Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces [1st ed.] 3319427563, 978-3-319-42756-0, 978-3-319-42758-4

Table of contents :
Front Matter....Pages i-xvii
Front Matter....Pages 1-3
Definition of Symmetric Spaces....Pages 5-15
Spaces L p ,  1 ≤ p ≤ ∞ ....Pages 17-27
The Space L 1 ∩L ∞ ....Pages 29-40
The Space L 1 +L ∞ ....Pages 41-56
Front Matter....Pages 57-58
Embeddings L1 ∩ L∞ ⊆ X ⊆ L1 + L∞ ⊆ L0 ....Pages 59-70
Embeddings. Minimality and Separability. Property (A)....Pages 71-81
Associate Spaces....Pages 83-93
Maximality. Properties (B) and (C)....Pages 95-110
Front Matter....Pages 111-113
Lorentz Spaces....Pages 115-125
Quasiconcave Functions....Pages 127-137
Marcinkiewicz Spaces....Pages 139-150
Embedding \(\varLambda _{\widetilde{V }}^{0}\) ⊆ X ⊆ MV* ....Pages 151-167
Front Matter....Pages 169-170
Definition and Examples of Orlicz Spaces....Pages 171-182
Separable Orlicz Spaces....Pages 183-193
Duality for Orlicz Spaces....Pages 195-205
Comparison of Orlicz Spaces....Pages 207-216
Intersections and Sums of Orlicz Spaces....Pages 217-235
Back Matter....Pages 237-259

Citation preview

Developments in Mathematics

Ben-Zion A. Rubshtein Genady Ya. Grabarnik Mustafa A. Muratov Yulia S. Pashkova

Foundations of Symmetric Spaces of Measurable Functions Lorentz, Marcinkiewicz and Orlicz Spaces

Developments in Mathematics VOLUME 45 Series Editors: Krishnaswami Alladi, University of Florida, Gainesville, FL, USA Hershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel

More information about this series at http://www.springer.com/series/5834

Ben-Zion A. Rubshtein • Genady Ya. Grabarnik Mustafa A. Muratov • Yulia S. Pashkova

Foundations of Symmetric Spaces of Measurable Functions Lorentz, Marcinkiewicz and Orlicz Spaces

123

Ben-Zion A. Rubshtein Mathematics Ben Gurion University of the Negev Be’er Sheva, Israel

Genady Ya. Grabarnik Mathematics and Computer Science St. John’s University New York, NY, USA

Mustafa A. Muratov Mathematics and Computer Sciences V.I. Vernadsky Crimean Federal University Simferopol, Russian Federation

Yulia S. Pashkova Mathematics and Computer Sciences V.I. Vernadsky Crimean Federal University Simferopol, Russian Federation

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-42756-0 ISBN 978-3-319-42758-4 (eBook) DOI 10.1007/978-3-319-42758-4 Library of Congress Control Number: 2016953731 Mathematics Subject Classification (2010): 46E30, 46E35, 26D10, 26D15, 46B70, 46B42, 46B10, 47G10 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Inna, Tanya, Andrey Fany, Yaacob, Laura, Golda Ajshe, Elvira, Enver, Lenur Anna, Ludmila, Sergey

Foreword

This book is the first part of the textbook Symmetric Spaces of Measurable Functions. It contains the main definitions and results of the theory of symmetric (rearrangement invariant) spaces. Special attention is paid to the classical spaces Lp , Lorentz, Marcinkiewicz, and Orlicz spaces. The book is intended for master’s and doctoral students, researchers in mathematics and physics departments, and as a general manual for scientists and others who use the methods of the theory of functions and functional analysis.

vii

Preface

This book is the first, basic, part of a more advanced textbook Symmetric Spaces of Measurable Functions. It contains an introduction to the theory, including a detailed study of Lorentz, Marcinkiewicz, and Orlicz spaces. The theory of symmetric (rearrangement invariant) function spaces goes back to the classical spaces Lp , 1  p  1. The theory was intensively developed during the last century, mainly in the context of general Banach lattices. It presents many interesting and deep results having important applications in various areas of function theory and functional analysis. The theory has a great many applications in interpolation of linear operators, ergodic theory, harmonic analysis, and various areas of mathematical physics. The authors of this book (at different years and in different countries) have studied and taught the theory of symmetric spaces. They discovered independently the following surprising fact: despite the abundance of monographs, there was no book suitable for our purposes either in the Russian mathematical literature or in the mathematical literature of the rest of the world. In fact, we wished to have a book with a relatively small volume that met the following criteria: 1. The book should contain basic concepts and results of the general theory of symmetric spaces with the main focus on a detailed exposition of classical spaces Lp ; 1  p  1, and Lorentz, Marcinkiewicz, and Orlicz spaces, as well. 2. The book should be accessible to master’s students, doctoral students, and researchers in mathematics and physics departments who are familiar with the basics of the measure theory and functional analysis in the framework of standard university courses. 3. The material of the book should correspond to a one-semester special course of lectures (about 4 months or 17–18 weeks). 4. The presentation should not require any additional source except standard references on basic concepts and theorems of measure theory and functional analysis.

ix

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Preface

In our opinion, this book, offered now to the reader, completely meets the above requirements. We can point out three main sources from which the material of the book was adopted. First is a monograph by S. G. Krein, J. I. Petunin, E. M. Semenov, Interpolation of Linear Operators. The second source is two volumes of J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I. Sequence Spaces and Classical Banach Spaces II. Function Spaces. Third, the part devoted to Orlicz spaces is based on a nice exposition of this theme in the book by G. A. Edgar, L. Sucheston, Stopping Times and Directed Processes. Our book includes four parts comprising seventeen chapters. This allows us to divide the corresponding one-semester lecture course into 4 months or 17 weeks, and rigorously restricts, in turn, the volume of material. As a result a great many important related topics have not been included in the main part of the book. The reader can find this additional material in the exercises at the end of each part and in the section called “Complements” at the end of the book. Throughout the main exposition, we deal only with symmetric spaces on the half-line RC D Œ0; 1/, while the symmetric spaces on the interval Œ0; 1 and the symmetric sequence spaces are considered in the exercises and complements. Each of the four parts begins with an overview and then is divided into chapters. Each part concludes with exercises and notes. Complements are located at the end of the book together with references and an index. Complements and exercises are intended for independent study. The list of references contains some historical material, the books and articles from which we took terminology, results, and their proofs, and also a bibliography for further rending. The list of references is not, of course, comprehensive, but it points out, we hope, the most of important directions of the theory. Be’er Sheva, Israel New York, NY, USA Simferopol, Russia Simferopol, Russia

Ben-Zion A. Rubshtein Genady Ya. Grabarnik Mustafa A. Muratov Yulia S. Pashkova

Contents

Part I Symmetric Spaces. The Spaces Lp , L1 \ L1 , L1 C L1 1

Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Distribution Functions, Equimeasurable Functions. . . . . . . . . . . . . . . . . 1.2 Generalized Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Decreasing Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Integrals of Equimeasurable Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Example. Lp , 1  p  1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 9 11 12 13 14

2

Spaces Lp ; 1  p  1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Hölder’s and Minkowski’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Completeness of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Separability of Lp , 1  p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 21 23 24

3

The Space L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Intersection of the Spaces L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Space L01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Approximation by Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Measure-Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Approximation by Simple Integrable Functions . . . . . . . . . . . . . . . . . . . .

29 29 30 33 35 38

4

The Space L1 C L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Maximal Property of Decreasing Rearrangements . . . . . . . . . . . . 4.2 The Sum of L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Embeddings L1  L1 C L1 and L1  L1 C L1 . The Space R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 45 49 51 55

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Part II Symmetric Spaces. The Embedding Theorem. Properties .A/; .B/; .C/ 5

Embeddings L1 \ L1  X  L1 C L1  L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Embedding Theorem L1 \ L1  X  L1 C L1 . . . . . . . . . . . . . 5.3 The Space L0 and the Embedding L1 C L1  L0 . . . . . . . . . . . . . . . . .

59 59 61 66

6

Embeddings. Minimality and Separability. Property .A/ . . . . . . . . . . . . . . 6.1 Embedded Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Intersection and the Sum of Two Symmetric Spaces . . . . . . . . . . 6.3 Minimal Symmetric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Minimality and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Separability and Property .A/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 75 76 79

7

Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Dual and Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Maximal Property of Products f  g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Examples of Associate Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Comparison of X1 and X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 85 90 92

8

Maximality. Properties (B) and (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1 The Second Associate Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2 Maximality and Property .B/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.3 Embedding X  X11 and Property .C/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.4 Property .AB/. Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Part III Lorentz and Marcinkiewicz Spaces 9

Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Definition of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Maximality. Fundamental Functions of Lorentz Spaces . . . . . . . . . . . 9.3 Minimal and Separable Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Four Types of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 119 120 124

10

Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Fundamental Functions and Quasiconcave Functions . . . . . . . . . . . . . . 10.2 Examples of Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The Least Concave Majorant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Quasiconcavity of Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 128 130 135 136

11

Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Maximal Function f  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Definition of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Duality of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . 11.4 Examples of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 143 144 147

Contents

12

Embedding 0  X  MV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e V 12.1 The Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Renorming Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Examples of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . 12.4 Comparison of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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151 151 156 157 162 163 166

Part IV Orlicz Spaces 13

Definition and Examples of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Orlicz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Fundamental Functions of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Examples of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 173 177 178

14

Separable Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Young Classes Y˚ and Subspaces H˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Separability Conditions for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The .2 / Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Examples of Orlicz Spaces with and Without the .2 / Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 185 190

15

Duality for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 The Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Duality for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Duality and the .2 / Condition. Reflexivity . . . . . . . . . . . . . . . . . . . . . . . .

195 195 197 200 205

16

Comparison of Orlicz Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Comparison of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Embedding Theorem for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Coincidence Theorem for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . 16.4 Zygmund Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 209 212 213

17

Intersections and Sums of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 The Intersection and the Sum of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . 17.2 The Spaces L˚ C L1 and L \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 The Spaces L˚ C L1 and L \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 The Spaces Lp \ Lq and Lp C Lq , 1  p  q  1 . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217 217 220 223 224 229 234

Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Symmetric Spaces on General Measure Spaces . . . . . . . . . . . . . . . . . . . . 2 Symmetric Spaces on Œ0; 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Symmetric Sequence Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

237 237 239 241

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4 5 6 7 8

The Spaces Lp ; 0 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weak Sequential Completeness. Property .AB/ . . . . . . . . . . . . . . . . . . . . The Least Concave Majorant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Minimal Part M0V of the Marcinkiewicz Space MV . . . . . . . . . . . . Lorentz Spaces Lp;q and Orlicz–Lorentz Spaces . . . . . . . . . . . . . . . . . . . .

243 245 246 247 248

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5

Fig. 1.6 Fig. 1.7

Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 2.1 Fig. 2.2 Fig. Fig. Fig. Fig. Fig.

3.1 3.2 3.3 3.4 3.5

The function f D 1A and its distribution function . . . . . . . . . . . . . . . . . 1 The function f .x/ D 2 and its distribution function . . . . . . . . . . . . . . x x  1.0;1/ .x/ and its The function f .x/ D 1x distribution the function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The function f .x/ D j sin xj and its distribution function . . . . . . . . . . 1 Equimeasurable functions f .x/ D 2 and x 1  1Œ1; C1/ .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g.x/ D .x  1/2 Equimeasurable functions f .x/ D x  1Œ0; 1 .x/, g.x/ D .1  x/  1Œ0; 1 .x/, and h.x/ D .1  j2x  1j/  1Œ0;1 .x/ . . . . x Equimeasurable functions f .x/ D 1, g.x/ D , 1Cx 1 and h.x/ D .1 C sin x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hypograph and epigraph of a generalized inverse function . . . . . . . Constancy intervals and jumps for f  and f  . . . . . . . . . . . . . . . . . . . . . Integrals of f  and jf j as area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   ˚.a/ C ˚.b/ aCb  ........................... Inequality ˚ 2 2 p q x y The functions and as areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p q The cases f  .1/ > 0 and f  .1/ D 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The inequality .f C g/  f  C g fails . . . . . . . . . . . . . . . . . . . . . . . . . . . The case f  .1/ D f  .c/, c < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case in which f  .1/ is not achieved, c D 1 . . . . . . . . . . . . . . . . . The common distribution function of step functions f and g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 7

7 7

8 8

8 10 12 13 18 18 31 32 34 34 36

xv

xvi

List of Figures

Fig. 3.6 Fig. 3.7

Equimeasurable functions that are and are not of the form f ı  ,  2 A.m/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequences an and bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 39

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8

Functions f  , jf j and v a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A constant-value interval of f  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The cases of jf j  f  .1/ and f  .a0 / D f  .a/ D f  .1/ . . . . . . . . . Relation between norms kgkL1 and khkL1 and kf kL1 . . . . . . . . . . . . . Value b D f  .1/ and function .f   b/C . . . . . . . . . . . . . . . . . . . . . . . . . Equality (4.2.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper cutoff of f  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Norm kf  fn kL1 CL1 as area. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

42 43 44 47 48 49 50 51

Fig. Fig. Fig. Fig. Fig.

5.1 5.2 5.3 5.4 5.5

Fundamental functions of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental functions of L1 C L1 and L1 \ L1 . . . . . . . . . . . . . . . . The sequence fak g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions hnk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions v.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60 61 62 66 69

Fig. 6.1 Fig. 6.2

'X .b  a/ D k1Œa;b kX in the case 'X .0C/ > 0 . . . . . . . . . . . . . . . . . . . . Inequality (6.4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78

Fig. Fig. Fig. Fig.

9.1 9.2 9.3 9.4

Two cases of Lorentz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 'L1 \L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper cutoffs minff  ; ng and differences gn D f   min.f  ; n/ . . Right cutoffs f   1Œ0;n/ and hn D f   f   1Œ0;n/ . . . . . . . . . . . . . . . . . . .

116 120 121 122

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12

V.x/ is quasiconcave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convex p quasiconcave V.x/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ D x2 C 1  1.0;1/ is both quasiconcave and convex . . . . . . . 0 .e V/ is the closed convex hull of 0 .V/ . . . . . . . . . . . . . . . . . . . . . . . . . . The least concave majorant e V of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ and e V.x/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A finite “interval of nonconcavity” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A maximal finite “interval of nonconcavity”. . . . . . . . . . . . . . . . . . . . . . . The case aV < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The case aV < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ D max.1; x/  1.0;1/ .x/ and e V.x/ D .x C 1/  1.0;1/ .x/ . . . . . Quasiconvex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128 128 129 129 131 131 132 132 134 134 135 137

Fig. Fig. Fig. Fig.

11.1 11.2 11.3 11.4

V.x/ for MV D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ for MV D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ for MV D L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.x/ and V .x/ for MV D L1 C L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 148 148 149

Fig. 12.1

f D

m P iD1

ci  1Œ0;bi  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

List of Figures

xvii

p x; x  0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Fig. 12.2

V.x/ D V .x/ D

Fig. Fig. Fig. Fig. Fig. Fig. Fig.

Orlicz functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Orlicz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˚ and ˚ 1 in (13.3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L˚ D Lp if ˚.x/ D xp , 1  p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orlicz function ˚ for which L˚ D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orlicz function for which L˚ D L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . Orlicz function ˚ for which L˚ D L1 C L1 . . . . . . . . . . . . . . . . . . . . . . R1 f  dm as area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13.1 13.2 13.3 13.4 13.5 13.6 13.7

Fig. 13.8

172 172 177 178 179 179 180 181

a

Fig. 13.9

'L˚  'L1 CL1  2'L˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9

Derivatives of conjugate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugate functions ˚.x/ and  .y/ as areas. . . . . . . . . . . . . . . . . . . . . . . Legendre’s parameter m D ˚ 0 .x/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Legendre transform in the case of derivative jump . . . . . . . . . . . Jump of ˚ 0 and linearity interval of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearity interval of ˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constancy interval of derivative ˚ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left and right tangent lines at y D m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˚ 0 and  0 when a˚ > 0 and b˚ < 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

196 197 198 199 199 200 200 201 204

Fig. Fig. Fig. Fig. Fig.

16.1 16.2 16.3 16.4 16.5

Functions ˚c .x/ D ecx  1; c  0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚c1 .x/ and ˚c2 .x/ for c1 < c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivatives of ˚˛ .x/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚˛ .x/; 0  ˛ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˚ 0 and its greatest convex minorant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 211 214 214 215

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10

Functions ˚1 _ ˚2 ; ˚1 ^ ˚2 ; .˚1 ^ ˚2 /Ï . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚; ˚1 , ˚ ^ ˚1 and .˚ ^ ˚1 /Ï . . . . . . . . . . . . . . . . . . . . . Functions ˚; ˚0 and their inverse functions . . . . . . . . . . . . . . . . . . . . . . . Functions  and  _ ˚1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚; ˚1 ; ˚ ^ ˚1 and the greatest convex minorant . . . . . Function .˚ ^ ˚1 /Ï . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ; ˚1 , and  _ ˚1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚p .x/ and ˚p0 .y/ as areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions ˚p _ ˚q ; ˚p ^ ˚q and .˚p ^ ˚q /Ï . . . . . . . . . . . . . . . . . . . . . Functions ˚p;q ; ˚p;1 ; ˚1;q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

218 221 221 222 223 224 224 225 226 227

Part I

Symmetric Spaces. The Spaces Lp, L1 \ L1, L1 C L1

In this part we begin to study symmetric spaces. We give basic definitions and consider the main examples of symmetric spaces. They are the spaces Lp , 1  p  1, L1 \ L1 , and L1 C L1 . We consider real functions on RC D Œ0; 1/, measurable with respect to the usual Lebesgue measure m on Œ0; 1/. A symmetric space is a normed space X of such functions, which form a Banach ideal lattice with a symmetric (rearrangement invariant) norm k  kX . Being Banach means being complete with respect to the norm k  kX . Functions that are equal almost everywhere with respect to m are identified in X. The fact that X is an “ideal lattice” means that jf j  jgj; g 2 X H) f 2 X; kf kX  kgkX : “Symmetry” means invariance of X with respect to passing to equimeasurable functions. In more detail, the “upper” distribution function f of a nonnegative measurable function f is defined by upper Lebesgue sets ff > tg: f .t/ D mff > tg D mfx W f .x/ > tg; t 2 RC : Two functions f and g are equimeasurable if jf j D jgj , i.e., jf j and jgj have the same distribution. Since RC has infinite measure m, there are cases in which jf j  C1 (for example, f .x/ D x). Therefore, instead of jf j , it is convenient to use the “decreasing rearrangement” f  of the function jf j. The function f  is a (unique) decreasing rightcontinuous function that is equimeasurable to jf j if f  .x/ is finite on .0; 1/. The definition of the symmetric spaces can be reformulated in terms of decreasing rearrangements f  as follows. A nonzero Banach space .X; k  kX / of measurable functions on RC is said to be a symmetric space if f   g ; g 2 X H) f 2 X; kf kX  kgkX :

2

I

Symmetric Spaces. The Spaces Lp , L1 \ L1 , L1 C L1

The first and main examples, which essentially determined the development of the theory of symmetric spaces, were the Banach spaces Lp , 1  p  1. These spaces are equipped with the norm

kf kLp D

80 Z1 ˆ ˆ ˆ yg; y 2 RC are the upper Lebesgue sets of f . Definition 1.1.1. The function f W Œ0; 1/ ! Œ0; 1, defined by (1.1.1), is called the (upper) distribution function of f . The following statement describes elementary properties of the distribution function. © Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_1

5

6

1 Definition of Symmetric Spaces

Proposition 1.1.2. The function f is decreasing1 and right-continuous. Proof. The function f is decreasing, since y1 < y2 H) ff > y1 g ff > y2 g H) f .y1 /  f .y2 /: The right-continuity of f follows from the fact that yn # y H) ff > yn g " ff > yg H) f .yn / " f .y/: t u The right-continuity arises from the choice of the strict inequality > in (1.1.1). While the function y ! f .y/ is right-continuous, the function y ! mff  yg is left-continuous (Figs. 1.1, 1.2, 1.3, and 1.4). Examples 1.1.3. 1. Let f D 1A be an indicator (characteristic function) of the set A 2 Fm :  1; x 2 AI 1A .x/ D 0; x 62 A: Then f D mA  1Œ0;1/ , i.e.,  f .x/ D

mA; 0  x < 1I 0; x  1:

Fig. 1.1 The function f D 1A and its distribution function

Note that f .x/ D 1 if mA D 1 and 0  x < 1. 1 1 2. Let f .x/ D 2 . Then f .x/ D p (Fig. 1.2). x x

1 Here and throughout the book we use the terms decreasing and strictly decreasing functions and do not use the terms nondecreasing and nonincreasing functions.

1.1 Distribution Functions, Equimeasurable Functions

Fig. 1.2 The function f .x/ D

3. Let f .x/ D

7

1 and its distribution function x2

x 1  1.0;1/ .x/. Then f .x/ D (Fig. 1.3). 1x xC1

Fig. 1.3 The function f .x/ D

x  1.0;1/ .x/ and its distribution the function 1x

4. Let f .x/ D j sin xj (Fig. 1.4). Then f .x/ D



1; 0  x < 1I 0; x  1:

Fig. 1.4 The function f .x/ D j sin xj and its distribution function

Definition 1.1.4. Nonnegative functions f and g are called equimeasurable if f D g , i.e., mff > yg D mfg > yg for all y 2 Œ0; 1/:

8

Fig. 1.5 Equimeasurable functions f .x/ D

1 Definition of Symmetric Spaces

1 1 and g.x/ D  1Œ1; C1/ .x/ x2 .x  1/2

Fig. 1.6 Equimeasurable functions f .x/ D x  1Œ0; 1 .x/, g.x/ D .1  x/  1Œ0; 1 .x/, and h.x/ D .1  j2x  1j/  1Œ0;1 .x/

Fig. 1.7 Equimeasurable functions f .x/ D 1, g.x/ D

x 1 , and h.x/ D .1 C sin x/ 1Cx 2

1.2 Generalized Inverse Functions

9

Examples 1.1.5. 1. Let A; B 2 Fm . Then the indicators 1A and 1B are equimeasurable if and only if mA D mB. 1 1 1 2. Let f .x/ D 2 and g.x/ D  1Œ1; C1/ .x/. Then f .x/ D g .x/ D p , i.e., 2 x .x  1/ x f and g are equimeasurable (Fig. 1.5). 3. The functions f .x/ D x  1Œ0; 1 .x/, g.x/ D .1  x/  1Œ0; 1 .x/, and h.x/ D .1  j2x  1j/  1Œ0;1 .x/ are equimeasurable, and f D g D p h D g (Fig. 1.6). 4. The functions f .x/ D x, g.x/ D x2 and h.x/ D x are equimeasurable, and f .x/ D g .x/ D h .x/ D C1 for all x 2 Œ0; C1/. 5. The functions f .x/ D 1, g.x/ D equimeasurable (Fig. 1.7).

1 x , and h.x/ D .1 C sin x/ are 1Cx 2 

f .x/ D g .x/ D h .x/ D

C1; 0  x < 1I 0; x  1:

The last two examples illustrate negative effects of sets of infinite measure on testing for equimeasurability. If, as in Example 1.1.5.2, the function f is continuous and strictly decreasing from C1 to 0, then its distribution function coincides with the inverse function f 1 of f . The inverse function is also continuous and strictly decreasing from C1 to 0, and its inverse in turn is the original function f D 1 f . In general, a decreasing function f may have jumps and be constant on some intervals. Hence, its inverse 1 in general need not exist in the usual sense. f Nevertheless, we can use the so-called generalized inverse function f  of the decreasing function f .

1.2 Generalized Inverse Functions Let y D g.x/ be a decreasing nonnegative function on Œ0; 1/ that possibly takes the value C1. Consider the hypograph of g, ˚  0 .g/ D .x; y/ 2 R2 W x  0; g.x/  y  0 ;

10

1 Definition of Symmetric Spaces

Fig. 1.8 Hypograph and epigraph of a generalized inverse function

y

y=g(x) x=h(y) Γ0

Γ0

0

x

and the epigraph of g, ˚   0 .g/ D .x; y/ 2 R2 W x  0; y  g.xC/ ; in the coordinate system Oxy.2 Both sets 0 .g/ and  0 .g/ are closed and have the common boundary  D 0 .g/ \  0 .g/: The set  contains the graph of the function g,  ˚  .g/ D .x; y/ 2 R2 W x  0; y D g.x/ ; and also all vertical segments corresponding to jump points of g (Fig. 1.8). The function x D h.y/ is the generalized inverse function to y D g.x/ if  ˚ .x; y/ 2 R2 W y  0; h.y/  x  0 D 0 .g/; ˚  .x; y/ 2 R2 W y  0; x  h.yC/ D  0 .g/: The set  D 0 .g/ \  0 .g/ contains the graph  ˚  .h/ D .x; y/ 2 R2 W y  0; x D h.y/ as well as all horizontal segments corresponding to intervals on which g is constant. If such intervals exist for g, then h has jumps and is not well defined at the jump points. Nevertheless, there is a unique right-continuous generalized inverse function h D g1 . Moreover, g D h1 if g and h are right-continuous. 2 From now on, we use the notation g.x/ and g.xC/ for the left- and right-hand limits of the function g at a point x.

1.3 Decreasing Rearrangements

11

In this case, h.y/ D inffx  0 W g.x/  yg; g.x/ D inffy  0 W h.y/  xg:

(1.2.1)

The functions h and g in (1.2.1) uniquely determine each other. It is clear that if g  C1, then h  0, and conversely. We shall usually write inverse instead of generalized inverse.

1.3 Decreasing Rearrangements Let f W RC ! R be a real measurable function on RC D Œ0; 1/ and let jf j .y/ D mfjf j > yg 2 Œ0; C1; y 2 RC be the distribution function of the absolute value jf j. As can be seen from the above examples, it is possible to have jf j .y/  C1. In the sequel, we assume (unless otherwise stated) that jf j .y/ 6 C1. Since the function jf j is decreasing and right-continuous, it has a unique generalized inverse, which is also decreasing and right-continuous. This inverse function 1 will be denoted by f  , and it has the form jf j f  .x/ D inffy  0 W jf j .y/  xg:

(1.3.1)

Since jf j .y/ 6 C1, there exists y0  0 such that the value jf j .y0 / D mfjf j > y0 g < C1 is finite. This means that lim jf j .y/ D 0

y!1

and hence f  .x/ < 1 for all x > 0. Thus, by construction of f  W Œ0; C1/ ! Œ0; C1/, it is a decreasing rightcontinuous function such that jf j D f  . In other words, the functions f  and jf j are equimeasurable. Definition 1.3.1. The function f  is called a decreasing rearrangement of the function jf j. Note 1.3.2. Since both functions f  and f  D jf j are right-continuous, we have f  .f  .y//  y and f  .f  .x//  x for all x and y (Fig. 1.9).

(1.3.2)

12

1 Definition of Symmetric Spaces

Fig. 1.9 Constancy intervals and jumps for f  and f 

y = f *(x) y yg D 0g D kf kL1 :

Chapter 2

Spaces Lp ; 1  p  1

In this chapter we study the class of Lp spaces, 1  p  1, which is one of the most important classes of symmetric spaces. We begin with the Hölder and Minkowski inequalities and prove that Lp is a symmetric space for all 1  p  1. In the case 1  p < 1, we show that Lp is separable and describe its dual.

2.1 Hölder’s and Minkowski’s Inequalities We continue to study the spaces Lp introduced at the end of Chapter 1. Our first goal is to prove that .Lp ; k  kLp / is a complete normed space, i.e. , a Banach space. We begin with two basic inequalities. Proposition 2.1.1. 1. Let p  1, a > 0, and b > 0. Then .a C b/p  2p1 .ap C bp /: 2. (Young’s inequality) Let p; q > 1, a > 0, and b > 0. If ab 

(2.1.1) 1 1 C D 1, then p q

ap bq C : p q

(2.1.2)

Equality is achieved if and only if a D bq1 or b D ap1 . Proof. 1. Since the function ˚.x/ D xp is convex on .0; 1/ for p  1, it follows that   a C b p ap C bp  2 2 for all a > 0, b > 0 (Fig. 2.1). © Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_2

17

2 Spaces Lp ; 1  p  1

18 Fig.   2.1 Inequality ˚.a/ C ˚.b/ aCb  ˚ 2 2

y F (x ) = x p

F(b )

F(a ) + F(b ) 2 a+b F 2 F (a ) 0

y

a

a+b 2

b

x

y y = x p-1 x = y q-1

y = x p-1 x = y q-1 S1+ S2 > ab

b S1

S2

S2 S1

a

0

Fig. 2.2 The functions

S1+ S2 = ab

b

x

0

a

x

xp yq and as areas p q

This implies .a C b/p  2p1 .ap C bq /; for all a > 0 and b > 0. 2. Since .p  1/.q  1/ D 1, the functions y D xp1 and x D xq1 are mutually inverse. Consider their graphs (Fig. 2.2). We have Zx S1 D

p1

u 0

xp du D ; S2 D p

Zy v q1 du D 0

yq ; q

and xy  S1 C S2 D

yq xp C : p q t u

2.1 Hölder’s and Minkowski’s Inequalities

19

Theorem 2.1.2. Let 1  p; q  1 and

1 1 C D 1. p q

1. (Hölder’s inequality) Let f 2 Lp , g 2 Lq . Then fg 2 L1 and kfgkL1  kf kLp  kgkLq :

(2.1.3)

Equality is achieved if and only if ˛jf jp D ˇjgjq for some ˛  0, ˇ  0. 2. (Minkowski’s inequality) If f ; g 2 Lp , 1  p  1, then kf C gkLp  kf kLp C kgkLp :

(2.1.4)

Proof. 1. The inequality (2.1.3) clearly holds if kf kLp or kgkLq is equal to 0 or 1. Thus we may assume that 0 < kf kLp < 1 and 0 < kgkLq < 1. If p D 1, q D 1, then Z1 kfgkL1 D

Z1 jfgjdm D

0

Z1 jf j jgjdm  kgkL1

0

jf jdm D kf kL1  kgkL1 : 0

Let 1 < p; q < 1. By putting in Young’s inequality (2.1.2), aD

jf .x/j jg.x/j ; bD ; kf kLp kgkLq

we obtain jf .x/j jg.x/j 1   kf kLp kgkLq p

jf .x/j kf kLp

!p

1 C q

jg.x/j kgkLq

!q ; x 2 Œ0; 1/;

and hence Z1 0

jf .x/g.x/j dx 1  kf kLp  kgkLq p

Z1 0

jf .x/j kf kLp

!p

1 dx C q

Z1 0

jg.x/j kgkLq

!q dx D

1 1 C D 1; p q

i.e., Z1 kfgkL1 D

jf .x/g.x/j dx  kf kLp  kgkLq : 0

2. For values p D 1 and p D 1, the inequality (2.1.4) is obvious. Let 1 < p < 1. Then Z1

Z1 jf C gj dm 

0

Z1 jf j jf C gj

p

0

p1

dm C

jgj jf C gjp1 dm: 0

(2.1.5)

2 Spaces Lp ; 1  p  1

20

If f ; g 2 Lp , then by setting in (2.1.1) a D jf j and b D jgj, we have jf C gjp  2p1 .jf jp C jgjp /; which in turn implies .f C g/p 2 L1 . Taking into account the equality .p  1/q D p, we obtain .jf C gjp1 /q D jf C gj.p1/q D jf C gjp 2 L1 ; i.e., jf C gjp1 2 Lq . Now we apply Hölder’s inequality (2.1.3) to the functions jf j 2 Lp and jf C gjp1 2 Lq : Z1 jf j jf C gjp1 dm  kf kLp  k jf C gjp1 kLq

01 1 1q Z D kf kLp @ jf C gj.p1/q dmA

0

0

01 1 1q Z p  D kf kLp @ jf C gjp dmA D kf kLp  kf C gkLp q : 0

In a similar way, for jgj 2 Lp and jf C gjp1 2 Lq , we have Z1

p  jgj jf C gjp1 dm  kgkLp  kf C gkLp q :

0

By combining these two inequalities with (2.1.5), we obtain Z1 kf C

p gkLp

D

p  jf C gjp dm  .kf kLp C kgkLp /  kf C gkLp q :

0

Since p 

p D 1, we obtain also (2.1.4). q

t u

Minkowski’s inequality is just the triangle inequality in the space Lp . The equality kcf kLp D jcj  kf kLp ; c 2 R; f 2 Lp ; also holds, and kf kLp D 0 if and only if f D 0. Thus, for 1  p  1, .Lp ; k  kLp / is a normed space.

2.2 Completeness of Lp

21

2.2 Completeness of Lp Let 1  p < 1 and ffn g be a fundamental sequence in Lp , i.e., lim kfn  fm kLp D 0:

n;m!1

By passing, if necessary, to a subsequence, we may assume that f0 D 0; kfk  fk1 kLp
0. Then there exists an open set G such that m.A M G/ < ". The set G has the form GD

1 [ .ai ; bi /; 0  ai < bi ; i  1: iD1

Since S mG < 1, we can find n large enough that m.G M Gn / < "; where Gn D niD1 .ai ; bi /. We additionally may assume that ai and bi are rational numbers. Consider the subset F.0/  F0 consisting of all functions g of the form gD

n X

ci  1Œai ;bi 

iD0

with rational ci , ai , and bi . Since for every pair A; B 2 Fm , we have m.A M B/ < " H) k1A  1B kLp  "1=p ; the countable set F.0/ is dense in F0 , and hence in Lp by part 1 of the proposition. t u

2.4 Duality Let X be a symmetric space and X the dual Banach space. The space X consists of all linear continuous (bounded) functionals u W X ! R equipped with the norm kukX D supfju.f /j W kf kX  1g < 1: Let 1  p; q  1 and

1 1 C D 1. For g 2 Lq we define a linear functional ug on p q

Lp by setting Z1 ug .f / D

fgdm; f 2 Lp : 0

Theorem 2.4.1. Let 1  p; q  1 and

1 1 C D 1. Then ug 2 Lp for all g 2 Lq , p q

and the mapping W Lq 3 g ! ug 2 Lp is an isometric isomorphism of Lq into the dual space Lp of the space Lp . If 1  p < 1, then .Lq / D Lp , and for p D 1, the embedding .L1 /  L1 is strict.

2.4 Duality

25

1 1 C D 1. Hölder’s inequality (2.1.3) implies that the p q functional ug is continuous and ˇ1 ˇ ˇZ ˇ ˇ ˇ ˇ jug .f /j D ˇ fgdmˇˇ  kf kLp kgkLq ; ˇ ˇ

Proof. Let 1 < p; q < 1,

0

i.e., kug kLp  kgkLq :

(2.4.1)

Suppose that u 2 Lp . Since 1A 2 Lp for mA < 1, the equality .A/ D u.1A / defines a set function

W A ! .A/ D u.1A /; A 2 Fm ; mA < 1:

(2.4.2)

For every finite segment Œ0; n, the restriction of to the  -algebra Fm .0; n/ D fA 2 Fm W A  Œ0; ng is a  -additive set function, which is absolutely continuous with respect to the measure m. The Radon–Nikodym theorem states the existence of a unique function hn integrable on Œ0; n such that Z hn dm; A  Œ0; n; A 2 Fm :

.A/ D

(2.4.3)

A

Since RC D

1 S

Œ0; n/, we obtain a unique measurable function g such that

nD1

gjŒ0;n D hn for all n D 1; 2; : : : and

Z

u.1A / D .A/ D

gdm

(2.4.4)

A

for all A 2 Fm with m.A/ < 1. The linearity of the functional u yields that Z1 u.f / D

fgdm D ug .f /; f 2 F0 :

(2.4.5)

0

We choose now a sequence gn 2 F0 such that gn " jgj. Such a sequence can be constructed by setting gn D min.n; g.n/ /  1Œ0;n ;

2 Spaces Lp ; 1  p  1

26

where for all i D 1; 2; : : :, g.n/ .x/ D

i1 i1 i ; if  jg.x/j < n : n n 2 2 2

We define now fn D .gn /q1  sign.g/; n D 1; 2; : : : : Then fn 2 F0 and Z1 u.fn / D

Z1 q

fn gdm D 0

gqn dm D kgn kLq : 0

On the other hand, u.fn /  kukLp kfn kLp ; .q1/p

where jfn jp D gn This implies

D gqn , by the equality .q  1/p D q.

kfn kLp

01 11=p Z q=p D @ gqn dmA D kgn kLq 0

and q=p

q

kgn kLq  kukLp kgn kLq ; i.e., kgn kLq  kukLp : Thus, gqn " jgjq and Z1 q

gqn dm  kukL : p

0

By Levi’s theorem, Z1

Z1 gqn dm

lim

n!1 0

i.e., g 2 Lq and kgkLq  kukLp .

q

D

jgjq dm  kukL ; p

0

2.4 Duality

27

The functional ug coincides with u 2 Lp on F0 , and F0 is dense in Lq . Hence ug D u. The equality .L1 / D L1 and the fact that the embedding .L1 /  L1 is strict will be shown later in Examples 7.1.1 and 7.1.2. t u Corollary 2.4.2. The spaces Lp are reflexive for 1 < p < 1.

Chapter 3

The Space L1 \ L1

The space L1 \ L1 , which we study in this chapter, consists of all bounded integrable functions equipped with the norm k  kL1 \L1 D max.k  kL1 ; k  kL1 /. We show that .L1 \ L1 ; k  kL1 \L1 / is a symmetric space and describe the closure L01 of L1 \ L1 in L1 . Given two equimeasurable functions f and g, we treat an approximation of g in the L1 \ L1 -norm by shifted functions f ı  , where  is a measure-preserving transformation. Step functions and integrable simple functions are applied for this purpose.

3.1 The Intersection of the Spaces L1 and L1 We begin with a general construction. Let X1 and X2 be two symmetric spaces. We consider the norm kf kX1 \X2 D max.kf kX1 ; kf kX2 /; f 2 X1 \ X2 ; on the intersection X1 \ X2 . Below, in Section 6.2, we shall show that for every pair of symmetric spaces X1 and X2 , the space .X1 \ X2 ; k  kX1 \X2 / is symmetric. Now we consider only the special case X1 D L1 and X2 D L1 . Clearly, we have that kf kL1 \L1 D maxfkf kL1 ; kf kL1 g; f 2 L1 \ L1 ;

(3.1.1)

is a norm on L1 \ L1 . Both conditions of Definition 1.5.1 hold, because they hold for each of the spaces L1 and L1 .

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_3

29

3 The Space L1 \ L1

30

To verify the completeness of L1 \ L1 , consider a Cauchy sequence ffn g in L1 \ L1 . Then ffn g is a Cauchy sequence in both spaces L1 and L1 . Since L1 and L1 are Banach spaces, there exist functions f 2 L1 and g 2 L1 satisfying kfn  f kL1 ! 0 and kfn  gkL1 ; n ! 1: The norm convergence in L1 and L1 implies convergence in measure, fn ! f and fn ! g. Hence f D g 2 L1 \ L1 , and fn converges in L1 \ L1 . By the equalities (1.6.1) and (1.6.2), we have Z1 kf kL1 D

Z1 jf jdm D

0

f  dm

0

and kf kL1 D vrai sup jf j D f  .0/: So the norm on L1 \ L1 has the form 01 1 Z kf kL1 \L1 D max @ f  dm; f  .0/A ; f 2 L1 \ L1 ; 0

and hence satisfies conditions 1 and 2 of symmetric spaces. Thus, L1 \ L1 is a symmetric space. By (3.1.1), we have kf kL1  kf kL1 \L1 ; kf kL1  kf kL1 \L1 ; f 2 L1 \ L1 ; i.e., both embeddings L1 \ L1  L1 and L1 \ L1  L1 are continuous. It will be shown below (Proposition 3.5.1) that L1 \ L1 is dense in L1 in the norm k  kL1 , but it is not dense in L1 in the norm k  kL1 .

3.2 The Space L01 The closure of L1 \L1 in L1 coincides with the space L01 , which can be described by means of decreasing rearrangements f  as follows. Since the function f  decreases on .0; 1/, there exists a limit f  .1/ WD lim f  .x/ D inf f  .x/: x!1

x>0

We set (Fig. 3.1) L01 D ff 2 L1 W f  .1/ D 0g  L1

3.2 The Space L01

31

y

y

f * ( 0)

f *( 0)

f*

f*

f *( • ) > 0

f *( • ) = 0

f *( • ) 0

x

0

x

Fig. 3.1 The cases f  .1/ > 0 and f  .1/ D 0

and consider on the space L01 the norm induced from L1 : kf kL01 D f  .0/; f 2 L01 : We want to show that .L01 ; k  kL01 / is a symmetric space. For this purpose, we need the following important property of decreasing rearrangements. Proposition 3.2.1. Let f and g be measurable functions. Then .f C g/ .x1 C x2 /  f  .x1 / C g .x2 /

(3.2.1)

for all x1 ; x2 2 RC . Proof. The inclusion fjf C gj > y1 C y2 g  fjf j > y1 g [ fjgj > y2 g implies that .f Cg/ .y1 C y2 /  f  .y1 / C g .y2 / for all y1 ; y2 2 RC . By choosing y1 D f  .x1 / and y2 D g .x2 /, we obtain .f Cg/ .f  .x1 / C f  .x2 //  x1 C x2 and f  .x1 / C g .x2 /  .f C g/ .x1 C x2 / at all points of continuity of the functions f  , g , .f C g/ , and their inverses. Since all these functions are right-continuous, the inequality (3.2.1) is valid also for all x1 and x2 . t u

3 The Space L1 \ L1

32 y

y

y

1

1

1

1

1

f + g = ( f + g )*

g

f = f* = g*

0

y 2

1

0

x

x

1

0

f* + g*

x 0

1/2

1

x

Fig. 3.2 The inequality .f C g/  f  C g fails

Note 3.2.2. It follows from (3.2.1) that .f C g/ .2x/  f  .x/ C g .x/I x  0; or .f C g/ .x/  f 

x 2

C g

x 2

I x  0:

Nevertheless, the “triangle inequality” .f C g/  f  C g can be false in general (Fig. 3.2). For example, if we set f .x/ D .1  x/1Œ0;1/ .x/ and g.x/ D x1Œ0;1/ .x/; x  0; then f  D g D f and .f C g/ .x/ D 1Œ0;1/ .x/; f  .x/ C g .x/ D 2.1  x/1Œ0;1/ .x/; and hence .f C g/ .x/ > f  .x/ C g .x/;

1 < x < 1: 2

Proposition 3.2.3. .L01 ; k  kL01 / is a symmetric space. Proof. If f ; g 2 L01 , then .cf / .1/ D cf  .1/ D 0, c > 0, and hence by (3.2.1), .f C g/ .1/ D lim .f C g/ .2x/  lim .f  .x/ C g .x// D f  .1/ C g .1/ D 0; x!1

i.e., L01 is a linear subset in L1 .

x!1

3.3 Approximation by Step Functions

33

If kfn  f kL1 ! 0 and fn 2 L01 , then f  .1/  .f  fn / .1/ C fn .1/ D .f  fn / .1/  .fn  f / .0/ D kfn  f kL1 ! 0; and hence f 2 L01 . Thus, L01 is closed in L1 , and the space L01 is a Banach space as well as L1 . Conditions 1 and 2 of symmetric spaces are clear. t u We shall return to the space L01 at the end of the chapter and prove that L01 is the closure of L1 \ L1 in L1 .

3.3 Approximation by Step Functions Recall that by step functions, we mean functions of the form f D

X

ai  1Ai ;

i2I

where I is a finite or countable set of indices, ai are real numbers, and Ai are mutually disjoint subsets of RC . Proposition 3.3.1. Let f  f  .1/. Then for every " > 0, there exists a step function f" satisfying kf  f" kL1 \L1  ": Proof. Set a D f  .1/. For " > 0, we set an D a C " n; n D 0; 1; : : : ; and bn D f  .b C n/; and b0 D f  .b/ D a C ": Two cases are possible (Figs. 3.3 and 3.4): 1. f  .x/ D a D f  .1/ for all sufficiently large x. In this case, we set f" .x/ D an ; if an  f .x/ < anC1 ; n D 0; 1; 2; : : : : Then for all x, Z1 jf .x/  f" .x/j  " and Œf .x/  f" .x/dx  " c; 0

where c D inffx  0 W f  .x/ D f  .1/g.

3 The Space L1 \ L1

34 Fig. 3.3 The case f  .1/ D f  .c/, c < 1

y

f* a+2 e a+e a c

0 Fig. 3.4 The case in which f  .1/ is not achieved, cD1

x

y

f

a +e b1 b2 a b

0

b +1

b +2

x

2. f  .x/ > a D f  .1/ for all x  0. If f  .x/  a C ", we define the function f" in the same way as in case 1. For a < f .x/ < a C ", we use the sequence fbn g: f" .x/ D bnC1 for bnC1  f .x/ < bn , n D 0; 1; 2; : : :. Then kf  f" kL1  "; and Z1 Œf .x/  f" .x/dx  b " C Œ.a C "/  a  1 D .b C 1/ ": 0

By choosing

 "1 D min

" " ; c bC1



3.4 Measure-Preserving Transformations

35

instead of " > 0 at the beginning of the proof, we obtain that kf  f" kL1 \L1  ": This completes the proof. Note 3.3.2. 1. By the construction of f" and g" , if the functions f and g are equimeasurable, then f" and g" are also equimeasurable. 2. It is possible to choose the step function f" in such a manner that kf  f" kL1 \L1  " and f" .x/ > a D f  .1/ D f" .1/; x  0: 3. For every function f satisfying the assumptions of Proposition 3.3.1, there exists a sequence of step functions ffn g such that fn " f and lim kf  fn kL1 \L1 D 0:

n!1

3.4 Measure-Preserving Transformations The notion of equimeasurability is closely related to measure-preserving transformations of .RC ; m/. Recall that a transformation  of the measure space .RC ; m/ is called measurepreserving if for every set A 2 Fm , the set  1 A D fx 2 RC W .x/ 2 Ag is measurable, and m. 1 A/ D mA. In other words, the measure m ı  1 defined by .m ı  1 /A D m. 1 A/ coincides with the original measure m. If, in addition,  is invertible and the inverse transformation  1 is also measurepreserving, then m. 1 A/ D m. A/ D mA for every A  Fm . Denote the group of all invertible measure-preserving transformations of .RC ; m/ by A.m/. Let f W RC ! R and  2 A.m/. The function f ı  is defined by .f ı  /.x/ D f ..x// for all x. Proposition 3.4.1. For every nonnegative function f and  2 A.m/, the functions f and f ı  are equimeasurable.

3 The Space L1 \ L1

36

Proof. For every x > 0, f ı  .x/ D mf.f ı  / > xg D m. 1 ff > xg/ D mff > xg D f .x/; i.e., f and f ı  are equimeasurable.

t u

For the class of step functions, the converse statement is also valid. Proposition 3.4.2. Let f and g be two nonnegative step functions such that f .x/ > f  .1/ and g.x/ > g .1/ for all x > 0. If f and g are equimeasurable, then there exists  2 A.m/ such that g D f ı . P P Proof. Let f D ai  1Ai ; g D bj  1Bj . Without loss of generality, we may i2I

j2J

assume that ai1 ¤ ai2 for i1 ¤ i2 and bj1 ¤ bj2 for j1 ¤ j2 : Then all values of ai and bj correspond to the jump points of the common distribution function  D f D g . By changing index sets if necessary, we may assume that I D J and ai D bi for all i 2 I (Fig. 3.5). For every jump point ai of the distribution function .x/,

i.e., RC

f  .1/ D g .1/ < .ai /  .ai / D mAi D mBi < 1; S S D Ai D Bi are two partitions of RC into disjoint sets of finite i2I

i2I

measure. By choosing invertible measure-preserving mappings i W Ai ! Bi , we obtain an invertible measure-preserving transformation  W RC ! RC such that  jAi D i ; i 2 I. The choice of  yields g D f ı . t u Note 3.4.3. The conditions f .x/ > f  .1/ and g.x/ > g .1/, x > 0, are essential in Proposition 3.4.2. For example, let f D 1Œ0;C1/ ; g D

1  1Œ0;1/ C 1Œ1;C1/ : 2

Then the functions f and g are equimeasurable, f  D f D g . However, f ı  D f ¤ g for all  2 A.m/. Fig. 3.5 The common distribution function of step functions f and g

h = h f = hg h ( ai _ ) h ( ai )

0 f * ( •) = g * ( •)

} mA = m B i

ai = bi

i

x

3.4 Measure-Preserving Transformations

37

The assumption that f and g are step functions is also essential, and Proposition 3.4.2 fails, in general, if f and g are not step functions. Example 3.4.4. Let f .x/ D .1x/1Œ0;1 and f1 .x/ D x1Œ0;1 , f2 D f1 ı2 , f3 D f1 ı3 , where measure-preserving transformations 2 and 3 are defined as follows:  1 8 < 2x; x 2 0; 2 ;  2 .x/ D 2x  1; x 2 12 ; 1 ; : x; x 2 Œ1; 1/;   8 3x; x 2 0; 13 ;  ˆ ˆ < 3x  1; x 2  13 ; 23  ; 3 .x/ D ˆ 3x  2; x 2 23 ; 1 ; ˆ : x; x 2 Œ1; 1/; Transformations 1 and 2 are measure-preserving. However, they are not invertible, i.e., 1 ; 2 62 A.m/. The functions f ; f1 ; f2 ; f3 are equimeasurable, f  D f D f1 D f2 D f3 ; and fi ¤ fi ; i D 1; 2; 3. It is clear that there is no  2 A.m/ such that f2 D f ı  or f3 D f ı  , and there is no  2 A.m/ such that f3 D f2 ı  or f2 D f3 ı  (see Fig. 3.6). Theorem 3.4.5. Let f and g be two equimeasurable functions such that f  a and g  a, where a D f  .1/ D g .1/. Then for every " > 0, there exists  2 A.m/ such that kf  g ı kL1 \L1 < ": Proof. Let f and g satisfy the assumptions above. For every " > 0, by Proposition 3.3.1, we can construct equimeasurable step functions f" and g" such that kf  f" kL1 \L1 

" " and kg  g" kL1 \L1  : 2 2

y

y

y

1

1

1

0

1

x 0

1

f2

f1

f

y

1

x 0

f3

1/2

1

x 0

1/3 2/3 1

Fig. 3.6 Equimeasurable functions that are and are not of the form f ı  ,  2 A.m/

x

3 The Space L1 \ L1

38

By Proposition 3.4.2, we obtain  2 A.m/ such that f" D g" ı : For every measurable function h, Z1 kh ı kL1 D

Z1 jh ı jdm D

0

jhjd.m ı 

1

Z1 /D

0

jhjdm D khkL1 ; 0

and also kh ı  kL1 D vrai sup jh ı j D vrai sup jhj D khkL1 : Hence, kg ı   g" ı  kL1 \L1 D kg  g" kL1 \L1 and kf  g ı  kL1 \L1  ".

t u 



Note 3.4.6. The assumption f  a, g  a, a D f .1/ D g .1/ in Theorem 3.4.5 is essential. However, we can formally pass from f and g to f D max.f ; a/; g D max.g; a/ and apply the functions f " and g" , which approximate f and g. On the other hand, the assumption can be entirely removed from the theorem using a wider class of transformations, as in Exercise 8.

3.5 Approximation by Simple Integrable Functions Recall that F1 denotes the set of all simple integrable functions and F0 consists of all simple functions from F1 with bounded support. Every f 2 F1 has the form f D

n X

ai  1Ai ; ai 2 R;

mAi < 1; 1  i  n; n 2 N:

iD1

If, in addition, there exists a > 0 such that Ai  Œ0; a, 1  i  n, then f 2 F0 . It is clear that F0  F1  L1 \ L1 : We shall denote by clX .Y/ the closure of a subset Y  X in X with respect to the norm k  kX .

3.5 Approximation by Simple Integrable Functions

39

Proposition 3.5.1. 1. clL1 \L1 .F0 / D L1 \ L1 . 2. clL1 .F0 / D clL1 .L1 \ L1 / D L1 . 3. clL1 .F0 / D clL1 .L1 \ L1 / D L01 , where by definition, L01 D ff 2 L1 W f  .1/ D 0g: Proof. 1. First, we show that every f 2 L1 \ L1 can be approximated by functions from F0 in the norm k  kL1 \L1 . We shall use functions g" , constructed in Proposition 3.3.1. Since L1 \ L1 is a symmetric space, we may assume that f D f   0. 1 For a given " > 0, we choose sequences fbn g1 nD0 and fan gnD0 such that (Fig. 3.7) b0 D 0; a0 D f  .0/; bn " 1; an D f  .bn /; an # 0; and an  anC1  "; bnC1  bn  1; n  0: We set g" D

1 X

an  1Œbn1 ;bn / and g";k D

nD1

k X

an  1Œbn1 ;bn / :

nD1

Then g";k 2 F0 ; kf  g" kL1  "; kg"  g";k kL1  ak # 0 and Z1 kf  g";k kL1 

fdm # 0; bk

Fig. 3.7 Sequences an and bn

y a

0

a

1

f*

a

2

a

3

0

b

1

b

2

b

3

x

3 The Space L1 \ L1

40

since the function f is integrable. So for sufficiently large k, kf  g";k kL1 \L1  2": Thus F0 is dense in L1 \ L1 . 2. Follows directly from Theorem 2.3.1. 3. For every " > 0 and f 2 L01 , we shall use the functions g" and g";k constructed in part 1. For those functions, kf  g" kL1  "; kg"  g";k kL1  ak ; where ak D f  .bk / ! f  .1/ D 0 for k ! 1. Since g";k 2 F0 , it follows that f  2 clL1 .F0 /. This means that L01  clL1 .F0 /  clL1 .L1 \ L1 /: The converse inclusions follow from part 1 and F0  L1 \ L1  L01 :

t u

Note 3.5.2. Recall that the spaces L1 \ L1 and L01 , as well as L1 , are nonseparable. Indeed, for all 0 < a < b, we have k1Œ0;b  1Œ0;a kL1 \L1  k1Œ0;b  1Œ0;a kL1 D 1; while 1Œ0;a 2 F0  L1 \ L1  L01 for all a > 0.

Chapter 4

The Space L1 C L1

In this chapter, we study the sum L1 C L1 of the spaces L1 and L1 . We show that L1 C L1 equipped with a natural norm k  kL1 CL1 is a symmetric space. The R1 norm k  kL1 CL1 can be written in the form 0 f  dm using the maximal property of decreasing rearrangements f  . We also describe embeddings of L1 and L1 into L1 C L1 and the closure R0 of L1 in L1 C L1 .

4.1 The Maximal Property of Decreasing Rearrangements The following maximal property of decreasing rearrangements is the basis of their numerous applications. In particular, we shall use the property in the proof of Theorem 4.2.1, below. Theorem 4.1.1. For every a > 0 Za

f  dm D sup

0

8 f  .1/, then there exists a subset A 2 Fm such that mA D a and Za 0

f  dm D

Z jf j dm:

(4.1.2)

A

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_4

41

4 The Space L1 C L1

42

Proof. We may assume that a > 0. If a D mA D 1, then Z

Z1 jf j dm 

Z1 jf j dm D

0

A

f  dm;

0

and equality is achieved with A D RC . Thus, we may assume that a < 1. 1. First we check that Z

ZmA jf j dm  f  dm:

(4.1.3)

0

A

Let a D mA < 1. Consider the function (Fig. 4.1) v a .y/ D minfa; jf j .y/g y  0: By applying the formula Z1

Z1 jf jdm D

0

jf j dm 0

to the function jf j  1A , we obtain Z

Z1 jf jdm D

Z1 jf j  1A dm D

0

A

jf j1A dm: 0

Since jf j1A  mA D a and jf j1A  jf j , it follows that jf j 1A  v a and Z1

Z1 jf j1A dm 

0

Za v dm D a

0

f  dm:

0

Hence, (4.1.3) is valid.

y

y

y

x = v a( y )

x =h ( y ) f

y = f *( x ) f * (a–) b = f *( a )

0

b

b

a

Fig. 4.1 Functions f  , jf j and v a

x

0

a

x 0

a

x

4.1 The Maximal Property of Decreasing Rearrangements

43

2. We show now that in the case f  .a/ > f  .C1/, there exists a set A 2 Fm such that mA D x and the equality (4.1.2) holds. Set b D f  .a/. Since b > f  .1/, the sets ff  > yg and fjf j > yg have the same finite measure a D jf j .b/; and the sets ff   bg and fjf j  bg have the same finite measure a D mff   bg D mfjf j  bg: We have a  a  a, since both mutually inverse functions f  and jf j are rightcontinuous. Since the measure m is continuous, there exists a set A such that ff  > bg  A  ff   bg and mA D a: For such a set A, the equality (4.1.2) holds (Fig. 4.2). 3. Set f  .1/ D c and consider the case b D f  .a/ D f  .1/ D c. In this case, jf j  c, and for every " > 0, the set fjf j  c  "g has infinite measure. Therefore, we can choose a set A such that mA D a and " .jf j/jA  c  : a Then Z jf jdm  A

Z 

" dm D cx  " D c a

A

Za

f  dm  ";

0

i.e., the equality (4.1.1) is valid. Fig. 4.2 A constant-value interval of f 

y

y = f * (x ) b = f * (a)

0

a

a

a

x

4 The Space L1 C L1

44

4. Set a0 D inffx W f  .x/ D f  .1/g and consider the case 0 < a0  a < 1, i.e., f  .x/ > c for x < a0  a and f  .x/ D c for x  a0 ; where as above, c D f  .1/. In this case, Za



Za0

f dm D 0

f  dm C .a  a0 /a:

0

Consider the decomposition jf j D jf j  1fjf j>cg C jf j  1fjf jcg of jf j (Fig. 4.3) and the decomposition f  D f   1.0;a0 / C f   1Œa0 ;1/ of f  . Fig. 4.3 The cases of jf j  f  .1/ and f  .a0 / D f  .a/ D f  .1/

y

f*=c·1[0,•]

c = f * (•) c _e f

0

A

x

y

y = f *(x) c = f * (•) 0

a

0

a

x

4.2 The Sum of L1 and L1

45

By applying the second part of the proof to jf j1fjf j>ag , we obtain a set A1 such that A1  fjf j > cg; mA1 D a0 ; and Za0

Z jf j dm D

f  dm:

0

A1

On the other hand, by applying the third part of the proof to jf j  1fjf jag and " > 0, we obtain a set A2 such that A2  fjf j  cg; mA2 D a  a0 ; and

Z jf j dm  .a  a0 /c  ": A2

Thus, for the set A D A1 [ A2 , we have Zx

Z jf j dm  A

f  dm  "and mA D a:

0

t u

This concludes the proof of the theorem.

4.2 The Sum of L1 and L1 We begin with a general construction using two symmetric spaces, X1 and X2 . By definition, the sum X1 C X2 consists of all functions f admitting a representation f D f1 C f2 ; f1 2 X1 ; f2 2 X2 :

(4.2.1)

Since X1 \ X2 ¤ f0g, the representation of f in the form of (4.2.1) is not unique. Indeed, f D .f1 C r/ C .f2  r/; f1 2 X1 ; f2 2 X2 for every function r 2 X1 \ X2 . On the other hand, if f D f1 C f2 D f 1 C f 2 ; f1 ; f 1 2 X1 ; f2 ; f 2 2 X2 ; then r D f1  f 1 D f2  f2 2 X1 \ X2 .

4 The Space L1 C L1

46

Thus, the space X1 C X2 consists of all pairs .f1 ; f2 / 2 X1 X2 , where two pairs .f1 ; f2 / and .f 1 ; f 2 / are regarded as equivalent if f1 f 1 D r and f2 f 2 D r for some r 2 X1 \ X2 . Formally, X1 C X2 is identified with the quotient space .X1 X2 /=L of the space X1 X2 by the subspace L D f.r; r/; r 2 X1 \ X2 g  X1 X2 ; i.e., X1 C X2 ' .X1 X2 /=L: The direct product X1 X2 is equipped with the norm k.f1 ; f2 /kX1 X2 D kf1 kX1 C kf2 kX2 ;

(4.2.2)

and the norm of a class .f1 ; f2 / C L 2 .X1 X2 /=L is k.f1 ; f2 / C Lk.X1 X2 /=L D

inf fkf1 C rkX1 C kf2  rkX2 g:

r2X1 \X2

(4.2.3)

Thus, X1 C X2 becomes a normed space with the norm kf kX1 CX2 D inffkf1 kX1 C kf2 kX2 ; f D f1 C f2 ; f1 2 X1 ; f2 2 X2 g;

(4.2.4)

where the infimum is taken over all representations of f of the form (4.2.1). Since X1 and X2 are Banach spaces and L is a closed subspace of the Banach space X1 X2 , the quotient space X1 C X2 ' .X1 X2 /=L with the norm (4.2.4) is also a Banach space. It can be shown that the sum X1 C X2 of arbitrary symmetric spaces X1 and X2 is a symmetric space (see Section 6.2 and Exercise 16). Now we consider a special case in which X1 D L1 and X2 D L1 . Theorem 4.2.1. The set L1 C L1 D ff D g C h; g 2 L1 ; h 2 L1 g equipped with the norm kf kL1 CL1 D inffkgkL1 C khkL1 W f D g C h; g 2 L1 ; h 2 L1 g:

(4.2.5)

is a symmetric space, and Z1 kf kL1 CL1 D 0

f  .s/ds D sup

8 0, there exist a measurable subset E  RC and a measure-preserving isomorphism  between .RC ; m/ and .E; mjE / such that kf1  g ı  kL1 \L1 < "; where f1 .x/ D max.f .x/; f  .1// and .g ı  /.x/ D g..x//; x 2 .0; 1/:

Notes

55

b. Let f and g be two measurable functions on RC such that f   g . Then for all " > 0, there exist a measurable subset E  RC , a measure-preserving isomorphism  W RC ! E, and a measurable function ˛ W RC ! Œ1; 1 such that kf  ˛  .g ı  /kL1 \L1 < ": c. For every f 2 L1 C L1 , there exists a sequence of measurable subsets En  RC such that mEn < 1 and .f  1En / ! f  almost everywhere.

Notes The definition of symmetric spaces on .RC ; m/ used in Section 1.5 and throughout the book is from [34, Section 2.4]. It is based on notions of ideal Banach lattices and symmetry (rearrangement invariance). Good presentations of the general theory of Banach lattices and normed ideal spaces of measurable functions can be found in [31, 35, 36, 75]. For general ordered topological linear spaces, see [22, Chapters 1–3], [65, Chapter 2]. The decreasing rearrangements f  of measurable functions f were introduced by G.H. Hardy, J.E. Littlewood, and G. Pólya; see, for example, [27, Section 10.12]. Some authors use stricter definitions of symmetric spaces. Lindenstrauss and Tzafriri [36] assume that every symmetric space is either minimal or maximal and has property .C/ (see Part II for terminology). Bennett and Sharpley [3, Chapter 2] include de facto the Fatou property in the definition. Most of the standard textbooks on functional analysis and measure theory contain a detailed description of the spaces Lp , 1  p  1. We prefer [17, 25, 31, 53, 59, 60]. The spaces Lp , 0 < p < 1, briefly described in Complement 3, were considered in [1, 15]; see also [63, Chapter 1], [64, Chapter 1]. The spaces L1 \L1 and L1 CL1 are a special case of the intersection X\Y and the sum X C Y of two general Banach spaces X and Y. The spaces X \ Y and X C Y are well defined once some suitable continuous embeddings X ! L and Y ! L into a topological linear space L are chosen and fixed (see [34, Section 1.3]). Results of Section 3.4 can be strengthened using noninvertible measurepreserving transformations [34, Theorems 2.1 and 4.11]. See also Exercise 8. The maximal property of decreasing rearrangements is due to G.H. Hardy, J.E. Littlewood, and G. Pólya; see [27, Section 10.13]. The definition of symmetric spaces X.RC ; m/ on .RC ; m/ is easily generalized to the case of symmetric spaces X.˝; / on general measure spaces .˝; / with finite or infinite  -finite measures . See [3, Chapter 1], [31, Chapter 4], [34, Section 2.8]. The general symmetric spaces X.˝; / on nonatomic measurable spaces .˝; / are considered in Complement 1. The notion of standard symmetric spaces

56

4 The Space L1 C L1

X.RC ; m/ corresponding to X.˝; / defined there is rather useful, especially when the measure space .˝; / is separable. The isomorphism of Lebesgue measure spaces used in this case was proved in [62]. Besides the case .˝; / D .RC ; m/, two special cases are of particular interest: a. .˝; / D .Œ0; a; ma /, where ma D mjŒ0;a is the usual Lebesgue measure on the interval. b. .˝; / D .N; ]/, where ] is the counting measure on the integers N. The symmetric spaces X.Œ0; 1/ and the symmetric sequence spaces X.N/ are described in Complements 2 and 3, respectively. Some related results are given also in Exercises 9, 10, 11, and 12. Note that the situation can be more intricate if .˝; / has atoms ! 2 ˝ with unequal values .f!g/ > 0; see [3, Section 2.2]. General nonseparable measure spaces .˝; / are described, for example, in [23, Chapters 32–34].

Part II

Symmetric Spaces. The Embedding Theorem. Properties .A/; .B/; .C/

In this part, we study general properties of symmetric spaces, such as separability, minimality, maximality, and reflexivity. The first important result is the basic embedding theorem, asserting that L1 \ L1  X  L1 C L1 for every symmetric space X. Both inclusions are continuous, and 'X .1/kf kL1 CL1  kf kX  2'X .1/kf kL1 \L1 ; f 2 X; where 'X .t/ D k1Œ0; t kX is the fundamental function of X. The closure X0 D clX .L1 \ L1 / of the space L1 \ L1 in a symmetric space X is itself a symmetric space, called the minimal part of the space X. In the case X0 D X, the symmetric space X is called minimal. For example, the spaces Lp , 1  p < 1 and L1 \ L1 are minimal, while L1 and L1 C L1 are not minimal. The minimal part of L1 C L1 has the form R0 D .L1 C L1 /0 D ff 2 L1 C L1 W f  .1/ D 0g ¤ L1 C L1 and L01 D L1 \ R0 ¤ L1 . Every separable symmetric space X is minimal, and its fundamental function 'X satisfies 'X .0C/ D 0. The converse is also true: every minimal symmetric space X satisfying 'X .0C/ D 0 is separable. Separable symmetric spaces can be characterized by the following property: .A/

ffn g  X; fn # 0 H) kfn kX # 0:

The space X is said to have order continuous norm if it has the property .A/. A continuous linear functional u on X (i.e., an element of the dual Rspace X ) has an integral form if u D ug for some g 2 L1 C L1 , where ug .f / D fg dm for all f 2 X.

58

II

Symmetric Spaces. The Embedding Theorem. Properties .A/; .B/; .C/

Although the dual space X in general need not be a symmetric space, the set X D fg 2 L1 C L1 W ug 2 X g equipped with the norm kgkX1 D kug kX is a symmetric space. The space X1 is called the associate space of X. For every symmetric space, there are continuous embeddings 1

X0  X  X11 ; where X11 D .X1 /1 is the second associate space of X. In the case X D X11 , the symmetric space X is called maximal. The spaces Lp ; 1  p  1, L1 \ L1 , and L1 C L1 are maximal, and the spaces R0 D .L1 C L1 /0 and L01 D L1 \ R0 are not maximal. The natural embedding of X into X11 is not necessarily isometric. The equality kf kX D kf kX11 , f 2 X is equivalent to the following property: .C/

ffn g  X; fn " f 2 X H) kf kX D sup kfn kX : n

The norm k  kX is called semicontinuous in this case. Every maximal symmetric space has the following property: .B/

ffn g  X; fn " and sup kfn kX < 1 H) fn " f for some f 2 X: n

If, in addition to .B/, X has the property .C/, then the embedding X into X11 is an isometric isomorphism between X and X11 . The converse is also true, i.e., the equality .X; k  kX / D .X11 ; k  kX11 / is equivalent to the following property:

.BC/

ffn g  X; fn "; supn kfn kX < 1 H); fn " f and kfn kX " kf kX for some f 2 X:

This property is usually called the Fatou property. Property .A/ is equivalent to separability of X, which in turn is equivalent to the equality fug ; g 2 X1 g D X . Every reflexive symmetric spaces X has property .AB/, i.e., it satisfies both .A/ and .B/: .AB/

ffn g  X; fn "; supn kfn kX < 1 H); fn " f and kfn  f kX ! 0 for some f 2 X:

Moreover, a symmetric space X is reflexive if and only if both X and X1 have property .AB/.

Chapter 5

Embeddings L1 \ L1  X  L1 C L1  L0

In this chapter, we prove the main embedding theorem for symmetric spaces. The theorem asserts that for every symmetric space X, there are continuous embeddings L1 \ L1  X  L1 C L1 and inequalities 2k  kL1 \L1  .'X /1 .1/ k  kX   k  kL1 CL1 : The space L0 of all measurable functions and the embedding L1 C L1  L0 are also considered.

5.1 Fundamental Functions First we show that every symmetric space X contains all functions of the form 1A ; A 2 Fm ; mA < 1: Indeed, let f 2 X, f ¤ 0. There exist a > 0 and B 2 Fm such that jf j  a  1B ; 0 < mB < 1: Therefore, 1B 2 X. We can choose measure-preserving transformations i 2 A.m/, i D 1; 2; : : : ; n, such that n [

i B A:

iD1

By setting i B D Bi , we have the functions 1B ; 1B1 ; : : : ; 1Bn , which are equimeasurable to 1B , and

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_5

59

5 Embeddings L1 \ L1  X  L1 C L1  L0

60

1B 2 X H) 1A 

n X

1Bi 2 X and 1A 2 X:

iD1

Thus 1A 2 X for all A 2 Fm with mA < 1. In particular, 1Œ0;x 2 X for all x  0. Definition 5.1.1. The function 'X .x/ D k1Œ0;x kX ; x  0: is called the fundamental function of the symmetric space X. If x D mA < 1, then the functions 1Œ0;x and 1A are equimeasurable. Hence, 'X .x/ D k1A kX for all A 2 Fm with x D mA < 1: Clearly, the function 'X is increasing and 'X .0/ D 0; and 'X .x/ > 0 for x > 0: We shall show below that 'X is continuous at every point x > 0. Since 'X is increasing, the right-hand limit 'X .0C/  0 exists. Note that both cases 'X .0C/ D 0 and 'X .0C/ > 0 are possible (Fig. 5.1). Examples 5.1.2. 1: The space Lp ; 1  p < 1 (Fig. 5.1): 0 'Lp .x/ D @

Z1

1 1p

0 x 1 1p Z 1 j1Œ0;x jp dmA D @ dmA D x p ; x  0:

0

0

2: The space L1 :

y

y jL

jL

1

1

0

y p

,

jL

1 < p 1: 4: The space L1 C L1 : 

Z1 'L1 CL1 .x/ D

1Œ0;x/ .s/ds D min.x; 1.0;1/ .x// D 0

x; 0  x  1I 1; x > 1:

5.2 The Embedding Theorem L1 \ L1  X  L1 C L1 We show that L1 \ L1 and L1 C L1 are the smallest and the largest symmetric spaces. Theorem 5.2.1. For every symmetric space X, L1 \ L1  X  L1 C L1

(5.2.1)

5 Embeddings L1 \ L1  X  L1 C L1  L0

62

and 2kf kL1 \L1  kf kX ; f 2 L1 \ L1 ; kf kX  kf kL1 CL1 ; f 2 X:

(5.2.2)

Proof. First, we check the embedding L1 \ L1  X. For f 2 L1 \ L1 , we set ak D f  .k/ and Ak D Œk; k C 1/; k D 0; 1; : : : : Then RC D

1 S

Ak is a partition of RC into disjoint sets Ak of measure 1 (Fig. 5.3)

kD0

such that akC1 < f .x/  ak ; x 2 Ak : Consider the functions f D

1 X

akC1  1Ak ; f D

kD0

1 X

ak  1Ak

kD0

and for n  1, fn D

n X

ak  1Ak :

kD0

Then f  f   f ; and f n " f as n ! 1:

y

a0 a1

f*

a2 a ak+k1 0

1

Fig. 5.3 The sequence fak g

2

3

k

k +1

x

5.2 The Embedding Theorem L1 \ L1  X  L1 C L1

63

The sequence ff n g is fundamental in X, since 1 X

akC1 D kf kL1  kf kL1

kD0

and for every m  1, kf nCm  f n kX D

nCm X

nCm X

ak k1Ak kX D 'X .1/

kDnC1

ak  'X .1/

kDnC1

1 X

ak ! 0;

kDnC1

as n ! 1. Thus there exists g 2 X such that kg  f n kX ! 0; n ! 1: Since f n jŒ0;n D f jŒ0;n for all n, we have gjŒ0;n D f jŒ0;n for all n, i.e., g D f 2 X, whence f   f and f 2 X. Thus, L1 \ L1  X and ! 1 1 X X ak  'X .1/ a0 C akC1 kf kX  kf kX  'X .1/ kD0

kD0

 'X .1/ .kf kL1 C kf kL1 /  2'X .1/kf kL1 \L1 : Now we prove the embedding X  L1 C L1 and check inequalities (5.2.2). We begin with functions f of a special form: f D

n1 X

ak  1Ak ; where ak  0; mAk D

kD0

n1 1 [ ; Ak D Œ0; 1: n kD0

For every cyclic permutation of indices f0; 1; 2; : : : ; n  1g, j W k ! k C j.mod n/; consider the function fj D

n1 X

aj .k/  1Ak ; 0  j  n  1:

kD0

The functions f0 ; f1 ; : : : ; fn1 are equimeasurable, and hence kfj kX D kf0 kX ; j D 0; 1; : : : ; n  1:

(5.2.3)

5 Embeddings L1 \ L1  X  L1 C L1  L0

64

On the other hand, n1 X

fj D

jD0

n1 X

! ak  1Œ0;1 :

kD0

Therefore, n1 X kD0

! ak

X n1 X n1 f  kfj kX D nkf0 kX ; 'X .1/ D j jD0 jD0 X

and kf0 kX 

! n1 1X ak  'X .1/ D kf0 kL1  'X .1/: n kD0

Thus, kf0 kX  'X .1/  kf0 kL1

(5.2.4)

for every function f D f0 of the special form (5.2.3). Let g 2 X. We estimate the norm kgkL1 CL1 by the norm kgkX . Since Z1 kgkL1 CL1 D

g dm;

0

we may assume, without loss of generality, that g D g and gjŒ1;1/ D 0. We can approximate g by functions gn of the form gn D

n1 X

ak  1Ak ;

kD0

where n D 2m , m D 1; 2; : : :,   

kC1 k kC1 ; Ak D ; : ak D g n n n Since g is decreasing, we have gn " g, and lim kgn kL1 D kgkL1 . n!1 Since the functions gn have the form (5.2.3), we can apply the inequality (5.2.4). Therefore, kgn kL1  .'X .1//1 kgn kX :

5.2 The Embedding Theorem L1 \ L1  X  L1 C L1

65

Setting n ! 1, we obtain kgkL1  .'X .1//1 kgkX : Hence for all f 2 X, Z1 kf kL1 CL1 D

f  dm D kf   1Œ0;1 kL1  .'X .1//1 kf   1Œ0;1 kX

0

 .'X .1//1 kf kX : Thus, both embeddings (5.2.1) and inequalities (5.2.2) hold.

t u

Note 5.2.2. In the second part of the previous proof we can use a function g supported on an arbitrary interval Œ0; a with a > 0 instead of g supported on Œ0; 1. Then the inequality 'X .1/kgkL1  kgkX becomes 'X .a/kf   1Œ0;a kL1  akf   1Œ0;a kX ; f 2 X; a > 0:

(5.2.5)

Proposition 5.2.3. Let X be a symmetric space, fn ; f 2 X and kfn  f kX ! 0 as n ! 1. Then fn ! f in measure. Proof. Let fn ; f 2 X and kfn  f kX ! 0 as n ! 1. Theorem 5.2.1 implies 'X .1/kfn  f kL1 CL1  kfn  f kX : Hence, hn D fn  f tends to 0 in norm k  kL1 CL1 , i.e., Z1 khn kL1 CL1 D

hn dm ! 0; n ! 1:

0

Suppose that contrary to our claim, fhn g does not tend to 0 in measure. Then for some positive " > 0, hn D mfhn > "g 6! 0; n ! 1: Then there exist a subsequence fhnk g and 0 < c < 1 such that hnk ."/ > c (Fig. 5.4). Hence hnk .c/  " for all k D 1; 2; : : : and

5 Embeddings L1 \ L1  X  L1 C L1  L0

66 Fig. 5.4 Functions hnk

y

h*n h*n ( c )

k

k

e 0

Z1 khnk kL1 CL1 D

hnk dm

Zc 

0

c

(e)

h

h*n

x

k

hnk dm  hnk .c/  c  "  c > 0

0

for all k D 1; 2; : : :. This contradicts the assumption khnk kL1 CL1 ! 0.

t u

5.3 The Space L0 and the Embedding L1 C L1  L0 Now we consider the space L0 of all real measurable functions on .RC ; m/. The space contains L1 C L1 and hence all symmetric spaces. We show that the space L0 is a complete metric space with respect to a natural metric d0 , defined as follows: Let v.x/ be a bounded concave function on RC such that v.0/ D 0 and v.x/ > 0 for all x > 0, and let be a Borel probability measure on RC that is equivalent to the Lebesgue measure m. We set Z1 p0 .f / D

v.jf j/d ; f 2 L0 : 0

The function v is semi-additive, i.e., v.a C b/  v.a/ C v.b/ for all a  0, b  0. Therefore, the functional p0 satisfies the triangle inequality p0 .f C g/  p0 .f / C p0 .g/; f ; g 2 L0 :

(5.3.1)

5.3 The Space L0 and the Embedding L1 C L1 L0

Since v.x/ increases and

67

v.x/ decreases, we have for x  0, x

v.cx/  v.x/; for 0  c  1 and v.cx/  cv.x/; for c  1 and p0 .cf /  p0 .f /; for jcj  1 and p0 .cf /  jcjp0 .f /; for jcj > 1; for all c 2 R, f 2 L0 . Thus, Z1 d0 .f ; g/ D p0 .f  g/ D

v.jf  gj/d ; f ; g 2 L0

(5.3.2)

0

is a translation-invariant metric in the linear space L0 . The operations of addition and multiplication by a scalar are continuous on L0 in the topology induced by the metric d0 . Two functions f and g that are equal almost everywhere are identified in L0 , whence d0 .f ; g/ D 0 ” f D g in L0 : Theorem 5.3.1. The metric space .L0 ; d0 / is complete. Proof. Let ffn g be a fundamental sequence in L0 , i.e., lim d0 .fn ; fm / D 0:

n;m!1

There exists an increasing sequence of indices fnk g1 kD1 such that Z1 d0 .fn ; fnk / D

v.jfn  fnk j/d  0

1 ; 2k

for all k and n > nk . Hence the series 1 X kD1

01 1 Z 1 X @ v.jfnkC1  fnk j/d .x/A d0 .fnkC1 ; fnk / D kD1

0

(5.3.3)

5 Embeddings L1 \ L1  X  L1 C L1  L0

68

converges, and the series 1 X

gk .x/;

gk .x/ D v.jfnkC1  fnk j/;

kD1

converges almost everywhere to a function h.x/ D

1 P

gk .x/.

kD1

The function h is finite almost everywhere and 01 1 Z1 Z 1 X @ gk .x/d .x/A : h.x/d .x/ D kD1

0

(5.3.4)

0

Indeed, let hn .x/ D

n X

gk .x/:

kD1

Then hn " h and the integral Z1

Z1 h.x/d .x/ D lim

n!1

0

01 1 Z 1 X @ gk .x/d .x/A ; hn .x/d .x/ D kD1

0

0

equals the sum of the series (5.3.3). This implies that h 2 L1 . /. Hence h is finite almost everywhere, and (5.3.4) holds. Since both functions v and v 1 are continuous and increasing, almost everywhere 1 P convergence of gk .x/ implies almost everywhere convergence of kD1 1 X fn1 .x/ C .fnkC1 .x/  fnk .x//: kD1

Denote the sum by f .x/. Then fnk ! f almost everywhere and fnk ! f in measure

, since the measure is finite. Thus d0 .fnk ; f / ! 0 as k ! 1. The inequality d0 .fn ; f /  d0 .fn ; fnk / C d0 .fnk ; f / provides convergence d0 .fn ; f / ! 0; n ! 1.

t u

Thus, we have established that .L0 ; d0 / is a complete metric space and the metric d0 is translation-invariant: d0 .f ; g/ D d0 .f  g; 0/; f ; g 2 L0 : Such spaces are called F-spaces.

5.3 The Space L0 and the Embedding L1 C L1 L0

69

y

y

x v (x ) = x+1

1

0

v( x ) = min(x,1)

1

0

x

1

x

Fig. 5.5 Functions v.x/

The function v.x/ is usually chosen as v.x/ D (Fig. 5.5). The measure can be chosen by setting

.A/ D

x or v.x/ D min.x; 1/ xC1

1 X 1 m.A \ Œn  1; n/; A 2 B: n 2 nD1

Then Zn 1 X 1 d0 .f ; g/ D v.jf  gj/dm; 2n nD1 n1

where 1

X 1 d

D  1Œn1;n : dm 2n nD1 The topology in the linear space .L0 ; d0 / is defined by the system of neighborhoods fU" .0/; " > 0g, where  

  1 U" .0/ D f 2 L0 W m 0; \ fjf j > "g < " : " Note that the system, and hence the considered topology, do not depend on the choice of v.x/ and . Convergence in this topology coincides with so-called stochastic convergence.

5 Embeddings L1 \ L1  X  L1 C L1  L0

70

Definition 5.3.2. A sequence ffn g of a measurable functions is said to converge .st/

stochastically to a function f (fn ! f ) if m.A \ fjfn  f j > "g/ ! 0; n ! 1

(5.3.5)

for all " > 0 and all A 2 Fm with mA < 1. .st/

Thus d0 .fn ; f / ! 0 if and only if ffn g ! f i.e., fn jA ! f jA in measure on each subset A of finite measure. It is known that the stochastic convergence topology on L0 is not locally convex, and hence it cannot be normed. It is clear that convergence in measure implies stochastic convergence. The converse is not true in general. The embedding L1 C L1  L0 is continuous, since convergence in norm in L1 C L1 implies convergence in measure and hence stochastic convergence (Proposition 5.2.3). By combining this fact with Theorem 5.2.1, we obtain the following corollary. Corollary 5.3.3. The natural embedding of a symmetric space X into L0 is continuous, i.e., convergence in norm k  kX in X implies stochastic convergence. Example 5.3.4. Note that convergence almost everywhere does not imply in general convergence Indeed, let fn D 1Œn1;n . Then fn ! 0 almost everywhere,

 in measure. 1 D 1 and fn D 1Œ0;1 , i.e., ffn g does not converge to 0 in measure. while m fn > 2 Hence, the sequence ffn g does not converge to 0 in any symmetric space X.

Chapter 6

Embeddings. Minimality and Separability. Property .A/

In this chapter we study minimal and separable symmetric spaces. The minimal part X0 of a symmetric space X is the closure of L1 \ L1 in X, and X is minimal if X0 D X. We show that every separable symmetric space is minimal, and the converse is true under the additional condition X .0C/ D 0. We consider also an important property .A/, which is equivalent to separability.

6.1 Embedded Symmetric Spaces Let X1 and X2 be two symmetric spaces. Proposition 6.1.1. Let X1  X2 . Then the natural embedding i W X1 3 f ! i.f / D f 2 X2 is continuous (bounded). Proof. Let ffn g be a sequence in X1 such that kfn  f kX1 ! 0 and kfn  gkX2 ! 0; f 2 X1 ; g 2 X2 : Then by Proposition 5.2.3, fn ! f and fn ! g in measure. Hence, f D i.f / D g 2 X1 . This means that the embedding operator i is closed. By the closed graph theorem, i is a continuous (bounded) operator. t u Note 6.1.2. Continuity of the embedding X1  X2 means that kf kX2  ckf kX1 ; f 2 X1 ; for some c > 0. © Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_6

71

6 Embeddings. Minimality and Separability. Property .A/

72

The open mapping theorem implies that X1 is closed in X2 if and only if the embedding is open, i.e., kf kX1  c1 kf kX2 ; f 2 X1 for some real c1 > 0. For example, R0 D clL1 CL1 .L1 \ L1 / is closed in L1 C L1 , while L1 \ L1 is dense in L1 . The embedding L1 \ L1  L1 is continuous, but it is not open. Now we show that the closure of a symmetric space in any other symmetric space is also a symmetric space. Proposition 6.1.3. Let Y be a nonzero linear subspace of a symmetric space X satisfying the following conditions: 1. jgj  jf j and f 2 Y H) g 2 Y. 2. f 2 Y H) f ı  2 Y for any  2 A.m/. Then the closure clX .Y/ of the subset Y in X equipped with the norm induced by k  kX is a symmetric space. Proof. Let 0  g  f 2 clX .Y/ and let ffn g be a sequence in Y such that lim kf  fn kX D 0:

n!1

Let gn D minfg; fn g. Then 0  g  gn D .g  fn /C  jf  fn j and lim kg  gn kX  lim kf  fn kX D 0:

n!1

n!1

Since gn  fn 2 Y, we have also gn 2 Y and g 2 clX .Y/. Thus clX .Y/ is a normed ideal lattice. Since Y satisfies condition 2, the closure clX .Y/ also satisfies the condition. Indeed, if f 2 clX .Y/, ffn g  Y, is such that fn ! f in X, and g D f ı  for some  2 A.m/, then gn D fn ı  2 Y and kgn  gkX D kfn ı   f ı  kX D kfn  f kX ! 0: Thus, gn ! g in X and g 2 clX .Y/.

6.2 The Intersection and the Sum of Two Symmetric Spaces

73

We now check that clX .Y/ is symmetric. We may assume without loss of generality that clX .Y/ D Y, i.e., Y is closed in X. Let f  0, g  0 be two equimeasurable functions and f 2 Y. We show that g 2 Y. First consider the case a D f  .1/ D g .1/ D 0. Then by Proposition 3.3.1, we can construct sequences of step functions ffn g and fgn g such that fn " f ; gn " g; lim kgn  gkL1 \L1 D 0; n!1

and the functions fn and gn are equimeasurable for all n. Then fn 2 Y, since fn  f . For these step functions fn and gn according to Proposition 3.4.2, there exist n 2 A.m/ such that gn D fn ı n . Condition 2 implies that gn 2 Y. Since Y is closed in X and lim kgn  gkX D 0, we have also g 2 Y. n!1

Consider now the case a D f  .1/ D g .1/ > 0. The functions f  1ff >ag and g  1fg>ag are equimeasurable, and they satisfy the assumption of Proposition 3.3.1. We have f 2 Y, whence f  1ff >ag 2 Y, and as was shown above, g  1fg>ag 2 Y. On the other hand, the set ff > bg has infinite measure for every b < a, and thus 1ff >bg 2 Y. By reducing if necessary the set A D ff > bg, we may assume that its complement A D RC n A also has infinite measure. Since there exists  2 A.m/ such that  1 A D A, we have by condition 2 that 1A 2 Y and 1Œ0;1/ D 1A C 1A 2 Y. This shows that Y contains all bounded functions and in particular, the function g  1fgag . Thus, g D g  1fg>ag C g  1fgag 2 Y: t u Corollary 6.1.4. The closure clX .X1 / of a symmetric space X1 in any other symmetric space X is also a symmetric space.

6.2 The Intersection and the Sum of Two Symmetric Spaces Let X1 and X2 be two symmetric spaces and kf kX1 \X2 D max.kf kX1 ; kf kX2 /; f 2 X1 \ X2 :

Proposition 6.2.1. .X1 \ X2 ; k  kX1 \X2 / is a symmetric space. Proof. It is clear that k  kX1 \X2 is a norm. We show now the completeness of this space.

6 Embeddings. Minimality and Separability. Property .A/

74

Let ffn g be a Cauchy sequence in X1 \ X2 . Then the sequence ffn g is a Cauchy sequence in both X1 and X2 . Since the spaces X1 and X2 are Banach spaces, there exist functions f 2 X1 and g 2 X2 such that kfn  f kX1 ! 0; kfn  gkX2 ! 0; n ! 1: By Proposition 5.2.3, fn ! f and fn ! g in measure, and hence f D g 2 X1 \ X2 and kfn  f kX1 \X2 ! 0. The other conditions of symmetric spaces are obvious for X1 \ X2 . t u The sum X1 C X2 of two symmetric spaces X1 and X2 consists of all functions f of the form f D g C h; g 2 X1 ; h 2 X2 : We set kf kX1 \X2 D inffkgkX1 C khkX1 W f D g C h; g 2 X1 ; h 2 X2 g:

(6.2.1)

Proposition 6.2.2. The space .X1 C X2 ; k  kX1 CX2 / is a symmetric space. Proof. Let X1 and X2 be as above. 1: We show now that X1 C X2 equipped with the norm k  kX1 CX2 is a Banach space. For this purpose, it is convenient to represent the space as a quotient space .X1 X2 /=L of the direct product X1 X2 D f.f1 ; f2 /; f1 2 X1 ; f2 2 X2 g equipped with the norm k.f1 ; f2 /kX1 X2 D kf kX1 C kf2 kX2 ; by the subspace L D f.f ; f /; f 2 X1 \ X2 g  X1 X2 : If two pairs .f1 ; f2 / and .g1 ; g2 / belong to the same coset of L in X1 X2 , then f1 C f2 D g1 C g2 ; i.e., the coset corresponds to the same element of the space X1 C X2 . On the other hand, every representation of f 2 X1 C X2 of the form f D f1 C f2 ; f1 2 X1 ; f2 2 X2 ; determines a pair .f1 ; f2 / from the same coset of L.

6.3 Minimal Symmetric Spaces

75

The norm kf kX1 CX2 defined by equality (6.2) is by definition just the norm in the quotient space .X1 X2 /=L. Thus, X1 C X2 ' .X1 X2 /=L is a normed space with the norm (6.2.1). Since X1 and X2 are Banach spaces, the space X1 X2 is also a Banach space. Since the subspace L is closed, the quotient space .X1 X2 /=L and the sum X1 C X2 are also Banach spaces. 2: Now we show that the space X1 C X2 is a normed ideal lattice. Let g 2 X1 C X2 and jf j  jgj. For a fixed " > 0, we find g1 2 X1 and g2 2 X2 such that g D g1 C g2 and kg1 kX1 C kg2 kX2  kgkX1 CX2 C ": Let us set hD

f  1fg¤0g : g

Then f D g1 h C g2 h and jg1 hj  kg1 k; jg2 hj  kg2 k: Hence, g1 h 2 X1 , g2 h 2 X2 , and f 2 X1 C X2 . Moreover, kf kX1 CX2  kg1 hkX1 C kg2 hkX2  kg1 kX1 C kg2 kX2  kgkX1 CX2 C "; which in turn implies kf kX1 CX2  kgkX1 CX2 : 3: The symmetry property of the norm k  kX1 CX2 follows directly from those properties of k  kX1 and k  kX2 using Theorem 3.4.5 (Exercise 8). t u

6.3 Minimal Symmetric Spaces Let X0 D clX .L1 \ L1 / be the closure of L1 \ L1 in X with respect to the norm kf kX0 D kf kX ; f 2 X0 : By Corollary 6.1.4, the space .X0 ; k  kX0 / is a symmetric space.

6 Embeddings. Minimality and Separability. Property .A/

76

Definition 6.3.1. The space X0 is called the minimal part of X. A symmetric space X is called minimal if the equality X0 D X holds. Note 6.3.2. Theorem 2.3.1 implies that the set F0 of all simple functions with bounded support is dense in L1 \ L1 . Hence, X0 D clX .F0 /, and X is minimal if and only if the set F0 is dense in X. Since F0  F1  L1 \ L1 , the same is true for the set F1 of all simple integrable functions. Examples 6.3.3. 1. L1 \ L1 is minimal. Indeed, .L1 \ L1 /0  L1 \ L1 by definition, and .L1 \ L1 /0 is a symmetric space. Hence, L1 \ L1  .L1 \ L1 /0 by Theorem 5.2.1. 2. The spaces Lp , 1  p < 1, are minimal, since the set L1 \ L1 is dense in each of them (Theorem 2.3.1). Thus L0p D Lp for 1  p < 1. 3. L1 C L1 is not minimal. Its minimal part .L1 C L1 /0 has the form .L1 C L1 /0 D R0 ; where R0 D ff 2 L1 C L1 W f  .1/ D 0g (see Proposition 4.3.1). 4. L1 is not minimal. Its minimal part L01 has the form L01 D R0 \ L1 (see Proposition 3.5.1).

6.4 Minimality and Separability Consider the fundamental function 'X .x/ D k1Œ0;x kX ; x  0; of a symmetric space X. The function is increasing; hence the right-hand limit at zero, '.0C/ D lim 'X .x/  0; x!0C

always exists. Thus two cases are possible: either 'X .0C/ D 0 or 'X .0C/ > 0. Returning to Examples 6.3.3 we see that the symmetric spaces R0 , L01 D R0 \ L1 , and also Lp , Lp \ L1 , 1  p < 1 are minimal. Here 'R0 .0C/ D 'Lp .0C/ D 0;

6.4 Minimality and Separability

77

while 'R0 \L1 .0C/ D 'Lp \L1 .0C/ > 0: Thus minimal spaces may be separable (for example, the spaces Lp ; 1  p < 1 and R0 ) and nonseparable (for example, L1 \ L1 and L01 ). Theorem 6.4.1. Let X be a minimal symmetric space. 1. If 'X .0C/ > 0, then X  L1 , and X is nonseparable. 2. If 'X .0C/ D 0, then X 6 L1 , and X is separable. Proof. 1: If 'X .0C/ D c > 0, then for 0 < a < b (Fig. 6.1), k1Œ0;b  1Œ0;a kX D k1Œ0;ba kX D 'X .b  a/  c > 0: Thus f1Œ0;a W a > 0g is a discrete uncountable subset in X, i.e., the space X is nonseparable. For every function f 2 X and xn # 0 (Fig. 6.2), kf  kX  kf   1Œ0;xn  kX  f  .xn /'X .xn /  c  f  .xn /:

(6.4.1)

If c D 'X .0C/ > 0, then the sequence ff  .xn /g is bounded. Hence f  .0/ < 1, i.e., f 2 L1 . Thus, X  L1 . Conversely, the inclusion X  L1 implies that 'X .0C/ D c > 0. Indeed, if X  L1 , then by Proposition 6.1.1, there exists a constant k > 0 such that kf kL1  kkf kX ; f 2 X: Let f D 1Œ0; 1  . Then n

1 D k1Œ0; 1  kL1  kk1Œ0; 1  kX D k'X n

n

By taking n ! 1, we obtain 'X .0C/ 

  1 : n

1 > 0. k

y

y j (x )= 1[0, x ] X

j (b _a)

1[ a,b ]

X

1

X

c 0

b _a

x

Fig. 6.1 'X .b  a/ D k1Œa;b kX in the case 'X .0C/ > 0

0

a

b

x

6 Embeddings. Minimality and Separability. Property .A/

78 Fig. 6.2 Inequality (6.4.1)

y f* (0)

X Õ L•

j

x

f * (xn )

f*

c

0

xn

x

2: Let now 'X .0C/ D 0. Then X 6 L1 , which follows from the proof of part 1. Further, for every f 2 F1 , we have lim kf  f  1Œ0;N kX D 0:

N!1

Indeed, for f D

n X

ai  1Ai ; ai 2 R; mAi < 1; 1  i  n 2 N;

iD1

we can set a D kf k1 D max .jai j/; and A D 1in

1 [

Ai :

iD1

Then kf  f  1Œ0;N kX  ak1A\.N;1/ kX D a'X .m.A \ .N; 1/// ! 0; N ! 1; since m.A \ ŒN; 1// ! 0 for N ! 1 and 'X .0C/ D 0. Thus, if X is a minimal symmetric space, then the set ff  1Œ0;N ; f 2 F1 g  F0 is dense in X as well as F1 . For every function f 2 F0 supported in the interval Œ0; N and for every " > 0, Lusin’s theorem provides that there exists a continuous function f" such that jf" j  jf j and mff" ¤ f g < ":

6.5 Separability and Property .A/

79

Then, kf  f" kX  2kf kL1 k1ff" ¤f g kX  2kf kL1 'X ."/ ! 0; " ! 0: The continuous function f" on Œ0; N can be uniformly approximated by polynomials with rational coefficients. Thus in the case of clX .F1 / D X and 'X .0C/ D 0, the countable set fp  1Œ0;N ; p is a polynomial with rational coefficients, N 2 Ng t u

is dense in X. Corollary 6.4.2. Let X be a minimal symmetric space X. Then 'X .0C/ > 0 ” X  L1 : If in addition, X ¤ L1 , then 'X .0C/ > 0 H) X  L01 D L1 \ R0 :

6.5 Separability and Property .A/ Consider now conditions of separability for symmetric spaces. First we show that the second part of Theorem 6.4.1 can be converted, i.e., that separability of X implies minimality of X and also 'X .0C/ D 0. Second, separability of X is equivalent to the following important property .A/. Definition 6.5.1. A symmetric space X is said to have property .A/ (order continuous norm) if .A/

ffn g  X; and fn # 0 H) kfn kX # 0; n ! 1:

Property .A/ can be written in the following form .A0 /. Proposition 6.5.2. Property .A/ is equivalent to the following property: .A0 /:

ffn g  X; fn # 0 and .fn  fnC1 /fnC1 D 0 H) kfn kX # 0:

Proof. .A/ ) .A0 / is obvious. Suppose .A0 / holds. Let ffn g  X; fn # 0 and " > 0. Then by setting gn D .fn  "f1 /C and hn D f1  1fgn 6D0g ; we have gn # 0; hn # 0 and .hn  hnC1 /hnC1 D 0:

6 Embeddings. Minimality and Separability. Property .A/

80

Property .A0 / implies khn kX # 0 as n ! 1. On the other hand, fn D fn  1fgn 6D0g C .fn  fn  1fgn 6D0g /  hn C "f1 ; whence kfn kX  khn kX C "kf1 kX ; and then kfn kX # 0 as n ! 1.

t u

Theorem 6.5.3. Let X be a symmetric space. Then the following are equivalent: 1. X is separable. 2. X is minimal and 'X .0C/ D 0. 3. X has property .A/. Proof. 2 H) 1 was proved in Theorem 6.4.1. 1 H) 3. Let X be separable but suppose X fails to satisfy .A/. Then by Proposition 6.5.2, X fails to satisfy .A0 /. This means that there exists a sequence fgn g such that gn 2 X; gn # 0; .gn  gnC1 /gnCk D 0; k  1; and lim kgn kX D c > 0:

n!1

The sequence fgn g is not fundamental. Otherwise, it must be convergent in norm k  kX to some g 2 X and gn ! g in measure. Since gn # 0, we have g D 0, which contradicts kgn kX  c > 0. Since fgn g is not fundamental in norm k  kX , we can find b > 0 and an increasing sequence of indices nk " 1 such that kgnk  gnkC1 kX > b: By normalizing the sequence fgnk  gnkC1 g, we obtain a new sequence hk D

gnk  gnkC1 ; k  1; kgnk  gnkC1 kX

such that khk kX D 1; hk  hl D 0; k ¤ l; h D sup hk 2 X: k

Here h 2 X, since jhj 

1 jf j. b

6.5 Separability and Property .A/

81

For every sequence ˛ D f˛k g 2 f1; 1gN , consider the function h.˛/ D

1 X

˛j hj 2 X;

jD1

where h.˛/ 2 X, since jh.˛/j  jhj. Then the set fh.˛/ W ˛ 2 f1; 1gN g is an uncountable discrete subset of X. This fact contradicts the separability of the space X. Thus X has property .A/. 3 H) 2. If xn # 0, then 1Œ0;xn  # 0, and by property .A/, 'X .xn / D k1Œ0;xn  kX ! 0; n ! 1: Thus 'X .0C/ D 0. Let 0  f 2 X and fn D min.f ; n/  1Œ0;n . Then .f  fn / # 0 and .A/ implies kf  fn kX ! 0. Since fn 2 L1 \ L1 , then f 2 clX .L1 \ L1 / D X0 , i.e., the space X is minimal. t u Note 6.5.4. The minimality of the space X is equivalent to the combination of the following two conditions: 1. kf   min.f  ; n/kX ! 0; n ! 1, for all f 2 X. 2. kf   f   1Œ0;n kX ! 0; n ! 1, for all f 2 X. Those conditions mean that every function f 2 X is the limit of its upper and right cutoff functions in norm k  kX .

Chapter 7

Associate Spaces

In this chapter, we study associate Rspaces X1 of symmetric spaces X. The space X1 is defined by the duality hf ; gi D fg dm, f 2 X, g 2 X1 , and the norm k  kX1 is induced by the canonical embedding of X1 into the dual space X of X. We show that .X1 ; k  kX1 / is a symmetric space and that the canonical embedding of X1 into X is surjective if and only if the space X is separable, i.e., X has property .A/.

7.1 Dual and Associate Spaces Let X be a symmetric space and let X be its dual Banach space. The space X consists of all linear continuous (bounded) functionals u W X ! R on X equipped with the norm kukX D supfju.f /j W kf kX  1g < 1: Every symmetric space X is a Banach ideal lattice. Hence its dual space X is also a Banach ideal lattice with the natural order u  v ” u.f /  v.f / for all f 2 X; f  0: For every u 2 X , one can define functionals juj, uC , and u such that u D uC  u ; juj D uC C u : The functional juj is the least element in X such that juj  u  juj and juj.f /  0 for all f 2 X, f  0.

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_7

83

84

7 Associate Spaces

In some cases, the dual space X of a symmetric space X is itself a symmetric space, or to be more precise, it can be identified in a natural way with a symmetric space. Consider two typical examples. Example 7.1.1. Let X D L1 . Then L1 D L1 by Theorem 2.4.1. More precisely, for every function g 2 L1 , there is ug 2 L1 , defined by Z1 ug .f / D

fgdm; f 2 L1 : 0

Then ug 2 L1 and kug kL1 D kgkL1 . Indeed, if u 2 L1 , the equality

u .A/ D u.1A /; A 2 Fm determines a  -additive set function u on RC , which is absolutely continuous with respect to the measure m. By the Radon–Nikodym theorem, there exists g such that Z

u .A/ D

gdm A

for all A 2 Fm with mA < 1, and hence Z1 u.f / D

Z1 fd u D

0

fgdm D ug .f / 0

for all f 2 L1 .m/. Thus, the embedding W L1 3 g ! ug 2 L1 is an isometric isomorphism between L1 and L1 . Example 7.1.2. Let X D L1 . For every function g 2 L1 , there exists a functional ug 2 L1 , Z1 ug W L1 3 f !

fgdm 2 R; 0

such that the mapping W L1 3 g ! ug 2 L1 is a linear isometry of L1 into L1 , i.e., kug kL1 D kgkL1 . However, in this case, .L1 / D fug ; g 2 L1 g does not coincide with the space L1 . In other words, not all functionals u 2 L1 have the form ug ; g 2 L1 .

7.2 The Maximal Property of Products f  g

85

Indeed, since clL1 .F0 / D L01 ¤ L1 ; we can choose u1 2 L1 such that u1 .1Œ0;1/ / D 1 and u1 .f / D 0 for all f 2 F0 . Z1 If u1 D ug1 for some g1 2 L1 , then we would have g1 dm D 1 and Z1

0

fg1 dm D 0 for all f 2 F0 , which is false. Thus, the embedding .L1 /  L1 is

0

strict (see also Theorem 2.4.1). Considering a symmetric space X, we would like to characterize the part of the space X that consists of all functionals ug 2 X ; g 2 L1 C L1 , while all such functions g form a symmetric space. Let X be a symmetric space. Consider the set X1 D fg 2 L0 W ug 2 X g;

(7.1.1)

where Z1 ug .f / D

fgdm; f 2 X:

(7.1.2)

0

We define the norm of g 2 X1 , by setting kgkX1 D kug kX :

(7.1.3)

Thus, the space X1 consists of all measurable functions g for which kgkX1

8ˇ 1 ˇ 9 ˇ 0. N In the same way as in Theorem 4.1.1, we obtain sets Ek , k D 1; 2; : : : ; N, such that mEk D a and Ek  fx 2 .0; 1/ W g .ka/  g.x/  g ..k  1/a/g; 1  k  N; and also sets Ek , k D N C 1; N C 2; : : :, such that mEk D a and Ek  fx 2 .0; 1/ W g .1/  "  g.x/  g .1/g; k > N: The sets Ek , k D 1; 2; : : :, can be chosen here disjoint and with

1 [ kD1

Ek D Œ0; 1/:

Since the sets Ek have the same measure mEk D a, there exist invertible measurepreserving mappings k W Ek ! Œ.k  1/a; ka/; k D 1; 2; : : : : Define  2 A.m/ by  jEk D k and consider the function va .x/ D

1 X kD1

which is equimeasurable to f  .x/.

f  .k .x//  1Ek .x/;

88

7 Associate Spaces

Using va , we have Z1 Z1 1 Z X sup .f ı  /  g dm  va g dm D f  ı k  gdm

2A.m/

0

kD1 E

0

a

k

N X

f  .ka/g .ka/

kD1

C

1 Z X

f  ı   Œg .1/  "dm:

kDNC1 E

k

If 1 Z X

Z1



f ı  dm D

kDNC1 E

f  dm D 1;

c

k

the inequality (7.2.5) holds. If the latter integral is finite, then Z1 Z1 1 X   sup .f ı  /  g dm  a f .ka/g .ka/  " f  dm:

2A.m/

kD1

0

It is easy to see that a

1 P

c

f  .ka/g .ka/ is an integral sum for the function f  g on

kD1

.0; 1/. By passing to the limit as a ! 0, we obtain Z1 Z1 Z1   .f ı  /  g dm  f g dm  " f  dm: sup

2A.m/

0

c

c

This implies (7.2.5), and hence the equality (7.2.1) holds. The case c D 1 in which g .x/ > g .1/ for all x  0 is considered in a similar way. Note that the case c D 0 corresponds to constant g , for which inequality (7.2.5) is trivial. Since in this case g  g almost everywhere, one has g > g  " on a set of infinite measure. t u Note that by putting g D 1A in (7.2.1), we obtain Theorem 4.1.1 (the maximal property of f  ) as a particular case of Proposition 7.2.1.

7.2 The Maximal Property of Products f  g

89

Theorem 7.2.2. .X1 ; k  kX1 / is a symmetric space. Proof. It is clear that .X1 ; k  kX1 / is a normed ideal lattice. Proposition 7.2.1 implies kgkX1 D sup

81 0, and hence convergence in (7.2.8) is uniform on the sets fA 2 Fm W mA < cg, c > 0. Since the set function gn is absolutely continuous with respect to m, the set function is also absolutely continuous with respect to m. Hence, by the Radon– Nikodym theorem, there exists a locally integrable function g such that

90

7 Associate Spaces

Z

.A/ D

gdm D u.1A /; A 2 Fm ; mA < 1; A

whence Z1 ug .f / D

fgdm D u.f /; f 2 X; 0

i.e., u D ug 2 .X1 /. Thus .X1 / is closed in X . The proof is complete.

t u

Definition 7.2.3. The symmetric space .X1 ; k  kX1 / is called the associate space of a symmetric space X.

7.3 Examples of Associate Spaces Examples 7.3.1. 1: Lp , 1  p  1. Let 1  q  1 be such that

1 1 C D 1. p q

Theorem 2.4.1 states that the mapping W Lq 3 g ! ug 2 .Lq /  Lp is an isometric isomorphism (see also Examples 7.1.1 and 7.1.2). Moreover, for 1  p < 1, we have .Lq / D Lp , and for p D 1, the embedding .L1 /  L1 is strict. 2: L1 \ L1 and L1 C L1 . For every function f 2 L1 C L1 and g 2 L1 \ L1 , ˇ ˇ1 ˇ ˇZ ˇ ˇ ˇ fgdmˇ  kf kL CL  kgkL \L : 1 1 1 1 ˇ ˇ ˇ ˇ 0

Indeed, if f D u C v, u 2 L1 , v 2 L1 , then ˇ1 ˇ ˇZ ˇ ˇ ˇ ˇ fgdmˇ  kukL  kgkL C kvkL  kgkL 1 1 1 1 ˇ ˇ ˇ ˇ 0

 .kukL1 C kvkL1 /kgkL1 \L1  kf kL1 CL1  kgkL1 \L1 : The inequality (7.3.1) implies .L1 C L1 /1 L1 \ L1 ; .L1 \ L1 /1 L1 C L1

(7.3.1)

7.3 Examples of Associate Spaces

91

and k  k.L1 CL1 /1  k  kL1 \L1 ; k  k.L1 \L1 /1  k  kL1 CL1 : On the other hand, kgk.L1 CL1 /1

ˇ ˇ1 ˇ ˇZ ˇ ˇ ˇ fgdmˇ D sup ˇ ˇ kf kL1 CL1 1 ˇ ˇ 0 ˇ ˇ1 ˇ1 ˇ1 0 ˇ ˇ ˇZ ˇZ ˇ ˇ ˇ ˇ  max @ sup ˇˇ fgdmˇˇ ; sup ˇˇ fgdmˇˇA kf kL1 1 ˇ ˇ kf kL1 1 ˇ ˇ 0

0

D max.kgkL1 ; kgkL1 / D kgkL1 \L1 ; and in a similar way, kf k.L1 \L1 /1

ˇ ˇ1 ˇ ˇZ ˇ ˇ ˇ ˇ D sup ˇ fgdmˇ  kf kL1 CL1 : kgkL1 \L1 1 ˇ ˇ 0

Thus, the following proposition holds. Proposition 7.3.2. 1. ..L1 C L1 /1 ; k  k.L1 CL1 /1 / D .L1 \ L1 ; k  kL1 \L1 /; 2. ..L1 \ L1 /1 ; k  k.L1 \L1 /1 / D .L1 C L1 ; k  kL1 CL1 /. It is an essential fact that the associate space X1 of X is completely determined by the minimal part X0 D clX .F0 / D clX .L1 \ L1 / of X. Proposition 7.3.3. .X0 /1 D X1 . Proof. By Theorem 5.2.1, F0  L1 \ L1  X, and by Corollary 6.1.4, X0 D clX .F0 / D clX .L1 \ L1 / is a symmetric space. Levi’s theorem implies that both the norms kgkX1 and kgk.X0 /1 can be obtained by means of functions f 2 F0 as kgkX1 D kgk.X0 /1

8ˇ 1 ˇ 9 ˇ 0. 3: The space L1 \ L1 is minimal and maximal: .L1 \ L1 /0 D L1 \ L1 D .L1 C L1 /1 D .L1 \ L1 /11 by Proposition 7.3.2. The space L1 \ L1 is nonseparable and 'L1 \L1 .0C/ > 0. 4: The space L1 C L1 is maximal but not minimal: .L1 C L1 /0  L1 C L1 D .L1 \ L1 /1 D .L1 C L1 /11 :

8.2 Maximality and Property .B/

97

The minimal part of L1 CL1 has the form .L1 CL1 /0 D R0 by Proposition 4.3.1 and Example 6.3.3. Moreover, R10 D .L1 C L1 /1 D L1 \ L1 : The space L1 C L1 is nonseparable, while R0 D clL1 CL1 .F0 / is separable, 'R0 .0C/ D 'L1 CL1 .0C/ D 0.

8.2 Maximality and Property .B/ Let X be a symmetric space. We consider in greater detail the embedding X  X11 and conditions of maximality X D X11 . Definition 8.2.1. A symmetric space X is said to have property .B/ (X is monotonically complete) if .B/ W

ffn g  X; 0  fn "; sup kfn kX < 1 H) fn " f n

for some f 2 X:

Theorem 8.2.2. Let X be a symmetric space. The following are equivalent: 1. X D X11 as sets. 2. X D Y1 as sets for some symmetric space Y. 3. X has property .B/ and X0 D .X11 /0 . . Proof. 1 ” 2. If X D Y1 for some symmetric space Y, then X1 D Y11 and X11 D Y111 D Y1 D X: If X D X11 , then X D Y1 for Y D X1 . 2 H) 3. Let X D Y1 and ffn g  X be such that 0  fn " and sup kfn kX < 1. n

Since both X and Y1 are Banach spaces, the norms kkX and kkY1 are equivalent by the open mapping theorem. Hence sup kfn kY1 D c < 1. n

Since kfn kY1 D sup

81 1: Hence for a large enough n0 , we have f  1Œ0;n0  2 L1 \ X; kf  1Œ0;n0  kX > 1: Since B1 is closed in L1 , there exists h 2 L1 separating f  1Œ0;n and B1 , i.e., Z1

Z1 hf  1Œ0;n dm > 1 and

hgdm  1 for all g 2 B1 :

0

0

For every f 2 X with kf kX > 1, we have found a function h0 D h  1Œ0;n0  2 L1 \ L1  X1 ; for which Z1 uh0 .f / D

fh0 dm > 1; 0

while kuh0 kX D sup

D sup

81 0. X D L1 as sets ” 'X0 .0C/ < 1.

10. Embedding theorems for symmetric sequence spaces. Let X D X.N/ be a symmetric sequence space in the sense of Complement 3. Show that a. l1  X.N/  l1 . b. If 'X.N/ .1/ D 1, then both embeddings are contractions. c. lim 'Xn.n/ > 0 ” X.N/ D l1 as sets. n!1

d.  'X .1/ < 1 ”

X D c0 as sets; if 1N 2 6 X X D l1 as sets; if 1N 2 X:

11. Minimality and separability for symmetric spaces on .0; 1/. Let X D X.0; 1/ be a symmetric space on Œ0; 1 in the sense of Complement 2. Show that a. The following conditions are equivalent: • X is minimal, i.e., X D clX .L1 /. • lim kf  min.f ; n/kX D 0 for every function f 2 X. n!1

•

lim kf   min.f  ; n/kX D 0 for every function f 2 X.

n!1

b. The following conditions are equivalent: • X is separable. • X is minimal and X ¤ L1 as sets. • X has property .A/. c. All the spaces Lp .0; 1/; 1  p  1, are minimal and maximal, while L1 .0; 1/ is the only minimal nonseparable symmetric space on Œ0; 1.

Exercises

107

12. Minimality and separability for symmetric sequence spaces. Let X D X.N/ be a symmetric sequence space in the sense of Complement 3. Show that a. The following conditions are equivalent: • X is minimal, i.e., X D clX .l1 /. • X is separable. • X has property .A/. • lim kf   f   1f1;2;:::;ng kX D 0 for every function f 2 X. n!1

b. The spaces lp ; 1  p < 1, are minimal, maximal, and separable. c. The space l1 is maximal but not minimal. Its minimal part .l1 /0 D cll1 .l1 / coincides with c0 . d. The space c0 is minimal, separable, but not maximal: c10 D l1 and .c0 /11 D l11 D l1 : 13. Embedding theorems and property .C/. Let X D X.0; 1/ be a symmetric space on Œ0; 1/, X0 D clX .L1 \ L1 / and let 11 X D .X1 /1 be the second associate space of X. Show that a. Every minimal (separable or nonseparable) symmetric space X has property .C/. b. Let X0  X  X11 be the canonical embeddings of X (Theorem 8.1.1). Then kf kX0 D kf kX D kf kX11 ; f 2 X0 ; even if X fails to have property .C/. c. For all f 2 L1 \ L1 , 'X .1/  kf kL1 \L1  kf kX : If 'X .1/ D 1, then both embeddings L1 \ L1  X and X  L1 C L1 are contractions. Hint: Use the inequality 'X .1/  kf kL1 CL1  kf kX ; f 2 X and the duality of L1 \ L1 and L1 C L1 , also the relations .X0 /1 D X1 and the equality 'X .1/  'X1 .1/ D1. d. Let X be a symmetric space. Then clX .L1 \ L1 / D clX11 .L1 \ L1 /: With regard to Theorem 8.2.2, the symmetric space X is maximal if and only if X has property .B/.

108

8 Maximality. Properties (B) and (C)

14. Order completeness. Recall that a linear ideal lattice X is said to be order complete if every orderbounded subset E  X has least upper bound sup E and greatest lower bound inf E in X. This means that if the set fg 2 X W g  f for all f 2 Eg is not empty, it contains the least element sup E, and also, if the set fg 2 X W g  f for all f 2 Eg is not empty, it contains the greatest element inf E. Show that a. The space L0 D L0 .0; 1/ is order complete. Hint: Prove that for every bounded (countable or uncountable) family ff˛ ; ˛ 2 A g of measurable functions f˛ on Œ0; 1/, there are two measurable functions f D ess sup f˛ and f D ess inf f˛ ˛

˛

such that for every measurable function g on Œ0; 1/, f˛  g a.e. for each ˛ ” f  g a.e. and f˛  g a.e. for each ˛ ” f  g a.e. The functions f and f are defined uniquely almost everywhere. b. Let X be a symmetric space on Œ0; 1/ and E  X. Then the natural embedding i W X ! L0 preserves the bounds inf and sup. That is, supL0 i.E/ D i.supX E/ and infL0 i.E/ D i.infX E/ for all order-bounded subsets E  X. c. Every symmetric space X is order complete. 15. Order convergence. .o/

Recall that a sequence ffn g in X is said to be order convergent to f 2 X (fn ! f in X) if there exist two sequences fhn g and fgn g in X such that fhn g increases, fgn g decreases, while sup hn D f D inf gn : n

n

Show that a. Order convergence in the space L0 coincides with almost everywhere convergence. For every symmetric space X, the order convergence in X implies convergence in measure and almost everywhere convergence. .o/

b. fn ! f in X if and only if fn ! f almost everywhere and the set ffn ; n  1g is order bounded in X. Moreover, f D sup hn D inf gn ; n

n

where gn D sup fk and hn D inf fk . That is, f D lim sup fn D lim inf fn . kn

kn

Notes

109

c. If X has property .A/, then order convergence implies convergence in norm in X. d. If X has property .B/ and fn ! f in norm k  kX , then there exists a subsequence .o/

ffnk g such that fnk ! f in X. 16. Symmetric norms on ideal lattices. Let X be a normed ideal lattice of measurable functions on Œ0; 1/. Recall that the norm k  kX is called symmetric if f 2 X ” f  2 X and kf kX D kf  kX for all f 2 X. Show that a. The norm k  kX on X is symmetric if and only if kf ı  kX D kf kX for all f 2 X and all measure-preserving isomorphisms  of RC into itself. Hint: Use Exercise 8. b. Let Y be a linear ideal subset of the symmetric space X. If f 2 Y; g 2 X; f  D g H) g 2 Y; then the closure clX Y is a symmetric space with respect to the norm induced by k  kX . c. Let X1 and X2 be two symmetric spaces on Œ0; 1/, and suppose the space X1 CX2 consists of all functions f of the form f D f1 C f2 ; f1 2 X1 ; f2 2 X2 ; and kf kX1 CX2 D inffkf1 kX1 C kf2 kX2 ; f1 2 X1 ; f2 2 X2 g: Then k  kX1 CX2 is a symmetric norm on X1 C X2 and .X1 C X2 ; k  kX1 CX2 / is a symmetric space (see Section 1.6).

Notes The proof of Theorem 5.2.1 is taken from [34, Section 2.4]. The constant C D 2 in the first part of inequalities (5.2.2) can be reduced to C D 1. A way to remove the constant in (5.2.2) is described in Exercise 13. It uses the second part of (5.2.2) and the duality between L1 \ L1 and L1 C L1 . Properties .A/, .B/, and .C/ were introduced by Kantorovich [31, Section 10.4], and Nakano [51]. The proof of Theorem 6.5.3 connecting minimality, separability, and property .A/ can be found in [31, Section 10.4] and in [36, Section 1.a]. This and other related results are considered there for general Banach lattices. The proof of Theorem 7.4.1 is also an adaptation of [31, Section 10.4] to the case of symmetric spaces.

110

8 Maximality. Properties (B) and (C)

The maximal property of products f  g used in Section 7.2 for description of associate spaces X1 is due to G.H. Hardy, J.E. Littlewood, and G. Polya; see [27, Section 10.13]. The maximal Hardy–Littlewood property may fail on general measure spaces .˝; / (see Complement 1), for example, if the space .˝; / has two atoms a; b 2 ˝ with unequal positive measures .fag/ ¤ .fbg/. Notions of resonant and strongly resonant measure spaces are useful for clarifying the situation [3, Section 2.2]. Property .C/ was introduced by Nakano [51]. Norming subspaces of X and conditions of the canonical embedding X ! X11 being an isometry are studied in [42, 51]; see also [31, Section 10.4] and [36, Section 1.b]. Property .B/ is equivalent to order reflexivity of X, [8]. Property .AB/ and other equivalent properties are considered in [31, Section 10.4] and [36, Section 1.c]; see also Complement 5. The Fatou property .BC/ mentioned in Section 8.3, and the so-called weak Fatou property are studied in detail in [3, Chapter 2], [78, Section 15.65], [79, Chapters 14–16]; see also Exercise 13. Every symmetric space is order complete and is also a so-called K-space [31, 32, 36, 75]. A simple explanation of the fact is given in Exercise 14. Symmetric spaces with property .AB/ are called KB-spaces [75]. These and other properties of symmetric spaces have been widely studied by many authors (see [9, 31, 35, 36, 42, 46, 75]). The property of order completeness considered in Exercise 14 was studied in detail in [36, Section 1.a] for general Banach lattices. Order convergence may be considered on general ordered spaces. Some related results in the case of symmetric spaces are given in Exercise 15. The relationship between .o/-convergence and other types of convergence are based on properties .A/ and .B/ (see [31, 32, 65, 75]). Other interesting results on symmetric spaces can be found in [2, 4, 7, 10, 11, 14, 16, 29, 47, 56, 66].

Part III

Lorentz and Marcinkiewicz Spaces

In this part we study two important classes of symmetric spaces: Lorentz spaces W and Marcinkiewicz spaces MV . Lorentz spaces W arise as generalizations of spaces L1 and L1 using suitable weight functions W. Both norms k  kL1 and k  kL1 can be described as the Stieltjes integral Z1 kf kW D

f  dW;

0

where in the first case, the weight function W.x/ D x corresponds to the usual Lebesgue measure, and in the second case, the weight function W.x/ D 1.0;1/ .x/ determines Dirac’s delta measure ı0 concentrated at 0. An arbitrary function W increasing and concave on Œ0; 1/ satisfying W.0/ D 0 may be chosen as a weight. The Banach space W D ff 2 L0 W kf kW < 1g with the norm k  kW is a symmetric space. It is called a Lorentz space with Lorentz weight W. Marcinkiewicz spaces MW with weight functions W arise as associate spaces of the Lorentz spaces W , i.e., 1W D MW and M1W D W , and the norm k  k1 D W k  kMW can be written as 0 kf kMW

1 D sup @ W.x/ x

Zx 0

1 f  dmA D kW f  kL1 ;

112

III

x 1 where W .x/ D and f  .x/ D W.x/ x

Zx

Lorentz and Marcinkiewicz Spaces

f  dm is the so-called maximal function

0

of f . For example, if W.x/ D 1.0;1/ .x/, we have kf kMW

1 0 x Z Z1  D sup @ f dmA D f  dm D kf kL1 ; x

0

0

and if W.x/ D x, then 0 kf kMW

1 D sup @ x x

Zx

1 f  dmA D f  .0/ D f  .0/ D kf kL1 :

0

Marcinkiewicz spaces are introduced for a wider class of weight functions than Lorentz spaces. Namely, the requirement of concavity W is replaced by the following weaker condition: both functions W and W are increasing. Such functions are called quasiconcave. The significance of the class of quasiconcave functions is clarified by the following results. A function V is the fundamental function 'X of a symmetric space X if and only if it is quasiconcave. The equality V D 'X holds just for Marcinkiewicz spaces X D MV . Further, if X is a symmetric space with a fundamental function V, e V is the least x ; x > 0, then concave majorant of the quasiconcave function V, and V .x/ D V.x/ 0 e  X0  X  X11  MV ; V

where 0 kf k D kf k0  kf kX0 D kf kX ; f 2 e V e V e V

and kf kX  kf kX11  kf kMV ; f 2 X: The function V and the function V itself need not be concave. However, there exists a norm k  k0 that is equivalent to the norm k  k and 'X D V. Thus the e V 0 minimal part e of the Lorentz space e V is the smallest, and the Marcinkiewicz V space MV is the largest, symmetric space with the given fundamental function 'X D V.

III

Lorentz and Marcinkiewicz Spaces

113

On the other hand, every symmetric space .X; k  kX / can be renormed with a new norm k  k0 that is equivalent to the original k  kX and has a concave fundamental function. p Note also that in the case V.x/ D V .x/ D x, x  0, we have V  L2  MV ; where the embeddings are strict.

Chapter 9

Lorentz Spaces

In this chapter, we study Lorentz spaces W . It is shown that every Lorentz space W with concave weight function W is a maximal symmetric space. We also describe conditions of minimality and separability of Lorentz spaces.

9.1 Definition of Lorentz Spaces First we consider weight functions W of Lorentz spaces W . Definition 9.1.1. A function W W RC ! RC is called a weight function (or Lorentz weight function) if 1. W.0/ D 0 and W.x/ > 0 for x > 0. 2. W.x/ is concave on .0; 1/. Recall that a function W W RC ! RC is called concave if W.˛ x1 C .1  ˛/x2 /  ˛W.x1 / C .1  ˛/W.x2 / for all x1 ; x2 2 RC and ˛ 2 Œ0; 1. Positivity and concavity of W ensure that it is increasing and continuous on .0; 1/. The limit lim W.x/  W.0/ D 0 exists, and both cases W.0C/ D 0 x!0C

and W.0C/ > 0 are possible (Fig. 9.1). The Lorentz space W with a weight function W is the set of all measurable functions f W RC ! R such that Z1 kf kW WD

f  dW < 1:

(9.1.1)

0

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_9

115

116

9 Lorentz Spaces y

y W ( x)

W( x ) W (0 +) >0

W(0 +) = 0

0

0

x

x

Fig. 9.1 Two cases of Lorentz functions

Here f  is the decreasing rearrangement of jf j, and

R1

f  dW is the improper

0

Riemann–Stieltjes integral of the decreasing function f  with respect to the increasing function W on RC . Since W is concave it is absolutely continuous on .0; 1/. The equality

W .Œ0; x/ D W.x/; x 2 RC

(9.1.2)

uniquely determines a Borel measure W on RC , which is absolutely continuous with respect the Lebesgue measure m on .0; 1/ and has an atom at the point x D 0 if W.0C/ > 0. Thus, Z1 kf kW D



Z1



f d W D f .0/W.0C/ C 0

f  W 0 dm;

(9.1.3)

0

where the latter integral is an improper Riemann integral of the decreasing function f  W 0 . The derivative W 0 of W exists almost everywhere on RC , and Zx W.x/ D W.0C/ C

W 0 dm; x > 0:

0

Theorem 9.1.2. The space .W ; k  kW / is a symmetric space, and Z1 kf kW D



Z1

f dW D 0

W ı f  dm; f 2 W :

(9.1.4)

0

Proof. Since W 0 is decreasing almost everywhere and by Proposition 7.2.1, kf kW

9 81 = 0 H) ˚.x/ D 0; for 0  x  V.0C/;

(10.5.1)

V.1/ < 1 H) ˚.x/ D 1; for x  V.1/

(10.5.2)

and

(see Fig. 5.5). This motivates the following definition. Definition 10.5.1. A function ˚ W Œ0; 1/ ! Œ0; 1 is called quasiconvex if the (generalized) inverse function V D ˚ 1 is quasiconcave (Fig. 10.12). Of course, the arguments (10.5.1) and (10.5.2) are taken into account in this definition. All properties of quasiconvex functions are easily reduced from the corresponding properties of quasiconcave functions.

10.5 Quasiconvex Functions

137

y

•

y

V (•) V (0+)

F (x)

V (x)

0

x

0

V (0+)

V (•)

x

Fig. 10.12 Quasiconvex functions

The following helpful result is an analogue of Theorem 10.3.1. Theorem 10.5.2. For every quasiconvex function ˚ W Œ0; 1/ ! Œ0; 1, there exists e such that ˚ e 1 is the least concave majorant for the a greatest convex minorant ˚ 1 quasiconcave function ˚ , and  e .x/ x  0: e x  ˚.x/  ˚ ˚ 2

(10.5.3)

Chapter 11

Marcinkiewicz Spaces

In this chapter we study Marcinkiewicz spaces MV constructed by a quasiconcave  weight V. The space MV is equipped Z xwith the norm kf kMV D kV f kL1 , x 1 , and f  .x/ D f  dm is the maximal Hardy–Littlewood where V .x/ D V.x/ x 0 function of f . We show that .MV ; k  kMV / is a maximal symmetric space. The associate space 1W of every Lorentz space W with a concave weight W is a Marcinkiewicz space and k  kMV D k  k1 . Conversely, the associate space M1V W coincides with a Lorentz space W with a concave weight W that is equivalent to V.

11.1 The Maximal Function f  Marcinkiewicz spaces MV are defined by quasiconcave weight functions V. The norm k  kMV in the space MV can be written in the form kf kMV D kV f  kL1 ; f 2 MV ; x , x > 0 and f  is the maximal Hardy–Littlewood function of V.x/ f . For f 2 L1 C L1 , we set where V .x/ D

f



1 .x/ D x

Zx

f  dm; x > 0:

0

First, we describe some elementary properties of f  .

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_11

139

140

11 Marcinkiewicz Spaces

Proposition 11.1.1. Let f 2 L1 C L1 and g 2 L1 C L1 . Then 1 is quasiconcave if f ¤ 0. f  2. f   f  , f  .0/ D f  .0/ and f  .1/ D f  .1/. 3. If jf j  jgj, then f   g and f   g . 4. .f C g/  f  C g . 1. f  is continuous and decreasing on .0; 1/, and

Proof. 1: Since the function f  is nonnegative and decreasing, the function Zx F.x/ D

f  dm

0

F.x/ is decreasing and x

is increasing and concave on .0; 1/. Hence f  .x/ D x 1 continuous, and D  is quasiconcave. F.x/ f Assertions 2 and 3 follow directly from the definition. 4: The maximal property (Theorem 4.1.1) implies that Zx

.f C g/ dm D sup

0

8 0, define  a ; x  0: U a .x/ D U.x/  min 1; x Proposition 11.1.3. Let X be a symmetric space, and U W RC ! RC a nonzero measurable function such that U a 2 X for some a > 0. Let also XU D ff 2 L1 C L1 W U  f  2 Xg and kf kXU D kU  f  kX ; f 2 XU : Then .XU ; k  kXU / is a symmetric space.

142

11 Marcinkiewicz Spaces

Proof. Properties of maximal functions f  (Proposition 11.1.1) provide that .XU ; k  kXU / is a linear normed space. Since for a > 0, 1 Œ0;a .x/ D

1 x

Z

x 0

  1 ; x > 0; 1Œ0;a dm D min a; x

we have a k1Œ0;a kXU D kU1 Œ0;a kX D kU kX

and U a 2 X implies 1Œ0;a 2 XU . Hence XU ¤ f0g. The norm k  kXU is monotonic, since for f 2 L1 C L1 and g 2 XU , we have jf j  jgj H) f   g H) Uf   Ug H) kf kXU  kgkXU and f 2 XU : The norm k  kXU is symmetric, since for g 2 XU , f  D g H) f  D g H) kf kXU  kgkXU and f 2 XU : Thus, XU is a symmetric normed ideal lattice. It remains to prove that .XU ; k  kXU / is a Banach space. For this, it is enough to show that if ffk g is a sequence in XU with 1 X

kfk kXU < 1;

kD1

then the series

1 P

fk converges in norm k  kXU .

kD1

We have 1 X

kfk kXU D

kD1

1 X

kU  fk kX < 1:

kD1

Since the normed space X is complete, the series 1 P kD1

1 P kD1

U  fk converges in X, i.e.,

U  fk 2 X. By Proposition 11.1.2, we have

U

1 X kD1

! fk

U

1 X kD1

fk D

1 X kD1

U  fk 2 X:

11.2 Definition of Marcinkiewicz Spaces

That is,

1 P

143

fk 2 XU and

kD1

1 X fk kD1

XU

1 ! 1 1 X X X  D U  fk U  fk  kfk kXU :  kD1

kD1

X

X

kD1

Thus, .XU ; k  kXU / is a Banach space.

t u

11.2 Definition of Marcinkiewicz Spaces x  1.0;1/ .x/. V.x/ Using the space XU in the case U D V and X D L1 , we obtain the space

Let V be a quasiconcave function on RC and V .x/ D

MV D .L1 /V D ff 2 L1 C L1 W V f  2 L1 g; with the norm kf kMV D kV f  kL1 ; f 2 MV : Theorem 11.2.1. The space MV D ff 2 L1 C L1 W kf kMV < 1g equipped with the norm 0 kf kMV D kV f  kL1

1 D sup @ V.x/ 0 0. ˚ is increasing: x1  x2 ” ˚.x1 /  ˚.x2 /. ˚ is convex: ˚.˛x1 C .1  ˛/x2 /  ˛˚.x1 / C .1  ˛/˚.x2 /, 0  ˛  1. ˚ is left-continuous: ˚.x/ D ˚.x/, x  0. The first condition provides the nontriviality of ˚ and enables us to set a˚ D supf˚ D 0g < 1 and b˚ D inff˚ D 1g > 0

(13.1.1)

such that a˚ D inff˚ > 0g  b˚ D supf˚ < 1g; since ˚ is increasing. The Orlicz function ˚ is increasing and convex by conditions 2 and 3, and hence the derivative ˚ 0 exists almost everywhere on .0; 1/. Moreover, there is a unique increasing left-continuous function W Œ0; 1/ ! Œ0; 1, such that D ˚ 0 almost everywhere on Œ0; 1/ and

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_13

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13 Definition and Examples of Orlicz Spaces

Zx ˚.x/ D

.u/du; 0  x  b˚ ; 0

while .x/ D 1 for x > b˚ . The convexity of ˚ yields that ˚ is continuous on Œ0; b˚ /. Therefore, the leftcontinuity condition 4 is needed only at the point x D b˚ . It is reduced to ˚.b˚ / D ˚.b˚ / in the case ˚.b˚ / < 1. Note that both cases ˚.b˚ / < 1 and ˚.b˚ / D 1 are possible if ˚.b˚ / < 1 (Fig. 13.1). Since ˚ is increasing on Œ0; 1/, it has a left-continuous (generalized) inverse function ˚ 1 . This function is finite, concave, continuous on .0; 1/, and ˚ 1 .0/ D 0 (Fig. 13.2). Thus, a˚ D ˚ 1 .0C/ D lim ˚ 1 .x/; b˚ D ˚ 1 .1/ D lim ˚ 1 .x/;

(13.1.2)

x!1

x!0C

while a jump of ˚ 1 is possible only at zero. Thus ˚ is an Orlicz function if and only if ˚ 1 is a Lorentz weight function. y

y

y aF > 0

aF = 0

bF = •

bF = • F

0

• y bF < • F (bF ) < •

F

x

aF

0

F

F

x

b

0

•

aF > 0 bF < • F (bF ) = •

F

aF

x 0

bF

x

Fig. 13.1 Orlicz functions y

y

F-1

F

y

-1

y F-1

F-1 bF

bF

aF

aF 0

x

0

Fig. 13.2 Inverse Orlicz functions

x

0

x

0

x

13.2 Orlicz Spaces

173

13.2 Orlicz Spaces For every Orlicz function ˚, we define a functional Z1 I˚ .f / D

˚.jf j/ dm 2 Œ0; 1

(13.2.1)

0

and set

   f  1 2 Œ0; 1 kf kL˚ D inf a > 0 W I˚ a

(13.2.2)

for every measurable function f W RC ! R. We put here inff;g D 1. We shall show that the set L˚ D ff 2 L0 W kf kL˚ < 1g with the norm k  kL˚ is a symmetric space. The space .L˚ ; k  kL˚ / is called an Orlicz space. We begin with several properties of the functional I˚ , which follow directly from the convexity of ˚. Proposition 13.2.1. Let f ; g 2 L0 and 0 < ˛ < 1. Then 1. 2. 3. 4.

I˚ .˛f C .1  ˛/g/  ˛I˚ .f / C .1  ˛/I˚ .g/. I˚ .˛f /  ˛I˚ .f /. The set B˚ D ff 2 L˚ W I˚ .f /  1g is convex. If f 2 L˚ and kf kL˚  1, then f 2 B˚ .

Proof. Let f ; g 2 L0 and 0 < ˛ < 1. 1. Since ˚ is convex, we have for 0  ˛  1, I˚ .˛f C .1  ˛/g/ D

Z1 ˚.j˛f C .1  ˛/gj/ dm 0

Z1  .˛˚.jf j/ C .1  ˛/˚.jgj/ dm D ˛I˚ .f / C .1  ˛/I˚ .g/: 0

2. Since ˚ is convex and ˚.0/ D 0, we have ˚.˛x/  ˛˚.x/ and Z1 I˚ .˛f / D

˚.j˛f j/ dm  ˛ 0

for 0  ˛  1.

Z1 ˚.jf j/ dm D ˛I˚ .f / 0

174

13 Definition and Examples of Orlicz Spaces

3. For f ; g 2 B˚ and 0 < ˛ < 1, we also have by 1, I˚ .˛f C .1  ˛/g/  ˛I˚ .f / C .1  ˛/I˚ .g/  ˛ C .1  ˛/ D 1: Therefore, ˛f C .1  ˛/g 2 B˚ , i.e. the set B˚ is convex. 4. Since I˚ and kkL˚ are determined by jf j, we may assume that f  0. If kf kL˚  Z1 f 1, there exists a sequence an # 1 such that ˚. / dm  1. Since the function an 0

f ˚ is left-continuous, ˚. / " ˚.f / almost everywhere. By Levi’s theorem, an Z1 1 0

f ˚. / dm ! an

Z1 ˚.f / dm; 0

i.e., f 2 B˚ .

t u

Theorem 13.2.2. The space .L˚ ; k  kL˚ / is a symmetric space. Proof. First we show that k  kL˚ is a norm on L˚ . Let f ; g 2 L˚ and f  0; g  0. If f ¤ 0 and g ¤ 0, we set ˛D

kf kL˚ ; kf kL˚ C kgkL˚

1˛ D

kgkL˚ : kf kL˚ C kgkL˚

Then by Proposition 13.2.1, 3 and 4, we have hD˛

g f C .1  ˛/ 2 B˚ ; kf kL˚ kgkL˚

and hence khkL˚



f Cg D kf k C kgk L˚ L˚

D

L˚

k f C g kL˚  1: kf kL˚ C kgkL˚

Thus, the triangle inequality kf C gkL˚  kf kL˚ C kgkL˚ holds in L˚ . In addition, if f 2 L˚ and ˛ ¤ 0, then  k˛f kL˚ D inf a > 0 W I˚



j˛jjf j a



1

13.2 Orlicz Spaces

175



 D j˛j inf a1 > 0 W I˚

jf j a1



 1 D j˛jkf kL˚ ;

i.e. ˛f 2 L˚ and k˛f kL˚ D j˛jkf  kL˚ . f  1 for all a > 0. For all b > 0, we obtain If kf kL˚ D 0, then I˚ a     Z1   b f jf j 1  I˚ D ˚ dm  ˚ mfjf j  bg: a a a 0

  b ! 1 as a ! 0, and hence mfjf j  bg D 0 for all b > 0, i.e., f D 0. a Thus, k  kL˚ is a norm on L˚ .   f < 1 for some a > 0, and Let f 2 L˚ and jgj  jf j. Then I˚ a

Then ˚

 jgj  jf j H) ˚

jgj a



 ˚

jf j a

 H) I˚

g a

 I˚

  f < 1; a

whence g 2 L˚ and kgkL˚  kf kL˚ . Thus, .L˚ ; k  kL˚ / is a normed ideal lattice. Next we verify that the space .L˚ ; k  kL˚ / is complete. Let ffn g be a Cauchy sequence in L˚ . Then there exists a subsequence ffnk g for which kfnk  fnk1 kL˚ 

1 : 2k

We set gD

1 X

jfnk  fnk1 j:

kD1

Since ˚ is convex and left-continuous, we have ! 1 1 X X 1 ˚.g/ D ˚ jfnk  fnk1 j  ˚.2k jfnk  fnk1 j/ k 2 kD1 kD1 and Z1 I˚ .g/ D 0

Z1 1 X 1 ˚.g/ dm  ˚.2k jfnk  fnk1 j/ dm  1: k 2 kD1 0

(13.2.3)

176

13 Definition and Examples of Orlicz Spaces

Therefore, kgkL˚  1, and g 2 L˚ . Consequently, the series (13.2.3) converges almost everywhere, and the series 1 X .fnk  fnk1 / C fn0 kD1

also converges almost everywhere to a function f such that jf  fn0 j  g: Thus, f 2 L˚ and kfnk  f kL˚ ! 0 as k ! 1. Since ffn g is a Cauchy sequence and has a convergent subsequence ffnk g, the sequence ffn g itself converges in L˚ , and kfn  f kL˚ ! 0 as n ! 1. Thus, .L˚ ; k  kL˚ / is a Banach space. It remains to verify that .L˚ ; k  kL˚ / is symmetric. For every f 2 L0 , we have Z1 I˚ .f / D

Z1 ˚.jf j/ dm D

˚ıjf j du D

0

0

Z1 D

Z1

mfjf j > ˚

1

mf˚.jf j/ > xg dx 0

Z1 .x/g dx D

0

Z1 mfjf j > yg d˚.y/ D

0

jf j d˚: 0

Let f 2 L˚ and let jgj be equimeasurable to jf j, i.e., jf j D jgj . Then Z1 I˚ .f / D

Z1 jf j d˚ D

0

jgj d˚ D I˚ .g/: 0

Hence g 2 L˚ and kgkL˚ D kf kL˚ . So, .L˚ ; k  kL˚ / is a symmetric space.

t u

Note 13.2.3. It should be emphasized that attempts to use the set Y˚ D ff 2 L0 W I˚ .f / < 1g and I˚ .f / instead L˚ and kf kL˚ can be unsuccessful in general. The sets Y˚ , called Young classes in general, need not be linear spaces for certain Orlicz functions ˚. It is the case that Y˚ ¤ 2Y˚ ; see Examples  14.4.2 below. jf j , a > 0, instead of ˚.jf j/, enables us A nice trick with using functions ˚ a to remove these difficulties. The “good” case, in which Y˚ D L˚ and I˚ .f / D kf kL˚ , will be studied in detail in Chapter 14 below.

13.3 Fundamental Functions of Orlicz Spaces

177

13.3 Fundamental Functions of Orlicz Spaces Now we turn to the fundamental functions 'L˚ of Orlicz spaces L˚ . Proposition 13.3.1. Let L˚ be an Orlicz space with the Orlicz function ˚. Then  1  ; x > 0; 'L˚ .x/ D ˚ 1 x1

(13.3.1)

and also 1 1 .1//1 ; 'L˚ .1/ D a1 .0C//1 : 'L˚ .0C/ D b1 ˚ D .˚ ˚ D .˚

(13.3.2)

Proof. For every x > 0 and a > 0, we have  I˚

1  1Œ0;x a





Z1 D

˚ 0

   Zx   1 1 1  1Œ0;x dm D ˚ dm D x ˚ : a a a 0

So

   1 'L˚ .x/ D k1Œ0;x kL˚ D inf a > 0 W x ˚ 1 a (   1 )  1  1 1 1 1 D inf a > 0 W a  ˚ : D ˚ x x This implies (13.3.2), since ˚ 1 is increasing and continuous on .0; 1/ (Fig. 13.3). t u The following is a direct consequence of Proposition 13.3.1. Corollary 13.3.2. Every Orlicz function ˚ is uniquely determined by the fundamental function 'L˚ of L˚ . Namely, Fig. 13.3 ˚ and ˚ 1 in (13.3.2)

y

y

bF F

F -1 aF

0

aF

bF

x

0

x

178

13 Definition and Examples of Orlicz Spaces

 1  1 ˚ 1 .x/ D 'L˚ ; x > 0; x while ˚ D .˚ 1 /1 is the inverse function of ˚ 1 . Corollaries 12.2.2, 13.3.2 and formulas (13.3.2) yield the following result. Corollary 13.3.3. 1. b˚ < 1 ” L˚  L1 ” 'L˚ .0C/ > 0. 2. a˚ > 0 ” L˚ L1 ” 'L˚ .1/ < 1.

13.4 Examples of Orlicz Spaces Examples 13.4.1. 1. The spaces Lp ; 1  p < 1. Let ˚.x/ D xp , 1  p < 1 (Fig. 13.4). Then 8 9 8 9 Z1 p Z1 < = < = jf j p p dm  1 D inf a > 0 W jf j dm  a kf kL˚ D inf a > 0 W : ; : ; ap 0

9 8 1 1p 01 > ˆ Z = < p A @ jf j dm  a D kf kLp ; D inf a > 0 W > ˆ ; :

0

0

1

i.e., L˚ D Lp , k  kL˚ D k  kLp and 'L˚ .x/ D 'Lp .x/ D ˚ 1 .x/ D x p . Here I˚ .f / D kf kL˚ and Y˚ D L˚ D Lp . 2. The space L1 (Fig. 13.5). Let  ˚.x/ D

Fig. 13.4 L˚ D Lp if ˚.x/ D xp , 1  p < 1

0; 0  x  1; 1; x > 1:

y

y F

0

y =x p =1

1

x

0

F

y =x

p

1< p < •

1

x

13.4 Examples of Orlicz Spaces

179

Fig. 13.5 Orlicz function ˚ for which L˚ D L1

y

•

F

0 Fig. 13.6 Orlicz function for which L˚ D L1 \ L1

1

x

y

•

F

1

0

1

Then kf kL˚ D kf  kL˚ D inffa > 0 W f   ag D f  .0/ D kf kL1 ; i.e., L˚ D L1 , k  kL˚ D k  kL1 and 'L˚ .x/ D 'L1 .x/ D 1.0;1/ .x/. Here Y˚ D ff W jf j  1g, i.e., Y˚ 6D L1 and Y˚ 6D 2Y˚ . 3. The space L1 \ L1 (Fig. 13.6). Let  x; 0  x  1; ˚.x/ D 1; x > 1: 

Z1 ˚

For every a > 0, the inequality 0

Z ff  >ag

f a

 dm  1 means that

f dm  1; a

x

180

13 Definition and Examples of Orlicz Spaces

and f   a. Hence kf kL˚ D kf  kL˚

D max

8 9 Z1 < = D inf a > 0 W f  dm  a and f   a : ;

81 0, we have  ˚

f a



 D

 f  1  1ff  >ag : a

Thus (Fig. 13.8)  I˚

f a





D ff  >ag

Since the function a ! unique point a0 for which

Z

R1

R1

 Z1 f  1 dm D f  dm: a a

f  dm is decreasing and continuous, there exists a

a

f  dm D a0 . For such a0 , we have

a0

Fig. 13.7 Orlicz function ˚ for which L˚ D L1 C L1

y

F

0

x-1

1

x

13.4 Examples of Orlicz Spaces

Fig. 13.8

1 R

181

y

f  dm as area

a

a

f*

0  I˚

f a0

x

 D 1 and a0 D kf kL˚ :

By setting g0 D .f   a0 /  1ff  >a0 g and h0 D min.f  ; a0 /; we obtain a representation of f  of the form f  D g0 C h0 , where g0 2 L1 , h0 2 L1 , and kg0 kL1 D kh0 kL1 D a0 : On the other hand, let f D g C h, g 2 L1 , h 2 L1 , and a D max.kgkL1 ; khkL1 /: Then 1a khkL1  1 and     Z1  Z1  jhj f jgj jgj I˚  ˚ C dm  ˚ C 1 dm a a a a 0

Z1 D 0

0

jgj dm  1: a

This means that kf kL˚  a D max.kgkL1 ; khkL1 //:

182

13 Definition and Examples of Orlicz Spaces

We have shown that L˚ D L1 C L1 and kf kL˚ D inffmax.kgkL1 ; khkL1 /; f D g C h; g 2 L1 ; h 2 L1 g:

(13.4.1)

By comparing (13.4.1) and (4.2.5), we see that the norms k  kL˚ and k  kL1 CL1 are equivalent, namely, kf kL˚  kf kL1 CL1  2kf kL˚ :

(13.4.2)

Therefore, the corresponding fundamental functions are equivalent, and 'L˚  'L1 CL1  2'L˚ :

(13.4.3)

On the other hand, ˚ 1 .x/ D .x C 1/  1Œ0;1/ , whence 'L˚ .x/ D .˚ 1 .x1 //1 D

x ; x  0; xC1

while 'L1 CL1 D min.x; 1/. We have x 2x  min.x; 1/  ; x  0: xC1 xC1 Thus the inequality (13.4.3) cannot be improved (see Fig. 13.9). Here Y˚ 6D L˚ D L1 C L1 . Indeed, by putting f D 2  1Œ1;1/ , we have f 2 L1  L˚ , although Z I˚ .f / D

1

0

˚.2  1Œ1;1/ /dm D 1;

i.e., f 62 Y˚ . Fig. 13.9 'L˚  'L1 CL1  2'L˚ y

2

2j

LF

1 j 1/ 2 0

1

LF

x

Chapter 14

Separable Orlicz Spaces

In this chapter, we study conditions of separability for Orlicz spaces L˚ . We consider Young classes Y˚ , the subspaces H˚ , and their embeddings H˚  Y˚  L˚ . We show that the equality H˚ D Y˚ D L˚ is equivalent to separability of L˚ . This and other equivalents of separability studied earlier in Chapters 6 and 7 can be expressed in term of an Orlicz function ˚. The .2 / condition is described to this end in detail.

14.1 Young Classes Y˚ and Subspaces H˚ The main purpose of this section is to describe conditions of minimality and separability for Orlicz spaces L˚ . To this end, we return to our study of Young classes Y˚ and corresponding subspaces H˚ . First, we want to know when the equality H˚ D Y˚ D L˚ holds. Recall that 8 9 Z1 < = Y˚ D f 2 L˚ W I˚ .f / D ˚.jf j/dm < 1 : : ;

(14.1.1)

0

By Proposition 13.2.1, Y˚ is a convex subset of L˚ that does not necessarily coincide with L˚ . To clarify when Y˚ D L˚ , consider the one-parameter family f˛Y˚ ; 0 < ˛ < 1g:

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_14

183

184

14 Separable Orlicz Spaces

For 0 < ˛1 < ˛2 < 1, we have     f f f 2 ˛1 Y˚ ” I˚ < 1 H) I˚ < 1 ” f 2 ˛2 Y˚ ; ˛1 ˛2 and thus the family is increasing. Consider also the set

   \ f < 1 for all ˛ > 0 D H˚ WD f 2 L0 W I˚ ˛Y˚ ˛ 0 0, the sequence ˚ a zero almost everywhere. Since h1 D f  f1 2 H˚ and hn  h1 , we have  I˚

h1 a

 D

a > 0,  lim I˚

n!1

 Hence I˚

hn a

hn a

˚ 0

 Consequently, the sequence ˚



Z1

hn a

h1 a

 dm < 1:

 approaches zero in norm of L1 , i.e., for all





Z1 D lim

˚

n!1 0

hn a

 dm D 0:

 < 1 for all sufficiently large n and khn kL˚ < a for all a > 0. Thus, lim khn kL˚ D lim kf  fn kL˚ D 0;

n!1

n!1

i.e., f 2 clL˚ .F0 / D L0˚ . Thus, H˚  L0˚ . We have shown that H˚  L0˚ and L0˚  H˚ . That is, H˚ D L0˚ .

t u

14.2 Separability Conditions for Orlicz Spaces

189

The first part of Proposition 14.2.1 can be refined for f 2 H˚ as follows. Proposition 14.2.3. If f 2 H˚ , then kf kL˚ D 1 ” I˚ .f / D 1:

(14.2.1)

Proof. Proposition 14.2.1 implies I˚ .f / D 1 H) kf kL˚ D 1 for all f 2 L˚ . We show that for f 2 H˚ , the reverse implication is also true. We may assume that b˚ D 1, for otherwise, by Theorem 14.2.2, H˚ D f0g. Let f 2 H˚ and kf kL˚  D 1. Choose a sequence 0 < an < 1 such that an " 1. jf j jf j Then # ˚.jf j/ almost everywhere, since the function ˚ is # 0 and ˚ an an finite, increasing, and continuous.   f f The equality kf kL˚ D 1 implies > 1 by > 1, and I ˚ a a n L˚ n Proposition 14.2.1, 3.   f < 1 for all n, since f 2 H˚ . Therefore, the On the other hand, I˚ an     f jf j is dominated by ˚ and converges sequence of integrable functions ˚ an a1 almost everywhere to ˚.jf j/ 2 L1 . Consequently,  1 < I˚

f an





Z1 D

˚ 0

 Z1 jf j dm # ˚.jf j/dm D I˚ .f /; an 0

i.e., I˚ .f /  1. On the other hand, kf kL˚ D 1 H) I˚ .f /  1; and thus I˚ .f / D 1.

t u

Now we consider other equivalents for L˚ D H˚ . Since L˚ ¤ f0g, we have L˚ D H˚ H) H˚ ¤ f0g ” b˚ D 1; and hence L˚ D H˚ ” L˚ D L0˚ and b˚ D 1:

190

14 Separable Orlicz Spaces

Since b˚ D 1 ” 'L˚ .0C/ D 0; the equality L˚ D H˚ means that the symmetric space X D L˚ satisfies condition 2 of Theorem 6.5.3. Thus Theorem 6.5.3 can be rewritten for Orlicz spaces as follows. Corollary 14.2.4. For every Orlicz space L˚ , the following are equivalent: 1. 2. 3. 4.

L˚ L˚ L˚ L˚

D H˚ . is minimal and 'L˚ .0C/ D 0. is separable. has property .A/.

14.3 The .2 / Condition The equivalent conditions of Corollary 14.2.4 can be expressed in terms of Orlicz functions ˚. Definition 14.3.1. An Orlicz function ˚ satisfies the .2 /-condition if a˚ D 0, b˚ D 1, and ˚.2x/ < 1: 0 0 and x0 2 .0; 1/ such that ˚.2x/ < k˚.x/

(14.3.3)

for all 0 < x < x0 . The .2 .1// condition: There exist k > 0 and x0 2 .0; 1/ such that ˚.2x/ < k˚.x/

(14.3.4)

for all x > x0 . It is clear that .2 / ) .2 .0// and .2 / ) .2 .1//. Conversely, (14.3.3) and (14.3.4) imply (14.3.2), i.e., a˚ D 0 and b˚ < 1.

14.3 The .2 / Condition

191

˚.2x/ is continuous on .0; 1/, the inequalities (14.3.3), ˚.x/ (14.3.4) imply (14.3.1). Since the function

Theorem 14.3.2. Each of the equivalent conditions in Corollary 14.2.4 is also equivalent to the .2 / condition. Proof. The .2 / condition implies that ˚.2x/  k˚.x/ for some k > 0 and all x 2 .0; 1/. Hence I˚ .2f /  kI˚ .f / for all f . Suppose that f 2 2Y˚ . Then I˚

  f < 1 and I˚ .f / < 1, i.e., f 2 Y˚ . 2

Therefore, 2Y˚ D Y˚ , i.e., L˚ D H˚ . Conversely, let equivalent conditions 1–4 in Corollary 14.2.4 hold. Then 'L˚ .0C/ D 0 and hence b˚ D 1. If a˚ > 0, then 1Œ0;1/ 2 L˚ and L1  L˚ . This contradicts the minimality of L˚ , since L˚ D L0˚  .L1 C L1 /0 D R0 63 1Œ0;1/ : So we may assume that 0 < ˚.x/ < 1 for all x > 0. If the .2 / condition does not hold, then there exist xn 2 .0; 1/ such that ˚.2xn / > 2n ˚.xn / > 0; n D 1; 2; : : : : Let An ; n D 1; 2; : : :, be disjoint intervals Œ0; 1/, with measures m.An / D

1 ; n D 1; 2; : : : : 2n ˚.xn /

The function f D

1 X

xn  1An

nD1

satisfies Z1 I˚ .f / D

˚.jf j/ dm D 0

1 1 X X ˚.xn / 1 D D1 D 1 D 1; 2n ˚.xn / nD1 2n ˚.xn / nD1 nD1

i.e., f 2 Y˚ , while 2f 62 Y˚ . We have Y˚ ¤ 2Y˚ , which contradicts H˚ D L˚ .

t u

14.4 Examples of Orlicz Spaces with and Without the .2 / Condition Examples 14.4.1. Consider again the Orlicz functions from Examples 14.1.3. 1. The function ˚.x/ D xp for x  0 with 1  p < 1 satisfies the .2 / condition, since a˚ D 0, b˚ D 1, and ˚.2x/ .2x/p D sup D 2p < 1: p ˚.x/ x x2.0; 1/ x2.0; 1/

0 < sup In this case,

L˚ D Lp and L˚ D Y˚ D H˚ : 2. The function  ˚.x/ D

0; 0  x  1; 1; x > 1;

does not satisfy either the .2 .0// condition or the .2 .1// condition, since a˚ D 1 > 0 and b˚ D 1 < 1. In this case, L˚ D L1 ; Y˚ D ff 2 L1 W kf kL1  1g; H˚ D f0g: 3. The function  ˚.x/ D

x; 0  x  1; 1; x > 1;

satisfies the .2 .0// condition and does not satisfy the .2 .1// condition.

14.4 Examples of Orlicz Spaces with and Without the .2 / Condition

193

In this case, L˚ D L1 \ L1 ; Y˚ D ff 2 L1 W kf kL1  1g; H˚ D f0g: 4. The function ˚.x/ D .x  1/  1Œ1;1/ .x/ satisfies the .2 .1// condition and does not satisfy the .2 .0// condition. In this case, L˚ D L1 C L1 ; H˚ D L0˚ D R0 : Consider now the case a˚ > 0, b˚ < 1 without the .2 / condition. Example 14.4.2. Let ˚.x/ D ex  x  1, x  0. For this function, b˚ D 1 and a˚ D 0. Since e2x  2x  1 D 1; x!1 ex  x  1

e2x  2x  1 D 4; x!0C ex  x  1 lim

lim

the .2 .0// condition holds, but the .2 .1// condition does not hold. Hence, Y˚ ¤ 2Y˚ . We show that for the function  1  2 ln x; for 0 < x  1I f .x/ D 0; for x > 1; f 2 Y˚ , but 2f 62 Y˚ . Indeed, Z1

Z1 I˚ .f / D

˚.jf j/dm D 0

1

e 2 ln x C

0

 1 1 ln x  1 dx D ; 2 2

while Z1 I˚ .2f / D

Z1 ˚.j2f j/dm D

0

  ln x  e C ln x  1 dx D C1:

0

Thus, f 2 Y˚ , and 2f 62 Y˚ . By Theorem 14.2.2, the space L˚ is not minimal, H˚ D L0˚ ¤ L˚ . Hence L˚ is nonseparable.

Chapter 15

Duality for Orlicz Spaces

In this chapter, we consider associate spaces L1˚ of Orlicz spaces L˚ . Using the Legendre transform of an Orlicz function ˚, we define the conjugate Orlicz function  just as the Legendre transform of ˚. We prove that the spaces L1˚ and L coincide as sets and k  kL  k  kL1  2k  kL . The duality between L˚ and L is studied ˚ in detail.

15.1 The Legendre Transform In Sections 15.1 and 15.2, we construct the conjugate Orlicz function  for an Orlicz function ˚. By definition, ˚ and  are conjugate if they are the Legendre transforms of each other. We begin now with a brief description of the Legendre transform. Given two Orlicz functions ˚ and  , consider the functional equation ˚.x/ C  .y/ D xy:

(15.1.1)

The functions ˚ and  are finite, increasing, and convex on Œ0; b˚ / and Œ0; b / respectively, where b˚ D sup ˚ 1 and b D sup  1 . The convexity of the functions ˚ and  implies that the derivatives ˚ 0 and  0 exist almost everywhere and that they are increasing functions. Let y D y.x/. Then we obtain from (15.1.1), using differentiation with respect to x, that ˚ 0 .x/ C  0 .y/y0 D y C xy0 ;

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_15

195

196

15 Duality for Orlicz Spaces

or ˚ 0 .x/  y D y0 .x   0 .y//:

(15.1.2)

Similarly, putting x D x.y/, we obtain ˚ 0 .x/x0 C  0 .y/ D x C yx0 ; or  0 .y/  x D x0 .y  ˚ 0 .x//:

(15.1.3)

Equalities (15.1.2) and (15.1.3) mean that ˚ 0 .x/ D y ”  0 .y/ D x

(15.1.4)

for all x 2 Œ0; b˚ / and y 2 Œ0; b /. Definition 15.1.1. Two Orlicz functions ˚ and  are called conjugate if their derivatives ˚ 0 and  0 are (generalized) inverse functions of each other (Fig. 15.1). Proposition 15.1.2 (Young Inequality). Let ˚ and  be two conjugate Orlicz functions; let D ˚ 0 and D  0 be their left-continuous derivatives. Then xy  ˚.x/ C  .y/; x > 0; y > 0;

(15.1.5)

and equality is achieved if and only if either y D .x/ or x D .y/, i.e., if and only if the point .x; y/ belongs to the union of the graphs of ˚.x/ and  .y/. Also, b D ˚ 0 .1/; b˚ D  0 .1/:

Fig. 15.1 Derivatives of conjugate functions

y

y

bF

y=F'(x) x=Y'(y)

aF > 0 bF < •

0

aF

bF

aF x

y=Y'(x) x=F'(y)

aF > 0 bF < •

x

15.2 The Geometric Interpretation y

197

y

y =F'(x) x =Y'(y)

y

y =F'(x) x =Y'(y)

y

y

y Y(y) 0

F(x)

Y(y) x

y =F'(x) x =Y'(y)

Y(y)

F(x) x

x 0

x 0

F(x) x

x

Fig. 15.2 Conjugate functions ˚.x/ and  .y/ as areas

Consider the function of two variables F.x; y/ D xy  ˚.x/   .y/: It is clear that F achieves its maximum value max F D 0 at a point .x; y/ if and only if y D ˚ 0 .x/ and x D  0 .y/. Hence, ˚.x/ D supfxy   .y/g

(15.1.6)

 .y/ D supfxy  ˚.x/g:

(15.1.7)

y

and

x

These equations mean that ˚ and  are the Legendre transforms of each other. Each of them can be used to determine the conjugacy of ˚ and  . Young’s inequality (Proposition 15.1.2) follows directly from (15.1.6) and (15.1.7). See also Fig. 15.2.

15.2 The Geometric Interpretation Consider the family of all lines y D mx C b that are tangent to the graph of the function y D ˚.x/. If the derivative ˚ 0 .x/ exists at a point x0 , the tangent line at the point .x0 ; ˚.x0 // satisfies ˚.x0 / D mx0 C b; m D ˚ 0 .x0 /: If the function ˚ 0 .x/ is invertible, b D ˚.x0 /  mx0 D ˚. 0 .m//  m 0 .m/ D  .m/:

198

15 Duality for Orlicz Spaces

Fig. 15.3 Legendre’s parameter m D ˚ 0 .x/

y

y=Φ(x) y=mx+b

Φ(x0)

x0

0

x

b= –Ψ(m)

Thus, the family of tangent lines to the graph of y D ˚.x/ can be parametrically represented as y D mx   .m/;

(15.2.1)

where the parameter m is equal to the value of derivative ˚ 0 .x/ at the point x (Fig. 15.3). The function ˚ itself is uniquely determined by the family (15.2.1). Indeed, rewriting (15.2.1) in the form F.x; y; m/ D y  mx C  .m/ D 0 and differentiating with respect to m, we obtain @F D x C  .m/ D 0: @m By taking into account (15.2.1), we obtain y D x. 0 /1 .x/   .. 0 /1 .x//: This is just the Legendre transform of  , i.e., y D ˚.x/. We now clarify what happens at the points of discontinuity of ˚ 0 and  0 . Recall that ˚ 0 and  0 are assumed to be left-continuous. Let m1 D ˚ 0 .x0  0/ D ˚ 0 .x0 / < ˚ 0 .x0 C 0/ D m2 ;

15.2 The Geometric Interpretation

199

Fig. 15.4 The Legendre transform in the case of derivative jump

y

y=m2x+b2 y=m1x+b1 y=F(x)

F(x0) x

0 b1

x

0

b2

Fig. 15.5 Jump of ˚ 0 and linearity interval of 

y m

x x = Y(y )

2

Y (m ) 2

y m

1

Y (y) Y(m ) 1

0

x0

x

0

m

1

m

2

y

i.e., the left and the right tangent lines at the point .x0 ; ˚.x0 // do not coincide (Figs. 15.4 and 15.5). Then for every y 2 Œm1 ; m2 ,  .y/ D  .m1 / C x0 .y  m1 /; i.e.,  .y/ is linear on the interval Œm1 ; m2 . Conversely, let ˚ be linear on an interval Œx1 ; x2  (Figs. 15.6 and 15.7), ˚.x/ D mx C b; where m D

˚.x1 /x2  ˚.x2 /x1 ˚.x2 /  ˚.x1 / and b D : x2  x1 x2  x1

Then  0 .y1 /  x1 < x2   0 .y2 / for all y1 < m < y2 , i.e.,  0 .m/ D  0 .m/ <  0 .mC/; and m is a discontinuity point of  0 (Fig. 15.8).

200

15 Duality for Orlicz Spaces

Fig. 15.6 Linearity interval of ˚

y y = F (x ) y = mx + b

F(x2)

F(x1) x1

0

Fig. 15.7 Constancy interval of derivative ˚ 0

x2

x

y x = Y'( y) y = F' (x ) m

0

x1

x2

x

15.3 Duality for Orlicz Spaces Using Young’s inequality, we can describe now the “duality” of Orlicz spaces L˚ and L . Corollary 15.3.1. Let ˚ and  be conjugate Orlicz functions. Then fg 2 L1 for all f 2 L˚ and g 2 L , and ˇ1 ˇ ˇZ ˇ ˇ ˇ ˇ fgdmˇ  2kf kL  kgkL : ˚  ˇ ˇ ˇ ˇ 0

(15.3.1)

15.3 Duality for Orlicz Spaces Fig. 15.8 Left and right tangent lines at y D m

201 x

x =Y'(m+) ( y− m) + Y (m) x

= Y( y )

x = Y' (m) ( y−m) + Y (m)

Y (m)

m

0

y

Proof. By (15.1.2), we have ˇ ˇ1 ˇ Z1 ˇZ ˇ ˇ ˇ fgdmˇ  .˚.jf j/ C  .jgj//dm  I˚ .f / C I .g/: ˇ ˇ ˇ ˇ 0

0

By applying this inequality to the functions 1 kf kL˚ kgkL



Z1 jfgjdm  I˚ 0

f g and , we have kf kL˚ kgkL

f kf kL˚



 C I

g kgkL

  2: t u

Recall that the associate space X1 D L1˚ of a symmetric space X D L˚ is defined as L1˚ D fg 2 L1 C L1 W kgkL1 < 1g; ˚

where kgkL1

˚

8ˇ 1 ˇ 9 ˇ 1, then  I˚

 1 1 jf j  I˚ .f /; b b

and therefore, by the definition of kgkL1 , we have ˚

 I˚ .f / C I

 Z1 1 1 ab jgj D D b D I˚ .f /: jfgjdm  a a a 0

 1 jgj D 0. a   1 jgj  1, i.e., kgkL  a D kgkL1 . Thus in both cases, I ˚ a 

Hence, I

15.3 Duality for Orlicz Spaces

203



 1 Now let g 2 F1 be a function such that  jgj is not almost everywhere a finite. This means that  .x/ can be infinite for some x, i.e., b D ˚ 0 .1/ < 1. Then ˚ 0 .x/ " b as x ! 1, and ˚.x/  b x for all x. In this case, jgj  ab almost everywhere. Indeed, if mfjgj > ab g > 0, there exists a measurable set 1  1A , we have A  fjgj > ab g such that 0 < mA < 1. For the function f D b mA   1 mA  1; I˚ .f / D ˚ b mA since kf kL˚  1. Consequently, Z1 a D kgkL1 

jfgjdm >

˚

0

ab mA D a: b mA

This contradiction shows that jgj  ab almost everywhere. a an n jgj is Taking an " 1 with an < 1, we have jgj  an b < b , whence  a a finite. In this case, we can prove as above that I

a g n  an < 1: a

  Since an " 1 and  is left continuous, we obtain I ga  1. For every g 2 L , we choose a sequence gn in F1 such that gn " jgj. Since I

gn kgn kL1

!  1 and kgn kL1  kgkL1 D a; ˚

˚

 we have I

1 gn a



  1 and hence I

˚

 1 g  1. Thus, kgkL  a D kgkL1 . ˚ a

t u

Replacing ˚ and  in Theorem 15.3.2, we have L˚ D L1 . Thus every Orlicz space L˚ D L11 ˚ is maximal (Theorem 8.2.2). Corollary 15.3.3. 1. .L˚ ; k  kL˚ / has property .B/. 2. .L˚ ; k  kL1 / has property .BC/. 

The norm in the space L1˚ can also be estimated using the functional I . Proposition 15.3.4. kgkL1  1 C I .g/; g 2 L . ˚

204

15 Duality for Orlicz Spaces

Proof. kgkL1

˚

ˇ ˇ1 ˇ ˇZ Z1 ˇ ˇ ˇ ˇ D sup ˇ fgdmˇ  sup .˚.jf j/ C  .jgj//dm kf kL˚ 1 ˇ ˇ kf kL˚ 1 0

D

0

sup .I˚ .f / C I .g//  1 C I .g/:

kf kL˚ 1

t u Note 15.3.5. Replacing ˚ and  in Proposition 15.3.4, we have kf kL1  1 C I˚ .f /; f 2 L˚ ; 

and taking

f instead of f , we have c    f for all c > 0: kf kL1  c 1 C I˚  c

Moreover, it can be proved that   f : D inf c 1 C I˚ c>0 c 

kf kL1



This formula for the norm k  kL1 on L˚ does not involve the conjugate Orlicz  function  . Corollary 15.3.6. Let ˚ and  be two conjugate Orlicz functions with a˚ D ˚ 1 .0C/ and b˚ D sup ˚ 1 . Then 1. a˚ D  0 .0C/ and b˚ D  0 .1/. 2. L˚ L1 ” a˚ > 0 ”  0 .0C/ > 0 ” L  L1 . 3. L˚  L1 ” b˚ < 1 ”  0 .1/ < 1 ” L L1 . Consider now an Orlicz function ˚ with b˚ D 1 such that ˚.x/ < 1 for all x  0 (Fig. 15.9). Fig. 15.9 ˚ 0 and  0 when a˚ > 0 and b˚ < 1

y

x

x = Y'( y) y = F'(x)

y = F'(x)

x = Y'(y) 0

aF

x

0

bF

y

15.4 Duality and the .2 / Condition. Reflexivity

205

This condition is equivalent to 'L˚ .0C/ D 0, since the fundamental function 'L˚ of the Orlicz space L˚ can be found by (13.3.1), and 'L˚ .0C/ D b1 ˚ . Let b˚ D 1. By Theorem 14.2.2, the minimal part L0˚ D clL˚ .L1 \ L1 / D clL˚ .F0 / of L˚ coincides with H˚ , i.e., L0˚ D H˚ ; and the space is separable. Applying Theorem 6.5.3 to this space, we obtain the following theorem. Theorem 15.3.7. Let ˚ and  be conjugate Orlicz functions and b˚ D 1. Then H1˚ D L1˚ and H˚ D .H1˚ / D .L1˚ /:

15.4 Duality and the .2 / Condition. Reflexivity Applying Theorem 6.5.3 to the space L˚ , we obtain from Theorems 14.3.2 and 15.3.2 the following result. Theorem 15.4.1. The following are equivalent: 1. 2. 3. 4.

˚ satisfies the .2 / condition. L˚ is separable. L˚ D H˚ . L˚ D .L /, where  is the conjugate function for ˚.

By combining this theorem with the results of Section 8.4, we obtain the following corollary. Corollary 15.4.2. Let ˚ be  be two conjugate Orlicz functions. The following are equivalent: 1. 2. 3. 4. 5.

Both ˚ and  satisfy the .2 / condition. Both L˚ and L are separable. .L˚ / D L ; .L / D L˚ ; L˚ is reflexive. L is reflexive.

Chapter 16

Comparison of Orlicz Spaces

In this chapter, we study embedding of Orlicz spaces in terms of corresponding Orlicz functions. We characterize the class of Orlicz functions corresponding to the same Orlicz space. The Zygmund classes are considered as examples of Orlicz spaces.

16.1 Comparison of Orlicz Spaces Let ˚1 and ˚2 be two Orlicz functions. Definition 16.1.1. We say that: 1. ˚1 majorizes ˚2 at 0 (˚1 0 ˚2 ) if there exist positive numbers a; b; x0 such that ˚2 .x/  b ˚1 .ax/ for all 0  x  x0 : 2. ˚1 majorizes ˚2 at 1 (˚1 1 ˚2 ) if there exist positive numbers a; b; x0 such that ˚2 .x/  b ˚1 .ax/ for all x  x0 : 3. ˚1 majorizes ˚2 (˚1 ˚2 ) if ˚1 0 ˚2 and ˚1 1 ˚2 . We show that one can set b D 1 in this definition. Proposition 16.1.2. 1. ˚1 0 ˚2 if and only if ˚2 .x/  ˚1 .a1 x/; 0  x  x1 for some a1 > 0 and x1 > 0. 2. ˚1 1 ˚2 if and only if ˚2 .x/  ˚1 .a2 x/; x  x2 for some a2 > 0 and x2 > 0.

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_16

207

208

16 Comparison of Orlicz Spaces

Proof. 1. Suppose that ˚2 .x/  b ˚1 .ax/; 0  x  x0 ; for some a > 0, b > 0, and x0 > 0. If b  1, then ˚2 .x/  b ˚1 .ax/  ˚1 .ax/; 0  x  x0 : If b > 1, then by putting y D ˚2 .y/ D ˚2

x b

x , we have b



1 1 ˚2 .x/  b˚1 .ax/ D ˚1 .aby/ b b

for all 0  x  x0 . That is, ˚2 .y/  ˚1 .a1 y/; 0  y  y1 ; x0 where a1 D ab and y1 D , i.e., ˚1 0 ˚2 . b The converse is obvious. 2. The proof is similar to that of 1.

t u

Proposition 16.1.3. The condition ˚1 ˚2 can be written as ˚2 .x/  b˚1 .ax/; x  0 for some b > 0 and a > 0. Proof. Let ˚1 ˚2 . Then by Proposition 16.1.2, ˚2 .x/  ˚1 .a1 x/; 0  x  x1 ; and ˚2 .x/  ˚1 .a2 x/; x  x2 ; for some positive a1 , a2 , x1 , and x2 . We may assume without loss of generality that ˚2 .x2 / < 1. Otherwise, we x2 choose x2 such that 0  x2  x2 and ˚2 .x2 / < 1. Then by setting a2 D a2  , we x2 have ˚2 .x/  ˚2 .x 

x2 x2 /  ˚1 .a2 x  / D ˚1 .a2 x/ x2 x2

for all x  x2 . Similarly, we may assume that ˚1 .a1 x1 / > 0.

16.2 The Embedding Theorem for Orlicz Spaces

209

Now we set a D max.a1 ; a2 / and bD

8 < 1;



: max

x1 xx2

 for x2  x1 ; ˚2 .x/ ; for x1 < x2 : ˚1 .ax/

Then ˚2 .x/  b˚1 .ax/ for all x  0.

t u

16.2 The Embedding Theorem for Orlicz Spaces Let L˚1 and L˚2 be two Orlicz spaces. Our aim is to characterize the embedding L˚1  L˚2 in terms of Orlicz functions ˚1 and ˚2 . Recall that the natural embedding L˚1  L˚2 is always bounded, i.e., kf kL˚2  ckf kL˚1 ; f 2 L˚1 ; for some c > 0 (Proposition 6.1.1). Theorem 16.2.1. Let ˚1 and ˚2 be two Orlicz functions and 'L˚1 , 'L˚2 the corresponding fundamental functions of Orlicz spaces L˚1 and L˚2 . Then the following are equivalent: 1. 2. 3. 4. 5.

˚1 ˚2 ; L˚1  L˚2 ; kf kL˚2  akf kL˚1 for some a > 0; 'L˚2  a'L˚1 for some a > 0; ˚2 .x/  ˚1 .ax/ for some a > 0 and all x > 0.

Proof. 1 H) 2. By Proposition 16.1.3 the condition ˚1 ˚2 can be written as ˚2 .x/  b˚1 .ax/; x  0; for some b > 0 and a > 0. If f 2 L˚1 , then for some c > 0,   Z1   f jf j I˚1 D ˚1 dm < 1: c c 0

Consequently,  I˚2

f ac





Z1 D

˚2 0

   Z1   jf j jf j f dm  b ˚1 dm D bI˚1 < 1; ac c c 0

210

16 Comparison of Orlicz Spaces

i.e., f 2 L˚2 . Thus L˚1  L˚2 . 2 H) 3 follows from Proposition 6.1.1 and Theorem 13.2.2. 3 H) 4. By taking f D 1Œ0;x in 3, we obtain 4. 4 H) 5. Using (13.3.1) for fundamental functions of Orlicz spaces, we have  1  1 1  1  ˚2 x  a ˚11 x1 ; x > 0: By taking x1 D ˚2 .y/, we obtain ˚11 .˚2 .y//  a˚21 .˚2 .y// D ay; y > 0; or ˚2 .y/  ˚1 .ay/; y > 0: 5 H) 1 is obvious.

t u

Note 16.2.2. Relation 1 ” 5 in Theorem 16.2.1 shows that we can take b D 1 in Proposition 16.1.3 by increasing a if necessary. On the other hand, if ˚2 .x/  b˚1 .x/; x > 0; for some b > 0, then ˚1 ˚2 . The converse may be false in general. In other words, it is not always possible to take a D 1 in Proposition 16.1.3 by increasing the factor b. Examples 16.2.3. 1. Consider the family of functions (Fig. 16.1) ˚c .x/ D ecx  1; x  0; c > 0: For 0 < c1 < c2 < 1, we have  ˚c1 .x/  ˚c2 .x/ D ˚c1

 c2 x ; x  0; c1

i.e., ˚c1 ˚c2 and ˚c2 ˚c1 . However, there is no constant b > 0 such that ˚c2 .x/  b˚c1 .x/ for all x  0. 2. Consider the family of functions (Fig. 16.2)  ˚c .x/ D

0; 0  x  cI 1; x > c:

16.2 The Embedding Theorem for Orlicz Spaces Fig. 16.1 Functions ˚c .x/ D ecx  1; c  0

211

y

ec2 x –1

ec1x –1

0

Fig. 16.2 Functions ˚c1 .x/ and ˚c2 .x/ for c1 < c2

x

y

y

•

•

Fc

Fc

1

1

c

0

1

For 0 < c1 < c2 < 1, we have ˚c2 .x/  ˚c1 .x/ D ˚c2

x



0

c

1

c

2

x

 c1 x ; x  0; c2

i.e., ˚c1 ˚c2 and ˚c2 ˚c1 . However, there is no constant b > 0 such that ˚c1 .x/  b˚c2 .x/ for all x  0. Note 16.2.4. Using the embedding theorem (Theorem 12.1.3) and the comparison theorem for Lorentz and Marcinkiewicz spaces (Theorem 12.4.1), the above Theorem 16.2.1 can be refined as follows. Let ˚1 and ˚2 be two Orlicz functions and V1 D 'L˚1 and V2 D 'L˚2 the fundamental functions of Orlicz spaces L˚1 and L˚2 . Since all Orlicz spaces are maximal (Theorem 8.2.2), Theorem 12.1.3 implies e V 1  L˚1  MV1 ; e V 2  L˚2  MV2 ; x for x > 0, i D 1; 2. where e V i is the least concave majorant of Vi , and Vi .x/ D Vi .x/ We claim that e V 1  e V 2 ” L˚1  L˚2 ” MV1  MV2 :

212

16 Comparison of Orlicz Spaces

Indeed, each of the embeddings holds if and only if V2  aV1 for some a > 0. (The first and the third hold by Theorem 12.4.1, and the second by Theorem 16.2.1.) It is worthwhile considering some particular cases in which one of the Orlicz spaces L˚1 or L˚2 in the inclusion L˚1  L˚2 coincides with L1 or L1 . Examples 16.2.5. 1. Let L˚1 D L1 , ˚1 .x/ D x, x  0. In this case, L1  L˚2 ” ˚20 .1/ < 1: 2. Let L˚2 D L1 , ˚2 .x/ D x, x  0. In this case, L˚1  L1 ” ˚10 .0C/ > 0: 3. Let L˚1 D L1 , ˚1 .x/ D 1.0;1/ .x/. In this case, L1  L˚2 ” a˚2 > 0: 4. Let L˚2 D L1 , ˚2 .x/ D 1.0;1/ .x/. In this case, L˚1  L1 ” b˚1 < 1:

16.3 The Coincidence Theorem for Orlicz Spaces Theorem 16.2.1 yields a simple way to verify the equality L˚1 D L˚2 by means of functions ˚1 and ˚2 . Definition 16.3.1. Two Orlicz functions ˚1 and ˚2 are called equivalent (˚1  ˚2 ) if ˚1 ˚2 and ˚2 ˚1 . Theorem 16.3.2. Let ˚1 and ˚2 be two Orlicz functions. The following are equivalent: 1. ˚1  ˚2 . 2. L˚1 D L˚2 as sets. 3. k  kL˚1 and k  kL˚2 are equivalent, i.e., a1 kf kL˚1  kf kL˚2  a2 kf kL˚1 for all f and some a1 > 0, a2 > 0. 4. 'L˚1 and 'L˚2 are equivalent in the following sense: a1 'L˚1 .x/  'L˚2 .x/  a2 'L˚1 .x/ for all x  0 and some a1 > 0, a2 > 0.

16.4 Zygmund Classes

213

5. ˚1 and ˚2 are equivalent in the following sense: ˚1 .a1 x/  ˚2 .x/  ˚1 .a2 x/; x  0 for some a1 > 0 and a2 > 0. Note 16.3.3. The constants a1 and a2 in the above conditions 3, 4, and 5 are the same. For example, condition 5 is equivalent to 1 1 1 ˚ .y/  ˚21 .y/  ˚11 .y/; y > 0; a2 1 a1 where y D ˚2 .x/ . Hence,   1   1   1 1 1 1 1 1 1 a1 ˚1  ˚2  a2 ˚1 ; x > 0; x x x i.e., a1 'L˚1 .x/  'L˚2 .x/  a2 'L˚1 .x/; x > 0: Corollary 16.3.4. With the notation of Note 16.2.4, e V 1 D e V 2 ” L˚1 D L˚2 ” MV1 D MV2 : Examples 16.3.5. 1. Let L˚1 D L1 , ˚1 .x/ D x, x  0. In this case, L1 D L˚2 as sets ” ˚20 .0C/ > 0 and ˚20 .1/ < 1: 2. Let L˚1 D L1 , ˚1 .x/ D 1.0;1/ .x/. In this case, L1 D L˚2 as sets ” a˚2 > 0 and b˚2 < 1:

16.4 Zygmund Classes Consider a one-parameter family of functions ˚˛ ; 0  ˛ < 1, defined by 8 0  x  1; < 0; ˚˛ .x/ D Rx ˛ : .ln u/ du; x > 1: 1

Since the derivatives ˚˛0 .x/ D ln˛ x  1Œ1;1/ .x/; x  0;

214

16 Comparison of Orlicz Spaces y

y

y

1

0

F'a

F'a

F'0

(a

00 0

1

x

0

1

x

are increasing, the ˚˛ are convex on Œ0; 1/ (Fig. 16.3). Thus ˚˛ are Orlicz functions. We denote the corresponding Orlicz spaces by Z˛ D L˚˛ ; 0  ˛ < 1; and call them Zygmund classes. If ˛ D 0, we have ˚0 .x/ D .x  1/  1Œ1;1/ .x/, i.e., Z0 D L1 C L1 (Example 13.4.1) (Fig. 16.4). For every 0  ˛  ˇ < 1 and p  1, we have ˚˛ .x/  ˚ˇ .x/  xp ; x  e: Therefore, by Theorem 16.2.1, L1 C L1 D Z0 Z˛ Zˇ Lp ; 0  ˛  ˇ < 1; p  1: Since a˚˛ D 1 for all 0  ˛ < 1, we have also Z˛ L1 Corollary 13.3.3). Thus Z˛ Lp for all 1  p  1. In the case of ˛  1, it is convenient to use the functions ˚ ˛ .x/ D x.ln x/˛  1Œ1;1/ .x/; x  0:

(see

16.4 Zygmund Classes

215

Since for ˛  1, 0

0 ˚ ˛ .x/ D ˚˛0 .x/ C ˛˚˛1 .x/

and ˚ ˛ .1/ D ˚˛ .1/ D 0, we have ˚ ˛ .x/ D ˚˛ .x/ C ˛˚˛1 .x/: This shows that for ˛  1, ˚ ˛ is an Orlicz function, and ˚˛ .x/  ˚ ˛ .x/  .˛ C 1/˚˛ .x/; x  e; i.e., ˚˛  ˚ ˛ . The Orlicz spaces L˚ ˛ , ˛  1 are usually denoted by L˚ ˛ D L ln˛ L: For 0  ˛ < 1, the function ˚ ˛ is not an Orlicz function, but one can use its greatest convex minorant .˚ ˛ /Ï . For example, if ˛ D 0, then the function ˚0 is the greatest convex minorant for ˚ 0 (Fig. 16.5). In spite of this, the notation Z˛ D L ln˛ L is often used for all ˛  0. To describe the associate spaces Z1˛ of Zygmund classes Z˛ , it is convenient to use the Orlicz functions ˚˛ . Indeed, the inverse function of ˚˛0 D ln˛ x  1Œ1;1/ .x/ has the form 1=˛

˛0 .x/ D ex ; where Zx ˛ .x/ D

1=˛

eu du; x  0; 0

is the conjugate of the Orlicz function ˚˛ . Fig. 16.5 ˚ 0 and its greatest convex minorant

y

y

F

F

0

1

0

0

1

1

x

0

1

x

216

16 Comparison of Orlicz Spaces

We obtain a one-parameter family of Orlicz spaces Z1˛ D L˛ ; 0  ˛ < 1; such that for all 0  ˛  ˇ < 1 and p  1, L1 \ L1 D Z10  Z1˛  Z1ˇ  Lp : In particular, L˛  L1 for all ˛  0, since ˛0 .0C/ D 1 > 0. Note 16.4.1. Finally, we mention two important properties of the classes Z˛ and Z1˛ . 1. Z˛ satisfies the .2 .1// condition, but it does not satisfy .2 .0//. 2. Z1˛ satisfies the .2 .0// condition, but it does not satisfy .2 .1//. In particular, all Z˛ and all Z1˛ , are nonseparable. The minimal part of Z˛ has the form Z0˛ D L0˚˛ D H˚˛ D Z˛ \ R0 :

Chapter 17

Intersections and Sums of Orlicz Spaces

In this chapter, we study the intersections and the sums of Orlicz spaces. In particular, for every pair of mutually conjugate Orlicz functions ˚ and  , the Orlicz spaces .L˚ \ L1 ; L C L1 / and .L˚ C L1 ; L \ L1 / are described. In particular, the spaces Lp C Lq and Lp \ Lq , 1  p; q  1, are considered in greater detail.

17.1 The Intersection and the Sum of Orlicz Spaces Let ˚1 and ˚2 be two Orlicz functions and let L˚1 and L˚2 be the corresponding Orlicz spaces. Then the symmetric spaces L˚1 \ L˚2 and L˚1 C L˚2 D ff 2 L1 C L1 W f D g C h; g 2 L˚1 ; h 2 L˚2 g are equipped with the norms kf kL˚1 \L˚2 D max.kf kL˚1 ; kf kL˚2 /

(17.1.1)

and kf kL˚1 CL˚2 D inffkgkL˚1 C khkL˚2 ; f D g C h; g 2 L˚1 ; h 2 L˚2 g;

(17.1.2)

respectively. We show that the symmetric spaces L˚1 \ L˚2 and L˚1 C L˚2 are themselves Orlicz spaces. To prove this, we consider the functions ˚1 _ ˚2 D max.˚1 ; ˚2 / and ˚1 ^ ˚2 D min.˚1 ; ˚2 /:

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4_17

217

218

17 Intersections and Sums of Orlicz Spaces

y

y

F2 F1

F1 F2

F1 Ÿ F2

F2 F1

F1 )

F2

F1Ÿ F2

~(

y

Ÿ

y2 y1

0

x 0

x

x1

0

x2

x

Fig. 17.1 Functions ˚1 _ ˚2 ; ˚1 ^ ˚2 ; .˚1 ^ ˚2 /Ï

The function ˚1 _ ˚2 is itself an Orlicz function, while the function ˚1 ^ ˚2 is not necessarily a convex function, i.e., it is not an Orlicz function in general (Fig. 17.1). Since the inverse functions ˚11 and ˚21 are concave, the function ˚11 _ ˚21 is quasiconcave (Proposition 10.2.3). Its least concave majorant .˚11 _ ˚21 /Ï is the inverse of the greatest convex minorant .˚1 ^ ˚2 /Ï of the quasiconvex function ˚1 ^ ˚2 . In this case, Theorem 10.5.2 implies .˚1 ^ ˚2 /

x 2

 .˚1 ^ ˚2 /Ï .x/  .˚1 ^ ˚2 /.x/; x  0:

(17.1.3)

Proposition 17.1.1. 1. L˚1 _˚2 D L˚1 \ L˚2 . 2. L.˚1 ^˚2 /Ï D L˚1 C L˚2 . Proof. 1. ˚1 _ ˚2 is an Orlicz function, and ˚i  ˚1 _ ˚2 , i D 1; 2. Hence, L˚i L˚1 _˚2 , i D 1; 2, and L˚1 \ L˚2 L˚1 _˚2 : On the other hand, ˚1 _ ˚2  ˚1 C ˚2 . Therefore, if f 2 L˚1 \ L˚2 , then  I˚1

f a1



 < 1 and I˚2

f a2

 0 and a2 > 0, whence       f f f  I˚1 C I˚2 0, I.˚1 ^˚2 /Ï

  f < 1; a

and by (17.1.3),    Z1 Z1  f f dm D ˚1 .˚1 ^ ˚2 /  1f˚1 ˚2 g dm 2a 2a 0

0



Z1 C

˚2 0

 f  1f˚2 1;

and .˚ ^ ˚1 /Ï is the greatest convex minorant of ˚ ^ ˚1 D min.˚; ˚1 /. The function .˚ ^ ˚1 /Ï has the form 8 ˆ 0  x  1; < 0; Ï ; 1 < x  x0 ; .˚ ^ ˚1 / .x/ D ˚.x0 / xx1 1 ˆ : ˚.x/; 0 x > x0 : Here x0 is chosen in such a way that the supporting line y D ˚.x0 /

x1 x0  1

17.2 The Spaces L˚ C L1 and L \ L1 y

F

V

F•

F

y

)

•

F•

F

V

y

F• ˜ (

y

221

F (1 ) 0

x

1

0

x

1

0

x

1 x0 x

0

Fig. 17.2 Functions ˚; ˚1 , ˚ ^ ˚1 and .˚ ^ ˚1 /Ï

y

x

x = F 0-1( y)

2 y = F ( x)

y = F 0 ( x)

1

x = F -1 ( y )

F(1 ) 0

2

1

x

0 F(1 )

y

Fig. 17.3 Functions ˚; ˚0 and their inverse functions

of the graph of y D ˚.x/ at the point x0 just passes through the point .0; 1/ (Fig. 17.2). Instead of .˚ ^ ˚1 /Ï , it is worthwhile using the Orlicz function (Fig. 17.3)  ˚0 .x/ D

0; 0  x  1I ˚.x  1/; x > 1:

Clearly, ˚ 1 .y/  ˚01 .y/ D ˚ 1 .y/ C 1  2˚ 1 .y/; y  ˚.1/; whence ˚

x 2

 ˚0 .x/  ˚.x/; x  1;

and .˚ ^ ˚1 /

x 2

 ˚0 .x/  .˚ ^ ˚1 /.x/:

Note 17.2.1. The norm k  kL˚0 has the form kf kL˚0 D inffmax.kgkL˚ ; khkL1 / W f D g C h; g 2 L˚ ; h 2 L1 g:

222

17 Intersections and Sums of Orlicz Spaces

Fig. 17.4 Functions  and  _ ˚1

y

y Y V F1

Y F1

0

x0

x

0

x0

x

It is equivalent but is not equal to the natural norm of L˚ C L1 defined by kf kL˚ CL1 D inffkgkL˚ C khkL1 W f D g C h; g 2 L˚ ; h 2 L1 g: It is clear that kf kL˚0  kf kL˚ CL1  2kf kL˚0 : For every Orlicz function  , the space L \ L1 coincides with the Orlicz space L _˚1 , where ˚1 .x/ D x; x  0 and  _ ˚1 D max.; ˚1 / (Fig. 17.4). Moreover,  _ ˚1 is an Orlicz function. Let now  D ˚ L be the conjugate Orlicz function of ˚. Then using the equalities L ˚1L D ˚1 and ˚1 D ˚1 and relations (17.1.4), we have .˚ ^ ˚1 /L D  _ ˚1 and . _ ˚1 /L .x/  .˚ ^ ˚1 /.x/  . _ ˚1 /L .2x/: By Proposition 17.1.2, .L \ L1 /1 D L˚ C L1 D L˚0 and .L˚ C L1 /1 D L \ L1 D L _˚1 :

17.3 The Spaces L˚ C L1 and L \ L1

223

17.3 The Spaces L˚ C L1 and L \ L1 Since ˚ 0 .0C/ > 0 H) L˚  L1 H) L˚ C L1 D L1 and ˚ 0 .1/ < 1 H) L˚ L1 H) L˚ C L1 D L˚ ; we may assume without loss of generality that ˚ 0 .0C/ D 0 and ˚ 0 .1/ D 1. By Proposition 17.1.1, L˚ C L1 D L.˚^˚1 /Ï , where the greatest convex minorant .˚ ^ ˚1 /Ï of ˚ ^ ˚1 D min.˚; ˚1 / has the form 

Ï

.˚ ^ ˚1 / .x/ D

˚.x/; 0  x  x0 ; x C ˚.x0 /  x0 ; x > x0 ;

and x0 is chosen in such a manner that y D x C ˚.x0 /  x0 is a supporting line of the graph of y D ˚.x/ at the point x0 (Fig. 17.5). If L˚ ¤ L1 , then there exists a point x0 2 .0; 1/ such that 0 < ˚ 0 .x0 / < 1: In this case, we may replace ˚.x/ by an equivalent function ˚.bx/ and assume that ˚ 0 .1/ D 1. Then (Fig. 17.6) 

Ï

.˚ ^ ˚1 / .x/ D

F

y

F F1

F F1 ~ V

y

(

y

˚.x/; 0  x  1; x C ˚.1/  1; x > 1:

)

V

F1 )

0

x

0

x

(

F x1

0

Fig. 17.5 Functions ˚; ˚1 ; ˚ ^ ˚1 and the greatest convex minorant

x1

x

224

17 Intersections and Sums of Orlicz Spaces

Fig. 17.6 Function .˚ ^ ˚1 /Ï

y y = F(x) F'(1) = 1

y = x + F (1 ) _ 1

F (1) 0 Fig. 17.7 Functions ; ˚1 , and  _ ˚1

F•

y

1

x

y

•

• Y V F•

Y

0

1

x

0

1

x

For every Orlicz function  , the space L \ L1 coincides with the Orlicz space L _˚1 (Fig. 17.7). If ˚ and  are two conjugate Orlicz functions, then by Proposition 17.1.2, .L \ L1 /1 D L˚ C L1 D L.˚^˚1 /Ï and .L˚ C L1 /1 D L \ L1 D L _˚1 :

17.4 The Spaces Lp \ Lq and Lp C Lq , 1  p  q  1 As noted above (Example 13.4.1(1)), the space Lp , 1  p < 1, is an Orlicz space with the Orlicz function ˚.x/ D xp . In this case, Lp D L˚ D H˚ and k  kLp D k  kL˚ . It will be convenient to use additional Orlicz functions ˚p .x/ D

xp ; x  0: p

17.4 The Spaces Lp \ Lq and Lp C Lq , 1  p  q  1

225

We have kf kL˚p

8 9 8 9  Z1 p Z1  < = < = jf j p jf j p D inf a > 0 W dm  1 D inf a > 0 W dm  a : ; : ; p ap p1=p 0

0

9 8 1 1p 01 > ˆ p Z  = < jf j A  a D 1 kf kLp : dm D inf a > 0 W @ > ˆ p1=p p1=p ; : 0

1 1 k  kL˚ D 1=p k  kLp . p1=p p The function ˚p corresponds to the normalizing condition ˚p0 .1/ D 1, instead of ˚.1/ D 1. The advantage of such normalization is that the conjugate power functions Thus, L˚p D L˚ D Lp and k  kL˚p D

0

y D ˚p0 .x/ D xp1 and x D ˚p0 0 .y/ D yp 1 ;

1 1 C 0 D 1; p p

are mutually inverse (.p  1/.p0  1/ D 1), i.e., the Orlicz functions ˚p and ˚p0 are conjugates of each other (Fig. 17.8). In the case p D 1, we have  xp 0; 0  x  1; ˚1 .x/ D lim ˚p .x/ D lim D p!1 p!1 p 1; x > 1: Note that ˚1 and ˚1 are conjugate Orlicz functions. For every pair of p; q 2 Œ1; 1, we consider the spaces Lp D L˚p , Lq D L˚q with Orlicz functions ˚p and ˚q . We are interested in using the spaces Lp \ Lq and Lp C Lq , which have the form Lp \ Lq D L˚p _˚q ; Lp C Lq D L.˚p ^˚q /Ï by Proposition 17.1.1. Fig. 17.8 Functions ˚p .x/ and ˚p0 .y/ as areas

y

_ y = xp 1 _ x = y p' 1 ( p _ 1 )( p' _ 1) = 1

y

F p' (y)

F p(x) 0

x

x

226

17 Intersections and Sums of Orlicz Spaces

For 1  p  q < 1, we have

8 1   qp p ˆ q ˆx ˆ ; < ; if 0  p .˚p _ ˚q /.x/ D max.˚p .x/; ˚q .x// D p 1   qp ˆ xq q ˆ ˆ : ; if x > ; q p

and

8 1   qp ˆ xq q ˆ ˆ ; < ; if 0  p .˚p ^ ˚q /.x/ D min.˚p .x/; ˚q .x// D q 1   qp ˆ xp q ˆ ˆ : ; if x > : p p

The greatest convex minorant .˚p ^ ˚q /Ï .x/ of .˚p ^ ˚q /.x/ has the form (Fig. 17.9) 8 q x ˆ ˆ ; 0  x  x1 ˆ ˆ ˆ < q q1 q q1 x ; x1 < x  x2 ; .˚p ^ ˚q /Ï .x/ D x1 x  ˆ q 1 ˆ p ˆ ˆ x ˆ : ; x > x2 ; p where

p.q  1/ x1 D q.p  1/

 p1 qp

p.q  1/ ; x2 D q.p  1/

 q1 qp

:

We omit here a routine procedure to find the common tangent line q1

y D x1 x 

q1 q x q 1

of the graphs of y D ˚p .x/ and y D ˚q .x/.

Fp Fq V

Fp

Fq

y

Fq

Fp )

Fp V Fq

y

Fp Fq V

Fq

Fp

~(

y

y2 y1

0

x 0

Fig. 17.9 Functions ˚p _ ˚q ; ˚p ^ ˚q and .˚p ^ ˚q /Ï

x

0

x1

x2

x

17.4 The Spaces Lp \ Lq and Lp C Lq , 1  p  q  1

227

Furthermore, instead of Orlicz functions .˚p _ ˚q /.x/ and .˚p ^ ˚q /Ï .x/, we shall use the following family of functions f˚p;q ; 1  p; q  1g: 8 xp ˆ 0  x  1I < ; ˚p;q .x/ D xpq 1 1 ˆ : C  ; x > 1; q p q 8 p < x ; 0  x  1I ˚p;1 .x/ D p : 1; x > 1;

.p; q < 1/;

.q D 1/;

8 < 0; 0  x  1I ˚1;q .x/ D xq 1 :  ; x > 1: q q

.p D 1/;

and ˚1;1 D ˚1 : Here ˚p;q is an Orlicz function for all p and q, and if p and q are finite, there 0 .1/ D 1 (Fig. 17.10). exists ˚p;q It is easy to verify that for all 1  p  q  1, ˚p _ ˚q Ð ˚p;q ; ˚p ^ ˚q Ð ˚q;p : Therefore, by Proposition 17.1.1, for all 1  p  q  1, Lp \ Lq D L˚p;q ; Lp C Lq D L˚q;p :

y

y

y Fp,q

xq q

p,q 1: C q p q 0

0

Using the conjugate exponents .p ; q /, 1 1 1 1 C 0 D 1; C 0 D 1; q p p q we obtain p1 1 q1 1 D 0; D 0; p q q p and hence

p;q

9 8 0 q > ˆ x > ˆ ˆ 0 < x  1I > = < 0; D ˚q0 ;p0 : D q0 > ˆ xp 1 1 > ˆ > ˆ ; : 0 C 0  0; x>1 p q p 0

0

If 1  p  q  1 (and hence 1  p  q  1), then .Lp \ Lq /1 D L1˚p;q D Lp;q D L˚ 0

D Lp0 C Lq0 ;

.Lp C Lq /1 D L1˚q;p D Lq;p D L˚ 0

D Lp0 \ Lq0 :

0 q ;p

and 0 p ;q

Exercises

229

Exercises 25. Orlicz spaces on general measure spaces. Let .˝; / be a measure space with a  -finite (finite or infinite) measure and let ˚ be an Orlicz function on Œ0; 1/. Let L˚ D L˚ .˝; / be the corresponding Orlicz space on .˝; /, that is, L˚ .˝; / D ff 2 L0 .˝; / W kf kL˚ .˝; / < 1g; where

  Z f  1 and I˚ .f / D ˚.jf j/d : D inf c > 0 W I˚ c 

kf kL˚ .˝; /

˝

Show that a. .L˚ .˝; /; k  kL˚ .˝; / / is a symmetric space on .˝; / in the sense of Complement 1. b. Let the measure be nonatomic and infinite. Then the standard space corresponding to the symmetric space L˚ .˝; / coincides with the Orlicz space L˚ .RC ; m/ on Œ0; 1/, i.e., L˚ .RC ; m/ D ff 2 L0 .˝; / W f  D g for some g 2 L˚ .˝; /g; and Z

Z ˚.jf j/d D ˝

˚.f  /dm and kf kL˚ .˝; / D kf  kL˚ .RC ;m/ :

RC

c. Let the Orlicz function ˚ be finite on .0; 1/ (b˚ D 1) and let 9 8   Z = < jf j d < 1 for all c > 0 H˚ .˝; / D f 2 L0 .˝; / W ˚ ; : c ˝

be the heart of L˚ .˝; /. Then H˚ .˝; / is a symmetric space on .˝; /. It has property .A/ and coincides with the minimal part .L˚ .˝; //0 of L˚ .˝; /, i.e., H˚ D clL˚ .F1 /, where F1 D F1 .˝; / denotes the set of all simple -integrable functions on .˝; /. 26. Duality for Orlicz spaces. Let L˚ D L˚ .˝; / be an Orlicz space on a  -finite measure space .˝; /, and let L˚ D .L˚ .˝; // denote the dual Banach space of L˚ . The corresponding associate space L1˚ D .L˚ .˝; //1 is defined by L1˚ D fg 2 L0 .˝; / W ug 2 L˚ g;

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17 Intersections and Sums of Orlicz Spaces

where

Z kgkL1 D kug kL˚ and ug .f / D

fgd ; f 2 L˚ :

˚

˝

Show that a. .L1˚ ; k  kL1 / is a symmetric space on .˝; /. ˚ b. L1˚ D L , where L D L .˝; / and  is the conjugate Orlicz function of the Orlicz function ˚. Also k  kL  k  kL1  2k  kL : ˚

c. If ˚ is finite on .0; 1/ (b˚ D 1), then L1˚ D H1˚ and .H1˚ / D H˚ , where W H1˚ 3 g ! ug 2 H˚ is an isometric isomorphism. d. If ˚ satisfies the .2 / condition, then .L1˚ / D .L / D L˚ : e. If both the functions ˚ and  satisfy the .2 / condition, then L˚ D H˚ , L D H , and both L˚ and L are reflexive. 27. Comparison of Orlicz spaces L˚ .˝; /. Let .˝; / be a  -finite measure space, and let ˚1 and ˚2 be two Orlicz functions. Using the relation “ ”, “ 0 ”, “ 1 ” from Chapter 16, show that a. If ˚1 ˚2 , i.e., ˚1 1 ˚2 and ˚1 0 ˚2 , then L˚1 .˝; /  L˚2 .˝; /. If is nonatomic and ˝ D 1, then L˚1 .˝; /  L˚2 .˝; / H) ˚1 ˚2 : In particular (Theorem 16.2.1), L˚1 .0; 1/  L˚2 .0; 1/ H) ˚1 ˚2 : b. Let the measure be finite. If ˚1 1 ˚2 , then L˚1 .˝; /  L˚2 .˝; /. If in addition, is nonatomic, then L˚1 .˝; /  L˚2 .˝; / ” ˚1 1 ˚2 : In particular, L˚1 .Œ0; 1/  L˚2 .Œ0; 1/ ” ˚1 1 ˚2 :

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c. Let the measure be atomic, i.e., .!/ > 0 for all ! 2 ˝. Let also ˚2 be finite (b˚2 D 1) and inff .!/; ! 2 ˝g > 0 ( has no arbitrarily small sets). If ˚1 0 ˚2 , then L˚1 .˝; /  L˚2 .˝; / and l˚1  l˚2 ” ˚1 0 ˚2 : 28. Comparison H˚ and L˚ . Let .˝; / be a  -finite measure space and ˚ an Orlicz function. Show that a. If ˚ satisfies the .2 / condition, then H˚ .˝; / D L˚ .˝; /. If the measure is nonatomic and ˝ D 1, then the equality H˚ .˝; / D L˚ .˝; / implies the .2 / condition. In particular (Theorem 15.4.1), H˚ .0; 1/ D L˚ .0; 1/ ” ˚ satisfies the .2 / condition: b. Let the measure be infinite. Then the .2 .1// condition implies that H˚ .˝; / D L˚ .˝; /. If, in addition, is nonatomic, then H˚ .˝; / D L˚ .˝; / implies the .2 .1// condition. In particular, H˚ .Œ0; 1; m1 / D L˚ .Œ0; 1; m1 / ” ˚ satisfies the .2 .1// condition: c. Let be atomic and have no arbitrarily small sets, i.e., inff .!/; ! 2 ˝g > 0. If the function ˚2 is finite (b˚2 D 1) and satisfies the .2 .0// condition, then H˚ .˝; / D L˚ .˝; /. In particular, for l˚ D L˚ .N/ and h˚ D H˚ .N/, h˚ D l˚ ” ˚ satisfies the .2 .0// condition: 29. The .2 / condition and ˚-mean convergence. Let L˚ D L˚ .˝; / be an Orlicz space. A sequence ffn g in L˚ Ris said to be ˚-mean convergent to f 2 L˚ if lim I˚ .fn  f / D 0 (Here I˚ .f / D ˚.jf j/d ). n!1

Show that

˝

a. lim kfn  f kL˚ D 0 ” lim I˚ .k.fn  f // D 0 for all k > 0. n!1

n!1

b. Convergence in norm is equivalent to ˚-mean convergence in L˚ .RC ; m/ iff ˚ satisfies the .2 / condition. c. Convergence in norm is equivalent to ˚-mean convergence in L˚ .0; 1/ iff ˚ satisfies the .2 .1// condition. d. Convergence in norm is equivalent to ˚-mean convergence in l˚ D L˚ .N/ iff ˚ satisfies the .2 .0// condition.

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30. Failure of the .2 .1// condition. Let L˚ D L˚ .Œ0; 1/ be an Orlicz space with Orlicz function ˚.x/ D ex  x  1; x  0, that satisfies the .2 .0// condition but fails to satisfy the .2 .1// condition. Show that a. If f .x/ D  ln x, then f 2 L˚ , f 2 2Y˚ , but f 62 Y˚ , i.e., Y˚ ¤ 2Y˚ and H˚ ¤ L˚ . b. If fn .x/ D  ln x  1Œ0; 1  .x/, then fn # 0, but kfn kL˚  1 for all n, i.e., L˚ fails to n have property .A/. c. If fn .x/ D  12 ln x  1Œ0; 1  .x/, then I˚ .fn / ! 0, but kfn kL˚  12 , i.e., ˚-mean n convergence does not imply convergence in norm k  kL˚ . d. If fn .x/ D  ln x1Œ 1 ;1 .x/, then fn 2 H˚ , fn " f 62 H˚ , although sup kfn kL˚ < 1, n

n

i.e., H˚ fails to have property .B/.

31. Relations between ˚ 1 and 'L˚ . Recall that the fundamental function of an Orlicz space L˚ D L˚ .RC ; m/ has the form  1  'L˚ .x/ D ˚ 1 x1 ; x > 0: Show that a. Let ˚1 and ˚2 be two Orlicz functions. Then the following conditions are equivalent: • • • •

L˚1 D L˚2 as sets. ˚1  ˚2 in the sense of Definition 16.3.1. The functions ˚11 and ˚21 are equivalent. The fundamental functions 'L˚1 and 'L˚2 are equivalent. Here two functions '1 and '2 are regarded as equivalent if c1 '1 .x/  '2 .x/  c2 '1 .x/ for all x and some c1 > 0 and c2 > 0.

b. Let  be the conjugate Orlicz function of ˚ and let W denote the least concave majorant of the quasiconcave function .˚ 1 / .x/ D

x ˚ 1 .x/

 1.0;1/ .x/:

Let also 1 D W 1 be the generalized inverse function of W. Then L1 D L1˚ as sets, and the Orlicz functions 1 and  are equivalent in the sense of Definition 16.3.1. c. X D L2 is the only Orlicz space for which X D X1 .

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32. Spaces Lp;q . Consider the spaces Lp;q , p; q 2 Œ1; 1, defined in Complement 12 by

Lp;q

8 9 01 1 1q ˆ > Z < = q  q p @ A D f 2 L0 W kf kLp;q D Œf .x/ d.x /

: ; 0

for 1  p < 1, 1  q < 1, and  Lp;1 D f 2 L0 W kf kLp;1

D sup x f .x/ < 1 ; 1 p



x>0

for 1  p < 1 (q D 1). In the case p D q D 1, we set L1;1 D L1 . Show that a. k  kLp;q is a symmetric quasinorm for all p; q 2 Œ1; 1 and it is a norm if q  p. b. The space .Lp;q ; k  kLp;q / is complete. c. If p > 1, the quasinorm k  kLp;q is equivalent to the norm k  k0Lp;q , where kf k0Lp;q

D kf



kLp;q f 2 Lp;q ; f



1 .x/ D x

Z

x

f  dm:

0

In this case, .Lp;q ; k  k0Lp;q / is a symmetric space. R  R Hint: Apply Jensen’s inequality ˚ fdm  ˚.f /d with the convex function '.x/ D xq and the probability measure 1

1

d .u/ D t. p 1/ d.u.1 p / / on the interval Œ0; t. Check that 1q 0 t q1  Z Zt .q1/.p1/ q1 p p @ f .u/duA  t .f  .u//q u p du p1 0

0

and that the inequality implies kf  kLp;q  for all 1 < p < 1 and 1  q < 1.

p kf kLp;q p1

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17 Intersections and Sums of Orlicz Spaces

Notes The basic material on Orlicz spaces presented in Part IV is taken from [18, Chapter 2]. These authors deal with Orlicz spaces L˚ .˝; / on general  -finite measure spaces .˝; /. Our presentation is adapted mainly to the symmetric spaces on .RC ; m/, while most of definitions and results related to the spaces L˚ .˝; / are formulated in Exercises 25–28. Besides the case L˚ .RC ; m/, the spaces L˚ .0; 1/ and l˚ D L˚ .N/ are of independent special interest. Good references for general Orlicz spaces are [3, Chapter 4.8], [33, Chapter 2], [35, Chapter 4], [36, Chapter 2], [79, Chapter 19]. See also [42, 44, 50, 57, 58, 76, 80]. The spaces L˚ were introduced first by Birnbaum and Orlicz [5] in 1931, and later refined in [33, 54, 81]. The Young classes were introduced in [77]. The subspaces H˚ called the hearts of L˚ originated in [42] and [49]. W. Orlicz introduced the spaces L˚ in [54, 55] by means of the conjugate function  of the original Orlicz function ˚. He used, in fact, the norm k  kL1  instead of k  kL˚ . The norm k  k˚ was introduced later by W.A.J. Luxemburg in [42]. Therefore, it is sometimes called the Luxemburg norm, while k  kL1 is called  the Orlicz norm on L˚ . Inequality (15.1.5) in Property 15.1.2 is due to Young [77]; see also [27, Section 4.8]. A more general form of Young’s inequality linked to the Legendre transform can be found in [6, 21] and [61]. The proof of Theorem 15.3.2 and other related results in Section 15.3 are taken from [18], Section 2.2. Proposition 15.3.4 can be refined as 1 kgkL1 D inf .1 C I˚ .kg//; g 2 L1˚ I ˚ k>0 k see Note 15.3.5. The above equality was proved in [33] in the case of finite measure . The .2 /-condition was used already in [54]. It yields the separability of the Orlicz space L˚ .˝; / on .˝; /, provided that the measure space is separable; see Complement 1. In this case, one can get many equivalents of separability by combining the corresponding results of Chapters 14 and 15 with property .A/ and other separability conditions obtained earlier for general symmetric spaces in Chapters 6 and 7. Zygmund’s classes Z˛ arise in the study of Hardy–Littlewood maximal functions. The spaces Z˛ were introduced in [73, 74] and [81]. The spaces R˛ for ˛ D k D 0; 1; 2; : : : were introduced by Fava; see [18, Section 2.2]. Edgar and Sucheston [18, Chapter 2] identified the spaces as the hearts of Zygmund classes Zk D L logk L, constructed as Orlicz spaces L˚ with suitable ˚.

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235

Note that one-parameter families of symmetric spaces fLp g, fZ˛ g, fR˛ g as well as the Lorentz spaces fe W ˛ g constructed in Exercise 24 form so-called scales of Banach spaces; see [34, Chapter 3]. The spaces Lp;q were introduced by G. Lorentz in [37]. A good presentation of Lp;q can be found in [72, Section 5.3]. An essentially wider class, called Orlicz–Lorentz spaces LW;˚ , is briefly described in Complement 8. The class includes both Orlicz and Lorentz spaces and has been intensively studied for the last two decades. We refer the reader to [13, 28, 30, 40, 43, 48].

Complements

1 Symmetric Spaces on General Measure Spaces Let .˝; / be an arbitrary space with a  -finite (finite or infinite) measure . The upper distribution function jf j .x/ D fu 2 Œ0; 1/ W jf .u/j > xg; x 2 RC ; is defined for every measurable function f W ˝ ! R as well as when .˝; / D .RC ; m/. If jf j .C1/ D lim jf j .x/ D 0, then there is a unique decreasing rightx!C1

continuous function f  on RC such that jf j D f  . The function f  is called the decreasing rearrangement of jf j. It should be emphasized that both jf j and f  are defined on RC , while f and jf j are defined on the space ˝ itself. A symmetric space X D X.˝; / on a measure space .˝; / is a nonzero ideal Banach lattice in L0 .˝; / with a symmetric (rearrangement invariant) norm k  kX.˝; / . This means that the following two conditions hold: 1. If f ; g 2 L0 .˝; /, jf j  jgj, and g 2 X, then f 2 X, and kf kX  kgkX . 2. If f ; g 2 L0 .˝; /, f  D g , and g 2 X, then f 2 X and kf kX D kgkX . For example, the spaces Lp D Lp .˝; /, 1  p  1, are symmetric spaces on .˝; /. Assume that the measure is nonatomic and ˝ D 1. Let X.˝; / be a symmetric space on .˝; /. It is convenient to consider, together with X.˝; /, the space X.RC ; m/. The space consists of all h 2 L0 .RC ; m/ such that f  D h for some f 2 X.˝; /, while khkX.RC ;m/ D kf kX.˝; / if f  D h and f 2 X.˝; /. In many important cases, this “standard” space X.RC ; m/ itself is a symmetric space on .RC ; m/. For example, if there exists a measure-preserving isomorphism

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4

237

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 W .˝; / ! .RC ; m/, it induces the isometric isomorphism f ! f ı  1 between X.˝; / and X.RC ; m/. Next assume that the measure is finite and nonatomic, let ˝ D a < 1. For every f 2 L0 .˝; /, the corresponding distribution function jf j is bounded, jf j .x/  a; x 2 RC : Hence, f  .x/ D 0 for every x > a, and one can regard f  as an element of L0 .Œ0; a; ma / for all f 2 L0 .˝; /, where ma D mjŒ0;a is the usual Lebesgue measure on Œ0; a. If there exists a measure-preserving isomorphism  W .˝; / ! .Œ0; a; ma /, then the standard space X.Œ0; a; ma / corresponding to a symmetric space X.˝; / on .˝; / is a symmetric space on .Œ0; a; ma /, and  induces an isometric isomorphism between X.˝; / and X.Œ0; a; ma /. In both the finite and infinite cases, the desired isomorphism  exists if .˝; / is a Lebesgue space. In particular, let ˝ be a Polish (complete separable metric) space and let be a Borel measure on ˝. Then .˝; / is a Lebesgue space. Thus in this case, every symmetric space X.˝; / on .˝; / can be identified with the corresponding standard space X.RC ; m/ or X.Œ0; a; ma /. Finally, assume that ˝ consists of an infinite number of atoms and that .!/ D 1 for all ! 2 ˝. Since is  -finite, ˝ is countable, and there exists a bijection  between ˝ and N. Thus, every symmetric space on .˝; / can be identified with a symmetric space on .N; ]/, where ] is the counting measure on N. Recall that a measure space .˝; / is said to be separable if the  -algebra F of all -measurable subsets contains a dense subset fAn ; n 2 Ng, i.e., for every " > 0 and A 2 F , there exists n such that .A4An / < ". A general measure space .˝; / may be nonseparable. A symmetric space X.˝; / on .˝; / is not necessarily isomorphic to the corresponding standard space. To see this, consider the (uncountable) direct product .˝; / D

Y

.˝t ; t /; where ˝t D Œ0; 1 and t D m1 D mjŒ0;1 :

0t1

The measure space is nonseparable, and for 1  p < 1, the space X.˝; / D Lp .˝; / is nonseparable, while the corresponding standard space X.Œ0; 1; m1 / D Lp .Œ0; 1; m1 / is separable. The conditions for separability of symmetric spaces on general measure spaces can be described as follows. • Let X.˝; / be a symmetric space on .˝; /, where is a nonatomic infinite measure, and let X.RC ; m/ be the corresponding standard space. Then the following conditions are equivalent: 1. X.˝; / is separable. 2. X.RC ; m/ is separable and .˝; / is separable. 3. X.˝; / has property .A/ and .˝; / is separable.

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4. X.RC ; m/ has property .A/ and .˝; / is separable. Analogous results hold for finite measure spaces.

2 Symmetric Spaces on Œ0; 1 Let .˝; / D .Œ0; 1; m1 /, where m1 is the usual Lebesgue measure on Œ0; 1. A symmetric space on the measure space .Œ0; 1; m1 / is a nonzero linear ideal Banach lattice X D X.0; 1/ of measurable functions on Œ0; 1 with a symmetric (rearrangement invariant) norm k  kX . The distribution function jf j corresponding to a function f 2 L0 .0; 1/ is bounded, 0  jf j  1. Therefore, the decreasing rearrangement f  of jf j can be considered as an element of the space L0 .0; 1/. It is convenient to use the natural embedding L0 .0; 1/ 3 f ! f  1Œ0;1 2 L0 .0; 1/ and the image ff  1Œ0;1 ; f 2 Xg  L0 .0; 1/: Most results for symmetric spaces on Œ0; 1 are easily deduced from the corresponding theorems on Œ0; 1/.

The Embedding Theorem • For every symmetric space X on Œ0; 1, there are the natural embeddings L1 .0; 1/  X  L1 .0; 1/  L0 .0; 1/: They are continuous, and k  kL1  k  kX  k  kL1 : For example, Lp .0; 1/, 1  p  1, are symmetric spaces on Œ0; 1, and for all 1  p  q  1, L1 .0; 1/  Lq .0; 1/  Lp .0; 1/  L1 .0:1/ and k  kL1  k  kLq  k  kLp  k  kL1 :

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Complements

Minimality and Separability A symmetric space X D X.0; 1/ is called minimal if L1 is dense in X, i.e., X coincides with its minimal part X0 D clX .L1 /. The space X D X.0; 1/ is minimal, either it is separable or X D L1 .0; 1/ as sets. The first case holds iff 'X .C0/ D 0, where 'X is the fundamental function of X. Moreover • The following conditions are equivalent: 1. 2. 3. 4.

X is separable. X is minimal and X ¤ L1 . X has property .A0 / (see. Proposition 6.5.2). .X1 / D X , where W X1 ! X is the natural embedding of the associate space X1 in the dual space X (cf., Theorems 6.5.3 and 7.4.1).

Maximality and Property .B/ The associate space X1 of a symmetric space X on Œ0; 1 is defined as 9 8 ˇ ˇ 1 ˇ ˇZ = < ˇ ˇ X1 D f 2 L1 W kf kX1 D sup ˇˇ fgdmˇˇ < 1 : ; : kgkX 1 ˇ ˇ 0

• .X1 ; k  kX1 / is a symmetric space on Œ0; 1, and X1 has both properties .B/ and .C/, i.e., the Fatou property. • The natural embeddings X0  X  X11 are continuous, and kf kX11  kf kX ; f 2 X and kf kX0 D kf kX D kf kX11 ; f 2 X0 : The strict inequality kf kX11 < kf kX is possible in the case that X fails to have property .C/ and f 62 X0 . A symmetric space X D X.0; 1/ is called maximal if X D X11 . • The following conditions are equivalent. 1. X D X11 as a set, i.e., X is maximal. 2. X D Y1 for some symmetric space Y. 3. X has property .B/ (cf., Theorem 8.2.2 and Exercise 13).

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• The following conditions are equivalent. 1. X D X11 and k  kX D k  kX11 . 2. X has the property ffn g  X; 0  fn " and sup kfn kX < 1 H) fn " f and

.BC/

n

jfn kX " kf kX for some f 2 X: .st/

3. If fn ! f in L0 and sup kfn kX  1, then kf kX  1 (cf. Theorem 8.3.5). n

Property .BC/ means that X has both properties .B/ and .C/. It is equivalent to the Fatou property (cf. Theorem 8.3.5 and Note 8.3.6).

3 Symmetric Sequence Spaces Consider the symmetric spaces X.N/. Let l0 D RN D ff D ff .n/g1 nD1 ; 1 < f .n/ < C1 for all ng be the space of all real sequences equipped with the natural (coordinatewise) algebraic operations and partial order. For each f D ff .n/g1 nD1 2 l0 , we set jf j .x/ D ]fk 2 N W jf .n/j > xg; x 2 RC ; where ].A/ is the cardinality of A  N. If the distribution function jf j W RC ! N [ f1g is finite, then the decreasing rearrangement f  D ff  .n/g1 nD1 of the sequence jf j can be written as f  .n/ D inffx 2 RC W jjf j .x/j  ng; n 2 N: It is obvious that f  is well defined only if f 2 l1 , where l1 D ff D ff .n/g1 nD1 2 l0 W kf kl1 D sup f .n/ < 1g n

is the space of all bounded sequences. Here, either f and f  belong to the space c0 D fg D fg.n/g1 nD1 2 l1 W lim g.n/ D 0g; n!1

or limf .n/ D inf f  .n/ D lim f  .n/ > 0: n

n

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Note that in the first case, there exists a permutation  W N ! N such that f  D jf j ı  . A symmetric sequence space X.N/ is a nonzero linear ideal Banach sublattice of the space l1 in which the norm k  kX.N/ is symmetric (rearrangement invariant), i.e., f 2 X.N/; g 2 l1 ; f  D g H) g 2 X.N/; kf kX.N/ D kgkX.N/ : The above-mentioned spaces c0 , l1 , and the spaces lp ; 1  p < 1, lp D ff D ff .n/g1 nD1 W kf klp < 1g; with kf klp WD

1 X

! 1p jf .n/j

p

;

1  p < 1;

nD1

are examples of symmetric sequence spaces.

Embedding Theorem For every symmetric sequence space X.N/, there are embeddings l1  X.N/  l1  l0 : They are continuous and k  kl1  k  kX.N/  k  kl1 : For example, we have l1  lp  lq  c0  l1 for all 1  p  q < 1 and k  kl1  k  klp  k  klq  k  kl1 : Moreover, cllq .lp / D lq for 1  p  q < 1 and cll1 .lp / D c0 for 1  p < 1.

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Minimality and Separability A symmetric sequence space X D X.N/ is called minimal if l1 is dense in X, i.e., X.N/ coincides with its minimal part X0 .N/ D clX.N/ .l1 /. • For every symmetric sequence space X.N/, the following conditions are equivalent: 1. X.N/ is minimal. 2. X.N/ is separable. 3. X.N/ has the following property .A/: .A/

ffk g  X; 0  fk # 0 ) kfk kX.N/ ! 0;

k ! 1:

4. .X1 .N// D X .N/. Note that for symmetric sequence spaces, minimality is equivalent to separability, since the condition 'X .0C/ D 0 is useless for symmetric spaces on N. It is clear that property .A/ in this case is equivalent to the following: .A1 /: For every f D ff .n/g 2 X.N/, one has kf   fN kX.N/ ! 0, N ! 1, where   f .n/; n  NI  fN .n/ D 0; n > N: The associate space X1 .N/ has the form ˇ1 ˇ ) ( ˇX ˇ ˇ ˇ 1 1 X D g D fg.n/gnD1 2 l1 W ˇ f .n/g.n/ˇ < 1 for all f 2 X ˇ ˇ nD1

and kgkX1

ˇ (ˇ 1 ) ˇX ˇ ˇ ˇ D sup ˇ f .n/g.n/ˇ W kf kX  1 : ˇ ˇ nD1

The maximality condition and the embedding theorem X0  X  X11 are easily adapted to the case of symmetric sequence spaces.

4 The Spaces Lp ; 0 < p < 1 As in the case p  1, we set for 0 < p < 1, 8 9 01 1 1p ˆ > Z < = p Lp D f W kf kLp D @ jf j dmA < 1 : ˆ > : ; 0

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In contrast to the case p  1, the functional f ! kf kLp is not a norm for 0< p< 1. Instead of the triangle inequality, we have here the following weaker inequality: kf C gkLp  C.kf kLp C kgkLp /; f ; g 2 Lp ; 1

where C D 2 p 1 > 1 is a constant that cannot be reduced to 1. Thus, kkLp is a quasinorm and .Lp ; kkLp / is a quasinormed space for 0 < p < 1. On the other hand, .Lp ; k  kLp / is a linear ideal lattice, which is complete with respect to the quasinorm kkLp . The symmetry condition (rearrangement invariance) kf kLp D kf  kLp ; f 2 Lp is obviously satisfied. Sometimes, instead of the quasinorm kf kLp , it is more convenient to use the functional Z1 p

np .f / D

jf jp dm D kf kLp ; 0

which is also symmetric (rearrangement invariant). For each 0 < p < 1, the function u.x/ D xp is concave, and hence it is semiadditive on Œ0; 1/, i.e., u.x C y/  u.x/ C u.y/; x; y 2 Œ0; 1/. Therefore, for each 0 < p < 1, the functional np satisfies the triangle inequality np .f C g/  np .f / C np .g/;

f ; g 2 Lp :

However, np is not a norm, since np .cf / D jcjp np .f / with 0 < p < 1. The corresponding metric dp .f ; g/ D np .f  g/ D kf  gkpp is translation-invariant, i.e., dp .f ; g/ D dp .f C h; g C h/;

f ; g; h 2 Lp :

• .Lp ; dp / is a complete separable metric space for each 0 < p < 1. In spite of the fact that np is not a norm, the operations of addition and scalar multiplication are continuous in the metric dp . Thus for 0 < p < 1, the space Lp is an F-space, i.e., a complete linear topological space Lp with translation-invariant metric dp , in which the operations of addition and multiplication by a scalar are continuous.

Complements

245

Recall that the space L0 with the metric d0 described in Section 6.3 has analogous properties. Like the space L0 , the linear topological spaces Lp , 0 < p < 1 are not normable, and even are not locally convex.

5 Weak Sequential Completeness. Property .AB/ Recall that a Banach space X is said to be weakly sequentially complete if it is sequentially complete in the weak topology  .X; X /. This means that if ffn ; n  1g  X and for each u 2 X there exists a finite limit lim u.fn /, then lim u.fn / D n!1

n!1

u.f / for some f 2 X and all u 2 X , i.e., fn ! f 2 X in the weak topology  .X; X /. If a Banach space X has a predual Banach space Y D X , i.e, Y D X, then the Banach–Steinhaus theorem implies that X is sequentially complete in the weak topology  .X; X /. But it does not imply in general sequential completeness in the weak topology  .X; X /. If the Banach space X is reflexive, the topologies  .X; X / and  .X; X / coincide. Thus, every reflexive Banach space X is weakly sequentially complete. The spaces Lp , 1 < p < 1, are reflexive, and therefore, they are weakly sequentially complete. The space L1 has the predual space L1 , but L1 is not weakly sequentially complete. The space L1 has no Banach predual space, but it is weakly sequentially complete. In the case that X is a symmetric space, properties .A/ and .B/ give an effective criterion of weak sequential completeness. • In every symmetric space X, the following conditions are equivalent: 1. X is separable and maximal; 2. X is minimal, maximal and 'X .0C/ D 0; 3. X has property .AB/: If 0  fn 2 X, fn " and sup kfn kX < 1, then there n

exists f 2 X such that kfn  f kX ! 0, n ! 1. 4. X is weakly sequentially complete.

It is clear that property .AB/ means that X has both properties .A/ and .B/. As it is shown in Section 8.4 (Theorem 8.4.3), a symmetric space X is reflexive if and only if property .AB/ holds in X, and in X1 as well. Note that .A/ implies .C/ in X, and the spaces X and X1 always have both properties .B/ and .C/. Properties .A/ and .AB/ in symmetric spaces can be reformulated by means of the sequence space N l1 D ff D ff .n/g1 nD1 2 R W kf kl1 D sup f .n/ < 1g n

and its closed subspace c0 D ff D ff .n/g1 nD1 2 l1 W lim f .n/ D 0g: n!1

246

Complements

• Property .A/ is equivalent to the fact that X contains no closed subspace X1 that is isomorphic to l1 in the sense of the theory of Banach spaces. • Property .AB/ is equivalent to the fact that X contains no closed subspace X1 that is isomorphic to c0 in the sense of the theory of Banach spaces.

6 The Least Concave Majorant Let V W RC ! RC be an arbitrary nonnegative function. To construct its least concave majorant e V, we consider the subgraph 0 .V/ D f.x; y/ 2 R2 W x  0; 0  y  V.x/g:

A

We denote by 0 .V/ the closed convex hull of the set 0 .V/. Let e V W RC ! V/ D 0 .V/. Œ0; C1 be a function on RC such that 0 .e Only the two following cases are possible: either e V.x/  C1 or e V.x/ < 1 for C e all x 2 R . In the second case, V is just the least concave majorant for V. We have

A

e V.x/ D inf U.x/;

x 2 RC ;

where inf is taken over all concave majorants U of the function V. On the other hand, let us consider a finite system of mass .m1 ; m2 ; : : : ; mn /, concentrated at the points .x1 ; x2 ; : : : ; xn /, such that m1 > 0; i D 1; 2; : : : ; n, and m1 C m2 C    C mn D 1. Denote by x D m1 x1 C m2 x2 C    C mn xn the center of mass of the system and set V.x/ D sup.m1 V.x1 / C m2 V.x2 / C    C mn V.xn //; where sup is taken over all systems of the above type. For n D 1, we get V.x/  V.x/; x 2 RC : Since e V is a concave function, we have e V.x/ 

n X iD1

and hence e V.x/  V.x/ for all x 2 RC .

m1e V.x1 /;

Complements

247

On the other hand, it is easy to verify that the function V.x/ itself is concave. Therefore, e V  V  V implies that e V D V. For the quasiconcave function V, its least concave majorant e V exists and is equivalent to it, namely 12 e V V e V (see Theorem 10.3.1). For an arbitrary positive function V we can find sufficient conditions under which V and e V are equivalent. The conditions can be formulated by means of dilation indices of the function V. Let p.V/ D lim

ln x

x!C1

ln b V.x/

D sup x>1

ln x ln b V.x/

and q.V/ D lim

x!C0

ln x ln x D inf : 0 0, then e V D L1 and MV D L1 , i.e., the space MV itself is minimal and separable.

8 Lorentz Spaces Lp;q and Orlicz–Lorentz Spaces Let

Lp;q

8 9 01 1 1q ˆ > Z < = q  q p @ A D f 2 L0 W kf kLp;q D Œf .x/ d.x /

: ; 0

Complements

249

for 1  p < 1, 1  q < 1, and 

1 Lp;1 D f 2 L0 W kf kLp;1 D sup x p f  .x/ < 1 ; x>0

for 1  p < 1 (q D 1). In the case p D q D 1, we set L1;1 D L1 . Clearly, Lp;p D Lp for all p, and 1

'Lp;q D k1Œ0; x kLp;q D x p D 'Lp .x/; x  0: Therefore, just the index p is regarded as the main index, and the index q q determines the appropriate weight function W.x/ D x p . • The family Lp;q increases with q, that is, Lp;q1  Lp;q2 and kf kLp;q1  kf kLp;q2 for 1  q1  q2  1. Thus for each fixed p, the space Lp;1 is the least space in the family, and it is the 1

Lorentz space W with the weight function W D x p . q If 1  q  p < 1, then the weight function W.x/ D x p is concave, and it can be verified that • .Lp;q ; k  kLp;q / is a symmetric space for all 1  q  p < 1. q

In the case 1  p < q < 1, the weight function W.x/ D x p is convex, and the triangle inequality fails for kkLp;q . However, kkLp;q is a quasinorm and .Lp;q ; kkLp;q / is a quasi-Banach space. Moreover, if 1 < p < q  1, then there exists a symmetric norm k  k Lp;q that is equivalent to the quasinorm k  kLp;q .  • Let kf k kLp;q , f 2 Lp;q , where f  .x/ D Lp;q D kf

1 x

Rx 0

f  dm; x > 0. Then

.Lp;q ; k  k Lp;q / is a symmetric space, and the following Hardy’s inequality holds: kf kLp;q  kf k Lp;q 

p kf kLp;q p1

(see Exercise 32). Note that for q D 1 and p > 1, the symmetric space .Lp;1 ; k  k Lp;1 / is just the 1

1

Marcinkiewicz space MV with V.x/ D x1 p , x  0, where V .x/ D x p D 'MV .x/, x  0. The spaces Lp;q , 1  p; q  1, admit the following very wide generalization. Let ˚ and W be two nonnegative functions on RC . Let L˚;W D ff 2 L0 W kf kL˚;W < 1g;

250

Complements

where kf kL˚;W

9 8 Z1    = < jf j dW  1 : D sup c > 0 W ˚ ; : c 0

• If ˚ is an Orlicz function, W is concave on Œ0; 1/, and W.0/ D 0, then .L˚;W ; kkL˚;W / is a symmetric space with fundamental function 'L˚;W .x/ D .˚ 1 ..W.x//1 //1 : If W.x/ D x, then L˚;W D L˚ is an Orlicz space. If ˚.x/ D x; x  0, then L˚;W D W is a Lorentz space. q The space Lp;q is obtained with ˚.x/ D xq and W.x/ D x p . In fact, k  kL˚;W is a norm or a quasinorm that admits an equivalent symmetric norm for a very wide class of pairs .˚; W/. These spaces are called Orlicz–Lorentz spaces.

References

1. Adasch, N., Ernst, B., Keim, D.: Topological Vector Spaces. Lecture Notes in Mathematics, vol. 639. Springer, Berlin (1978) 2. Astashkin, S.: Rademacher functions in symmetric spaces. J. Math. Sci. 169(6), 725–886 (2010) 3. Bennett, C., Sharpley, R.: Interpolation of Operators. Pure and Applied Mathematics, vol. 129. Academic Press Professional, Boston (1988) 4. Bergh, J., Löfström, J.: Interpolation spaces: an introduction, Grundlehren der mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976) 5. Birnbaum, Z., Orlicz, W.: Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Stud. Math. 3, 1–67 (1931) 6. Borwein, J., Lewis, A.: Convex analysis and nonlinear optimization: theory and examples. Springer Science & Business Media, New York (2010) 7. Boyd, D.: Indices of function spaces and their relationship to interpolation. Can. J. Math 21(121), 1245–1251 (1969) 8. Bukhvalov, A., Veksler, A., Geiler, V.: Normed lattices. J. Sov. Math. 18(4), 516–551 (1982) 9. Bukhvalov, A., Veksler, A., Lozanovskii, G.: Banach lattices: some aspects of their theory. Russ. Math. Surv. 34(2), 159–212 (1979). [In Russian: Uspehi Mat. Nauk 34 2(206), (1979), 137–183] 10. Burenkov, V.: Function spaces: main integral inequalities connected with the spaces lp. Moscow Publishing House of the University of Friendship of Nations (1989) 11. Calderón, A.: Spaces between l1 and l1 and the theorem of marcinkiewicz. Stud. Math. 26(3), 273–299 (1966) 12. Calderon, A., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88(1), 85–139 (1952) 13. Cerdà, J., Hudzik, H., Kami´nska, A., Mastyło, M.: Geometric properties of symmetric spaces with applications to Orlicz–Lorentz spaces. Positivity 2(4), 311–337 (1998) 14. Chong, K., Rice, N.: Equimeasurable Rearrangements of Functions. Queen’s Papers in Pure and Applied Mathematics, vol. 28. National Library of Canada (1971) 15. Day, M.: The spaces lp with 0 < p < 1. Bull. Am. Math. Soc. 46(10), 816–823 (1940) 16. Day, P.: Rearrangements of measurable functions. Ph.D. Thesis, California Institute of Technology (1970) 17. Dunford, N., Schwartz, J.: Linear operators: general theory. In: Linear Operators. Interscience Publishers, New York (1958) 18. Edgar, G., Sucheston, L.: Stopping Times and Directed Processes. Encyclopedia of Mathematics and its Applications, vol. 47. Cambridge University Press, New York (1992)

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19. Edmunds, D., Evans, W.: Hardy Operators, Function Spaces and Embeddings. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg (2013) 20. Eisenstadt, B., Lorentz, G., et al.: Boolean rings and Banach lattices. Ill. J. Math. 3(4), 524–531 (1959) 21. Fenchel, W.: On conjugate convex functions. Can. J. Math 1, 73–77 (1949) 22. Fremlin, D.: Topological Riesz Spaces and Measure Theory. Cambridge University Press, London/New York (1974) 23. Fremlin, D.: Measure theory III: measure algebras. Reader in Mathematics. University of Essex, Colchester (2002) 24. Greenberg, H., Pierskalla, W.: A review of quasi-convex functions. Oper. Res. 19(7), 1553– 1570 (1971) 25. Halmos, P.: Measure Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York (1974) 26. Halperin, I.: Function spaces. Can. J. Math 5, 273–288 (1953) 27. Hardy, G., Littlewood, J., Pólya, G.: Inequalities. Cambridge university press, Cambridge (1952) 28. Hudzik, H., Kami´nska, A., Mastylo, M.: On the dual of Orlicz–Lorentz space. Proc. Am. Math. Soc. 130(6), 1645–1654 (2002) 29. Johnson, W., Maurey, B., Schechtman, G., Tzafriri, L.: Symmetric Structures in Banach Spaces. American Mathematical Society: Memoirs of the American Mathematical Society, vol. 217 American Mathematical Society (1979) 30. Kami´nska, A.: Some remarks on Orlicz-Lorentz spaces. Math. Nachr. 147(1), 29–38 (1990) 31. Kantorovich, L., Akilov, G.: Functional Analysis in Normed Spaces. International Series of Monographs in Pure and Applied Mathematics, vol. 46, xiii, 773 p. Pergamon Press, Oxford/London/New York/Paris/Frankfurt (1964); Translated from the Russian by D. E. Brown. Edited by A. P. Robertson (English) 32. Kantorovich, L., Vulikh, B., Pinsker, A.: Functional analysis in semi-ordered spaces, 548 pp. Gosudarstv. Izdat. Tekhn.-Teor. Lit., Moskva-Leningrad (1950); In Russian: Functional Analisys. Nauka, Moscow (1977) 33. Krasnosel’skij, M., Rutitskij, Y.: Convex Functions and Orlicz Spaces, Translated from the Russian. P. Noordhoff Ltd., Groningen (1961). [In Russian: Gosizdat phis-math lit., Moscow (1958)] 34. Krein, S.G., Petunin, Y.I., Semenov, E.M.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. American Mathematical Society, Providence (1982). In Russian: Nauka, Moscow (1978) 35. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces: sequence spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, New York (1977) 36. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces: function spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97. Springer, New York (1979) 37. Lorentz, G.G.: Some new functional spaces. Ann. Math. 51(2), 37–55 (1950) (English) 38. Lorentz, G.: On the theory of spaces . Pac. J. Math. 1(3), 411–429 (1951) 39. Lorentz, G.: Relations between function spaces. Proc. Am. Math. Soc. 12(1), 127–132 (1961) 40. Lorentz, G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing, New York (1986) 41. Lorentz, G.G.: Mathematics from Leningrad to Austin. In: Rudolph, A., Lorentz. G.G. (eds.) Lorentz’ Selected Works in Real, Functional, and Numerical Analysis. Contemporary Mathematicians, vol. 2, xxviii, 648 p. Birkhäuser, Boston (English) 42. Luxemburg, W.A.J.: Banach function spaces. Getal en Figuur. 6, 70 p. Van Gorcum & Comp, Assen/Delft (1955) (English) 43. Luxemburg, W., Zaanen, A.: Riesz Spaces, vol. I. North-Holland Publishing Co./American Elsevier Publishing Co./North-Holland Mathematical Library, Amsterdam/London/New York (1971) 44. Maligranda, L.: Orlicz Spaces and Interpolation. Department de Matemática, University of Estadual de Campinas, Campinas (1989)

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45. Marcinkiewicz, J.: Sur l’interpolation d’opérations. C. R. Acad. Sci. Paris 208, 1272–1273 (1939) 46. Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin/Heidelberg (2012) 47. Mityagin, B.: An interpolation theorem for modular spaces. Mat. Sb. 108(4), 473–482 (1965) 48. Montgomery-Smith, S.: Comparison of Orlicz–Lorentz spaces. Stud. Math. 103(2), 161–189 (1992) 49. Morse, M., Transue, W.: Functionals f bilinear over the product a  b of two pseudo-normed vector spaces: II. admissible spaces a. Ann. Math. 51(2) 576–614 (1950) 50. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, New York (1983) 51. Nakano, H.: Modulared Semi-Ordered Linear Spaces. Maruzen Co. Ltd., Tokyo (1950) 52. Natanson, I.P.: Theory of Functions of a Real Variable. Courier Dover Publications, Mineola (2016). In Russian: Nauka, Moscow (1974) 53. Neveu, J.: Mathematical Foundations of the Calculus of Probability. Holden-Day Series in Probability and Statistics. Holden-Day, San Francisco (1965) 54. Orlicz, W.: Uber eine gewisse Klasse von Räumen vom typus b. Bull. Int. Acad. Pol. Sér. A Kraków (8–9) 207–220 (1932) 55. Orlicz, W.: Über Räume (LM ) (German). Bull. Int. Acad. Polon. Sci. A 1936, 93–107 (1936) 56. Pick, L., Kufner, A., John, O., Fucík, S.: Function Spaces, vol. 1. Walter de Gruyter (2012) 57. Pick, L., Kufner, A., John, O., Fucík, S.: Function Spaces, vol. 1. Walter de Gruyter, Berlin/Boston (2012) 58. Rao, M., Ren, Z.: Applications of Orlicz Spaces. Marcel Dekker, New York (2002) 59. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic, New York (1980) 60. Riesz, F., Sz.-Nagy, B.: Lectures in Functional Analysis (Leçons d’analyse Fonctionnelle), vol. 110. Académie des Sciences de Hongrie, Budapest (1952) 61. Rockafellar, R.: Convex Analysis. Princeton Mathematical Series, vol. XVIII, 451 pp. Princeton University Press, Princeton (1970) (English) 62. Rokhlin, V.: On the fundamental ideas of measure theory. American Mathematical Society Translations, vol. 71. American Mathematical Society, Providence (1952). [Math. Sb. 25 (1949), 107–150] 63. Rolewicz, S.: Metric Linear Spaces. D. Reidel, Dordrecht (1985) 64. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973) 65. Schaefer, H.: Banach Lattices and Positive Operators. Springer, New York/Heidelberg (1974); Die Grundlehren der mathematischen Wissenschaften, Band 215 66. Schwarz, H.: Banach Lattices and Operators, Teubner-Texte zur Mathematik, Band 71. B.G. Teubner, Leibzig (1984) 67. Semenov, E.: A scale of spaces with an interpolation property. Dokl. Akad. Nauk SSSR 148, 1038–1041 (1963) 68. Semenov, E.: Imbedding theorems for Banach spaces of measurable functions. Dokl. Akad. Nauk SSSR 156, 1292–1295 (1964) 69. Semenov, E.: Interpolation of linear operators in symmetric spaces. Dokl. Akad. Nauk SSSR 164(4), 746–749 (1965) 70. Sharpley, R.: Spaces  ˛ (x) and interpolation. J. Funct. Anal. 11(4), 479–513 (1972) 71. Stein, E.: Singular Integrals and Differentiability Properties of Functions, vol. 2. Princeton University Press, Princeton (1970) 72. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Mathematical Series: Monographs in Harmonic Analysis, vol. 1. Princeton University Press, Princeton (1971) 73. Titchmarsh, E.: Additional note on conjugate functions. J. Lond. Math. Soc. 1(3), 204–206 (1929) 74. Titchmarsh, E.: On conjugate functions. Proc. Lond. Math. Soc. 2(1), 49–80 (1929)

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Index

Symbols ˚1 ˚2 , 212 ˚1 ˚2 , 207 ˚1 0 ˚2 , 207 ˚1 1 ˚2 , 207 -algebra B, 5 B.RC /, 5 Fm , 5 Fm .0; n/, 25 Fm .RC /, 5 a˚ , 171 b˚ , 171 b , 196

Convergence ˚-mean, 231 almost everywhere, 54 essentially uniform, 22 in measure, 54 order, 108 stochastic, 70 Cutoff function right f   1Œ0;n , 81 upper min.f  ; n/, 81 D dilation indices p.V/, q.V/, 247

.st/

fn ! f , 70 gn # g, 21 gn " g, 21

B Banach lattice, 14 ideal , 14

C Class Young Y˚ , 176, 183 Zygmund Z˛ , 214 Condition .2 .0//, 190 .2 .1//, 190 .2 /, 190

E Embedding, see also Theorem embedding e  e , 162 V1 V2

W X ! X , 103 L1 C L1  .L1 \ L1 /1 , 91 L1 \ L1  .L1 C L1 /1 , 91 L˚1  L˚2 , 209 M.V1 /  M.V2 / , 162 X  X11 , 165 X1  X2 , 71 .L1 / L 1 , 90 W L1 ! L 1 , 85 W L1 ! L  1 , 84 W Lq ! L  p , 90 W MW !  W , 146 W X1 ! X , 92, 104  W .X / ! .X1 / , 104 1 W X11 ! .X1 / , 104 i W X ! X11 , 104

© Springer International Publishing Switzerland 2016 B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of Measurable Functions, Developments in Mathematics 45, DOI 10.1007/978-3-319-42758-4

255

256 F Factor space .X1  X2 /=L, 74 Function .˚ ^ ˚1 /Ï , 220 .˚1 ^ ˚2 /Ï , 218 .˚p ^ ˚q /Ï , 226 1A , 6 F.x; y; m/, 198 U a , 141 V, 127 V , 127 Vmax , 129 Vmin , 129 ˚, 136, 171 ˚ L , 219 ˚ 1 , 172 ˚0 , 221 ˚1 _ ˚2 , 217 ˚1 ^ ˚2 , 217 ˚˛ ; 0  ˛ < 1, 213 ˚1 , 220 ˚p , 225 ˚p _ ˚q , 226 ˚p ^ ˚q , 226 ˚1;1 , 227 ˚1;q , 227 ˚p;1 , 227 ˚p;q , 227  , 195 p;q , 228 min.f ; n/, 50 min.f  ; n/, 50 ˚ ˛ , 214 , 171 , 196 e V , 129, 130, 246 e ˚ , 136 f  , 11 f  , 139 f 1 , 9 f" , 33 W, 115 Lorentz, 115 weight, 115 characteristic, 6 concave, 115 decreasing rearrangement, 11 distribution f , 5 epigraph  0 .g/, 10 finite, 23, 38 fundamental, 60 'W , 119 'L1 , 61

Index 'Lp ; 1  p < 1, 60 'L˚ , 177 'Lp \L1 ; 1  p < 1, 77 'MV , 143 'MV , 143 'R0 , 77 'X1 , 135 'X , 60 'L1 CL1 , 61 'L1 \L1 , 61 generalized inverse, 10 graph  .g/, 10 hypograph 0 .g/, 10 least concave majorant, 130 locally integrable, 49 maximal, 139 Orlicz, 171 conjugate, 196 quasiconcave, 127 quasiconvex, 136 semiadditive, 66 separating, 101 simple integrable, 23, 38 step, 33, 36 Functional I˚ W L0 ! RC , 173 p0 W L0 ! RC , 66 u W X ! R, 24, 83 ug W L1 ! R, 84 ug W L1 ! R, 84 Functions equimeasurable, 7

G Group A.m/, 35 I Ideal lattice symmetric, 109 Inequality Hölder, 18 Minkowski, 18 Young, 17, 196

M Maximal property f  g , 86 f  , 41 Measure

u , 84

Index

257 .A0 /, 79 on .0; 1/, 240 .B/, 97 on .0; 1/, 240 .BC/, 102 on .0; 1/, 240 .C/, 99 .F/, 102 Fatou, 102 maximality on .0; 1/, 240 minimality on .0; 1/, 106, 240 on N, 107, 243 separability on .0; 1/, 106, 240 on N, 107, 243 weakly sequentially complete, 245

m, 5 m ı  1 , 35

W , 116 Metric d0 .f ; g/, 67 translation-invariant, 67

N Norm k.f1 ; f2 /kX1 X2 , 46 kf k0 , 102 kf kW , 116 kf kL01 , 31 kf kL1˚ , 201 kf kL1 CL1 , 46, 182 kf kL1 \L1 , 29, 180 kf kL˚ , 173 kf kL1 , 14 kf kLp , 178 kf kLp ; 1  p < 1, 14 kf kL˚0 , 221 kf kL˚1 CL˚2 , 217 kf kL˚1 \L˚2 , 217 kf kL˚p , 225 kf kL1 , 179 kf kLL˚ CL1 , 222 kf kMV , 143 kf kX1 CX2 , 46 kf kX1 \X2 , 29 kgk1W , 144 kgkMW , 144 kgkX1 , 85 khkX11 , 95 kuk X , 24, 83 ug  , 84 L1 ug  , 84 L1 Lorentz, 115 monotonic, 14 monotonically complete, 97 order complete, 108 order continuous, 79 order semicontinuous, 99

P Property .A/, 79 on .0; 1/, 106 on N, 107, 243 .AB/, 103, 245

S Set B˚ , 173 BL˚ , 186 BY˚ , 186 Va , 104  , 10  .e V /, 131  .g/, 10  0 .g/, 10 0 .V/, 129 0 .Vn /, 129 0 .e V /, 129 0 .g/, 10 RC , 5 F0 , 152 F1 , 152 F.0/ , 24 F0 , 23, 38, 121 F1 , 23, 38 Lp ; 1  p  1, 14 R0 , 50 X0 , 75 Y˚ , 176, 183 clX .L1 \ L1 /, 75 clX .Y/, 72 indicator, 6 Sets upper Lebesgue, 5 Space M0V , 247 X1 .N/, 243 ..L1 C L1 /1 ; kk.L1 CL1 /1 /, 91 ..L1 \ L1 /1 ; kk.L1 \L1 /1 /, 91

258 Space (cont.) .W ; kkW /, 116 .RC ; m/, 13 .L01 ; kkL01 /, 31 .L0 ; d0 /, 67 .L1 C L1 /0 , 76, 97 .L1 C L1 /1 , 91 .L1 C L1 /11 , 96 .L1 C L1 ; kkL1 CL1 /, 46 .L1 \ L1 /0 , 76 .L1 \ L1 /1 , 91 .L1 \ L1 /11 , 96 .L˚ ; kkL˚ /, 173 .Lp C Lq /1 , 228 .Lp ; kkLp /; 1  p  1, 14 .Lp \ Lq /1 , 228 .L˚1 C L˚2 /1 , 220 .L˚1 \ L˚2 /1 , 220 .L˚ C L1 /1 , 224 .L˚ C L1 /1 , 222 .L \ L1 /1 , 222 .L \ L1 /1 , 224 .MV ; kkMV /, 143 .X0 /1 , 91 .X0 ; kkX0 /, 75 .X1 ; kkX1 /, 89 .X11 /0 , 97 .X1 C X2 ; kkX1 CX2 /, 74 .X1 \ X2 ; kkX1 \X2 /, 73 .XU ; kkXU /, 141 F-space, 54, 68, 244 0W , 121, 146 1W , 119 11 W , 119 1W , 144 W .0; 1/, 164 W1 C W2 , 164 W1 \ W2 , 164 e , 147 V H˚ , 187 H˚˛ , 216 H˚ .˝; /, 229 L ln˛ L, 215 L 1 , 84 L 1 , 85 L0˚ , 187 L01 , 76, 96 L0˚˛ , 216 L1˚ , 195 L1˚ , 230 L11 1 , 96 L11 p ; 1  p < 1, 96 L0 .0; 1/, 239

Index L0 , 66 L˚ C L1 , 223 L˚ C L1 , 220 L , 195 L \ L1 , 220 L \ L1 , 223 Lp .0; 1/, 239 Lp .0; 1/, 51 Lp C L1 , 53 Lp C L1 , 53 Lp C Lq , 224, 225 Lp ; 0 < p < 1, 243 Lp ; 0 < p < 1, 54 Lp \ L1 , 53 Lp \ L1 , 53 Lp \ Lq , 224, 225 L0p ; 1  p < 1, 76 L.˚1 ^˚2 /Ï , 218 L1 \ L1 , 29 L˚;W , 250 L˚1 _˚2 , 218 L˚1 C L˚2 , 217 L˚1 \ L˚2 , 217 L˚˛ ; 0  ˛ < 1, 214 L˚p , 225 L˚ .˝; /, 229 L˚ ˛ , 215 Lp;q , 233 Lp .˝; /, 237 M0V , 165 M1W , 119, 146 M1 , 147 e V MV .0; 1/, 164 MV , 143 R0 , 50, 76, 97 X.˝; / , 237 X.0; 1/, 239 X.N/, 241 X , 83 X1 , 85 X11 , 95 X1 C X2 , 46 X1  X2 , 46 X1 \ X2 , 29 Z1˛ , 215 Z0 , 214 Z˛ ; 0  ˛ < 1, 214 Z0˛ , 216 c0 , 241 l0 , 241 lp , 52, 242 .L1 /, 85 .L˚ /, 205

Index .L /, 205 .L1 /, 84 .Lq /, 90 .MW /, 146 .X1 /, 89 clW .F0 /, 146 clW .L1 \ L1 /, 146 clL1 CL1 .F0 /, 50 clL1 \L1 .F0 /, 39 clL1 .F0 /, 39 clL1 .F0 /, 39 clX .L1 \ L1 /, 75 clX .Y/, 39, 72 L1˚ , 201 Lp;q , 248 dual, 24, 83 L p ; 1  p  1, 25 X , 24 Lorentz on .0; 1/, 164 Lorentz W , 116 Lorentz Lp;q , 248 Lorentz–Orlicz L˚;W , 250 Marcinkiewicz on .0; 1/, 164 Marcinkiewicz MV , 143 Orlicz general measure space, 229 Orlicz L˚ , 173 quasi-Banach symmetric, 54 rearrangement invariant, 14

259 symmetric, 14 general measure space, 237 on Œ0; 1, 239 symmetric sequence, 241 weakly sequentially complete, 245 Subspace norming, 99 Symmetric space .X; kkX /, 13 associate, 90 general measure space, 237 maximal, 96 minimal, 76 minimal part, 76 on Œ0; 1, 239 reflexive, 103 second associate, 95 separable, 77

T Theorem embedding 0  X  MV , 153 e V L1 \ L1  X  L1 C L1 , 61 0 X  X  X11 , 96 on Œ0; 1, 106, 239 on N, 106, 242 Transform Legendre, 195 Transformation measure-preserving, 35 invertible, 35