; 1. A nondecreasing function ' W Œ0; a/ ! Œ0; 1/ is called quasi-concave on Œ0; a/ if '.t / D 0 ” t D 0; t is nondecreasing on .0; a/: '.t /

(7.9.11) (7.9.12)

Remark 7.9.11. Every nonnegative concave function on Œ0; a/ with a 2 .0; 1 that vanishes only at the origin is automatically quasi-concave on Œ0; a/. The converse is not true. For example, take a D 1, then the function ´ max¹1; t º if t 2 .0; 1/; '.t / WD 0 if t D 0; is quasi-concave but not concave on Œ0; a/. Remark 7.9.12. It follows from Remark 7.9.7 that if X is a rearrangement-invariant Banach function space, and 'X is its fundamental function, then 'X is quasi-concave on Œ0; .R//. The converse is in some sense true, too, namely, if ' is a quasi-concave function on Œ0; 1/, then there exists at least one rearrangement-invariant Banach function space X such that ' D 'X . We will finish this section with a simple but useful inequality. Lemma 7.9.13. Let X be a rearrangement-invariant Banach function space and let 'X be its fundamental function. Then, for every t 2 .0; 1/ \ Rng./ and every f 2 M.R; /, one has Z t

0

f  .s/ ds  'X .t /kf kX 0 :

(7.9.13)

Proof. Since t 2 .0; 1/ \ Rng./, there is a -measurable subset E of R such that .E/ D t . Thus, by the Hölder inequality (Theorem 6.2.6), Z 1 Z t f  .s/ ds D E .s/f  .s/ ds  k E kX kf kX 0 D 'X .t /kf kX 0 ; 0

0

as desired. The proof is complete.

264

Chapter 7 Rearrangement-invariant spaces

7.10 Endpoint spaces Definition 7.10.1. Let ' be a quasi-concave function on Œ0; .R//. Then, the collection M' D M' .R; / is defined as ´ μ M' WD g 2 M0 .R; /I

'.t /g  .t / < 1

sup t2.0;.R//

is called the Marcinkiewicz endpoint space. (We shall throughout use the shorter name Marcinkiewicz space as there is no confusion likely to arise. In fact, the Marcinkiewicz space Lp;1 which was introduced in Definition 3.18.3 is (at least for p 2 .1; 1/) a special case of an endpoint Marcinkiewicz space). Proposition 7.10.2. If ' is quasi-concave on Œ0; .R//, then the functional k  kM' M' DM' .R;/ , defined by kgkM' .R;/ WD

'.t /g  .t /;

sup

g 2 M0 .R; /;

(7.10.1)

t2.0;.R//

is a Banach function norm and the corresponding Marcinkiewicz space M' .R; / is a rearrangement-invariant Banach function space. Moreover, 'M' .t / D '.t /;

t 2 Œ0; .R//:

(7.10.2)

Proof. The functional g 7!

sup

¹g  .t /'.t /º;

g 2 M0 .R; /

0 0 arbitrarily. According to Lemma 6.3.6, there exists ı > 0 such that kf E kM' < ", whenever .E/ < ı. Thus, using that the measure space .R; / is resonant, for every t 2 .0; ı/ we have Z Z '.t / '.t / t  f .s/ ds D sup jf E j d t R 0 .E /Dt t Z '.t / t D sup .f E / .s/ ds t 0 .E /Dt Z '.r/ r  sup sup .f E / .s/ ds r 0 .E /Dt r >0 D

sup kf E kM'  ";

.E /Dt

which yields that lim t!0C '.t /f  .t / D 0. To prove an analogous result near infinity, consider an increasing sequence Rn of sets of finite measure whose union is R (which can be found because .R; / is -finite). Denote En D R n Rn . Then En # ;, so for every " > 0 there is k 2 N such that " sup '.t /.f Ek / .t / < : 2 t>0 For every t > 0 we have '.t /f  .t /  '.t /.f Ek / .t / C '.t /.f Rk / .t / Z '.t / t D '.t /.f Ek / .t / C .f Rk / .s/ ds t Z0 '.t /  jf j d:  '.t /.f Ek / .t / C t Rk By the property (P5) from Definition 6.1.5, every Banach functionR space is locally (i.e. on sets of finite measure) embedded to L1 . We thus have Rk jf j d
t0 . Thus, for t > t0 we have '.t /f  .t / < ", as required.

Now assume that a function f 2 M satisfies lim t!0C '.t /f  .t / D 0 and lim t!1 '.t /f  .t / D 0. Let .En /1 nD1 be any sequence of subsets of R such that En # ; -a.e. Choose " > 0 arbitrarily. Then we can find ı1 ; ı2 such that

274

Chapter 7 Rearrangement-invariant spaces

0 < ı1 < ı2 and for every t 2 .0; ı1 / [ .ı2 ; 1/ we have '.t /f  .t / < ". So, for every n 2 N, sup t2.0;ı1 /[.ı2 ;1/

'.t /.f En / .t / 

sup

'.t /f  .t /  ":

t2.0;ı1 /[.ı2 ;1/

Furthermore, we observe that lim t!1 f  .t / D 0. Indeed, if lim t!1 f  .t / D c > 0, we have f  .t /  c for every t > 0, which implies '.t /f  .t /  c'.t /; t > 0, contradicting the assumption lim t!1 '.t /f  .t / D 0. Denote

² A D x 2 RI jf .x/j 

³ " : 2'.ı2 /

Then .A/ < 1, which together with the fact that En # ; -a.e. implies limn!1 .En \ A/ D 0. The assumptions given on f obviously ensure that R1  0 f .s/ ds < 1, so we can find n0 2 N such that Z .En \A/ "ı1 f  .s/ ds < 2'.ı2 / 0 whenever n > n0 . Thus, for n > n0 we have sup '.t /.f En / .t /  '.ı2 /.f En / .ı1 /

t2Œı1 ;ı2 

 '.ı2 /.f En \A / .ı1 / C '.ı2 /.f En \Ac / .ı1 / Z '.ı2 / .En \A/  f .s/ ds C '.ı2 /.f En \Ac / .ı1 /  ":  ı1 0 We have just shown that whenever n > n0 , kf En kM' D sup '.t /.f En / .t /  "; t>0

which concludes the proof. (ii) Suppose that f 2 .M' /a . Then f 2 M' and M' ,! L1 (because lim t!1 '.t/ t > 0), i.e. f 2 L1 . Moreover, from the proof of part (i) it follows that lim t!0C '.t /f  .t / D 0. Conversely, assume that a function f 2 L1 satisfies lim t!0C '.t /f  .t / D 0. Let .En /1 nD1 be a sequence of subsets of R such that En # ; -a.e. A similar proof as in the part (i) yields lim

sup '.t /.f En / .t / D 0:

n!1 t2.0;1

Thus, to get f 2 .M' /a it only remains to show that lim sup '.t /.f En / .t / D 0;

n!1 t>1

275

Section 7.11 Almost-compact embeddings

which is, due to the assumption lim t!1

'.t/ t

> 0, the same as

lim sup t .f En / .t / D 0:

n!1 t>1

Using that f 2 L1 D .L1 /a , we obtain 

Z

t

.f En / .s/ ds lim sup t .f En / .t / D lim sup n!1 t>1 n!1 t>1 0 Z 1 .f En / .s/ ds D 0; D lim n!1 0

as required.

7.11 Almost-compact embeddings In this section we shall study an important relation between function spaces which is stronger than an ordinary embedding but weaker than a compact embedding. We have already once briefly met this relation, namely in connection with inclusions of type Lˆ1 ./ ,! E ˆ2 ./ between Orlicz spaces, where ˆ1 and ˆ2 are Young functions (see Remark 4.17.8). The almost-compact embeddings, called in literature also absolutely continuous embeddings, have recently proved its importance and usefulness and have been studied by several authors (let us mention, for instance, [54, 80, 206]). Our exposition is based on a mixture of results from [80, 204, 206]. Definition 7.11.1. Suppose that X and Y are Banach function spaces over a measure space .R; /. We say that X is almost-compactly embedded into Y if for every sequence ¹En º1 nD1 of -measurable subsets of R satisfying En ! ; -a.e., we have lim

sup kf En kY D 0:

n!1 kf k 1 X

We denote the almost-compact embedding of X into Y by 

X ,! Y: We first observe that, in the definition of an almost-compact embedding, the sequence ¹E n º can be¯taken nonincreasing. This assertion easily follows by replacing ®S ¹En º by k n Ek so we omit the proof. Theorem 7.11.2. Let X and Y be Banach function spaces over a -finite measure  space .R; /. Then X ,! Y if and only if lim

sup kf En kY D 0

n!1 kf k 1 X

holds for every sequence ¹En º1 nD1 satisfying En # ; -a.e.

276

Chapter 7 Rearrangement-invariant spaces

We next note that an almost-compact embedding is always preserved between swapped associate spaces. Theorem 7.11.3. Let X and Y be Banach function spaces over a -finite measure   space .R; /. Then X ,! Y if and only if Y 0 ,! X 0 . 

Proof. Suppose that X ,! Y . Let ¹En º1 nD1 be an arbitrary sequence of sets in R satisfying En # ; -a.e. Using the definition of the associate norm and the fact that Y 00 D Y , we get   Z 0 sup jfg En j d lim sup kg En kX D lim sup n!1 kgk

n!1 kgk

Y 0 1

Y 0 1



kf kX 1 R



Z

D lim

sup

D lim

sup kf En kY 00

D lim

sup kf En kY D 0;

n!1 kf k 1 X

sup

kgkY 0 1 R

jfg En j d

n!1 kf k 1 X n!1 kf k 1 X



i.e. Y 0 ,! X 0 , as required.   Now suppose that Y 0 ,! X 0 . From the first part of the proof we obtain X 00 ,! Y 00 ,  that is, by the Lorentz–Luxemburg theorem (Theorem 6.2.9), X ,! Y , as desired. The proof is complete. 

The following theorem provides a characterization of X ,! Y in terms of convergence -a.e. Theorem 7.11.4. Let X and Y be Banach function spaces over a -finite measure  space .R; /. Then X ,! Y if and only if for every sequence ¹fn º1 nD1 of measurable functions on R satisfying kfn kX  1 and fn ! 0 -a.e., one has kfn kY ! 0. 

Proof. Suppose that X ,! Y . First, we will construct a -measurable function g such that g > 0 on R and kgkY < 1. Let ¹Rn º1 nD1 be the sequence of sets of finite measure satisfying Rn " R. For every positive integer n, consider a function gn given by 1 1 gn D n   Rn : 2 1 C k Rn kY P Let us also define the function g by g D 1 nD1 gn . We have kgn kY D

1 1 1   k Rn kY < n : n 2 1 C k Rn kY 2

277

Section 7.11 Almost-compact embeddings

Thus

 n  1 n X  X X 1   gk   lim kgk kY  D 1: kgkY D lim  n!1  n!1  2k kD1

kD1

Y

kD1

Because, obviously, g > 0 on R, g has the required properties. Let ¹fn º1 nD1 be a sequence of -measurable functions on R satisfying kfn kX  1 and fn ! 0 -a.e. Choose " > 0 arbitrarily. Let En D ¹x 2 R W jfn .x/j  "g.x/º. Because fn ! 0 -a.e. and "g > 0 on R, for -a.e. x 2 R we have that x 2 En holds only for a finite number of positive integers n. This implies En ! ; -a.e. Observe that kfn kY D kfn En C fn Enc kY  kfn En kY C kfn Enc kY : 

The assumptions X ,! Y and kfn kX  1 give lim kfn En kY  lim

n!1

sup kh En kY D 0:

n!1 khk 1 X

Moreover, kfn Enc kY  k"gkY D "kgkY  ": Altogether, we have lim sup kfn kY  "; n!1

which holds for every " > 0. So, limn!1 kfn kY D 0. Conversely, suppose that for every sequence ¹fn º1 nD1 of -measurable functions on R satisfying kfn kX  1 and fn ! 0 -a.e., it holds kfn kY ! 0. Let ¹En º1 nD1 be a sequence of subsets of R satisfying En ! ; -a.e. Then we can find a sequence of functions ¹fn º1 nD1 such that kfn kX  1 and kfn En kY C

1 > sup kf En kY : n kf kX 1

Because En ! ; -a.e., we have fn En ! 0 -a.e. Due to the assumption, kfn En kY ! 0. Thus   1 D 0: lim sup kf En kY  lim kfn En kY C n!1 kf k 1 n!1 n X Our aim now is to study relations of an almost-compact embedding to other types of embeddings. In particular, in the following two theorems we will show that an almostcompact embedding is in general stronger than a usual (continuous) embedding but weaker than a compact embedding.

278

Chapter 7 Rearrangement-invariant spaces

Theorem 7.11.5. Suppose that .R; / is a -finite measure space and X and Y are  Banach function spaces over .R; / satisfying X ,! Y . Then X ,! Y . Proof. Let ¹fn º1 nD1 be a sequence in X such that kfn  f kX ! 0 for some f 2 X. To get a contradiction, assume that kfn  f kY 6! 0. Then we can find " > 0 and a subsequence ¹gk º1 of ¹fn º1 nD1 satisfying kgk  f kY  " for every k 2 N. kD1 of ¹gk º1 such that hl ! Because gk ! f in X, there is a subsequence ¹hl º1 lD1 kD1 

f -a.e. Using that X ,! Y , by Theorem 7.11.4 we obtain khl  f kY ! 0, which gives a contradiction. So, X ,! Y . Theorem 7.11.6. Suppose that .R; / is a -finite measure space and X and Y are  Banach function spaces over .R; / satisfying X ,!,! Y . Then X ,! Y . Proof. Let ¹fn º1 nD1 be a sequence in X such that kfn kX  1 for every n 2 N and fn ! 0 -a.e. To get a contradiction, assume that kfn kY 6! 0. Then there is " > 0 of ¹fn º1 and a subsequence ¹gk º1 nD1 satisfying kgk kY  " for every k 2 N. kD1 1 Because ¹gk ºkD1 is bounded in X and X ,!,! Y , we can find a subsequence of ¹gk º1 such that ¹hl º1 is convergent in Y . But hl ! 0 -a.e., so ¹hl º1 lD1 kD1 lD1 the limit must be 0. So, khl kY ! 0, which contradicts the assumption. Thus,  X ,! Y . Now we will use the results obtained so far to observe that in the cases that might be a possible interest, a Banach function space cannot be almost-compactly embedded into itself. Definition 7.11.7. We say that a -finite measure space .R; / is friendly if there exists a sequence ¹En º1 nD1 of -measurable subsets of R such that En # ; -a.e. and .En / > 0 for every n 2 N. Every sequence ¹En º1 nD1 having such properties will be called a friendly sequence. It is not hard to observe that a measure space is not friendly if and only if its nonatomic part has measure 0 and its atomic part consists of at most finitely many atoms. Remark 7.11.8. Measure spaces, which are not friendly, are exactly those on which Banach function spaces are almost-compactly embedded into itself. Indeed, an easy observation shows that in the definition of an almost-compact embedding, one can consider only friendly sequences. Thus, if the measure space .R; / is not friendly,  then for every pair of Banach function spaces X and Y we have X ,! Y and also 

Y ,! X. So, X ,! Y and Y ,! X, i.e. every two Banach function spaces X and Y  coincide and X ,! X.

Section 7.11 Almost-compact embeddings

279

Conversely, assume that .R; / is friendly and fix a friendly sequence ¹En º1 nD1 in R. The sets ¹En º can be with no loss of generality taken in such a way that .En / < 1 for every n 2 N. If X is a Banach function space over .R; /, one can 1 consider a sequence ¹fn º1 nD1 of functions in X defined by fn D kEn kX En . For every n 2 N, we have sup kf En kX  kfn kX D 1; kf kX 1



which shows that X ,! X cannot hold in this situation. Next theorem shows that, on atomic measure spaces, an almost-compact embedding coincides with a compact one. Theorem 7.11.9. Suppose that .R; / is a -finite completely atomic measure space.  Then for any pair of Banach function spaces X and Y over .R; /, X ,! Y holds if and only if X ,!,! Y . Proof. We have already observed that a compact embedding always implies the almostcompact one.  Assume that X ,! Y for some pair of Banach function spaces X and Y . Let ¹fn º1 nD1 be a bounded sequence in X. We will show that there is a subsequence ¹fnk º1 which converges pointwise to some function f (by the Fatou lemma, f 2 kD1 converges to f in the norm X). From Theorem 7.11.4 it follows that then ¹fnk º1 kD1 of the space Y , so X ,!,! Y , as required. Because the measure space .R; / is -finite, it has at most countably many atoms. We will consider only the case of infinitely many atoms, in the other case the proof proceeds similarly. the sequence containing all atoms of .R; /. Because ¹fn º1 Denote by ¹ak º1 nD1 kD1 is bounded in X, the sequence ¹fn .ak /º1 nD1 must be bounded for every k 2 N. We will construct a pointwise converging subsequence of ¹fn º1 nD1 by induction as follows: We formally set fn0 D fn . Suppose that, for some m 2 N [ ¹0º, we have already m 1 constructed the sequence ¹fnm º1 nD1 . Because .¹fn .amC1 /ºnD1 is bounded, we can mC1 1 m 1 mC1 find a subsequence ¹fn ºnD1 of ¹fn ºnD1 such that ¹fn .amC1 /º1 nD1 is convern 1 gent. Then the diagonal sequence ¹fn ºnD1 converges pointwise on the entire R, as required. Thus, in view of Theorem 7.11.9, in what follows we especially focus on almostcompact embeddings between Banach function spaces over a nonatomic measure space. The case when the measure of the space is 0 is trivial and it was discussed above (note that such a space is not friendly). Furthermore, in the following theorem we observe that there are no almost-compact embeddings between Banach function spaces over nonatomic infinite measure space.

280

Chapter 7 Rearrangement-invariant spaces

Theorem 7.11.10. Suppose that .R; / is a nonatomic -finite measure space with .R/ D 1. Then there is no pair of Banach function spaces X and Y over .R; /  such that X ,! Y . To prove the theorem we need the following auxiliary assertion. Lemma 7.11.11. Suppose that X is a Banach function space over a nonatomic finite measure space .R; / with .R/ D 1. Let c 2 .0; 1/. Then there are constants C1 ; C2 > 0 such that for every A  R satisfying .A/ D c, we have C1  k A kX  C2 : Proof. To get a contradiction, suppose that sup ¹k A kX W A  R; .A/ D cº D 1: Then we can find a sequence ¹An º1 nD1 of subsets of R such that .An / D c and 3 k An kX > n for every n 2 N. Because the measure space .R; / is nonatomic, 2 each An can be written as a union of n2 disjoint sets A1n ; A2n ; : : : ; Ann of measure nc2 . We have n2 X k Ain kX  k An kX > n3 ; iD1

so for every n 2 N there is i.n/ 2 ¹1; 2; : : : ; n2 º such that k Ai.n/ kX > n. We denote n

i.n/

Bn D A n . S Consider the set B D 1 nD1 Bn . Then .B/ 

1 X

.Bn / D c

nD1

1 X 1 < 1; n2

nD1

so, due to the property (P4) of Banach function spaces, k B kX < 1. On the other hand, for every n 2 N we have k B kX  k Bn kX > n; which is a contradiction. This justifies the existence of the constant C2 . Consider now the associate space X 0 . From the first part of the proof we know that there is a constant D > 0 such that k A kX 0  D, whenever A is a subset of R of measure c. Fix such a set A. Because the function kAAk 0 belongs to the unit ball of X X 0 , we have Z .A/ c k A kX D sup jf j d   : k A kX 0 D kf kX 0 1 A By setting C1 D

c D,

we get the result.

281

Section 7.11 Almost-compact embeddings

Proof of Theorem 7.11.10. Denote En D RnRn . Then En # ;. Moreover, .En / D 1 for every n 2 N. Combining this with the fact that .R; / is nonatomic, we get that for every n 2 N there is a set An  En with .An / D 1. Let X and Y be any pair of Banach function spaces over .R; /. According to Lemma 7.11.11, we can find positive constants C , D such thatfor every  n 2 N, k An kX  C and k An kY  D. The first inequality implies that  C1 An X  1, so    1 D  > 0:  sup kf En kY   An   C C kf kX 1 Y 

Therefore X ,! Y cannot be true. We shall now present a characterization of almost-compact embeddings on nonatomic measure spaces. It follows from this lemma that our definition of an almost-compact embedding is a generalization of the definition of absolutely continuous embedding, given in [80] for the case of rearrangement-invariant spaces over ..0; 1/; dx/. Lemma 7.11.12. Suppose that .R; / is a nonatomic measure space with 0 < .R/ < 1 and X and Y are Banach function spaces over .R; /. Then the following two statements are equivalent: 

(i) X ,! Y ; (ii) lim t!0C supkf kX 1 sup.E /t kf E kY D 0. Proof.

(i) ) (ii) Consider a function H defined by H.t / D

sup kf E kY ; t 2 .0; .R/:

sup

kf kX 1 .E /t

Clearly, H is nondecreasing on .0; .R/. Thus, it will be enough to prove lim

sup

sup

n!1 kf k 1 .E /a X n

kf E kY D 0

(7.11.1)

for some sequence ¹an º1 n # 0. Our proof will work for any such nD1 with aP sequence satisfying, moreover, that 1 nD1 an < 1. For every n 2 N we can find fn 2 X, En  R such that kfn kX  1, .En /  an and 1 (7.11.2) sup kf E kY < kfn En kY C : sup n kf kX 1 .E /an S Denote Fn D 1 kDn Ek . Then F1  F2  : : : and 

\ 1 nD1

 Fn

D lim .Fn /  lim n!1

n!1

1 X kDn

.Ek /  lim

n!1

1 X kDn

ak D 0:

282

Chapter 7 Rearrangement-invariant spaces 

This implies Fn # ; -a.e. Because En  Fn for every n 2 N and X ,! Y , we have lim kfn En kY  lim

n!1

sup kf En kY  lim

n!1 kf k 1 X

sup kf Fn kY D 0:

n!1 kf k 1 X

Using the inequality (7.11.2), we obtain (7.11.1). (ii) ) (i) Choose an arbitrary sequence ¹En º1 nD1 of subsets of R such that En # ; -a.e. Because .R/ < 1 we have \  1 En D 0: lim .En / D  n!1

nD1

Thus sup kf En kY  lim

lim

n!1 kf k 1 X

sup

sup

n!1 kf k 1 .E /D.E / X n

kf E kY D 0;



so X ,! Y . We shall now point out that L1 is almost-compactly embedded into every rearrangement-invariant Banach function space essentially different from itself. Theorem 7.11.13. Let .R; / be a nonatomic measure space satisfying 0 < .R/ < 1 and let X be a rearrangement-invariant Banach function space over .R; /. Denote by ' the fundamental function of X. Then the following statements are equivalent: (i) X ¤ L1 ; 

(ii) L1 ,! X; (iii) lim t!0C '.t / D 0. Proof.

(i) ) (iii) According to (i), there is a function f 2 X such that ess supR jf j D 1. Consider a sequence of sets .En /1 nD1 defined by En D ¹x 2 R W jf .x/j  nº: Because ess supR jf j D 1, we have .En / > 0, n 2 N. From the inequality n En  jf j we obtain k En kX  n1 kf kX . Hence, 0 D inf.E />0 k E kX D inf t>0 '.t /. Since the function ' is nondecreasing on .0; 1/, we get (iii).

(iii) ) (ii) The constant function 1 belongs to the unit ball of L1 and for every function f from the unit ball of L1 we have jf j  1 -a.e. This implies lim

sup

t!0C kf k

sup kf E kX D lim

L1 1 .E /t

sup k E kX D lim '.t / D 0:

t!0C .E /t

Using Lemma 7.11.12 we get the result.

t!0C

283

Section 7.11 Almost-compact embeddings

(ii) ) (i) Because every nonatomic measure space .R; / with .R/ > 0 is 

friendly, L1 ,! L1 cannot be true. Near the other endpoint space, L1 , we have an analogous result. Namely, L1 is an almost-compact target for every rearrangement-invariant Banach function space essentially different from itself. This assertion can be obtained from Theorem 7.11.13 by duality arguments. Theorem 7.11.14. Let .R; / be a nonatomic measure space satisfying 0 < .R/ < 1 and let X be a rearrangement-invariant Banach function space over .R; /. Denote by ' the fundamental function of X. Then the following statements are equivalent: (i) X ¤ L1 ; 

(ii) X ,! L1 ; (iii) lim t!0C

t '.t/

D 0.

We shall present an important necessary condition for an almost-compact embedding between two rearrangement-invariant spaces in terms of their fundamental functions. Lemma 7.11.15. Suppose that .R; / is a nonatomic measure space satisfying 0 < .R/ < 1 and let X and Y be rearrangement-invariant Banach function spaces over .R; /. Let S denote the set of nonnegative nonzero simple functions on R. Then the following conditions are equivalent. 

(i) X ,! Y ; (ii) lim t!0C supkf kX 1 kf  Œ0;t/ kYN D 0; (iii) lim t!0C supu2S Proof.

ku Œ0;t/ kYN ku Œ0;t/ kXN

D 0. 

(i) , (ii) Due to Lemma 7.11.12, X ,! Y is equivalent to lim

sup

sup kf E kY D 0:

t!0C kf k 1 .E /t X

Thus it is sufficient to show that, for every f 2 X and for every t 2 .0; .R//, sup kf E kY D kf  Œ0;t/ kYN :

.E /t

Fix f 2 X and t 2 .0; .R//. Whenever E is a measurable subset of R with .E/  t , we have kf E kY D k.f E / kYN D k.f E / Œ0;t/ kYN  kf  Œ0;t/ kYN ;

284

Chapter 7 Rearrangement-invariant spaces

and therefore

sup kf E kY  kf  Œ0;t/ kYN :

.E /t

For f 2 X and t 2 .0; .R//, we can find a measurable set F  R with .F / D t such that f  Œ0;t/ D .f F / (this follows from the proof of Lemma 7.3.6). Thus, we can write sup kf E kY D sup k.f E / kYN  k.f F / kYN D kf  Œ0;t/ kYN ;

.E /t

.E /t

which gives the reverse inequality. (ii) , (iii) We will show that for every t 2 .0; .R// sup kf  Œ0;t/ kYN D sup

kf kX 1

u2S

ku Œ0;t/ kYN ; ku Œ0;t/ kXN

which is obviously enough for the proof. Suppose that f 2 X, kf kX  1. Then we have   f f  f Œ0;t/  Œ0;t/ D Œ0;t/ ; kf kX kf kX which gives      f   kf  Œ0;t/ kYN   Œ0;t/   sup kg Œ0;t/ kYN ;  kf k X kgkX D1 YN so

sup kf  Œ0;t/ kYN 

kf kX 1

sup kf kX D1

kf  Œ0;t/ kYN :

The reverse inequality is obvious, thus sup kf  Œ0;t/ kYN D

kf kX 1

sup kf kX D1

kf  Œ0;t/ kYN :

Furthermore, sup kf kX D1



kf Œ0;t/ kYN D D D

     f  sup  Œ0;t/  N kf k

0¤f 2X

X



sup 0¤f 2X

sup 0¤f 2X

kf Œ0;t/ kYN kf kX kf  Œ0;t/ kYN : kf  kXN

Y

Section 7.11 Almost-compact embeddings

285

We need to show that sup 0¤f 2X

Because kf

k XN

kf  Œ0;t/ kYN kf  Œ0;t/ kYN D sup :  kf  kXN 0¤f 2X kf Œ0;t/ kXN

 kf  Œ0;t/ kXN , it must be sup

0¤f 2X

kf  Œ0;t/ kYN kf  Œ0;t/ kYN  sup :  kf  kXN 0¤f 2X kf Œ0;t/ kXN

On the other hand, whenever f 2 X, f ¤ 0, we can find a measurable set F such that .F / D t and f  Œ0;t/ D .f F / D .f F / Œ0;t/ . Then k.f F / Œ0;t/ kYN kg  Œ0;t/ kYN kf  Œ0;t/ kYN D  sup ; kf  Œ0;t/ kXN k.f F / kXN kg  kXN 0¤g2X which gives the reverse inequality. Finally, we observe that the supremum can be taken over the (smaller) set S instead of X n ¹0º. Indeed, for every f 2 X, f ¤ 0, we can find a sequence  ¹un º1 nD1 with un 2 S , .n 2 N/, and un " jf j. This implies un Œ0;t/ "  f Œ0;t/ , and thus kf  Œ0;t/ kYN kun Œ0;t/ kYN D ; n!1 ku Œ0;t/ k N kf  Œ0;t/ kXN n X lim

which gives the result. The next result yields a necessary condition for an almost-compact embedding in terms of fundamental functions. Theorem 7.11.16. If .R; / is a nonatomic measure space satisfying 0 < .R/ < 1 and X and Y are rearrangement-invariant Banach function spaces over .R; / 

such that X ,! Y , then 'Y .t / D 0; t!0C 'X .t / where 'X , 'Y are fundamental functions of X, Y respectively. lim

Proof. The function f D 1 on R belongs to S defined in the Lemma 7.11.15. Thus, for t 2 .0; .R/, kf  Œ0;t/ kY ku Œ0;t/ kY 'Y .t / D  sup :  'X .t / kf  Œ0;t/ kX u2S ku Œ0;t/ kX According to the lemma, we have lim

t!0C

'Y .t / D 0: 'X .t /

286

Chapter 7 Rearrangement-invariant spaces

Corollary 7.11.17. Let .R; / be a nonatomic measure space satisfying 0 < .R/ < 1 and let ' be a positive nondecreasing concave function on .0; .R//. Then it  does not hold that ƒ' ,! M' . Proof. ƒ' and M' have the same fundamental function ' so the necessary condition for almost-compact embedding from Theorem 7.11.16 cannot hold. Our next example shows that the necessary condition from Theorem 7.11.16 is not sufficient for an almost-compact embedding. Example 7.11.18. Suppose that X, Y are rearrangement-invariant Banach function spaces with fundamental functions 'X , 'Y , respectively. We will show that the conD 0 does not imply X ,! Y . In particular, it does not imply dition lim t!0C ''XY .t/ .t/ 

X ,! Y . Let p 2 .1; 1/. Denote by p 0 the conjugate index satisfying p1 C p10 D 1. We will consider for X the Marcinkiewicz space Lp;1 and for Y the Lorentz-Zygmund space Lp;1I1 over ..0; 1/; 1 /, consisting of all measurable functions f such that 1

kf kp;1 D sup f  .t /t p < 1 t2.0;1/

and

Z kf kp;1I1 D

1

0

1

f  .s/s p 1 ds D e  log s

Z

f  .s/

1 0

1

s p0 .e  log s/

ds < 1;

respectively. We note that the functionals k  kp;1 and k  kp;1I1 are equivalent to rearrangement-invariant Banach function norms. Next, we have Lp;1 6,! Lp;1I1 (see Theorem 9.5.14). Moreover, if we denote by ' the fundamental function of Lp;1 and by the fundamental function of Lp;1I1 , then, for every t 2 .0; 1/, we have 1

1

'.t / D sup .0;t/ .s/s p D t p ; s2.0;1/

while

Z .t / D

t 0

1 s

1 p0

.e  log s/

ds:

Because lim '.t / D lim

t!0C

t!0C

.t / D 0;

we can use the L’Hospital rule to get lim

t!0C

0 .t / .t / p D lim 0 D lim D 0: t!0 t!0 '.t / C ' .t / C e  log t

287

Section 7.11 Almost-compact embeddings

Example 7.11.19. Let .R; / be a nonatomic measure space such that .R/ D 1. Suppose that ˆ1 , ˆ2 are Young functions. Then the almost-compact embedding  Lˆ1 .R; / ,! Lˆ2 .R; / between the corresponding Orlicz spaces holds if and only if, for every  > 0, ˆ2 .t / D0 (7.11.3) lim t!1 ˆ1 .t / (cf. Theorems 4.17.7 and 4.17.9). Remark 7.11.20. Suppose that p 2 .1; 1/ and consider the Young function ˆ.t / D t p . Then the Orlicz space Lˆ coincides with the Lebesgue space Lp . The characterization of almost-compact embedding between Orlicz spaces from the previous example together with Theorems 7.11.13 and 7.11.14 shows that for 1  p; q  1, 

Lp ,! Lq holds if and only if q < p. Note that Lp ,! Lq if and only if q  p, while Lp ,!,! Lq is never true. In the rest of section we will present a complete characterization of all possible mutual almost-compact embeddings among the Lorentz and Marcinkiewicz endpoint spaces. We shall work for our typographical convenience on the measure space ..0; 1/; dx/. This of course can be done with no loss of generality and the results of this section can be easily extended to all nonatomic finite measure spaces. Suppose that ' is a quasi-concave function on .0; 1/. In the following text, 'Q det for every t 2 .0; 1/. notes the quasi-concave function satisfying '.t Q / D '.t/ Lemma 7.11.21. Let ' and be quasi-concave functions on .0; 1/. Suppose that there exist positive constants C1 , C2 such that C1 '.t / 

.t /  C2 '.t /

for every t 2 .0; 1/. Then M' D M . Proof. Assume that f 2 M' . Then f 2 M , because kf kM D sup t2.0;1/

.t /f  .t /  C2 sup '.t /f  .t / D C2 kf kM' < 1: t2.0;1/

The converse embedding follows from symmetry. Lemma 7.11.22. Suppose that ' is a quasi-concave function on .0; 1/. Let ˛ be the least nondecreasing concave majorant of '. Q Then M'0 D ƒ˛ . '.t/ t Proof. The function ˛ satisfies ˛.t/ 2  '.t/  ˛.t / for t 2 .0; 1/. Thus also 2  t  '.t / on .0; 1/. Due to Lemma 7.11.21, M' D M˛Q . ˛.t/ First, we will show that M'0  ƒ˛ . This is equivalent to ƒ0˛  M'00 D M' D M˛Q . But ƒ0˛ has the fundamental function ˛Q and M˛Q is the largest rearrangement-invariant space with this fundamental function, so ƒ0˛  M˛Q .

288

Chapter 7 Rearrangement-invariant spaces

On the other hand, because M' D M˛Q , we have M'0 D M˛Q0 . Using that M˛Q0 has the fundamental function ˛ and ƒ˛ is the smallest rearrangement-invariant space with fundamental function ˛, we obtain ƒ˛  M˛Q0 D M'0 . Lemma 7.11.23. Let ' and be positive nondecreasing concave functions on .0; 1/. Suppose that there exist positive constants C1 , C2 such that C1 '.t / 

.t /  C2 '.t /

(7.11.4)

for every t 2 .0; 1/. Then ƒ' D ƒ . Proof. According to Lemma 7.11.22, ƒ' D M'0Q and ƒ tion (7.11.4) gives Q /  C2 Q .t / C1 Q .t /  '.t

D M 0Q . The assump-

for every t 2 .0; 1/. So, due to Lemma 7.11.21, M'Q D M Q , thus also ƒ' D M'0Q D M 0Q D ƒ . Theorem 7.11.24. Suppose that ' and are positive nondecreasing concave functions on .0; 1/. Then the following four statements are equivalent. 

(i) ƒ' ,! ƒ ; 

(ii) M' ,! M ; 

(iii) ƒ' ,! M ; (iv) lim t!0C

.t/ '.t/

D 0.

Proof. According to Theorem 7.11.16, each of the conditions (i), (ii), (iii) implies (iv). (iv) ) (i) Due to Lemma 7.11.15, we only need to prove that lim sup

t!0C u2S

ku .0;t/ kƒ D 0; ku .0;t/ kƒ'

where S denotes the set of nonnegative nonzero simple functions on .0; 1/. Suppose that u 2 S . Given t 2 .0; 1/, we have 

u .0;t/ D

n X

ci .0;ti / ;

iD1

where ci > 0, i D 1; 2; : : : ; n and 0 < t1 <    < tn  t . Because .ti / .s/  sup ; '.ti / '.s/ 0 0; D 0;

< 0;

D 0; ˇ0 C

1 < p < 1; p D 1; either ˛1 C or ˛1 C

1 q 1 q

1 q

1 q

< 0;

< 0;

D 0; ˇ1 C

1 q

< 0;

or q D 1; ˛1 D 0; ˇ1 D 0; if

p D 1; ˛1 C

if

p D 1; ˛1 C

if

p D 1; ˛1 C 1 q

if

p D 1; ˛1 C

if

p D 1; either ˛1 C

1 q 1 q 1 q

> 0; D 0; ˇ1 C D 0; ˇ1 C

1 q 1 q

> 0;

1 q

< 0;

D 0;

< 0; 1 q

D 0; ˇ1 C

or q D 1; ˛1 D 0; ˇ1 D 0:

320

Chapter 9 Generalized Lorentz–Zygmund spaces

Proof. The assertion follows by an elementary calculation and is left to the reader. In the case (ii) one has to use (9.3.3).

9.5 Embeddings between GLZ spaces Our next aim is to study relations between the spaces Lp;q;A;B and L.p;q;A;B/ . Theorem 9.5.1. Let 1  p  1, 0 < q  1, A D .˛0 ; ˛1 /, B D .ˇ0 ; ˇ1 / 2 R2 , and assume that one of the conditions in (9.3.2) is satisfied. (i) If 1 < p  1, then

L.p;qIA;B/ D Lp;qIA;B :

(9.5.1)

(ii) The space L.1;1IA;B/ coincides with the space if

˛1 C 1 < 0; either ˛0 C 1 < 0;

L1;1IAC1;B L1;1I.0;˛1 C1/;.ˇ0 C1;ˇ1 /

if if

or ˛0 C 1 D 0; ˇ0 C 1 < 0; ˛1 C 1 < 0; ˛0 C 1 > 0; ˛1 C 1 < 0; ˛0 C 1 D 0; ˇ0 C 1 > 0;

L1;1I.0;˛1 C1/;.0;ˇ1 /;.1;0/ L1;1I.0;0/;.0;ˇ1 C1/

if if

L1;1I.0;˛1 C1/;.0;ˇ1 /

L1;1I.˛0 C1;0/;.ˇ0 ;ˇ1 C1/ L1;1I.0;0/;BC1 L1;1I.0;0/;.0;ˇ1 C1/;.1;0/

˛1 C 1 < 0; ˛0 C 1 D 0; ˇ0 C 1 D 0; ˛1 C 1 D 0; ˇ1 C 1 < 0; and either ˛0 C 1 < 0; or ˛0 C 1 D 0; ˇ0 C 1 < 0; ˛1 C 1 D 0; ˇ1 C 1 < 0; ˛0 C 1 > 0; ˛1 C 1 D 0; ˇ1 C 1 < 0; ˛0 C 1 D 0;

if if

ˇ0 C 1 > 0; ˛1 C 1 D 0; ˇ1 C 1 < 0; ˛0 C 1 D 0; ˇ0 C 1 D 0:

if

(iii) Let 1 < q  1. Then L1;qIAC1;B ¤ L.1;qIA;B/

if

˛0 C

1 1 > 0; ˛1 C < 0; q q

(9.5.2)

and L1;qI. 10 ; 10 /;BC1 ¤ L.1;qI. 1 ; 1 /;B/ q

q

q

if

q

ˇ0 C

1 1 > 0; ˇ1 C < 0: (9.5.3) q q

(iv) Let 0 < q < 1. Then L1;qIAC 1 ;B ¤ L.1;qIA;B/ q

if

˛0 C

1 1 > 0; ˛1 C < 0 q q

321

Section 9.5 Embeddings between Generalized Lorentz–Zygmund spaces

and L1;qI.0;0/;BC 1  L.1;qI. 1 ; 1 /;B/ q

Proof.

q

ˇ0 C

if

q

1 1 > 0; ˇ1 C < 0: q q

(i) Assume first that 1  q  1. Since p > 1, the Hardy inequality   Z t  1 1   11 A   t p q ` .t /``B .t / t 1  . p  q `A .t /``B .t /g.t / g.s/ ds (9.5.4) t   q 0

q

holds for every g 2 M C .0; 1/ (Theorem 3.11.8). Applied to g D f  , (9.5.4) implies Lp;qIA;B ,! L.p;qIA;B/ . Combined with (9.2.1), this yields (9.5.1). If 0 < q < 1, we use an analogous argument, applying [131, Theorem 2.2]. (ii) By the Fubini theorem, Z kf k.1;1IA;B/ D

1



Z

1

f .s/ 0

t

1 A

B



` .t /`` .t / dt ds:

s

Calculating the inner integral, we obtain the assertion. (iii) Both embeddings in (9.5.2) and (9.5.3) follow from the corresponding Hardy inequality (cf. [74, Lemmas 4.2 and 4.3]). The distinction of the spaces follows by comparing their fundamental functions (cf. Lemma 9.4.1). (iv) This follows from [131, Theorem 2.2]. Now we shall prove some auxiliary results which will be needed later. Lemma 9.5.2. Let 0 < q  1, and A; B 2 R2 . Assume that for each i 2 ¹0; 1º one of the following conditions holds: ˛i C

1 < 0; q

˛i C

1 1 D 0; ˇi C < 0; q q

q D 1; ˛i D 0; ˇi D 0:

Then for all f 2 L1;qIA;B , 1

1

kt  q `˛1 .t /``ˇ1 .t /f  .t /kq;.1;1/ . kt  q `˛0 .t /``ˇ0 .t /f  .t /kq;.0;1/ : Proof. Our assumptions imply that 1

1

kt  q `˛1 .t /``ˇ1 .t /kq;.1;1/  1  kt  q `˛0 .t /``ˇ0 .t /kq;.0;1/ :

(9.5.5)

Consequently, for all f 2 L1;qIA;B , 1

1

kt  q `˛1 .t /``ˇ1 .t /f  .t /kq;.1;1/  f  .1/ kt  q `˛1 .t /``ˇ1 .t /kq;.1;1/ 1

1

 f  .1/ kt  q `˛0 .t /``ˇ0 .t /kq;.0;1/  kt  q `˛0 .t /``ˇ0 .t /f  .t /kq;.0;1/ :

322

Chapter 9 Generalized Lorentz–Zygmund spaces

Corollary 9.5.3. Let all the assumptions of Lemma 9.5.2 be satisfied. Then L1;qIA;B D L1;qI˛0 ;ˇ0 .0; 1/: If moreover q D 1, then L1;1IA;B D L1;1I˛0 ;ˇ0 .0; 1/ D L1;1I.˛0 ;0/;.ˇ0 ;0/ :

(9.5.6)

The following result is a dual version of Lemma 9.5.2. Lemma 9.5.4. Let 0 < q  1, and A; B 2 R2 . Assume that for each i 2 ¹0; 1º one of the following conditions holds: 1 < 0; q 1 1 ˛i C D 0; ˇi C < 0; q q q D 1; ˛i D 0; ˇi D 0: ˛i C

Then for all f 2 L.1;qIA;B/ , 1

1

kt 1 q `˛0 .t /``ˇ0 .t /f  .t /kq;.0;1/ . kt 1 q `˛1 .t /``ˇ1 .t /f  .t /kq;.1;1/ : Proof. Using (9.5.5), we have for all f 2 L.1;qIA;B/ , 1

kt 1 q `˛0 .t /``ˇ0 .t /f  .t /kq;.0;1/ 1

D kt  q `˛0 .t /``ˇ0 .t /

Z

t

f  .s/ dskq;.0;1/ 0 Z 1 1  q ˛0 ˇ0  kt ` .t /`` .t /kq;.0;1/ f  .s/ ds 0 Z 1 1 f  .s/ ds  kt  q `˛1 .t /``ˇ1 .t /kq;.1;1/ 0 Z t  q1 ˛1 ˇ1   kt ` .t /`` .t / f .s/ dskq;.1;1/ 0

D kt

1 q1 ˛1

`

.t /``

ˇ1

.t /f  .t /kq;.1;1/ :

Corollary 9.5.5. Let A; B 2 R2 . Assume that for each i 2 ¹0; 1º one of the following conditions holds: ˛i < 0; ˛i D 0; ˇi < 0; ˛i D 0; ˇi D 0: Then L.1;1IA;B/ D L.1;1I.0;˛1 /;.0;ˇ1 // :

323

Section 9.5 Embeddings between Generalized Lorentz–Zygmund spaces

We shall now formulate modifications of Lemmas 9.3.1 and 9.4.1 and of Theorem 9.5.1 for the case when .R/ < 1. The proofs are analogous to the corresponding ones above and therefore omitted. Lemma 9.5.6. Let .R/ < 1, 0 < p; q  1, and ˛; ˇ 2 R. Let X be one of the spaces Lp;qI˛;ˇ , L.p;qI˛;ˇ / . Then X is nontrivial if and only if one of the following conditions holds: 8 ˆ ˆp < 1; ˆ ˆ