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

Developments in Mathematics

Ben-Zion A. RubshteinGenady Ya. GrabarnikMustafa A. MuratovYulia S. Pashkova

Foundations of Symmetric Spaces of Measurable FunctionsLorentz, Marcinkiewicz and Orlicz Spaces

Developments in Mathematics

VOLUME 45

Series Editors:Krishnaswami Alladi, University of Florida, Gainesville, FL, USAHershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel

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

http://www.springer.com/series/5834

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

Foundations of SymmetricSpaces of MeasurableFunctionsLorentz, Marcinkiewicz and Orlicz Spaces

123

Ben-Zion A. RubshteinMathematicsBen Gurion University of the NegevBe’er Sheva, Israel

Mustafa A. MuratovMathematics and Computer SciencesV.I. Vernadsky Crimean Federal UniversitySimferopol, Russian Federation

Genady Ya. GrabarnikMathematics and Computer ScienceSt. John’s UniversityNew York, NY, USA

Yulia S. PashkovaMathematics and Computer SciencesV.I. Vernadsky Crimean Federal UniversitySimferopol, Russian Federation

ISSN 1389-2177 ISSN 2197-795X (electronic)Developments in MathematicsISBN 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 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe 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 informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective 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 bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

ToInna, Tanya, AndreyFany, Yaacob, Laura, GoldaAjshe, Elvira, Enver, LenurAnna, Ludmila, Sergey

Foreword

This book is the first part of the textbook Symmetric Spaces of MeasurableFunctions. It contains the main definitions and results of the theory of symmetric(rearrangement invariant) spaces. Special attention is paid to the classical spacesLp, Lorentz, Marcinkiewicz, and Orlicz spaces.

The book is intended for master’s and doctoral students, researchers in mathe-matics and physics departments, and as a general manual for scientists and otherswho 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 ofMeasurable Functions. It contains an introduction to the theory, including a detailedstudy of Lorentz, Marcinkiewicz, and Orlicz spaces.

The theory of symmetric (rearrangement invariant) function spaces goes backto the classical spaces Lp, 1 � p � 1. The theory was intensively developedduring the last century, mainly in the context of general Banach lattices. It presentsmany interesting and deep results having important applications in various areas offunction theory and functional analysis. The theory has a great many applicationsin interpolation of linear operators, ergodic theory, harmonic analysis, and variousareas of mathematical physics.

The authors of this book (at different years and in different countries) havestudied and taught the theory of symmetric spaces. They discovered independentlythe following surprising fact: despite the abundance of monographs, there was nobook suitable for our purposes either in the Russian mathematical literature or in themathematical literature of the rest of the world.

In fact, we wished to have a book with a relatively small volume that met thefollowing criteria:

1. The book should contain basic concepts and results of the general theory ofsymmetric spaces with the main focus on a detailed exposition of classical spacesLp; 1 � p � 1, and Lorentz, Marcinkiewicz, and Orlicz spaces, as well.

2. The book should be accessible to master’s students, doctoral students, andresearchers in mathematics and physics departments who are familiar with thebasics of the measure theory and functional analysis in the framework of standarduniversity courses.

3. The material of the book should correspond to a one-semester special course oflectures (about 4 months or 17–18 weeks).

4. The presentation should not require any additional source except standardreferences on basic concepts and theorems of measure theory and functionalanalysis.

ix

x Preface

In our opinion, this book, offered now to the reader, completely meets the aboverequirements.

We can point out three main sources from which the material of the book wasadopted.

First is a monograph by S. G. Krein, J. I. Petunin, E. M. Semenov, Interpolationof Linear Operators.

The second source is two volumes of J. Lindenstrauss, L. Tzafriri, ClassicalBanach Spaces I. Sequence Spaces and Classical Banach Spaces II. FunctionSpaces.

Third, the part devoted to Orlicz spaces is based on a nice exposition of this themein the book by G. A. Edgar, L. Sucheston, Stopping Times and Directed Processes.

Our book includes four parts comprising seventeen chapters. This allows us todivide 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 themain part of the book. The reader can find this additional material in the exercisesat the end of each part and in the section called “Complements” at the end of thebook. Throughout the main exposition, we deal only with symmetric spaces on thehalf-line R

C D Œ0;1/, while the symmetric spaces on the interval Œ0; 1� and thesymmetric 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 endof 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 bibliographyfor further rending. The list of references is not, of course, comprehensive, but itpoints out, we hope, the most of important directions of the theory.

Be’er Sheva, Israel Ben-Zion A. RubshteinNew York, NY, USA Genady Ya. GrabarnikSimferopol, Russia Mustafa A. MuratovSimferopol, Russia Yulia S. Pashkova

Contents

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

1 Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1 Distribution Functions, Equimeasurable Functions. . . . . . . . . . . . . . . . . 51.2 Generalized Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Decreasing Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Integrals of Equimeasurable Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Example. Lp, 1 � p � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Spaces Lp; 1 � p � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Hölder’s and Minkowski’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Completeness of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Separability of Lp, 1 � p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 The Space L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 The Intersection of the Spaces L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The Space L01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Approximation by Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Measure-Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5 Approximation by Simple Integrable Functions . . . . . . . . . . . . . . . . . . . . 38

4 The Space L1 C L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 The Maximal Property of Decreasing Rearrangements . . . . . . . . . . . . 414.2 The Sum of L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Embeddings L1 � L1 C L1 and L1 � L1 C L1.

The Space R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

xi

xii Contents

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

5 Embeddings L1 \ L1 � X � L1 C L1 � L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 The Embedding Theorem L1 \ L1 � X � L1 C L1 . . . . . . . . . . . . . 615.3 The Space L0 and the Embedding L1 C L1 � L0 . . . . . . . . . . . . . . . . . 66

6 Embeddings. Minimality and Separability. Property .A/ . . . . . . . . . . . . . . 716.1 Embedded Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 The Intersection and the Sum of Two Symmetric Spaces . . . . . . . . . . 736.3 Minimal Symmetric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Minimality and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.5 Separability and Property .A/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.1 Dual and Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2 The Maximal Property of Products f �g� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.3 Examples of Associate Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.4 Comparison of X1 and X� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

8 Maximality. Properties (B) and (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1 The Second Associate Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2 Maximality and Property .B/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.3 Embedding X � X11 and Property .C/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.4 Property .AB/. Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Part III Lorentz and Marcinkiewicz Spaces

9 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.1 Definition of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1159.2 Maximality. Fundamental Functions of Lorentz Spaces . . . . . . . . . . . 1199.3 Minimal and Separable Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209.4 Four Types of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

10 Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12710.1 Fundamental Functions and Quasiconcave Functions . . . . . . . . . . . . . . 12710.2 Examples of Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12810.3 The Least Concave Majorant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13010.4 Quasiconcavity of Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . 13510.5 Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

11 Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13911.1 The Maximal Function f �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13911.2 Definition of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14311.3 Duality of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . 14411.4 Examples of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents xiii

12 Embedding �0

eV� X � MV�

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15112.1 The Embedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15112.2 The Renorming Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15612.3 Examples of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . 15712.4 Comparison of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . 162Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Part IV Orlicz Spaces

13 Definition and Examples of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17113.1 Orlicz Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17113.2 Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17313.3 Fundamental Functions of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 17713.4 Examples of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

14 Separable Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18314.1 Young Classes Y˚ and Subspaces H˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18314.2 Separability Conditions for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 18514.3 The .�2/ Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19014.4 Examples of Orlicz Spaces with and Without

the .�2/ Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

15 Duality for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19515.1 The Legendre Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19515.2 The Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19715.3 Duality for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20015.4 Duality and the .�2/ Condition. Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . 205

16 Comparison of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20716.1 Comparison of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20716.2 The Embedding Theorem for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . 20916.3 The Coincidence Theorem for Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . 21216.4 Zygmund Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

17 Intersections and Sums of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21717.1 The Intersection and the Sum of Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . 21717.2 The Spaces L˚ C L1 and L� \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22017.3 The Spaces L˚ C L1 and L� \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22317.4 The Spaces Lp \ Lq and Lp C Lq, 1 � p � q � 1 . . . . . . . . . . . . . . . . 224Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2371 Symmetric Spaces on General Measure Spaces . . . . . . . . . . . . . . . . . . . . 2372 Symmetric Spaces on Œ0; 1� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393 Symmetric Sequence Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

xiv Contents

4 The Spaces Lp; 0 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2435 Weak Sequential Completeness. Property .AB/ . . . . . . . . . . . . . . . . . . . . 2456 The Least Concave Majorant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2467 The Minimal Part M0

V of the Marcinkiewicz Space MV . . . . . . . . . . . . 2478 Lorentz Spaces Lp;q and Orlicz–Lorentz Spaces . . . . . . . . . . . . . . . . . . . . 248

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

List of Figures

Fig. 1.1 The function f D 1A and its distribution function . . . . . . . . . . . . . . . . . 6

Fig. 1.2 The function f .x/ D 1

x2and its distribution function . . . . . . . . . . . . . . 7

Fig. 1.3 The function f .x/ D x

1 � x� 1.0;1/.x/ and its

distribution the function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Fig. 1.4 The function f .x/ D j sin xj and its distribution function . . . . . . . . . . 7

Fig. 1.5 Equimeasurable functions f .x/ D 1

x2and

g.x/ D 1

.x � 1/2 � 1Œ1;C1/.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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/ . . . . 8

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

1C x,

and h.x/ D 1

2.1C sin x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Fig. 1.8 Hypograph and epigraph of a generalized inverse function . . . . . . . 10Fig. 1.9 Constancy intervals and jumps for f � and �f � . . . . . . . . . . . . . . . . . . . . . 12Fig. 1.10 Integrals of f � and �jf j as area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Fig. 2.1 Inequality ˚

�

a C b

2

�

� ˚.a/C ˚.b/

2. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Fig. 2.2 The functionsxp

pand

yq

qas areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Fig. 3.1 The cases f �.1/ > 0 and f �.1/ D 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Fig. 3.2 The inequality .f C g/� � f � C g� fails . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Fig. 3.3 The case f �.1/ D f �.c/, c < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Fig. 3.4 The case in which f �.1/ is not achieved, c D 1 . . . . . . . . . . . . . . . . . 34Fig. 3.5 The common distribution function of step functions f

and g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

xv

xvi List of Figures

Fig. 3.6 Equimeasurable functions that are and are not of theform f ı � , � 2 A.m/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Fig. 3.7 Sequences an and bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Fig. 4.1 Functions f �, �jf j and va . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Fig. 4.2 A constant-value interval of f � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Fig. 4.3 The cases of jf j � f �.1/ and f �.a0/ D f �.a/ D f �.1/ . . . . . . . . . 44Fig. 4.4 Relation between norms kgkL1

and khkL1 and kf kL1 . . . . . . . . . . . . . 47Fig. 4.5 Value b D f �.1�/ and function .f � � b/C . . . . . . . . . . . . . . . . . . . . . . . . . 48Fig. 4.6 Equality (4.2.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Fig. 4.7 Upper cutoff of f � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Fig. 5.1 Fundamental functions of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Fig. 5.2 Fundamental functions of L1 C L1 and L1 \ L1 . . . . . . . . . . . . . . . . 61Fig. 5.3 The sequence fakg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Fig. 5.4 Functions hnk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Fig. 5.5 Functions v.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Fig. 6.1 'X.b � a/ D k1Œa;b�kX in the case 'X.0C/ > 0 . . . . . . . . . . . . . . . . . . . . 77Fig. 6.2 Inequality (6.4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Fig. 9.1 Two cases of Lorentz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Fig. 9.2 'L1\L1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Fig. 9.3 Upper cutoffs minff �; ng and differences gn D f � � min.f �; n/ . . 121Fig. 9.4 Right cutoffs f � � 1Œ0;n/ and hn D f � � f � � 1Œ0;n/ . . . . . . . . . . . . . . . . . . . 122

Fig. 10.1 V.x/ is quasiconcave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Fig. 10.2 Convex quasiconcave V.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Fig. 10.3 V.x/ D p

x2 C 1 � 1.0;1/ is both quasiconcave and convex . . . . . . . 129Fig. 10.4 �0.eV/ is the closed convex hull of �0.V/ . . . . . . . . . . . . . . . . . . . . . . . . . . 129Fig. 10.5 The least concave majorant eV of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Fig. 10.6 V.x/ and eV.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Fig. 10.7 A finite “interval of nonconcavity” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Fig. 10.8 A maximal finite “interval of nonconcavity”. . . . . . . . . . . . . . . . . . . . . . . 132Fig. 10.9 The case aV < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Fig. 10.10 The case aV < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Fig. 10.11 V.x/ D max.1; x/ � 1.0;1/.x/ and eV.x/ D .x C 1/ � 1.0;1/.x/ . . . . . 135Fig. 10.12 Quasiconvex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Fig. 11.1 V.x/ for MV D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Fig. 11.2 V.x/ for MV D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Fig. 11.3 V.x/ for MV D L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Fig. 11.4 V.x/ and V�.x/ for MV D L1 C L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Fig. 12.1 f � DmP

iD1ci � 1Œ0;bi� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Fig. 4.8 Norm kf � fnkL1CL1as area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

List of Figures xvii

Fig. 12.2 V.x/ D V�.x/ D px; x � 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

Fig. 13.1 Orlicz functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Fig. 13.2 Inverse Orlicz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Fig. 13.3 ˚ and ˚�1 in (13.3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Fig. 13.4 L˚ D Lp if ˚.x/ D xp, 1 � p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Fig. 13.5 Orlicz function ˚ for which L˚ D L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Fig. 13.6 Orlicz function for which L˚ D L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . 179Fig. 13.7 Orlicz function ˚ for which L˚ D L1 C L1 . . . . . . . . . . . . . . . . . . . . . . 180

Fig. 13.81R

a�f �dm as area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Fig. 13.9 'L˚ � 'L1CL1� 2'L˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Fig. 15.1 Derivatives of conjugate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196Fig. 15.2 Conjugate functions ˚.x/ and �.y/ as areas. . . . . . . . . . . . . . . . . . . . . . . 197Fig. 15.3 Legendre’s parameter m D ˚ 0.x/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Fig. 15.4 The Legendre transform in the case of derivative jump . . . . . . . . . . . 199Fig. 15.5 Jump of ˚ 0 and linearity interval of � . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Fig. 15.6 Linearity interval of ˚ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Fig. 15.7 Constancy interval of derivative ˚ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200Fig. 15.8 Left and right tangent lines at y D m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Fig. 15.9 ˚ 0 and � 0 when a˚ > 0 and b˚ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Fig. 16.1 Functions ˚c.x/ D ecx � 1; c � 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Fig. 16.2 Functions ˚c1 .x/ and ˚c2 .x/ for c1 < c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 211Fig. 16.3 Derivatives of ˚˛.x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Fig. 16.4 Functions ˚˛.x/; 0 � ˛ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214Fig. 16.5 ˚0 and its greatest convex minorant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Fig. 17.1 Functions ˚1 _ ˚2; ˚1 ^ ˚2; .˚1 ^ ˚2/Ï . . . . . . . . . . . . . . . . . . . . . . . . 218Fig. 17.2 Functions ˚; ˚1, ˚ ^ ˚1 and .˚ ^ ˚1/Ï . . . . . . . . . . . . . . . . . . . . . 221Fig. 17.3 Functions ˚;˚0 and their inverse functions . . . . . . . . . . . . . . . . . . . . . . . 221Fig. 17.4 Functions � and � _ ˚1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222Fig. 17.5 Functions ˚; ˚1; ˚ ^ ˚1 and the greatest convex minorant . . . . . 223Fig. 17.6 Function .˚ ^ ˚1/Ï . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Fig. 17.7 Functions �;˚1, and � _ ˚1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224Fig. 17.8 Functions ˚p.x/ and ˚p0.y/ as areas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Fig. 17.9 Functions ˚p _ ˚q; ˚p ^ ˚q and .˚p ^ ˚q/

Ï . . . . . . . . . . . . . . . . . . . . . 226Fig. 17.10 Functions ˚p;q; ˚p;1; ˚1;q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Part ISymmetric Spaces. The Spaces Lp,

L1 \ L1, L1 C L1

In this part we begin to study symmetric spaces. We give basic definitionsand 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 Banachmeans being complete with respect to the norm k � kX. Functions that are equalalmost 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 equimeasurablefunctions.

In more detail, the “upper” distribution function �f of a nonnegative measurablefunction 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 thesame 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 “decreasingrearrangement” f � of the function jf j. The function f � is a (unique) decreasing right-continuous 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 decreas-ing rearrangements f � as follows. A nonzero Banach space .X; k�kX/ of measurablefunctions on R

C 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 thetheory of symmetric spaces, were the Banach spaces Lp, 1 � p � 1. These spacesare equipped with the norm

kf kLp D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

0

@

1Z

0

jf jpdm

1

A

1=p

; 1 � p < 1

vrai supx

jf .x/j; p D 1:

The symmetric property kf kLp D kf �kLp , f 2 Lp, is a direct consequence of thedefinition of the Lebesgue integral.

Using the Lp-spaces, one can construct a larger family of symmetric spaces

Lp \ Lq; Lp C Lq; p; q 2 Œ1;1�:

These spaces will be considered in Part IV as a special case of Orlicz spaces. Buttwo important members of this family, namely L1 \ L1 and L1 C L1, are studiedhere in Part I.

The spaces L1 \ L1 and L1 C L1 play a fundamental role in the theory ofsymmetric spaces. As will be established later, they are just the smallest and thelargest symmetric spaces, not only within the above family, but also among allsymmetric spaces.

The space L1 \ L1 is equipped with the norm

kf kL1\L1D maxfkf kL1 ; kf kL1

g

and consists of all integrable bounded functions on RC. It has the greatest norm

among the norms of all symmetric spaces X, provided that k1Œ0;1�kX D 1. The spaceL1 \ L1 coincides with the closure of simple integrable functions on R

C withbounded support.

The space L1 C L1 consists of all functions of the form

f D g C h; g 2 L1; h 2 L1

with the norm

kf kL1CL1D inffkgkL1 C khkL1

; f D g C h; g 2 L1; h 2 L1g:

This norm can be represented also as

kf kL1CL1D

1Z

0

f �dm D sup

8

<

:

Z

A

jf jdm;A � RC;mA D 1

9

=

;

:

I Symmetric Spaces. The Spaces Lp, L1 \ L1, L1 C L1 3

The latter equality shows that the space L1 C L1 consists precisely of all locallyintegrable functions on R

C. This fact follows from the following maximal propertyof decreasing rearrangements f �:

aZ

0

f �dm D sup

8

<

:

Z

A

jf jdm;A � RC;mA D a

9

=

;

; a > 0:

Chapter 1Definition of Symmetric Spaces

In this chapter we begin the study of symmetric spaces. The definition of thesymmetric spaces is based on two important notions: the distribution function �jf jand the decreasing rearrangement f � of a function f on R

C. Most of the chapter isdevoted to studying properties of �jf j and f �.

1.1 Distribution Functions, Equimeasurable Functions

Let m be the usual Lebesgue measure on the set of nonnegative real numbers RC DŒ0; 1/ [52]. This means that the restriction mjB of m to the Borel � -algebra B DB.RC/ is the unique Borel measure on R

C satisfying

m.Œ0; x�/ D x; x 2 RC:

The measure m is the natural extension of mjB defined on the � -algebra Fm DFm.R

C/ of all Lebesgue measurable subsets of RC.For every nonnegative measurable function f W RC ! R

C, we define the function

�f .y/ WD mff > yg; y 2 RC; (1.1.1)

where

ff > yg WD fx 2 RC W f .x/ > yg; y 2 R

C

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 calledthe (upper) distribution function of f .

The following statement describes elementary properties of the distributionfunction.

© Springer International Publishing Switzerland 2016B.-Z.A. Rubshtein et al., Foundations of Symmetric Spaces of MeasurableFunctions, 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 > y1g ff > y2g H) �f .y1/ � �f .y2/:

The right-continuity of �f follows from the fact that

yn # y H) ff > yng " ff > yg H) �f .yn/ " �f .y/:

utThe 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 isleft-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 setA 2 Fm:

1A.x/ D�

1; x 2 AI0; x 62 A:

Then �f D mA � 1Œ0;1/, i.e.,

�f .x/ D�

mA; 0 � x < 1I0; 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.

2. Let f .x/ D 1

x2. Then �f .x/ D 1p

x(Fig. 1.2).

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

1.1 Distribution Functions, Equimeasurable Functions 7

Fig. 1.2 The function f .x/ D 1

x2and its distribution function

3. Let f .x/ D x

1 � x� 1.0;1/.x/. Then �f .x/ D 1

x C 1(Fig. 1.3).

Fig. 1.3 The function f .x/ D x

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

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

�f .x/ D� 1; 0 � x < 1I0; 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/:


Fig. 1.5 Equimeasurable functions f .x/ D 1

x2and g.x/ D 1

.x � 1/2� 1Œ1;C1/.x/

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

1C x, and h.x/ D 1

2.1C sin x/

1.2 Generalized Inverse Functions 9

Examples 1.1.5. 1. Let A; B 2 Fm. Then the indicators 1A and 1B are equimeasur-able if and only if mA D mB.

2. Let f .x/ D 1

x2and g.x/ D 1

.x � 1/2 �1Œ1;C1/.x/. Then �f .x/ D �g.x/ D 1px

, i.e.,

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 �h D g (Fig. 1.6).

4. The functions f .x/ D x, g.x/ D x2 and h.x/ D px 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 x

1C x, and h.x/ D 1

2.1 C sin x/ are

equimeasurable (Fig. 1.7).

�f .x/ D �g.x/ D �h.x/ D� C1; 0 � x < 1I0; x � 1:

The last two examples illustrate negative effects of sets of infinite measure ontesting for equimeasurability.

If, as in Example 1.1.5.2, the function f is continuous and strictly decreasingfrom C1 to 0, then its distribution function coincides with the inverse function f �1of 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 ��1f in general need not exist in the usual sense.

Nevertheless, we can use the so-called generalized inverse function f � of thedecreasing function �f .

1.2 Generalized Inverse Functions

Let y D g.x/ be a decreasing nonnegative function on Œ0;1/ that possibly takes thevalue C1.

Consider the hypograph of g,

�0.g/ D ˚

.x; y/ 2 R2 W x � 0; g.x�/ � y � 0

�

;


Fig. 1.8 Hypograph andepigraph of a generalizedinverse function

0

Γ

Γ

x

y

0

0x=h(y)y=g(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 jumppoints.

Nevertheless, there is a unique right-continuous generalized inverse functionh D g�1. Moreover, g D h�1 if g and h are right-continuous.

2From now on, we use the notation g.x�/ and g.xC/ for the left- and right-hand limits of thefunction 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 inversefunction ��1

jf j will be denoted by f �, and it has the form

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 > y0g < C1

is finite. This means that

limy!1 �jf j.y/ D 0

and hence f �.x/ < 1 for all x > 0.Thus, by construction of f � W Œ0;C1/ ! Œ0;C1/, it is a decreasing right-

continuous function such that �jf j D �f � . In other words, the functions f � and jf j areequimeasurable.

Definition 1.3.1. The function f � is called a decreasing rearrangement of thefunction 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 (1.3.2)

for all x and y (Fig. 1.9).

Fig. 1.9 Constancy intervalsand jumps for f � and �f �

x

=

x x xx x

y y

0 0

yyy

yy y<

f h(x)* (y)

<f *

=x

x x

y

Specifically,

f �.�f �.y// D y for f �.x/ D y � y � y D f �.x�/and

�f �.f �.x// D x for �f �.y/ D x � x � x D �f �.y�/:It is clear that f �.�f �.y// D y and �f �.f �.x// D x at the points of continuity of f �and �f � .

1.4 Integrals of Equimeasurable Functions

Let f1 and f2 be two nonnegative equimeasurable functions. Since �f1 D �f2 is right-continuous, we have

mfa < f1 � bg D �f1 .b/ � �f1 .a/ D �f2 .b/ � �f2 .a/ D mfa < f2 � bg;for all 0 � a < b � 1.

The Lebesgue integral

1Z

0

fdm of the nonnegative measurable function f is

uniquely determined by the measures mfa < f � bg of all Lebesgue sets fa <

f � bg, 0 � a < b � 1. Hence

1Z

0

f1dm D1Z

0

f2dm

for every pair of nonnegative equimeasurable functions f1 and f2.In particular, for every measurable function f defined on Œ0;1/, the Lebesgue

integrals

1Z

0

jf jdm and

1Z

0

f �dm of nonnegative equimeasurable functions jf j and f �

are equal.

1.5 Definition of Symmetric Spaces 13

Fig. 1.10 Integrals of f � and�jf j as area

=

=

y

x

f

f

(x)

( y)

*

0

y

h

x

S

The Lebesgue integral

1Z

0

f �dm can be considered the (improper) Riemann

integral of the decreasing function f �. The integral coincides with the Riemann

integral

1Z

0

�jf jdm of the inverse �jf j of the function f �, and it is equal to the area

S of the hypograph f.x; y/ 2 R2 W x � 0; �jf j.x/ � y � 0g. Thus (Fig. 1.10),

S D1Z

0

jf jdm D1Z

0

f �dm D1Z

0

�jf jdm: (1.4.1)

1.5 Definition of Symmetric Spaces

Definition 1.5.1. A nonzero Banach space .X; k � kX/ of measurable functions on.RC;m/ is called symmetric if the following conditions hold:

1. If jf j � jgj and g 2 X, then f 2 X and kf kX � kgkX.2. If f and g are equimeasurable and g 2 X, then f 2 X and kf kX D kgkX.

Recall that all equalities and inequalities are understood up to sets of measurezero. Thus f D g (f � g) in X means that f .x/ D g.x/ (f .x/ � g.x/) for almostevery x 2 Œ0;1/ with respect to the measure m. We shall use the same notationfor individual functions and for the class of functions coinciding with f almosteverywhere.

A normed space .X; k�kX/ satisfying condition 1 is called a normed ideal lattice.The norm k � kX is called monotonic in this case.

Condition 2 is the symmetry or rearrangement invariance condition. It meansthat

f 2 X ” f � 2 X

and

kf kX D kf �kX; f 2 X;

where f � is the decreasing rearrangement of jf j.Thus, a symmetric space is a Banach ideal lattice with a symmetric (rearrange-

ment invariant) norm.Conditions 1 and 2 can be combined into one as follows:

f � � g� and g 2 X; H) f 2 X and kf kX � kgkX:

1.6 Example. Lp, 1 � p � 1

The main and most important examples of symmetric spaces are the spaces Lp,1 � p � 1.

For every measurable function f on RC D Œ0;1/, we set

kf kLp D

8

ˆ

ˆ

<

ˆ

ˆ

:

0

@

1Z

0

jf jpdm

1

A

1=p

; 1 � p < 1

vrai sup jf j; p D 1;

(1.6.1)

where by definition,

vrai sup jf j D inffc > 0 W mfjf j > cg D 0g:

The space Lp D Lp.RC;m/ consists of all measurable functions f such that

kf kLp < 1.Of course, all the functions that are equal almost everywhere on R

C are identifiedin Lp. So

f D 0 in Lp ” kf kLp D 0:

Theorem 1.6.1. For every 1 � p � 1, .Lp; k � kLp/ is a symmetric space.

It is well known that .Lp; k � kLp/ is a Banach space.Nevertheless we shall give a detailed proof of the fact in Chapter 2.

1.6 Example. Lp, 1 � p � 1 15

Now we note only that the first condition in the definition of symmetric spacesfollows directly from (1.6.1), while the symmetry condition is a consequence of theequality

kf kLp D

8

ˆ

ˆ

<

ˆ

ˆ

:

0

@

1Z

0

.f �/pdm

1

A

1=p

; 1 � p < 1

f �.0/; p D 1:

(1.6.2)

The equality of the integrals

1Z

0

jf jpdm and

1Z

0

.f �/pdm for equimeasurable

functions jf jp and .f �/p D .jf jp/� follows from (1.4.1). The equality kf kL1D f �.0/

follows from (1.3.1) and (1.6.1), namely

f �.0/ D inffy > 0 W �jf j.y/ � 0g D inffy > 0 W mfjf j > yg D 0g D kf kL1:

Chapter 2Spaces Lp; 1 � p � 1

In this chapter we study the class of Lp spaces, 1 � p � 1, which is one of the mostimportant classes of symmetric spaces. We begin with the Hölder and Minkowskiinequalities and prove that Lp is a symmetric space for all 1 � p � 1. In the case1 � 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 goalis 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 � 2p�1.ap C bp/: (2.1.1)

2. (Young’s inequality) Let p; q > 1, a > 0, and b > 0. If1

pC 1

qD 1, then

ab � ap

pC bq

q: (2.1.2)

Equality is achieved if and only if a D bq�1 or b D ap�1.

Proof. 1. Since the function ˚.x/ D xp is convex on .0;1/ for p � 1, it followsthat

�

a C b

2

�p

� ap C bp

2

for all a > 0, b > 0 (Fig. 2.1).


17

18 2 Spaces Lp; 1 � p � 1

Fig. 2.1 Inequality

˚

�

a C b

2

�

� ˚.a/C ˚.b/

2

0

FF

F

F

F F

2

a

( )a

( )a

( )x

( )b

( )b

a+b

ba+b

+

2

2

x

= x p

y

0 0

2 1S 2SS

2S1S

x x

y y

+ >

a

p-1y

a

ab 2S1S

1S

+ =

=x

xq-1y=

p-1y =x

xq-1y=

abb b

Fig. 2.2 The functionsxp

pand

yq

qas areas

This implies

.a C b/p � 2p�1.ap C bq/;

for all a > 0 and b > 0.2. Since .p � 1/.q � 1/ D 1, the functions y D xp�1 and x D xq�1 are mutually

inverse. Consider their graphs (Fig. 2.2).We have

S1 DxZ

0

up�1du D xp

p; S2 D

yZ

0

vq�1du D yq

q;

and

xy � S1 C S2 D xp

pC yq

q:

ut

2.1 Hölder’s and Minkowski’s Inequalities 19

Theorem 2.1.2. Let 1 � p; q � 1 and1

pC 1

qD 1.

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

kfgkL1 D1Z

0

jfgjdm D1Z

0

jf j jgjdm � kgkL1

1Z

0

jf jdm D kf kL1 � kgkL1:

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

a D jf .x/jkf kLp

; b D jg.x/jkgkLq

;

we obtain

jf .x/jkf kLp

� jg.x/jkgkLq

� 1

p

jf .x/jkf kLp

!p

C 1

q

jg.x/jkgkLq

!q

; x 2 Œ0;1/;

and hence

1Z

0

jf .x/g.x/j dx

kf kLp � kgkLq

� 1

p

1Z

0

jf .x/jkf kLp

!p

dx C 1

q

1Z

0

jg.x/jkgkLq

!q

dx D 1

pC 1

qD 1;

i.e.,

kfgkL1 D1Z

0

jf .x/g.x/j dx � kf kLp � kgkLq :

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

1Z

0

jf C gjpdm �1Z

0

jf j jf C gjp�1dm C1Z

0

jgj jf C gjp�1dm: (2.1.5)

20 2 Spaces Lp; 1 � p � 1

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 � 2p�1.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 gjp�1/q D jf C gj.p�1/q D jf C gjp 2 L1;

i.e., jf C gjp�1 2 Lq. Now we apply Hölder’s inequality (2.1.3) to the functionsjf j 2 Lp and jf C gjp�1 2 Lq:

1Z

0

jf j jf C gjp�1dm � kf kLp � k jf C gjp�1 kLq D kf kLp

0

@

1Z

0

jf C gj.p�1/qdm

1

A

1q

D kf kLp

0

@

1Z

0

jf C gjpdm

1

A

1q

D kf kLp � �kf C gkLp

�pq :

In a similar way, for jgj 2 Lp and jf C gjp�1 2 Lq, we have

1Z

0

jgj jf C gjp�1dm � kgkLp � �kf C gkLp

�pq :

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

kf C gkpLp

D1Z

0

jf C gjpdm � .kf kLp C kgkLp/ � �kf C gkLp

�pq :

Since p � p

qD 1, we obtain also (2.1.4). ut

Minkowski’s inequality is just the triangle inequality in the space Lp. Theequality

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 ffng be a fundamental sequence in Lp, i.e.,

limn;m!1 kfn � fmkLp D 0:

By passing, if necessary, to a subsequence, we may assume that

f0 D 0; kfk � fk�1kLp <1

2k; k D 1; 2; : : : : (2.2.1)

We show that the sequence ffng converges almost everywhere to a function f 2 Lp

and

limn!1 kfn � f kLp D 0:

To prove this, consider the series1P

kD1.fk � fk�1/, which has partial sums fn D

nP

kD1.fk � fk�1/, and set

gn DnX

kD1jfk � fk�1j and g D

1X

kD1jfk � fk�1j:

Then gn " g.1 The integrals

1Z

0

gpndm D kgnkp

Lp

are uniformly bounded, since by Minkowski’s inequality (2.1.4),

kgnkLp�

nX

kD1kfk � fk�1kLp �

nX

kD1

1

2k� 1:

Since gpn " gp, Levi’s theorem implies that the function gp is integrable (and hence

finite almost everywhere) and

1 � limn!1

1Z

0

gpndm D

1Z

0

gpdm D kgkpLp;

i.e., g 2 Lp.

1From now on, we shall use the notation gn " g if fgng is increasing and g D supn

gn almost

everywhere.In a similar way, gn # g means that fgng is decreasing and g D inf

ngn almost everywhere.

22 2 Spaces Lp; 1 � p � 1

Since

1X

kD1jfk.x/ � fk�1.x/j D g.x/ < 1

for almost all x 2 Œ0;1/, the original series1P

kD1.fk � fk�1/ converges almost

everywhere to a function f . Then fn DnP

kD1.fk � fk�1/ also tends to f almost

everywhere. Finally,

kf kpLp

D1Z

0

jf jpdm �1Z

0

gpdm D kgkpLp< 1;

i.e., f 2 Lp, and by (2.2.1), we have

kf � fnkLp �1X

kDnC1kfk � fk�1kLp �

1X

kDnC1

1

2kD 1

2n! 0; n ! 1:

Hence, Lp, 1 � p < 1, is a Banach space.For p D 1, convergence in L1 is essentially uniform convergence. This means

that

kfn � f kL1! 0; n ! 1;

if and only if there exists a set N of measure 0 such that fn.x/ ! f .x/ uniformly onR

C n N.This implies in turn the completeness of the space L1. Indeed, if ffng is a

fundamental sequence in L1, then it is a uniformly fundamental sequence on RCnN

for a suitable set N of measure 0,

limm;n!1 sup

x2RCnN

jfm.x/ � fn.x/j D 0:

Hence fn.x/ ! f .x/ on RC n N for a function f 2 L1 and k fn � f kL1

! 0;

n ! 1.Thus, for all p 2 Œ1;1�, .Lp; k � kLp/ is a Banach space.

2.3 Separability of Lp, 1 � p < 1 23

2.3 Separability of Lp, 1 � p < 1

Denote by F1 the set of all simple2 integrable functions, and by F0 the set of allsimple functions with bounded support. Functions from F1 have the form

f DnX

iD0ai � 1Ai ; ai 2 R; mAi < 1;

and if f 2 F0, the sets Ai are bounded, i.e., there exists a > 0 such that Ai � Œ0; a�for all i.

Theorem 2.3.1. Let 1 � p < 1. Then

1. F0 is dense in Lp in norm k � kLp .2. Lp is separable.

Proof. 1. Let 0 � f 2 Lp. Consider step functions f .n/ that approximate thefunction f from below. Let

f .n/.x/ D i � 12n

; ifi � 12n

� f .x/ <i

2n; i 2 N;

and

fn D min.n; f .n// � 1Œ0;n�:

Then fn 2 F0 and

kf � fnkpLp

D1Z

0

jf � fnjpdm �Z

ff �ngf pdm C

1Z

n

f pdm C n

2n: (2.3.1)

Since

1Z

0

f pdm < 1, the right side of (2.3.1) tends to 0 as n ! 1, and hence

kf � fnkLp ! 0; n ! 1:

For arbitrary functions f 2 Lp, we can use the decomposition

f D f C � f �; f C D max.f ; 0/; f � D � min.f ; 0/;

reducing to the case f � 0.

2A function is simple if it is a measurable step function with finitely many values.

24 2 Spaces Lp; 1 � p � 1

2. Let A 2 Fm, mA < 1, and " > 0. Then there exists an open set G such thatm.A M G/ < ". The set G has the form

G D1[

iD1.ai; bi/; 0 � ai < bi; i � 1:

Since mG < 1, we can find n large enough that m.G M Gn/ < "; whereGn D Sn

iD1.ai; bi/. We additionally may assume that ai and bi are rationalnumbers.Consider the subset F.0/ � F0 consisting of all functions g of the form

g DnX

iD0ci � 1Œai;bi�

with rational ci, ai, and bi.Since for every pair A; B 2 Fm, we have

m.A M B/ < " H) k1A � 1BkLp � "1=p;

the countable set F.0/ is dense in F0, and hence in Lp by part 1 of the proposition.ut

2.4 Duality

Let X be a symmetric space and X� the dual Banach space. The space X� consistsof 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 and1

pC 1

qD 1. For g 2 Lq we define a linear functional ug on

Lp by setting

ug.f / D1Z

0

fgdm; f 2 Lp:

Theorem 2.4.1. Let 1 � p; q � 1 and1

pC 1

qD 1. Then ug 2 L�

p for all g 2 Lq,

and the mapping

W Lq 3 g ! ug 2 L�p

is an isometric isomorphism of Lq into the dual space L�p of the space Lp. If

1 � p < 1, then .Lq/ D L�p , and for p D 1, the embedding .L1/ � L�1 is

strict.

2.4 Duality 25

Proof. Let 1 < p; q < 1,1

pC 1

qD 1. Hölder’s inequality (2.1.3) implies that the

functional ug is continuous and

jug.f /j Dˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� kf kLpkgkLq ;

i.e.,

kugkL�

p� kgkLq : (2.4.1)

Suppose that u 2 L�p . 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; n�g

is a � -additive set function, which is absolutely continuous with respect to themeasure m. The Radon–Nikodym theorem states the existence of a unique functionhn integrable on Œ0; n� such that

.A/ DZ

A

hndm; A � Œ0; n�; A 2 Fm: (2.4.3)

Since RC D

1S

nD1Œ0; n/, we obtain a unique measurable function g such that

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

u.1A/ D .A/ DZ

A

gdm (2.4.4)

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

u.f / D1Z

0

fgdm D ug.f /; f 2 F0: (2.4.5)

We choose now a sequence gn 2 F0 such that gn " jgj. Such a sequence can beconstructed by setting

gn D min.n; g.n// � 1Œ0;n�;

26 2 Spaces Lp; 1 � p � 1

where for all i D 1; 2; : : :,

g.n/.x/ D i � 12n

; ifi � 12n

� jg.x/j < i

2n:

We define now

fn D .gn/q�1 � sign.g/; n D 1; 2; : : : :

Then fn 2 F0 and

u.fn/ D1Z

0

fngdm D1Z

0

gqndm D kgnkq

Lq:

On the other hand,

u.fn/ � kukL�

pkfnkLp ;

where jfnjp D g.q�1/pn D gq

n, by the equality .q � 1/p D q.This implies

kfnkLp D0

@

1Z

0

gqndm

1

A

1=p

D kgnkq=pLq

and

kgnkqLq

� kukL�

pkgnkq=p

Lq;

i.e.,

kgnkLq � kukL�

p:

Thus, gqn " jgjq and

1Z

0

gqndm � kukq

L�

p:

By Levi’s theorem,

limn!1

1Z

0

gqndm D

1Z

0

jgjqdm � kukqL�

p;

i.e., g 2 Lq and kgkLq � kukL�

p.

2.4 Duality 27

The functional ug coincides with u 2 L�p on F0, and F0 is dense in Lq. Hence

ug D u.The equality .L1/ D L�

1 and the fact that the embedding .L1/ � L�1 is strictwill be shown later in Examples 7.1.1 and 7.1.2. utCorollary 2.4.2. The spaces Lp are reflexive for 1 < p < 1.

Chapter 3The Space L1 \ L1

The space L1 \ L1, which we study in this chapter, consists of all boundedintegrable 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 anapproximation of g in the L1 \ L1-norm by shifted functions f ı � , where � is ameasure-preserving transformation. Step functions and integrable simple functionsare 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 thespecial case X1 D L1 and X2 D L1.

Clearly, we have that

kf kL1\L1D 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 holdfor each of the spaces L1 and L1.


29

30 3 The Space L1 \ L1

To verify the completeness of L1 \ L1, consider a Cauchy sequence ffng inL1 \ L1. Then ffng is a Cauchy sequence in both spaces L1 and L1. Since L1 andL1 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 andfn ! 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

kf kL1 D1Z

0

jf jdm D1Z

0

f �dm

and

kf kL1D vrai sup jf j D f �.0/:

So the norm on L1 \ L1 has the form

kf kL1\L1D max

0

@

1Z

0

f �dm; f �.0/

1

A ; f 2 L1 \ L1;

and hence satisfies conditions 1 and 2 of symmetric spaces. Thus, L1 \ L1 is asymmetric 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 willbe 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 describedby means of decreasing rearrangements f � as follows.

Since the function f � decreases on .0;1/, there exists a limit

f �.1/ WD limx!1 f �.x/ D inf

x>0f �.x/:

We set (Fig. 3.1)

L01 D ff 2 L1 W f �.1/ D 0g � L1

3.2 The Space L01

31

0 0

0

0>f *

f *f *

f *

( ) 0=f *( )f *( )•

• •

( )0f *( )

x x

y y

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 kL01D 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 y2g � fjf j > y1g [ fjgj > y2gimplies 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. Sinceall these functions are right-continuous, the inequality (3.2.1) is valid also for all x1and x2. ut

0

= +( )= =+ +f fff* f*g* g*g g *g

0

1 1 1 1

2

1 1 0 1 0 11/2 xxx x

y y y y

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

2< x < 1:

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 limx!1.f C g/�.2x/ � lim

x!1.f�.x/C g�.x// D f �.1/C g�.1/ D 0;

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

3.3 Approximation by Step Functions 33

If kfn � f kL1! 0 and fn 2 L01, then

f �.1/ � .f � fn/�.1/C f �

n .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 spaceas well as L1. Conditions 1 and 2 of symmetric spaces are clear. ut

We shall return to the space L01 at the end of the chapter and prove that L01 isthe closure of L1 \ L1 in L1.

3.3 Approximation by Step Functions

Recall that by step functions, we mean functions of the form

f DX

i2I

ai � 1Ai ;

where I is a finite or countable set of indices, ai are real numbers, and Ai are mutuallydisjoint subsets of RC.

Proposition 3.3.1. Let f � f �.1/. Then for every " > 0, there exists a stepfunction 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,

jf .x/ � f".x/j � " and

1Z

0

Œf .x/ � f".x/�dx � " c;

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

Fig. 3.3 The casef �.1/ D f �.c/, c < 1

0 c

*f

ee

aa

a+2+

x

y

Fig. 3.4 The case in whichf �.1/ is not achieved,c D 1

0

2

1

+

x

y

+1b

a

e

b

b

af

b +2b

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 fbng: f".x/ D bnC1 for bnC1 � f .x/ <

bn, n D 0; 1; 2; : : :.Then

kf � f"kL1� ";

and1Z

0

Œf .x/ � f".x/�dx � b "C Œ.a C "/ � a� � 1 D .b C 1/ ":

By choosing

"1 D min

�

"

c;

"

b C 1

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 areequimeasurable, 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 existsa sequence of step functions ffng such that fn " f and

limn!1 kf � fnkL1\L1

D 0:

3.4 Measure-Preserving Transformations

The notion of equimeasurability is closely related to measure-preserving transfor-mations of .RC;m/.

Recall that a transformation � of the measure space .RC;m/ is called measure-preserving if for every set A 2 Fm, the set

��1A D fx 2 RC W �.x/ 2 Ag

is measurable, and m.��1A/ D mA.In other words, the measure m ı ��1 defined by

.m ı ��1/A D m.��1A/

coincides with the original measure m.If, in addition, � is invertible and the inverse transformation ��1 is also measure-

preserving, then

m.��1A/ 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 functionsf and f ı � are equimeasurable.

Proof. For every x > 0,

�f ı � .x/ D mf.f ı �/ > xg D m.��1ff > xg/ D mff > xg D �f .x/;

i.e., f and f ı � are equimeasurable. utFor 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 thereexists � 2 A.m/ such that g D f ı � .

Proof. Let f D P

i2Iai � 1Ai ; g D P

j2Jbj � 1Bj . Without loss of generality, we may

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 commondistribution function � D �f D �g.

By changing index sets if necessary, we may assume that I D J and ai D bi forall i 2 I (Fig. 3.5).

For every jump point ai of the distribution function �.x/,

f �.1/ D g�.1/ < �.ai�/ � �.ai/ D mAi D mBi < 1;

i.e., RC D S

i2IAi D S

i2IBi are two partitions of R

C into disjoint sets of finite

measure. By choosing invertible measure-preserving mappings �i W Ai ! Bi, weobtain an invertible measure-preserving transformation � W R

C ! RC such that

� jAi D �i; i 2 I. The choice of � yields g D f ı � . ut

Note 3.4.3. The conditions f .x/ > f �.1/ and g.x/ > g�.1/, x > 0, are essentialin Proposition 3.4.2. For example, let

f D 1Œ0;C1/; g D 1

2� 1Œ0;1/ C 1Œ1;C1/:

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 commondistribution function of stepfunctions f and g

}

==

_

af

f g

ia

* g( )•

( )

( )i

ah

h h h

h

( )•* ib

mi

A mi

B

i

=

= =

0 x

3.4 Measure-Preserving Transformations 37

The assumption that f and g are step functions is also essential, and Proposi-tion 3.4.2 fails, in general, if f and g are not step functions.

Example 3.4.4. Let f .x/ D .1�x/�1Œ0;1� and f1.x/ D x�1Œ0;1�, f2 D f1ı�2, f3 D f1ı�3,where measure-preserving transformations �2 and �3 are defined as follows:

�2.x/ D8

<

:

2x; x 2 �0; 12

�

;

2x � 1; x 2 � 12; 1�

;

x; x 2 Œ1; 1/;

�3.x/ D

8

ˆ

ˆ

<

ˆ

ˆ

:

3x; x 2 �0; 13

�

;

3x � 1; x 2 � 13; 23

�

;

3x � 2; x 2 � 23; 1�

;

x; x 2 Œ1; 1/;

Transformations �1 and �2 are measure-preserving. However, they are notinvertible, i.e., �1; �2 62 A.m/.

The functions f ; f1; f2; f3 are equimeasurable,

f � D f D f �1 D f �

2 D f �3 ;

and fi ¤ f �i ; 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 thereis 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 andg � 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 Proposi-tion 3.3.1, we can construct equimeasurable step functions f" and g" such that

kf � f"kL1\L1� "

2and kg � g"kL1\L1

� "

2:

0 0 0 0

1 1

y y

x x x

yy

x1

f f1f2 f3

1 1/2 1/3 2/31

11

1

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

By Proposition 3.4.2, we obtain � 2 A.m/ such that

f" D g" ı �:For every measurable function h,

kh ı �kL1 D1Z

0

jh ı � jdm D1Z

0

jhjd.m ı ��1/ D1Z

0

jhjdm D khkL1 ;

and also

kh ı �kL1D vrai sup jh ı � j D vrai sup jhj D khkL1

:

Hence,

kg ı � � g" ı �kL1\L1D kg � g"kL1\L1

and kf � g ı �kL1\L1� ". ut

Note 3.4.6. The assumption f � a, g � a, a D f �.1/ D g�.1/ in Theorem 3.4.5is 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 ofall simple functions from F1 with bounded support. Every f 2 F1 has the form

f DnX

iD1ai � 1Ai ; ai 2 R; mAi < 1; 1 � i � n; n 2 N:

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 thenorm 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 functionsfrom F0 in the norm k � kL1\L1

. We shall use functions g", constructed inProposition 3.3.1. Since L1 \ L1 is a symmetric space, we may assume thatf D f � � 0.For a given " > 0, we choose sequences fbng1

nD0 and fang1nD0 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" D1X

nD1an � 1Œbn�1;bn/ and g";k D

kX

nD1an � 1Œbn�1;bn/:

Then

g";k 2 F0; kf � g"kL1� "; kg" � g";kkL1

� ak # 0and

kf � g";kkL1 �1Z

bk

fdm # 0;

Fig. 3.7 Sequences an and bn

0

y

xb1

a3

a2

a

f*1

a0

2 3b b

since the function f is integrable. So for sufficiently large k,

kf � g";kkL1\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";kkL1

� 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:ut

Note 3.5.2. Recall that the spaces L1 \ L1 and L01, as well as L1, arenonseparable.

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 � L01for all a > 0.

Chapter 4The Space L1 C L1

In this chapter, we study the sum L1 C L1 of the spaces L1 and L1. We showthat L1 C L1 equipped with a natural norm k � kL1CL1

is a symmetric space. Thenorm k � kL1CL1

can be written in the formR 1

0f � dm using the maximal property

of decreasing rearrangements f �. We also describe embeddings of L1 and L1 intoL1 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 theirnumerous applications. In particular, we shall use the property in the proof ofTheorem 4.2.1, below.

Theorem 4.1.1. For every a > 0

aZ

0

f � dm D sup

8

<

:

Z

A

jf j dm W mA D a

9

=

;

: (4.1.1)

If, in addition, f �.a/ > f �.1/, then there exists a subset A 2 Fm such that mA D aand

aZ

0

f � dm DZ

A

jf j dm: (4.1.2)


41

42 4 The Space L1 C L1

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

Z

A

jf j dm �1Z

0

jf j dm D1Z

0

f � dm;

and equality is achieved with A D RC.

Thus, we may assume that a < 1.

1. First we check that

Z

A

jf j dm �mAZ

0

f � dm: (4.1.3)

Let a D mA < 1. Consider the function (Fig. 4.1)

va.y/ D minfa; �jf j.y/g y � 0:

By applying the formula

1Z

0

jf jdm D1Z

0

�jf jdm

to the function jf j � 1A, we obtain

Z

A

jf jdm D1Z

0

jf j � 1Adm D1Z

0

�jf j�1A dm:

Since �jf j�1A � mA D a and �jf j�1A � �jf j, it follows that �jf j1A � va and

1Z

0

�jf j�1A dm �1Z

0

vadm DaZ

0

f �dm:

Hence, (4.1.3) is valid.

0 0

( )

( )

=

x x

y y

a a 0 xa

b b

y

b

xf

y

f a*

( )= =y f hx*

f * (a–)

( )x v a y=

Fig. 4.1 Functions f �, �jf j and va

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 suchthat 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 samefinite 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 right-continuous.

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

A

jf jdm �Z

A

�

c � "

a

dm D cx � " DaZ

0

f �dm � ";

i.e., the equality (4.1.1) is valid.

Fig. 4.2 A constant-valueinterval of f �

0

=

x

y

a

y f x( )*

=b f a( )*

a a

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,

aZ

0

f �dm Da0Z

0

f �dm C .a � a0/a:

Consider the decomposition

jf j D jf j � 1fjf j>cg C jf j � 1fjf j�cg

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 ofjf j � f �.1/ andf �.a0/ D f �.a/ D f �.1/

0

0

0 x

x

y

y

a a

A

e

f*=c·1[0,•]

c= f * (•)

c= f * (•)

y = f *(x)

c _

f

4.2 The Sum of L1 and L1 45

By applying the second part of the proof to jf j�1fjf j>ag, we obtain a set A1 such that

A1 � fjf j > cg; mA1 D a0;

and

Z

A1

jf j dm Da0Z

0

f � dm:

On the other hand, by applying the third part of the proof to jf j � 1fjf j�ag and " > 0,we obtain a set A2 such that

A2 � fjf j � cg; mA2 D a � a0;

andZ

A2

jf j dm � .a � a0/c � ":

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

Z

A

jf j dm �xZ

0

f � dm � "and mA D a:

This concludes the proof of the theorem. ut

4.2 The Sum of L1 and L1

We begin with a general construction using two symmetric spaces, X1 and X2. Bydefinition, 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.

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 somer 2 X1 \ X2. Formally, X1 C X2 is identified with the quotient space .X1 X2/=Lof the space X1 X2 by the subspace

L D f.r;�r/; r 2 X1 \ X2g � 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 kf1kX1 C kf2kX2 ; (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 infr2X1\X2

fkf1 C rkX1 C kf2 � rkX2g: (4.2.3)

Thus, X1 C X2 becomes a normed space with the norm

kf kX1CX2 D inffkf1kX1 C kf2kX2 ; f D f1 C f2; f1 2 X1; f2 2 X2g; (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 specialcase 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 L1gequipped with the norm

kf kL1CL1D inffkgkL1 C khkL1

W f D g C h; g 2 L1; h 2 L1g: (4.2.5)

is a symmetric space, and

kf kL1CL1D

1Z

0

f �.s/ds D sup

8

<

:

Z

A

jf jdm W mA D 1

9

=

;

: (4.2.6)

4.2 The Sum of L1 and L1 47

Proof. First we prove the equality

kf kL1CL1D

1Z

0

f �dm:

For every A 2 Fm with mA < 1 and every representation of f of the form

f D g C h; g 2 L1; h 2 L1;

we haveZ

A

jf jdm � kgkL1 C khkL1� mA:

Whence by Theorem 4.1.1,

1Z

0

f �dm D sup

8

<

:

Z

A

jf jdm; mA D 1

9

=

;

� kgkL1 C khkL1;

and hence

1Z

0

f �dm � kf kL1CL1:

On the other hand, let b D f �.1�/, g D f � h, where (Figs. 4.4 and 4.5)

h D sign.f / � min.jf j; b/:

Fig. 4.4 Relation betweennorms kgkL1

and khkL1and kf kL1

0

L

L

h

f

g

h

A

1

x

y

•


00 1 1x x

y y

b

f ** f b( )+_

Fig. 4.5 Value b D f �.1�/ and function .f � � b/C

Then jgj D .jf j � b/C. Since the functions .jf j � b/C and .f � � b/C areequimeasurable,

kgkL1 D1Z

0

.jf j � b/Cdm D1Z

0

.f � � b/Cdm D1Z

0

f �dm � b D1Z

0

f �dm � khkL1:

Thus,

1Z

0

f �dm D kgkL1 C khkL1� kf kL1CL1

:

The equality (4.2.6) is proved.Since L1 and L1 are Banach spaces, the space L1 C L1 with the norm (4.2.5)

is a Banach space.Let jf j � jf j and f 2 L1 C L1. Then for every " > 0, we can find g 2 L1 and

h 2 L1 such that f D g C h and

kgkL1 C khkL1� kf kL1CL1

C ":

We set g D gv and h D hv, where

v D f

f� 1ff ¤0g:

Then g 2 L1, h 2 L1, and f D g C h 2 L1 C L1, with

kf kL1CL1� kgkL1 C khkL1

� kgkL1 C khkL1� kf kL1CL1

C ";

i.e.,

kf kL1CL1� kf kL1CL1

:

We have shown that L1 C L1 is a Banach ideal lattice.The equality kf kL1CL1

D kf �kL1CL1for f 2 L1 C L1 follows from (4.2.5).

utThe condition f 2 L1 C L1 may be formulated in several equivalent ways.

4.3 Embeddings L1 � L1 C L1 and L1 � L1 C L1. The Space R0 49

0

( )

( )

( )

x

y y

a

= =

=

0 xa

af

y yf

hxf

y

x* ( )f x*

*( )af*

( )=hxf

y

Fig. 4.6 Equality (4.2.7)

Corollary 4.2.2. For every function f 2 L0, the following are equivalent:

1. f 2 L1 C L1.

2. f is locally integrable, i.e.,Z

A

jf jdm < 1, if mA < 1.

3. There exists a > 0 such that �jf j.a/ < 1 and

1Z

a

�jf jdm < 1.

4.

aZ

0

f �dm < 1 for some a > 0.

Proof. Let f 2 L0.1 ” 2 by Theorem 4.1.1.

1 ” 4 by the equality (4.2.6), since

a2Z

a1

f �dm < 1 for all 0 < a1 < a2 < 1.

3 ” 4, since for every a > 0,

aZ

0

f �dm D1Z

f �.a/

�jf jdm C af �.a/: (4.2.7)

This completes the proof of the corollary (Fig. 4.6). ut

4.3 Embeddings L1 � L1 C L1 and L1 � L1 C L1.The Space R0

The space L1 C L1 is important, because it is the “largest” symmetric space(see Theorem 5.2.1). Hence convergence in norm k � kL1CL1

is the weakest normconvergence that we consider. In particular,

L1 � L1 C L1 and L1 � L1 C L1;

and

kf kL1D f �.0/ �

1Z

0

f �dm D kf kL1CL1;

kf kL1 D1Z

0

f �dm �1Z

0

f �dm D kf kL1CL1:

Consider now the set R0 WD ff 2 L1 C L1 W f �.1/ D 0g.

Proposition 4.3.1. 1. clL1CL1.L1/ D L1 C L1, i.e., L1 is dense in L1 C L1.

2. R0 is a symmetric space and

clL1CL1.F0/ D clL1CL1

.L1 \ L1/ D clL1CL1.L1/ D R0:

Proof. 1. We show that every nonnegative function f 2 L1 C L1 can beapproximated in norm k � kL1CL1

by its cutoffs fn D min.f ; n/. Since L1and L1 C L1 are symmetric spaces, we can clearly assume that f D f � andfn D f �

n D min.f �; n/ (Fig. 4.7).Since f �.x/ < 1 for all x > 0, we have an D �f �.n/ # 0 as n ! 1, and

hence an < 1 for n large enough (Fig. 4.8). Thus,

kf � fnkL1CL1D

1Z

0

.f � fn/dm D1Z

n

�f �dm # 0; n ! 1:

2. We show that R0 is closed in L1 C L1. Let ffng be a sequence in R0, f 2 L1 CL1, and

kf � fnkL1CL1D

1Z

0

.f � fn/�dm ! 0; n ! 1:

Fig. 4.7 Upper cutoff of f �

0 x

y

an0 x

y

an

nn

=n min ( )f f n*,=

=

( )

( )

y

yx

f x*

f*h

Exercises 51

Fig. 4.8 Normkf � fnkL1CL1

as area

0 1

y

x x

_( ) ( )nf f x*

_( ) ( )nf f 1*

_( )nf f *

Then .f � fn/�.1/ ! 0 and .f � fn/�.x/ ! 0 as n ! 1 and x > 0. Theinequality (3.2.1) implies that

0 � f �.x1 C x2/ � f �n .x1/C .f � fn/

�.x2/:

Since f �n .1/ D 0 and .f � fn/�.x2/ ! 0 as x1 ! 1, we obtain

0 � f �.1/ � f �n .1/ D 0;

i.e., f � 2 R0.Every function f 2 R0 can be represented in the form f D g C h, where g 2 L1

and h 2 L01 D L1 \ R0.By Proposition 3.5.1, clL1 .F0/ D L1 and clL1

.F0/ D L01, i.e., g can beapproximated by functions from F0 in the norm k � kL1 , and h can be approximatedby functions from F0 in the norm k � kL1

. Hence, the function f D g C h can beapproximated by functions from F0 in the norm k � kL1CL1

, since the latter norm ismajorized by k � kL1 and k � kL1

. ut

Exercises

1. Spaces Lp.0; 1/; 1 � p � 1. Employing the notation and terminology ofComplement 2, show that

a. Lp D Lp.0; 1/, p 2 Œ1;1�, are symmetric spaces on the interval Œ0; 1�.b. There are continuous embeddings

L1.0; 1/ � Lp.0; 1/ � Lq.0; 1/ � L1.0; 1/

and

k � kL1� k � kLp � k � kLq � k � kL1

for all 1 � q � p � 1. The embeddings are strict for all 1 < q < p < 1.

c. clLq.Lp/ D Lq for all 1 � q < p � 1.d. The embeddings

L1 �\

1�p<1Lp and

[

1<p�1Lp � L1

are strict.

2. Spaces lp D Lp.N/; 1 � p � 1. Employing notation and terminology ofComplement 3, show that

a. lp D Lp.N/, 1 � p � 1, are symmetric sequence spaces.b. For 1 � q � p < 1, there are continuous embeddings

l1 � lq � lp � c0 � l1;

and

k � kl1 � k � klq � k � klp � k � kc0 D k � kl1 :

The embeddings are strict for all 1 < q < p < 1.c. cllp.lq/ D lp and cll1.lp/ D c0 for all 1 � q < p < 1.d. The embeddings

l1 �\

1<p�1lp and

[

1�p<1lp � c0

are strict.

3. Spaces Lp.0;1/; 1 � p � 1.Let Lp D Lp.0;1/. Show that

a. For all 1 � p � 1, the embeddings

L1 \ L1 � Lp � L1 C L1

are continuous and strict.b. For all 1 � p � 1, clLp.L1 \ L1/ D Lp and clL1CL1

.Lp/ D R0.c. For all 1 � p; q � 1,

Lp � Lq ” p D q:

d. The embeddings

L1 \ L1 �\

1<p<1Lp and

[

1<p<1Lp � L1 C L1

are strict. Also, Lp [ Lq ¤ Lp C Lq for p ¤ q.

Exercises 53

4. Spaces Lp \ L1 and Lp \ L1; 1 � p � 1 on .0;1/.Show that for all 1 � q � p � 1,

a.

L1 \ L1 � Lp \ L1 � Lq \ L1 � L1

and

k � kL1\L1 � k � kLp\L1 � k � kLq\L1 � k � kL1 :

b.

clL1 .L1 \ L1/ D clL1 .Lp \ L1/ D clL1 .Lq \ L1/ D L1:

c.

L1 \ L1 � Lq \ L1 � Lp \ L1 � L1

and

k � kL1\L1� k � kLq\L1

� k � kLp\L1� k � kL1

:

d. clL1.L1 \ L1/ D clL1

.Lq \ L1/ D clL1.Lp \ L1/ D L01, where L01 D

L1 \ R0 is the minimal part of L1.

5. Spaces Lp C L1 and Lp C L1 on .0;1/, 1 � p � 1.Show that for all 1 � q < p � 1,

a.

L1 � Lq C L1 � Lp C L1 � L1 C L1

and

k � kL1 � k � kLqCL1 � k � kLpCL1 � k � kL1CL1 :

b.

clLqCL1 .L1/ D Lq C L1

and for 1 � q < p < 1,

clLpCL1 .Lq C L1/ D Lq C L1; clL1CL1 .Lp C L1/ D R0:

c.

L1 � Lp C L1 � Lq C L1 � L1 C L1

and

k � kL1� k � kLpCL1

� k � kLqCL1� k � kL1CL1

:

d.

clLqCL1.L1/ D clLqCL1

.Lp C L1/ D Lq C L1:

6. Spaces Lp; 0 < p < 1. Employing notation and terminology of Complement 4,consider the spaces Lp.0;1/, Lp.0; 1/, lp D Lp.N/, 0 < p < 1. Show that for all0 < p < q < 1,

a. Lp are quasi-Banach symmetric spaces with quasinorms k � kLp .b. Lp are F-spaces with the metric dLp.f ; g/ D kf � gkp

Lp.

c. Lq.0; 1/ � Lp.0; 1/ and k � kLp.0;1/ � k � kLq.0;1/.d. lp � lq and k � klq � k � klp .

Hint: Use the inequality

ab � ap

pC bq

q

for all a > 0, b > 0, 0 < p < 1, and 1p C 1

q D 1. Note that here q D pp�1 < 0.

7. Convergence of decreasing sequences. Show that for every sequence ffng1nD1 in

L0 D L0.0;1/ and f 2 L0,

a. If fn ! f in measure, then f �n .x/ ! f �.x/) at every point x of continuity of the

function f .b. If jfnj � g, g�.1/ D 0, and fn.x/ ! f .x/ almost everywhere, then.fn � f /�.x/ ! 0 for all x > 0.

c. If jfnj � g, g 2 L1, and fn.x/ ! f .x/ almost everywhere on Œ0;1/, then

limn!1

Z

A

f �n dm D

Z

A

f �dm

for every measurable subset A � RC.

8. Approximation in norm k � kL1\L1. Show that

a. Let f and g be two nonnegative equimeasurable functions on RC. Then for all

" > 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 ! Œ�1; 1� suchthat

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 throughoutthe book is from [34, Section 2.4]. It is based on notions of ideal Banach latticesand symmetry (rearrangement invariance). Good presentations of the general theoryof Banach lattices and normed ideal spaces of measurable functions can be foundin [31, 35, 36, 75]. For general ordered topological linear spaces, see [22, Chapters1–3], [65, Chapter 2].

The decreasing rearrangements f � of measurable functions f were introduced byG.H. Hardy, J.E. Littlewood, and G. Pólya; see, for example, [27, Section 10.12].

Some authors use stricter definitions of symmetric spaces. Lindenstrauss andTzafriri [36] assume that every symmetric space is either minimal or maximal andhas 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 theorycontain 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 consideredin [1, 15]; see also [63, Chapter 1], [64, Chapter 1].

The spaces L1\L1 and L1CL1 are a special case of the intersection X\Y andthe sum XCY of two general Banach spaces X and Y. The spaces X\Y and XCYare well defined once some suitable continuous embeddings X ! L and Y ! Linto a topological linear space L are chosen and fixed (see [34, Section 1.3]).

Results of Section 3.4 can be strengthened using noninvertible measure-preserving 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 tothe case of symmetric spaces X.˝;/ on general measure spaces .˝;/ with finiteor 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


X.RC;m/ corresponding to X.˝;/ defined there is rather useful, especially whenthe measure space .˝;/ is separable. The isomorphism of Lebesgue measurespaces used in this case was proved in [62].

Besides the case .˝;/ D .RC;m/, two special cases are of particularinterest:

a. .˝;/ D .Œ0; a�;ma/, where ma D mjŒ0;a� is the usual Lebesgue measure on theinterval.

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/ aredescribed in Complements 2 and 3, respectively. Some related results are given alsoin Exercises 9, 10, 11, and 12.

Note that the situation can be more intricate if .˝;/ has atoms ! 2 ˝ withunequal values .f!g/ > 0; see [3, Section 2.2].

General nonseparable measure spaces .˝;/ are described, for example, in[23, Chapters 32–34].

Part IISymmetric 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 kL1CL1� 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 < 1and L1 \ L1 are minimal, while L1 and L1 C L1 are not minimal. The minimalpart 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 Xsatisfying 'X.0C/ D 0 is separable.

Separable symmetric spaces can be characterized by the following property:

.A/ ffng � X; fn # 0 H) kfnkX # 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 space X�) has

an integral form if u D ug for some g 2 L1 C L1, where ug.f / D R

fg dm for allf 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 setX1 D fg 2 L1 C L1 W ug 2 X�g equipped with the norm kgkX1 D kugkX� is asymmetric space. The space X1 is called the associate space of X.

For every symmetric space, there are continuous embeddings

X0 � X � X11;

where X11 D .X1/1 is the second associate space of X. In the case X D X11, thesymmetric space X is called maximal. The spaces Lp; 1 � p � 1, L1 \ L1, andL1 C L1 are maximal, and the spaces R0 D .L1 C L1/0 and L01 D L1 \ R0 arenot maximal.

The natural embedding of X into X11 is not necessarily isometric. The equalitykf kX D kf kX11 , f 2 X is equivalent to the following property:

.C/ ffng � X; fn " f 2 X H) kf kX D supn

kfnkX:

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

.B/ ffng � X; fn " and supn

kfnkX < 1 H) fn " f for some f 2 X:

If, in addition to .B/, X has the property .C/, then the embedding X into X11 isan isometric isomorphism between X and X11.

The converse is also true, i.e., the equality .X; k � kX/ D .X11; k � kX11 / isequivalent to the following property:

.BC/ffng � X; fn "; supn kfnkX < 1 H);

fn " f and kfnkX " 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 X1g D X�.Every reflexive symmetric spaces X has property .AB/, i.e., it satisfies both .A/

and .B/:

.AB/ffng � X; fn "; supn kfnkX < 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 haveproperty .AB/.

Chapter 5Embeddings L1 \ L1 � X � L1 C L1 � L0

In this chapter, we prove the main embedding theorem for symmetric spaces. Thetheorem asserts that for every symmetric space X, there are continuous embeddingsL1 \ L1 � X � L1 C L1 and inequalities 2k � kL1\L1

� .'X/�1.1/ k � kX �

� k � kL1CL1: 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[

iD1�iB A:

By setting �iB D Bi, we have the functions 1B; 1B1 ; : : : ; 1Bn , which are equimeasur-able to 1B, and


59

60 5 Embeddings L1 \ L1 � X � L1 C L1 � L0

1B 2 X H) 1A �nX

iD11Bi 2 X and 1A 2 X:

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 k1AkX 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):

'Lp.x/ D0

@

1Z

0

j1Œ0;x�jpdm

1

A

1p

D0

@

xZ

0

dm

1

A

1p

D x1p ; x � 0:

2: The space L1:

0

1

1< <

1 x 0 1 x 0 x

y

j

1

y

1

y

L1j •

•L , pp

jL

Fig. 5.1 Fundamental functions of Lp

5.2 The Embedding Theorem L1 \ L1 � X � L1 C L1 61

10 x

y

1

10 x

y

1

j L1

Ã

L•jL1+ L•

Fig. 5.2 Fundamental functions of L1 C L1 and L1 \ L1

'L1.x/ D vrai sup

RC

j1Œ0;x�j D 1.0;1/.x/ x � 0:

3: The space L1 \ L1 (Fig. 5.2):

'L1\L1.x/ D max.x; 1.0;1/.x// D

8

<

:

0; x D 0I1; 0 < x � 1Ix; x > 1:

4: The space L1 C L1:

'L1CL1.x/ D

1Z

0

1Œ0;x/.s/ds D min.x; 1.0;1/.x// D�

x; 0 � x � 1I1; 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 symmetricspaces.

Theorem 5.2.1. For every symmetric space X,

L1 \ L1 � X � L1 C L1 (5.2.1)

and

2kf kL1\L1� kf kX; f 2 L1 \ L1;

kf kX � kf kL1CL1; 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

1S

kD0Ak is a partition of RC into disjoint sets Ak of measure 1 (Fig. 5.3)

such that

akC1 < f .x/ � ak; x 2 Ak:

Consider the functions

f D1X

kD0akC1 � 1Ak ; f D

1X

kD0ak � 1Ak

and for n � 1,

f n DnX

kD0ak � 1Ak :

Then

f � f � � f ; and f n " f as n ! 1:

1

ak

a

f

k

k k

a2

a1

a

*

0

+

+1

1

2 30 x

y

Fig. 5.3 The sequence fakg


The sequence ff ng is fundamental in X, since

1X

kD0akC1 D kf kL1 � kf kL1

and for every m � 1,

kf nCm � f nkX DnCmX

kDnC1akk1Ak kX D 'X.1/

nCmX

kDnC1ak � 'X.1/

1X

kDnC1ak ! 0;

as n ! 1. Thus there exists g 2 X such that

kg � f nkX ! 0; n ! 1:

Since f njŒ0;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

kf kX � kf kX � 'X.1/

1X

kD0ak � 'X.1/

a0 C1X

kD0akC1

!

� 'X.1/ .kf kL1C 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 Dn�1X

kD0ak � 1Ak ; where ak � 0; mAk D 1

n;

n�1[

kD0Ak D Œ0; 1�: (5.2.3)

For every cyclic permutation of indices f0; 1; 2; : : : ; n � 1g,

�j W k ! k C j.mod n/;

consider the function

fj Dn�1X

kD0a�j.k/ � 1Ak ; 0 � j � n � 1:

The functions f0; f1; : : : ; fn�1 are equimeasurable, and hence

kfjkX D kf0kX; j D 0; 1; : : : ; n � 1:


On the other hand,

n�1X

jD0fj D

n�1X

kD0ak

!

� 1Œ0;1�:

Therefore,

n�1X

kD0ak

!

'X.1/ D�

�

�

�

�

�

n�1X

jD0fj

�

�

�

�

�

�

X

�n�1X

jD0kfjkX D nkf0kX;

and

kf0kX �

1

n

n�1X

kD0ak

!

� 'X.1/ D kf0kL1 � 'X.1/:

Thus,

kf0kX � 'X.1/ � kf0kL1 (5.2.4)

for every function f D f0 of the special form (5.2.3).Let g 2 X. We estimate the norm kgkL1CL1

by the norm kgkX.Since

kgkL1CL1D

1Z

0

g�dm;

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 Dn�1X

kD0ak � 1Ak ;

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

ak D g

�

k C 1

n

�

; Ak D

k

n;

k C 1

n

�

:

Since g is decreasing, we have gn " g, and limn!1 kgnkL1 D kgkL1 .

Since the functions gn have the form (5.2.3), we can apply the inequality (5.2.4).Therefore,

kgnkL1 � .'X.1//�1kgnkX:

Setting n ! 1, we obtain

kgkL1 � .'X.1//�1kgkX:

Hence for all f 2 X,

kf kL1CL1D

1Z

0

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

� .'X.1//�1kf kX:

Thus, both embeddings (5.2.1) and inequalities (5.2.2) hold. utNote 5.2.2. In the second part of the previous proof we can use a function gsupported 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 asn ! 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 kL1CL1� kfn � f kX:

Hence, hn D fn � f tends to 0 in norm k � kL1CL1, i.e.,

khnkL1CL1D

1Z

0

h�n dm ! 0; n ! 1:

Suppose that contrary to our claim, fhng does not tend to 0 in measure. Then forsome positive " > 0,

�h�

nD mfh�

n > "g 6! 0; n ! 1:

Then there exist a subsequence fh�nk

g and 0 < c < 1 such that �h�

nk."/ > c (Fig. 5.4).

Hence h�nk.c/ � " for all k D 1; 2; : : : and


Fig. 5.4 Functions hnk

c

hnk*

h

e

e

cnk* ( )

hh

nk*

( )0

y

x

khnk kL1CL1D

1Z

0

h�nk

dm �cZ

0

h�nk

dm � h�nk.c/ � c � " � c > 0

for all k D 1; 2; : : :.This contradicts the assumption kh�

nkkL1CL1

! 0. ut

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/. Thespace contains L1 C L1 and hence all symmetric spaces.

We show that the space L0 is a complete metric space with respect to a naturalmetric d0, defined as follows: Let v.x/ be a bounded concave function on R

C suchthat v.0/ D 0 and v.x/ > 0 for all x > 0, and let be a Borel probability measureon R

C that is equivalent to the Lebesgue measure m. We set

p0.f / D1Z

0

v.jf j/d; f 2 L0: (5.3.1)

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 The Space L0 and the Embedding L1 C L1 L0 67

Since v.x/ increases andv.x/

xdecreases, we have for x � 0,

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,

d0.f ; g/ D p0.f � g/ D1Z

0

v.jf � gj/d; f ; g 2 L0 (5.3.2)

is a translation-invariant metric in the linear space L0. The operations of additionand multiplication by a scalar are continuous on L0 in the topology induced by themetric 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 ffng be a fundamental sequence in L0, i.e.,

limn;m!1 d0.fn; fm/ D 0:

There exists an increasing sequence of indices fnkg1kD1 such that

d0.fn; fnk/ D1Z

0

v.jfn � fnk j/d � 1

2k;

for all k and n > nk.Hence the series

1X

kD1d0.fnkC1

; fnk/ D1X

kD1

0

@

1Z

0

v.jfnkC1� fnk j/d.x/

1

A (5.3.3)


converges, and the series

1X

kD1gk.x/; gk.x/ D v.jfnkC1

� fnk j/;

converges almost everywhere to a function h.x/ D1P

kD1gk.x/.

The function h is finite almost everywhere and

1Z

0

h.x/d.x/ D1X

kD1

0

@

1Z

0

gk.x/d.x/

1

A : (5.3.4)

Indeed, let

hn.x/ DnX

kD1gk.x/:

Then hn " h and the integral

1Z

0

h.x/d.x/ D limn!1

1Z

0

hn.x/d.x/ D1X

kD1

0

@

1Z

0

gk.x/d.x/

1

A ;

equals the sum of the series (5.3.3). This implies that h 2 L1./. Hence h is finitealmost everywhere, and (5.3.4) holds.

Since both functions v and v�1 are continuous and increasing, almost everywhere

convergence of1P

kD1gk.x/ implies almost everywhere convergence of

fn1 .x/C1X

kD1.fnkC1

.x/ � fnk.x//:

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. utThus, 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

xx

xv ( )

0 x 0

1

y

1 x

y

1

=+ 1 min(x,1)xv( ) =

Fig. 5.5 Functions v.x/

The function v.x/ is usually chosen as v.x/ D x

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

(Fig. 5.5).The measure can be chosen by setting

.A/ D1X

nD1

1

2nm.A \ Œn � 1; n�/; A 2 B:

Then

d0.f ; g/ D1X

nD1

1

2n

nZ

n�1v.jf � gj/dm;

where

d

dmD

1X

nD1

1

2n� 1Œn�1;n�:

The topology in the linear space .L0; d0/ is defined by the system of neighbor-hoods fU".0/; " > 0g, where

U".0/ D�

f 2 L0 W m

�

0;1

"

�

\ fjf j > "g�

< "

:

Note that the system, and hence the considered topology, do not depend on thechoice of v.x/ and .

Convergence in this topology coincides with so-called stochastic convergence.

Definition 5.3.2. A sequence ffng of a measurable functions is said to converge

stochastically to a function f (fn.st/�! 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.

Thus d0.fn; f / ! 0 if and only if ffng .st/�! f i.e., fnjA ! f jA in measure on eachsubset 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. Theconverse is not true in general.

The embedding L1 C L1 � L0 is continuous, since convergence in normin 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 thefollowing corollary.

Corollary 5.3.3. The natural embedding of a symmetric space X into L0 iscontinuous, 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 generalconvergence in measure. Indeed, let fn D 1Œn�1;n�. Then fn ! 0 almost everywhere,

while m

�

fn >1

2

D 1 and f �n D 1Œ0;1�, i.e., ffng does not converge to 0 in measure.

Hence, the sequence ffng does not converge to 0 in any symmetric space X.

Chapter 6Embeddings. Minimality and Separability.Property .A/

In this chapter we study minimal and separable symmetric spaces. The minimalpart X0 of a symmetric space X is the closure of L1 \ L1 in X, and X is minimalif X0 D X. We show that every separable symmetric space is minimal, and theconverse is true under the additional condition �X.0C/ D 0. We consider also animportant 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 ffng 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 / Dg 2 X1. This means that the embedding operator i is closed. By the closed graphtheorem, i is a continuous (bounded) operator. utNote 6.1.2. Continuity of the embedding X1 � X2 means that

kf kX2 � ckf kX1 ; f 2 X1;

for some c > 0.


71

72 6 Embeddings. Minimality and Separability. Property .A/

The open mapping theorem implies that X1 is closed in X2 if and only if theembedding is open, i.e.,

kf kX1 � c1kf kX2 ; f 2 X1

for some real c1 > 0.For example, R0 D clL1CL1

.L1 \ L1/ is closed in L1 C L1, while L1 \ L1 isdense 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 spaceis also a symmetric space.

Proposition 6.1.3. Let Y be a nonzero linear subspace of a symmetric space Xsatisfying 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 byk � kX is a symmetric space.

Proof. Let 0 � g � f 2 clX.Y/ and let ffng be a sequence in Y such that

limn!1 kf � fnkX D 0:

Let gn D minfg; fng. Then

0 � g � gn D .g � fn/C � jf � fnj

and

limn!1 kg � gnkX � lim

n!1 kf � fnkX D 0:

Since gn � fn 2 Y, we have also gn 2 Y and g 2 clX.Y/. Thus clX.Y/ is a normedideal lattice.

Since Y satisfies condition 2, the closure clX.Y/ also satisfies the condition.Indeed, if f 2 clX.Y/, ffng � Y, is such that fn ! f in X, and g D f ı � forsome � 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 ofgenerality 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 thatg 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 ffng and fgng such that

fn " f ; gn " g; limn!1 kgn � gkL1\L1

D 0;

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 inX and lim

n!1 kgn � gkX D 0, we have also g 2 Y.

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 wasshown above, g � 1fg>ag 2 Y.

On the other hand, the set ff > bg has infinite measure for every b < a, and thus1ff>bg 2 Y.

By reducing if necessary the set A D ff > bg, we may assume that itscomplement A D R

C n A also has infinite measure. Since there exists � 2 A.m/such that ��1A 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 functiong � 1fg�ag. Thus,

g D g � 1fg>ag C g � 1fg�ag 2 Y:

utCorollary 6.1.4. The closure clX.X1/ of a symmetric space X1 in any othersymmetric 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.


Let ffng be a Cauchy sequence in X1 \ X2. Then the sequence ffng is a Cauchysequence in both X1 and X2. Since the spaces X1 and X2 are Banach spaces, thereexist 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. utThe 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 X2g: (6.2.1)

Proposition 6.2.2. The space .X1 C X2; k � kX1CX2 / 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 � kX1CX2 is a Banachspace. 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 X2g

equipped with the norm

k.f1; f2/kX1�X2 D kf kX1 C kf2kX2 ;

by the subspace

L D f.f ;�f /; f 2 X1 \ X2g � 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 kX1CX2 defined by equality (6.2) is by definition just the norm inthe 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 X1X2 is also a Banach space.

Since the subspace L is closed, the quotient space .X1 X2/=L and the sumX1 C X2 are also Banach spaces.

2: Now we show that the space X1 C X2 is a normed ideal lattice.Let g 2 X1CX2 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

kg1kX1 C kg2kX2 � kgkX1CX2 C ":

Let us set

h D f

g� 1fg¤0g:

Then f D g1h C g2h and

jg1hj � kg1k; jg2hj � kg2k:

Hence, g1h 2 X1, g2h 2 X2, and f 2 X1 C X2. Moreover,

kf kX1CX2 � kg1hkX1 C kg2hkX2 � kg1kX1 C kg2kX2 � kgkX1CX2 C ";

which in turn implies

kf kX1CX2 � kgkX1CX2 :

3: The symmetry property of the norm k � kX1CX2 follows directly from thoseproperties of k � kX1 and k � kX2 using Theorem 3.4.5 (Exercise 8). ut

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.

Definition 6.3.1. The space X0 is called the minimal part of X. A symmetric spaceX 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 withbounded support is dense in L1 \ L1. Hence, X0 D clX.F0/, and X is minimalif and only if the set F0 is dense in X. Since F0 � F1 � L1 \ L1, the same is truefor the set F1 of all simple integrable functions.

Examples 6.3.3. 1. L1 \ L1 is minimal. Indeed, .L1 \ L1/0 � L1 \ L1 bydefinition, and .L1\L1/0 is a symmetric space. Hence, L1\L1 � .L1\L1/0by Theorem 5.2.1.

2. The spaces Lp, 1 � p < 1, are minimal, since the set L1 \ L1 is dense in eachof 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 atzero,

'.0C/ D limx!0C'X.x/ � 0;

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 < 1and 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 0:

Thus f1Œ0;a� W a > 0g is a discrete uncountable subset in X, i.e., the space X isnonseparable.

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, ifX � 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; 1n �. Then

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

�

1

n

�

:

By taking n ! 1, we obtain 'X.0C/ � 1

k> 0.

c

11

Xj

_b a ba

_b a( )b[ ]a,

( )X Xx0,x 1[ ]j

0 0x

y y

x

=

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

Fig. 6.2 Inequality (6.4.1)

c

XÕL•

x

f*

f*x( )

n

n

f* 0( )x

j

0

y

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

limN!1 kf � f � 1Œ0;N�kX D 0:

Indeed, for

f DnX

iD1ai � 1Ai ; ai 2 R; mAi < 1; 1 � i � n 2 N;

we can set

a D kf k1 D max1�i�n

.jaij/; and A D1[

iD1Ai:

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 minimalsymmetric space, then the set

ff � 1Œ0;N�; f 2 F1g � 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 kL1k1ff"¤f gkX � 2kf kL1

'X."/ ! 0; " ! 0:

The continuous function f" on Œ0;N� can be uniformly approximated by poly-nomials 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 Ngis dense in X. ut

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 contin-uous norm) if

.A/ ffng � X; and fn # 0 H) kfnkX # 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/: ffng � X; fn # 0 and .fn � fnC1/fnC1 D 0 H) kfnkX # 0:

Proof. .A/ ) .A0/ is obvious.Suppose .A0/ holds. Let ffng � 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:


Property .A0/ implies khnkX # 0 as n ! 1. On the other hand,

fn D fn � 1fgn 6D0g C .fn � fn � 1fgn 6D0g/ � hn C "f1;

whence

kfnkX � khnkX C "kf1kX;

and then kfnkX # 0 as n ! 1. utTheorem 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 sequencefgng such that

gn 2 X; gn # 0; .gn � gnC1/gnCk D 0; k � 1;

and

limn!1 kgnkX D c > 0:

The sequence fgng is not fundamental. Otherwise, it must be convergent in normk � kX to some g 2 X and gn ! g in measure. Since gn # 0, we have g D 0, whichcontradicts kgnkX � c > 0. Since fgng is not fundamental in norm k � kX, we canfind b > 0 and an increasing sequence of indices nk " 1 such that

kgnk � gnkC1kX > b:

By normalizing the sequence fgnk � gnkC1g, we obtain a new sequence

hk D gnk � gnkC1

kgnk � gnkC1kX; k � 1;

such that

khkkX D 1; hk � hl D 0; k ¤ l; h D supk

hk 2 X:

Here h 2 X, since jhj � 1

bjf j.

6.5 Separability and Property .A/ 81

For every sequence ˛ D f˛kg 2 f�1; 1gN, consider the function

h.˛/ D1X

jD1˛jhj 2 X;

where h.˛/ 2 X, since jh.˛/j � jhj. Then the set

fh.˛/ W ˛ 2 f�1; 1gNg

is an uncountable discrete subset of X. This fact contradicts the separability of thespace 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 � fnkX ! 0. Since fn 2 L1 \ L1, then f 2 clX.L1 \ L1/ D X0, i.e., the spaceX is minimal. utNote 6.5.4. The minimality of the space X is equivalent to the combination of thefollowing 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 andright cutoff functions in norm k � kX.

Chapter 7Associate Spaces

In this chapter, we study associate spaces X1 of symmetric spaces X. The space X1

is defined by the duality hf ; gi D R

fg dm, f 2 X, g 2 X1, and the norm k � kX1 isinduced by the canonical embedding of X1 into the dual space X� of X. We showthat .X1; k � kX1 / is a symmetric space and that the canonical embedding of X1 intoX� 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 equippedwith 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 alsoa 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 andjuj.f / � 0 for all f 2 X, f � 0.


83

84 7 Associate Spaces

In some cases, the dual space X� of a symmetric space X is itself a symmetricspace, or to be more precise, it can be identified in a natural way with a symmetricspace.

Consider two typical examples.

Example 7.1.1. Let X D L1. Then L�1 D L1 by Theorem 2.4.1. More precisely,

for every function g 2 L1, there is ug 2 L�1 , defined by

ug.f / D1Z

0

fgdm; f 2 L1:

Then ug 2 L�1 and kugkL�

1D kgkL1

. Indeed, if u 2 L�1 , 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

u.A/ DZ

A

gdm

for all A 2 Fm with mA < 1, and hence

u.f / D1Z

0

fdu D1Z

0

fgdm D ug.f /

for all f 2 L1.m/. Thus, the embedding

W L1 3 g ! ug 2 L�1

is an isometric isomorphism between L1 and L�1 .

Example 7.1.2. Let X D L1. For every function g 2 L1, there exists a functionalug 2 L�1,

ug W L1 3 f !1Z

0

fgdm 2 R;

such that the mapping W L1 3 g ! ug 2 L�1 is a linear isometry of L1 intoL�1, i.e., kugkL�

1

D kgkL1 . However, in this case, .L1/ D fug; g 2 L1g does notcoincide with the space L�1. In other words, not all functionals u 2 L�1 have theform 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 L�1 such that u1.1Œ0;1// D 1 and u1.f / D 0 for all f 2 F0.

If u1 D ug1 for some g1 2 L1, then we would have

1Z

0

g1dm D 1 and

1Z

0

fg1dm D 0 for all f 2 F0, which is false. Thus, the embedding .L1/ � L�1 is

strict (see also Theorem 2.4.1).

Considering a symmetric space X, we would like to characterize the part of thespace X� that consists of all functionals ug 2 X�; g 2 L1 C L1, while all suchfunctions 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

ug.f / D1Z

0

fgdm; f 2 X: (7.1.2)

We define the norm of g 2 X1, by setting

kgkX1 D kugkX� : (7.1.3)

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

kgkX1 D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W f 2 X; kf kX � 1

9

=

;

< 1: (7.1.4)

7.2 The Maximal Property of Products f �g�

In order to check that the space .X1; k � kX1 / is symmetric, we need the following“maximal” property of the products f �g�.

Proposition 7.2.1. Let f ; g 2 L0, f ; g � 0. Then

1Z

0

f �g�dm D sup�2A.m/

1Z

0

.f ı �/ � g dm: (7.2.1)

Proof. First we prove the inequality

1Z

0

fg dm �1Z

0

f �g� dm: (7.2.2)

Since g � 0, there exists a sequence gn of nonnegative simple integrablefunctions such that gn " g. Each gn can be written in the form

gn DNnX

kD1an;k � 1An;k ;

where an;k � 0, An;k 2 Fm, mAn;k < 1, and

An;1 � An;2 � : : : � An;Nn :

Then

g�n D

NnX

kD1an;k � 1Œ0;m.An;k/�: (7.2.3)

By inequality (7.2.3) and Theorem 4.1.1, we obtain

1Z

0

fgn dm DNnX

kD1an;k

Z

An;k

f dm �NnX

kD1an;k

m.An;k/Z

0

f � dm

D1Z

0

f �NnX

kD1an;k � 1Œ0;m.An;k � dm D

1Z

0

f �g�n dm �

1Z

0

f �g� dm:

By Levi’s theorem, we can pass to the limit as n ! 1 and obtain (7.2.2).Since every function of the form f ı � , � 2 A.m/, is equimeasurable with f , we

have by (7.2.2),

1Z

0

.f ı �/ � g dm �1Z

0

f �g� dm: (7.2.4)

By taking the supremum in (7.2.4) over all � 2 A.m/, we obtain

sup�2A.m/

1Z

0

.f ı �/ � g dm �1Z

0

f �g� dm:

Now we prove the reverse inequality

sup�2A.m/

1Z

0

.f ı �/ � g dm �1Z

0

f �g� dm: (7.2.5)

First consider the case that there exists 0 < c < 1 such that

g�.x/ > g�.1/; for 0 < x < c;

and

g�.x/ D g�.1/; for x � c:

For every natural number N, we set a D c

Nand let " > 0.

In the same way as in Theorem 4.1.1, we obtain sets Ek, k D 1; 2; : : : ;N, suchthat 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 with1[

kD1Ek D Œ0;1/:

Since the sets Ek have the same measure mEk D a, there exist invertible measure-preserving 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/ D1X

kD1f �.�k.x// � 1Ek.x/;

which is equimeasurable to f �.x/.


Using va, we have

sup�2A.m/

1Z

0

.f ı �/ � g dm �1Z

0

vag dm D1X

kD1

Z

Ek

f � ı �k � gdm

� aNX

kD1f �.ka/g�.ka/

C1X

kDNC1

Z

Ek

f � ı � � Œg�.1/ � "�dm:

If

1X

kDNC1

Z

Ek

f � ı �dm D1Z

c

f �dm D 1;

the inequality (7.2.5) holds. If the latter integral is finite, then

sup�2A.m/

1Z

0

.f ı �/ � g dm � a1X

kD1f �.ka/g�.ka/ � "

1Z

c

f �dm:

It is easy to see that a1P

kD1f �.ka/g�.ka/ is an integral sum for the function f �g� on

.0;1/. By passing to the limit as a ! 0, we obtain

sup�2A.m/

1Z

0

.f ı �/ � g dm �1Z

c

f �g�dm � "1Z

c

f �dm:

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 setof infinite measure. ut

Note that by putting g D 1A in (7.2.1), we obtain Theorem 4.1.1 (the maximalproperty of f �) as a particular case of Proposition 7.2.1.

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

8

<

:

1Z

0

f �g�dm W f 2 X; kf kX � 1

9

=

;

; (7.2.6)

whence .X1; k � kX1 / is symmetric.We show now that X1 is a Banach space. Since X� is a Banach space and the

embedding

W X1 3 g ! ug 2 X�

is isometric, it is sufficient to check that .X1/ is closed in X�.Let gn 2 X1 and u 2 X� be such that kugn � ukX� ! 0. Then

ugn.f / D1Z

0

gnfdm ! u.f /; f 2 X: (7.2.7)

In particular, with f D 1A we have

ugn.1A/ DZ

A

gndm ! u.1A/ (7.2.8)

for every measurable set A of finite measure.Consider the set functions

gn.A/ D ugn.1A/ DZ

A

gndm; n � 1;

which are finite for mA < 1. We obtain by (7.2.8) the limit set function

.A/ D u.1A/ D limn!1gn.A/;

which also satisfies j.A/j < 1 for all A with m.A/ < 1.Since the set functions gn are � -additive, is also � -additive.Indeed the convergence in (7.2.7) is uniform on the sets ff 2 X W kf kX � cg,

c > 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 setfunction is also absolutely continuous with respect to m. Hence, by the Radon–Nikodym theorem, there exists a locally integrable function g such that

.A/ DZ

A

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

whence

ug.f / D1Z

0

fgdm D u.f /; f 2 X;

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

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

7.3 Examples of Associate Spaces

Examples 7.3.1. 1: Lp, 1 � p � 1. Let 1 � q � 1 be such that1

pC 1

qD 1.

Theorem 2.4.1 states that the mapping

W Lq 3 g ! ug 2 .Lq/ � L�p

is an isometric isomorphism (see also Examples 7.1.1 and 7.1.2). Moreover, for1 � p < 1, we have .Lq/ D L�

p , and for p D 1, the embedding .L1/ � L�1is strict.

2: L1 \ L1 and L1 C L1. For every function f 2 L1 C L1 and g 2 L1 \ L1,

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� kf kL1CL1� kgkL1\L1

: (7.3.1)

Indeed, if f D u C v, u 2 L1, v 2 L1, then

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� kukL1 � kgkL1C kvkL1

� kgkL1

� .kukL1 C kvkL1/kgkL1\L1

� kf kL1CL1� kgkL1\L1

:

The inequality (7.3.1) implies

.L1 C L1/1 L1 \ L1; .L1 \ L1/1 L1 C L1

7.3 Examples of Associate Spaces 91

and

k � k.L1CL1/1 � k � kL1\L1; k � k.L1\L1/1 � k � kL1CL1

:

On the other hand,

kgk.L1CL1/1 D supkf kL1CL1

�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� max

0

@ supkf kL1�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

; supkf kL1

�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1

A

D max.kgkL1 ; kgkL1/ D kgkL1\L1

;

and in a similar way,

kf k.L1\L1/1 D supkgkL1\L1

�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� kf kL1CL1:

Thus, the following proposition holds.

Proposition 7.3.2. 1. ..L1 C L1/1; k � k.L1CL1/1 / D .L1 \ L1; k � kL1\L1/;

2. ..L1 \ L1/1; k � k.L1\L1/1 / D .L1 C L1; k � kL1CL1/.

It is an essential fact that the associate space X1 of X is completely determinedby 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 D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W f 2 F0; kf kX � 1

9

=

;

: (7.3.2)

Hence k � kX1 D k � k.X0/1 and X1 D .X0/1. ut

7.4 Comparison of X1 and X�

We shall study now the natural embedding

W X1 ! .X1/ � X�

and find conditions under which .X1/ D X�.Theorem 6.5.3 can be completed as follows.

Theorem 7.4.1. Let X be a symmetric space. The following are equivalent:

1. X is separable.2. X is minimal and 'X.0C/ D 0.3. X has property .A/ (order continuous norm).4. .X1/ D X�.

Proof. The equivalence of conditions 1, 2, and 3 was proved in Theorem 6.5.3.3 H) 4. Let u 2 X� and fn # 0. Condition 3 implies kfnkX ! 0 and u.fn/ ! 0.

Hence the set function u defined by

u.A/ D u.1A/; A 2 F ; mA < 1;

on all sets of finite measure is � -additive. Since u is absolutely continuous withrespect to m, the Radon–Nikodym theorem implies that

u.A/ DZ

A

gdm

for some measurable function g. Hence u D ug, i.e., u 2 .X1/ and X� D .X1/.4 H) 3. Let fn # 0 in X. Then for every nonnegative g 2 X1, we have

ug.fn/ D1Z

0

fngdm # 0; n ! 1;

since fng � f1g 2 L1.If .X1/ D X�, then u.fn/ ! 0 for all u 2 X�, i.e., fn ! 0 weakly in the

�.X;X�/ topology. Since fn # 0, it also converges to 0 in the norm of X.Indeed, suppose the contrary, i.e.,

a D infn

kfnkX > 0:

7.4 Comparison of X1 and X� 93

Since the sequence ffng decreases, it follows that

�

�

�

�

�

kX

iD1˛ifi

�

�

�

�

�

X

� fk � a; ˛i � 0;

kX

iD1˛i D 1

for every convex combination of functions ffn; n � 1g. The inequality kf kX � a isalso valid for all f belonging to the norm closure of the convex hull of ffn; n � 1g.However, this contradicts the weak convergence of ffng to zero. ut

Returning to Examples 7.3.1, we see that v.X1/ D X� for X D Lp, 1 � p < 1,and the embedding v.X1/ � X� is strict for X D L1, L1\L1, and L1CL1. Otherexamples will be considered in Chapters 11 and 15 for Lorentz, Marcinkiewicz, andOrlicz spaces.

Chapter 8Maximality. Properties (B) and (C)

In this chapter, we study the second associate space X11 of a symmetric spaceX. To describe properties of the natural embedding X � X11, we two considerimportant properties .B/ and .C/ of the space X. We show that X has property .C/if and only if the natural embedding X ! X11 is isometric. If in addition to .C/, Xhas property .B/, then X D X11, i.e., the symmetric space X is maximal, and thenatural embedding X ! X11 is an isometric isomorphism between X and X11.

8.1 The Second Associate Space

The associate space X1 of a symmetric space X is also a symmetric space byTheorem 7.2.2. Hence, we can consider the associate space

X11 D .X1/1

of the symmetric space X1. This space consists of all h 2 L1 C L1 such that

khkX11 D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

hgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W g 2 X1; kgkX1 � 1

9

=

;

< 1:

If f 2 X, then for every g 2 X1,

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� kf kX � kgkX1 :


95

96 8 Maximality. Properties (B) and (C)

This means that X � X11 and

kf kX11 � kf kX; f 2 X:

Thus, the following embedding theorem holds.

Theorem 8.1.1. Let X be a symmetric space. Then

X0 � X � X11; (8.1.1)

where X0 D clX.L1 \ L1/, and X11 is the second associate space of X. Moreover,

kf kX0 D kf kX; f 2 X0

and

kf kX11 � kf kX; f 2 X:

Recall that the symmetric space X0 is called the minimal part of X, and the spaceX is called minimal if X0 D X.

Definition 8.1.2. A symmetric space X is called maximal if X D X11.

The following examples show that both inclusions in Theorem 8.1.1 may bestrict.

Examples 8.1.3. 1: The spaces Lp, 1 � p < 1, are minimal and maximal,

L0p D Lp D L11p :

These spaces are separable and 'Lp.0C/ D 0.2: The space L1 is maximal but not minimal:

L01 � L1 D L11 D L111:

The minimal part of L1 has the form L01 D R0 \ L1 by Propositions 3.5.1and 4.3.1.

Both spaces L01 and L1 are nonseparable. The space L01 is minimal, while'L01

.0C/ D 'L1.0C/ > 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 L1CL1 has the form .L1CL1/0 D R0 by Proposition 4.3.1and Example 6.3.3. Moreover,

R10 D .L1 C L1/1 D L1 \ L1:

The space L1 C L1 is nonseparable, while R0 D clL1CL1.F0/ is separable,

'R0 .0C/ D 'L1CL1.0C/ 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 monoton-ically complete) if

.B/ W ffng � X; 0 � fn "; supn

kfnkX < 1 H) fn " f

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 ffng � X be such that 0 � fn " and sup

nkfnkX < 1.

Since both X and Y1 are Banach spaces, the norms k�kX and k�kY1 are equivalentby the open mapping theorem. Hence sup

nkfnkY1 D c < 1.

Since

kfnkY1 D sup

8

<

:

1Z

0

fngdm W 0 � g 2 Y; kgkY � 1

9

=

;

;

it follows that for every 0 � g 2 Y with kgkY � 1, the functions fng are integrableand

1Z

0

fngdm � supn

kf kY1 D c < 1:

By the Fatou–Lebesgue theorem, the function supn.fng/ D .sup

nfn/g is integrable,

and hence it is finite almost everywhere. Let f D supn

fn. Then

kf kY1 D sup

8

<

:

1Z

0

fgdm W 0 � g 2 Y; kgkY � 1

9

=

;

� c < 1;

i.e., f 2 Y1 D X. Hence, the space X D Y1 has property .B/.The equality X D X11 implies X0 D .X11/0.3 H) 1. Let f 2 X11 and f � 0. Then there is a sequence ffng such that 0 �

fn 2 F0 and fn " f . Since the embedding X � X11 is continuous and X0 D .X11/0 assets, the norms k � kX0 and k � k.X11/0 are equivalent by the open mapping theorem.

Hence for some c > 0,

kf kX � c � kf kX11 ; f 2 X0 D .X11/0:

Since ffng � F0 � X0 and f 2 X11, we have

kfnkX � c � kfnkX11 � c � kf kX11 < 1

for all n � 1. Therefore, supn kfnkX < 1, and property .B/ implies f 2 X.We have shown that X11 � X. So X11 D X. ut

Note 8.2.3. The condition X0 D .X11/0 clearly holds if X is closed in X11 in normk � kX11 and in particular in the case that the embedding X into X11 is isometric.

In fact, it can be shown (Exercise 13) that kf kX D kf kX11 for all f 2 X0, andX0 D .X11/0 for every symmetric space X. Thus Theorem 8.2.2 can be completedby the implication

X has property .B/ ” X is maximal:

8.3 Embedding X � X11 and Property .C/

We consider now the embedding X � X11 in greater detail.Theorem 8.1.1 implies that X � X11 and kf kX � kf kX11 for all f 2 X. The

distinction between the norms kf kX and kf kX11 can be clarified as follows.

8.3 Embedding X � X11 and Property .C/ 99

The norm kf kX can be found using the unit ball of the dual space X�. Namely,

kf kX D supfju.f /j W u 2 X�; kukX� � 1g: (8.3.1)

The norm kf kX11 in X11 D .X1/1 is defined by the unit ball of the space X1,using (7.1.4),

kf kX11 D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W g 2 X1; kgkX1 � 1

9

=

;

: (8.3.2)

Since the embedding

W X1 3 g ! ug 2 .X1/ � X�

is isometric, we have

kf kX11 D supfju.f /j W u 2 .X1/ � X�; kukX� � 1g: (8.3.3)

Comparing (8.3.1) and (8.3.3), it is convenient to use the following definition.

Definition 8.3.1. A subspace V � X� is called norming if

kf kX D supfju.f /j W u 2 V; kukX� � 1g:

Thus the embedding X � X11 is isometric if and only if .X1/ is a normingsubspace in the dual space X�.

We shall verify this property using the following property .C/.

Definition 8.3.2. A symmetric space X is said to have property .C/ (order-semicontinuous norm) if

.C/ W ffng;� X; 0 � fn " f 2 X H) supn

kfnkX D kf kX:

Theorem 8.3.3. Let X be a symmetric space. Then the following are equivalent.

1. The natural embedding X � X11 is isometric, i.e.,

kf kX D kf kX11 ; f 2 X:

2. .X1/ is a norming subspace in X�.3. X has property .C/.

Proof. 1 ” 2 has been checked above.2 H) 3. Let " > 0 and let ffng be a sequence in X such that 0 � fn " f 2 X.If .X1/ is a norming subspace in X�, then there exists 0 � g 2 X1 such that

kgkX1 D kugkX� D 1 and

ug.f / D1Z

0

fgdm � kf kX � ":

Since fng " fg and supn

kfngkL1 � kfgkL1 < 1, we have by the Fatou–Lebesgue

theorem that fg 2 L1 and

ug.fn/ D1Z

0

fngdm "1Z

0

fgdm D ug.f /:

Hence, limn!1

kfnkX � kf kX � " and supn

kfnkX D kf kX. This means that X has

property .C/.3 H) 2. Suppose that X has property .C/.First we prove that the set

B1 D ff 2 X \ L1 W kf kX � 1g

is closed in L1.Indeed, let ffng � B1 and lim

n!1 kfn � f kL1 D 0. Then f 2 L1, and by passing

if necessary to a subsequence, we may assume that fn ! f almost everywhere. Bysetting

gn D infk�n

jfkj;

we obtain gn 2 B1 such that gn " f .Property .C/ implies

kf kX D limn!1 kgnkX � lim

n!1kfnkX � 1:

Thus f 2 B1, and hence B1 is closed in L1.We show now that for every f 2 X with kf kX > 1, there exists h0 2 X1 such that

uh0 .f / D1Z

0

fh0dm > 1;

while kuh0kX� � 1.

8.3 Embedding X � X11 and Property .C/ 101

Indeed, since f � 1Œ0;n� " f , then by property .C/,

kf � 1Œ0;n�kX ! kf kX > 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.,

1Z

0

hf � 1Œ0;n�dm > 1 and

1Z

0

hgdm � 1 for all g 2 B1:

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

uh0 .f / D1Z

0

fh0dm > 1;

while

kuh0kX� D sup

8

<

:

1Z

0

h0gdm; g 2 X; kgkX � 1

9

=

;

D

D sup

8

<

:

1Z

0

h0gdm; g 2 B1

9

=

;

� 1:

Hence, .X1/ is a norming subspace in X�. utIt is important to note that property .B/ does not imply property .C/.

Example 8.3.4. The space X D L1 with the norm kf kL1D vrai sup

xjf .x/j is

maximal,

L1 D .L1/� D L11:

Consider a new norm k � k0 on L1 defined by

kf k0 D kf kL1C f �.1/:

Since f �.1/ � kf kL1, the norms k � kL1

and k � k0 are equivalent,

kf kL1� kf k0 � 2kf kL1

:

Since

kf k0 D f �.0/C f �.1/;

the norm k � k0 is symmetric and .L1; k � k0/ is a symmetric space. Moreover,X1 D L1 and k � kX1 D k � kL1 Hence, X11 D L1 and k � kX11 D k � kL1

.The norm k � k0 on L1 does not coincide with the norm k � kL1

. For example,k1Œ0;1/k0 D 2 and k1Œ0;1/kL1

D 1.On the other hand, restrictions of the norms k � k0 and k � kL1

on the subspaceL01 of L1 coincide.

We have 1Œ0;n/ " 1Œ0;1/, while k1Œ0;n/k0 D k1Œ0;n/kL1D 1 < 2 D k1Œ0;1/k0. The

space .L1; k � kL0 / fails to have property .C/.

By combining both properties .B/ and .C/, we obtain the following result.

Theorem 8.3.5. Let X be a symmetric space. Then the following are equivalent.

1. The embedding X � X11 is an isometry of .X; k � kX/ onto .X11; k � kX11 /.2. X has the following property:

.BC/ffng � X; 0 � fn " and sup

nkfnkX < 1 H) fn " f and

kfnkX " kf kX for some f 2 X:

Proof. Property .BC/means that the space X has both properties .B/ and .C/. Thus1 ” 2, by Theorems 8.3.3 and 8.2.2. utNote 8.3.6. Fatou’s property. All the equivalent conditions in Theorem 8.3.5 canbe reformulated as follows:

.F/ Wffng � X; fn ! f almost everywhere and sup

nkfnkX < 1 H) f 2 X

and kf kX � limn

kfnkX:

In the case X D L1, property .F/ is just the statement of Fatou’s theorem.Therefore, .F/ is known as Fatou’s property.

8.4 Property .AB/. Reflexivity 103

8.4 Property .AB/. Reflexivity

We consider now symmetric spaces for which

X0 D X D X11:

Definition 8.4.1. A symmetric space X is said to have property .AB/ if

.AB/ W 0 � fn "; fn 2 X; and supn

kfnkX < 1 H) fn " f 2 X and

kf � fnkX ! 0 for some f 2 X:

Note 8.4.2. By Theorems 7.4.1 and 8.2.2, the following are equivalent:

1. X has the property .AB/.2. X has both the properties .A/ and .B/.3. X is minimal and maximal, and 'X.0C/ D 0.4. X is separable and maximal.

Moreover, there are other important equivalents to property .AB/. Some of themare mentioned in Complement 5.

If the above conditions hold, then X0 D X D X11 and k � kX D k � kX11 , since .A/implies .C/.

Using .AB/ in X and in its associate space X1, we can describe the class ofreflexive symmetric spaces.

Recall that for every Banach space X, there is a canonical embedding

W X 3 f ! f 2 X��;

where f is a bounded linear functional on X�, defined by

f .u/ D u.f /; u 2 X�:

The embedding W X ! X�� is isometric. The space X is called reflexive if

.X/ D X��:

If X is reflexive, then the dual space X� is also reflexive.

Theorem 8.4.3. A symmetric space X is reflexive if and only if both the space Xand its associate space X1 have property .AB/.

Proof. Consider the canonical mappings

X

i

κ X∗∗

υ∗

X1

υ

X11 υ1X1∗ X∗

(8.4.1)

Here i and are canonical embeddings, and the mapping W X1 ! X� isdefined by

g.f / D1Z

0

fgdm; f 2 X; g 2 X1:

Further, 1 W X11 ! X1� is the analogous mapping for the space X1, and � W.X�/� ! .X1/� is the dual mapping for W X1 ! X�.

If X has property .B/, then i.X/ D X11.If X1 has property .A/, then 1.X1/ D X1�.If X has property .A/, then W X1 D X� is an isometry of X1 onto X�, and hence

� is an isometry of X�� onto X1�.If all three mappings i, 1, and .�/�1 are surjective, then .X/ D X��, i.e., the

space X is reflexive.Conversely, suppose that a symmetric space X is reflexive.We check that X has property .AB/.Let ffng be an increasing sequence of nonnegative functions in X that is bounded

in norm k � kX:

ffng � X; 0 � fn " and a D supn

kfnkX < 1:

Since X is identified with the second dual space X�� D .X�/� of X�, the ball

Va D ff 2 X W kf kX � ag

is weakly compact in X, i.e., it is compact in the weak topology �.X;X�/ D�.X��;X�/.

Therefore, the set of limit points of the sequence ffng is a nonempty subset of Va.For every limit point g of the sequence ffng, we can choose a subnet ffn˛ g that

converges weakly to g. We may assume that ffn˛ g is increasing and that for each n,there exists n˛ such that n˛ > n.

8.4 Property .AB/. Reflexivity 105

We set f D supn

fn. Then fn " f and f D limn!1 fn almost everywhere, whence

sup˛

fn˛ � supn

fn D f

and g D f 2 Va � X. Thus, f is the only weak limit point of the sequence ffng, andhence fn converges weakly to f in X.

We have already checked property .B/. To verify property .A/, we shall showthat the sequence ffng used above converges to f in the norm of X. Indeed, if thiswere false, then by passing if necessary to a subsequence, we could assume that

kf � fnkX � c > 0 (8.4.2)

for some c > 0 and all n D 1; 2; : : :. On the other hand, the weak convergence fn tof implies that there exists g of the form

g DkX

jD1�jfnj ; 0 � �j � 1;

kX

jD1�j D 1; k 2 N;

such that

kf � gkX <1

2c:

By choosing n0 > nj for all 1 � j � k, we have

0 � f � fn0 � f � g:

That, in turn, implies

kf � fn0kX <1

2c;

in contrast to (8.4.2).Thus, kfn � f kX ! 0 as n ! 1.We have shown that the space X has property .AB/. ut

Corollary 8.4.4. A symmetric space X is reflexive if and only if both X and X1 areseparable and maximal.

Example 8.4.5 (Lp, 1 � p � 1).

1: For 1 < p < 1, Lp are reflexive. The space Lp has property .AB/, and

L�p D .L1p/ D Lq; 1 < q D p

p � 1 < 1:

2: For p D 1, the space L1 has property .AB/, but the associate space L1 D .L1/1

does not have property .AB/ . The space L1 is not reflexive.3: For p D 1, the space L1 does not have property .A/, although the associate

space L1 D .L1/1 has property .AB/. The space L1 is not reflexive.

Exercises

9. Embedding theorems for symmetric spaces on Œ0; 1�.Let X D X.0; 1/ be a symmetric space on Œ0; 1� in the sense of Complement 2,

and let 'X denote the fundamental function of X.Show that

a. L1.0; 1/ � X � L1.0; 1/ � L0.0; 1/.b. If 'X.1/ D 1, then both embeddings L1.0; 1/ � X and X � L1 are contractions.c. X D L1 as sets ” 'X.0C/ > 0.d. X D L1 as sets ” '0

X.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

n!1'X.n/

n > 0 ” X.N/ D l1 as sets.

d.

'X.1/ < 1 ”�

X D c0 as sets; if 1N 62 XX 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

n!1 kf � min.f ; n/kX D 0 for every function f 2 X.

• limn!1 kf � � min.f �; n/kX D 0 for every function f 2 X.

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


• X is minimal, i.e., X D clX.l1/.• X is separable.• X has property .A/.• lim

n!1 kf � � f � � 1f1;2;:::;ngkX D 0 for every function f 2 X.

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

X11 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 arecontractions.Hint: Use the inequality

'X.1/ � kf kL1CL1� kf kX; f 2 X

and the duality of L1 \ L1 and L1 C L1, also the relations .X0/1 D X1 and theequality '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 ifX has property .B/.


14. Order completeness.Recall that a linear ideal lattice X is said to be order complete if every order-

bounded subset E � X has least upper bound sup E and greatest lower bound inf Ein X. This means that if the set fg 2 X W g � f for all f 2 Eg is not empty, itcontains the least element sup E, and also, if the set fg 2 X W g � f for all f 2 Egis 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˛; ˛ 2A 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.

Recall that a sequence ffng in X is said to be order convergent to f 2 X (fn.o/�! f

in X) if there exist two sequences fhng and fgng in X such that fhng increases, fgngdecreases, while

supn

hn D f D infn

gn:

Show that

a. Order convergence in the space L0 coincides with almost everywhere con-vergence. For every symmetric space X, the order convergence in X impliesconvergence in measure and almost everywhere convergence.

b. fn.o/�! 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 supn

hn D infn

gn;

where gn D supk�n

fk and hn D infk�n

fk. That is, f D lim sup fn D lim inf fn.

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

ffnk g such that fnk

.o/�! 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 allf 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 Xand 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 clXY is a symmetric space with respect to the norm inducedby k � kX.

c. Let X1 and X2 be two symmetric spaces on Œ0;1/, and suppose the space X1CX2

consists of all functions f of the form

f D f1 C f2; f1 2 X1; f2 2 X2;

and

kf kX1CX2 D inffkf1kX1 C kf2kX2 ; f1 2 X1; f2 2 X2g:

Then k � kX1CX2 is a symmetric norm on X1 C X2 and .X1 C X2; k � kX1CX2 / is asymmetric 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 inthe first part of inequalities (5.2.2) can be reduced to C D 1. A way to remove theconstant in (5.2.2) is described in Exercise 13. It uses the second part of (5.2.2) andthe 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]. Thisand 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 caseof symmetric spaces.


The maximal property of products f �g� used in Section 7.2 for descriptionof 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 generalmeasure spaces .˝;/ (see Complement 1), for example, if the space .˝;/ hastwo atoms a; b 2 ˝ with unequal positive measures .fag/ ¤ .fbg/. Notions ofresonant 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� andconditions 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, Chapters14–16]; 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. Symmetricspaces with property .AB/ are called KB-spaces [75]. These and other properties ofsymmetric 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 indetail in [36, Section 1.a] for general Banach lattices.

Order convergence may be considered on general ordered spaces. Some relatedresults in the case of symmetric spaces are given in Exercise 15. The relationshipbetween .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 IIILorentz 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

kf k�W D1Z

0

f �dW;

where in the first case, the weight function W.x/ D x corresponds to the usualLebesgue 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 k�W < 1g

with the norm k � k�W is a symmetric space. It is called a Lorentz space with Lorentzweight W.

Marcinkiewicz spaces MW with weight functions W arise as associate spaces ofthe Lorentz spaces �W , i.e., �1

W D MW and M1W D �W , and the norm k � k�1

WD

k � kMW can be written as

kf kMW D supx

0

@

1

W.x/

xZ

0

f �dm

1

A D kW�f ��kL1;

112 III Lorentz and Marcinkiewicz Spaces

where W�.x/ D x

W.x/and f ��.x/ D 1

x

xZ

0

f �dm is the so-called maximal function

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

kf kMW D supx

0

@

xZ

0

f �dm

1

A D1Z

0

f �dm D kf kL1 ;

and if W.x/ D x, then

kf kMW D supx

0

@

1

x

xZ

0

f �dm

1

A D f ��.0/ D f �.0/ D kf kL1:

Marcinkiewicz spaces are introduced for a wider class of weight functionsthan Lorentz spaces. Namely, the requirement of concavity W is replaced bythe following weaker condition: both functions W� and W are increasing. Suchfunctions are called quasiconcave. The significance of the class of quasiconcavefunctions is clarified by the following results.

A function V is the fundamental function 'X of a symmetric space X if and onlyif it is quasiconcave. The equality V D 'X holds just for Marcinkiewicz spacesX D MV�

.Further, if X is a symmetric space with a fundamental function V , eV is the least

concave majorant of the quasiconcave function V , and V�.x/ D x

V.x/; x > 0, then

�0

eV� X0 � X � X11 � MV�

;

where

kf k�eV

D kf k�0

eV

� kf kX0 D kf kX; f 2 �0

eV

and

kf kX � kf kX11 � kf kMV�; f 2 X:

The function V� and the function V itself need not be concave. However, thereexists a norm k � k0 that is equivalent to the norm k � k�

eVand 'X D V . Thus the

minimal part �0

eVof the Lorentz space �

eV is the smallest, and the Marcinkiewiczspace 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 newnorm k � k0 that is equivalent to the original k � kX and has a concave fundamentalfunction.

Note also that in the case V.x/ D V�.x/ D px, x � 0, we have

�V � L2 � MV ;

where the embeddings are strict.

Chapter 9Lorentz 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 alsodescribe 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

x!0C W.x/ � W.0/ D 0 exists, and both cases W.0C/ D 0

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

kf k�W WD1Z

0

f �dW < 1: (9.1.1)


115

116 9 Lorentz Spaces

0

=

x

y

0 x

y

W

W

x

0 0+

( )

( )

W x( )

W 0 0>+( )

Fig. 9.1 Two cases of Lorentz functions

Here f � is the decreasing rearrangement of jf j, and1R

0

f �dW is the improper

Riemann–Stieltjes integral of the decreasing function f � with respect to theincreasing function W on R

C.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,

kf k�W D1Z

0

f �dW D f �.0/W.0C/C1Z

0

f �W 0dm; (9.1.3)

where the latter integral is an improper Riemann integral of the decreasing functionf �W 0. The derivative W 0 of W exists almost everywhere on R

C, and

W.x/ D W.0C/CxZ

0

W 0dm; x > 0:

Theorem 9.1.2. The space .�W ; k � k�W / is a symmetric space, and

kf k�W D1Z

0

f �dW D1Z

0

W ı �f �dm; f 2 �W : (9.1.4)

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

kf k�W D kf kL1W.0C/C sup

�2A.m/

8

<

:

1Z

0

.jf j ı �/W 0dm

9

=

;

; (9.1.5)

9.1 Definition of Lorentz Spaces 117

it follows that

kf1 C f2k�W D W.0C/.f1 C f2/�.0/C

1Z

0

.f1 C f2/�W 0dm

D W.0C/kf1 C f2kL1C sup�2A.m/

8

<

:

1Z

0

.jf1 C f2j ı �/W 0dm

9

=

;

� W.0C/kf1kL1C sup�2A.m/

8

<

:

1Z

0

.jf1j ı �/W 0dm

9

=

;

CW.0C/kf2kL1C sup�2A.m/

8

<

:

1Z

0

.jf2j ı �/W 0dm

9

=

;

D kf1k�W C kf2k�W :

Other properties of the norm and all conditions of symmetric spaces follow directlyfrom (9.1.1).

Let us check the completeness of the space �W . First, we note that

�W � L1 C L1:

Indeed, if W 0 � 0, then �W D L1 � L1 C L1. If W 0.x/ > 0 for some x 2 .0;1/,then for every f 2 �W ,

xZ

0

f �dm � 1

W 0.x/

xZ

0

f �W 0dm < 1;

i.e., f 2 L1 C L1. Thus �W � L1 C L1.

Suppose that ffkg � �W and1P

kD1kfkk�W D C < 1. Since �W � L1 C L1, we

have

1X

kD1kfkkL1CL1

< 1:

Hence, the series1P

kD1jfkj converges almost everywhere to f 2 L1 C L1. Proposi-

tion 7.2.1 and equality (9.1.5) imply that for every � 2 A.m/,

1Z

0

NX

kD1.jfkj ı �/W 0dm �

NX

kD1

1Z

0

f �k W 0dm � C:

Then by Levi’s theorem,

1Z

0

.f ı �/W 0dm � C; � 2 A.m/

and

sup�2A.m/

1Z

0

.f ı �/W 0dm D1Z

0

f �W 0dm � C:

If W.0C/ D 0, then

1Z

0

f �dW D1Z

0

f �W 0dm < 1;

and therefore f 2 �W .

If W.0C/ > 0, then �W � L1, and convergence of1P

kD1kfkk�W implies

convergence of1P

kD1kfkkL1

. Hence kf kL1D f �.0/ < 1, and by (9.1.3),

kf k�W < 1, i.e., f 2 �W .

We have shown that if1P

kD1kfkk�W < 1, then the series

1P

kD1fk converges in norm

k � k�W to some f 2 �W . This means that �W is complete.Thus, the space .�W ; k � k�W / is a Banach space.Using substitutions x D �f �.y/, y D f �.x/, we have

kf k�W D1Z

0

f �dW D f �.x/W.x/j10 �1Z

0

W.x/df �.x/

D0Z

1W.x/df �.x/ D

1Z

0

W.�f �.y//dy D1Z

0

W ı �f �dm:

Thus (9.1.4) holds. ut

9.2 Maximality. Fundamental Functions of Lorentz Spaces 119

9.2 Maximality. Fundamental Functions of Lorentz Spaces

Proposition 9.2.1. Every Lorentz space .�W ; k � k�W / is a maximal symmetricspace, �W D �11

W and k � k�W D k � k�11W

.

Proof. According to Theorem 8.3.5, it is enough to verify property .BC/.Suppose that 0 � fn ", fn 2 �W , and sup

nkfnk�W D a < 1. By definition,

kfnk�W D1Z

0

f �n dW D

1Z

0

f �n dW ;

where the measure W on RC is defined by (9.1.2). In other words, fn 2 L1.W/

and

supn

1Z

0

f �n dW D a < 1:

By Fatou’s theorem, g D supn

f �n is integrable in measure W and

1 >

1Z

0

gdW D supn

1Z

0

f �n dW D a D sup

nkfnk�W :

Since fn is equimeasurable with f �n for all n, the function f D sup

nfn is equimeasur-

able with g D f �. Thus, f 2 �W and kf k�W D supn

kfnk�W . ut

Note 9.2.2. In Chapter 11 below we describe the associate space �1W of �W. It

coincides with the Marcinkiewicz space MW. Its associate space in turn coincideswith �11

W D M1W D �W. This equality also implies that all Lorentz spaces �W are

maximal.

It is easy to find the fundamental functions of Lorentz spaces �W :

'�W .x/ D1Z

0

.1Œ0;x�/�dW D

xZ

0

dW D W.x/

for all x > 0 and '�W .0/ D W.0/ D 0, i.e.,

'�W D W: (9.2.1)

Thus, for a given concave function W on RC with W.0/ D 0, we have a

symmetric space X D �W such that 'X D W.


Fig. 9.2 'L1\L1

0

V ( )x

1

1

y

x

Example 9.2.3. Let us set

W.x/ D .x C 1/ � 1.0;1/.x/; x � 0:

Then the Lorentz space �W with this concave weight function W has the norm

kf k�W D f �.0/C1Z

0

f �dm D kf kL1C kf kL1 ;

i.e., the Lorentz space coincides with the space L1 \ L1 equipped with the normk � kL1

C k � kL1 .On the other hand, the original norm maxfk�kL1

; k�kL1g induces the fundamentalfunction

'L1\L1 .x/ D maxfx; 1.0;1/.x/g; x � 0;

which is not concave and is even convex (see Fig. 9.2).

This example shows that the fundamental function 'X of the symmetric space Xis not necessarily concave.

Theorems 10.4.2 and 11.2.2 below describe the class of all fundamental functionsof all symmetric spaces.

9.3 Minimal and Separable Lorentz Spaces

In this section we study minimality and separability conditions for a Lorentz space�W . The conditions are expressed in terms of fundamental functions '�W D W,namely by means of the limits

W.0C/ D limx!1 W.x/ 2 Œ0;1/ and W.1/ D lim

x!1 W.x/ 2 .0;1�:

9.3 Minimal and Separable Lorentz Spaces 121

Recall that a symmetric space X is minimal if X coincides with its minimal part

X0 D clX.L1 \ L1/ D clX.F0/;

where F0 is the set of all simple functions with bounded support. The space X isseparable if and only if it is minimal and 'X.0C/ D 0 (Theorem 6.5.3).

Theorem 9.3.1. Let �W be a Lorentz space with a weight W.

1. The minimal part �0W of �W has the form

�0W D �W \ R0 D ff 2 �W W f �.1/ D 0g:

2. �W is minimal if and only if W.1/ D 1, where

W.1/ D 1 ” 1Œ0;1/ 62 �W ” L1 6� �W :

3. �W is separable if and only if W.1/ D 1 and W.0C/ D 0.

Proof. 1: By Proposition 4.3.1, .L1 C L1/0 D R0, whence

�0W � R0 \ �W D ff 2 �W W f �.1/ D 0g:

To prove the reverse inclusion, suppose that f 2 R0 \ �W .Since �0

W and �W \ R0 are symmetric spaces, we may assume without lossof generality that f D f �. Consider upper cutoff functions min.f �; n/ and thedifferences

gn D f � � min.f �; n/:

Since f � is decreasing, gn D .f � � n/ � 1ff �>ng are also decreasing, i.e., g�n D gn

(Fig. 9.3).

x x0 x0 0

y y y

n

f * min ( f *, n)

n

*gn

anan

an

= gn

Fig. 9.3 Upper cutoffs minff �; ng and differences gn D f � � min.f �; n/

Using the measure W defined by (7.1.2) on RC, we have

kf k�W D1Z

0

f �dW D1Z

0

f �dW < 1

and

kgnk�W D1Z

0

g�n dW D

1Z

0

gndW DanZ

0

gndW < 1;

where the values an are determined by the equality ff � � ng D Œ0; an�. Since an # 0and gn ! 0 almost everywhere, we have

0 � gn � f � 2 L1.RC; W/:

By Lebesgue’s dominated convergence theorem,

kgnk�W D kgnkL1.RC;W /D

anZ

0

gndW ! 0; n ! 1;

i.e., the function f � is approximated in norm by its upper cutoff functions min.f �; n/.Next, we show that f � is also a norm limit of its right cutoff functions f � � 1Œ0;n�,

n � 1 (Fig. 9.4). Replacing f � by min.f �; n/, we may assume that the function f �is bounded. Let

hn D f � � f � � 1Œ0;n/; n � 1:

The functions hn are decreasing and right-continuous on Œn;1/, and theirdecreasing rearrangements h�

n are

h�n .x/ D hn.x C n/; x 2 R

C; n � 1:

0 x

y y

n 0 xn

nb n

n

b

f*f .

01

,[ ]*

y

0 x

nn

bh*

Fig. 9.4 Right cutoffs f � � 1Œ0;n/ and hn D f � � f � � 1Œ0;n/

9.3 Minimal and Separable Lorentz Spaces 123

Recall that f 2 R0, i.e., f �.1/ D 0. Then

bn D h�n .0/ D hn.n/ D f �.n/ # 0; n ! 1;

and

h�n � f �, h�

n # 0; n ! 1:

Therefore,

khnk�W D1Z

0

h�n dW D

1Z

0

h�n dW ! 0; n ! 1:

In the case W.0C/ D W.0/ > 0, it is important that h�n .0/ D bn # 0, n ! 1.

This condition also ensures the uniform convergence of h�n # 0 on R

C. Hence

khnkL1D kh�

n kL1D f �.n/ D bn # 0; n ! 1;

i.e., hn ! 0 in norm of the space �W \ L1.Thus, f � 2 cl�W .F0/ for all f � 2 �W \ R0, i.e., �0

W � �W \ R0.

2: We have shown that

�0W D cl�W .F0/ D �W \ R0;

and

�0W D �W ” �W � R0:

The latter condition is equivalent to 1Œ0;1/ 62 �W , and hence to the condition

k1Œ0;1/k�W D1Z

0

dW D W.1/ D 1:

3: By Theorem 6.4.1, the space �W is separable if and only if it is minimal and

'�W .0C/ D W.0C/ D 0;

while �W is minimal if and only if

W.1/ D 1:

The theorem is proved. ut

9.4 Four Types of Lorentz Spaces

We shall summarize the results from previous sections in the following Table 9.1.Let us consider each of the four cases shown in more detail.Case (1). W.0C/ D 0, W.1/ D 1.Since W.1/ D 1, it follows by Theorem 9.3.1 that �W is minimal with

'�W .0C/ D W.0C/ D 0:

This implies separability of �W (Theorem 6.5.3). By Theorem 6.5.3, conditions .1/are equivalent to property .A/ and to the equality .�1

W/ D ��W .

Example. �W D L1 with W.x/ D x, x � 0.Case (2). W.0C/ > 0, W.1/ D 1.As in case .1/, the condition W.1/ D 1 ensures minimality of the Lorentz

space �W . However, �W is nonseparable, since the fundamental function '�W D Wsatisfies '�W .0C/ D W.0C/ > 0 (Theorem 6.5.3).

The condition W.0C/ > 0 implies

kf k�W � f �.0/W.0C/ D W.0C/kf kL1

for all f 2 �W . Hence, �W � L1.On the other hand,

k1.0;1/k�W D1Z

0

dW D W.1/ D 1;

i.e., 1.0;1/ 62 �W , and therefore �W 6D L1. Thus,

�W � L1 \ R0 D L01:

Example. �W D L1 \ L1 with W.x/ D .x C 1/ � 1.0;1/.x/, x � 0.Case (3). W.0C/ D 0, W.1/ < 1.

Table 9.1 Four types ofLorentz spaces

W.0C/ D 0 W.0C/ > 0W.1/ D 1 .1/ .2/

�W � R0; �W � L1 \ R0,

�W 6� L1, �W is minimal

�W is separable �W is nonseparable

W.1/ < 1 .3/ .4/

�W L1, �W H L1,

�W ¤ L1 �W is not minimal

�W is not minimal

9.4 Four Types of Lorentz Spaces 125

The space �W is not minimal, since W.1/ < 1 (Proposition 9.3.1). Moreover,

k1.0;1/k�W D W.0C/C1Z

0

dW D W.1/ < 1;

i.e., 1.0;1/ 2 �W , and hence L1 � �W .The condition W.0C/ D 0 implies �W 6D L1.Example. �W D L1 C L1 with W.x/ D minfx; 1g, x � 0. In this case,

kf k�W D1Z

0

f �dW D1Z

0

f �W 0dm D1Z

0


Case (4). W.0C/ > 0, W.1/ < 1.These conditions imply respectively �W � L1 and �W L1, i.e., �W D L1.

The inequalities

0 < W.0C/ � W.x/ � W.1/ < 1

imply

W.0C/kf kL1� kf k�W � W.1/kf kL1

;

where kf kL1D f �.0/.

Example. �W D L1 with W.x/ D 1.0;1/.x/, x � 0. Clearly, the space�W D L1 is not minimal, L01 D L1 \ R0.

Example 9.4.1. Spaces Lp, 1 < p < 1. The fundamental function of the space Lp

is 'Lp.x/ D x1p for x � 0.

We set Wp D 'Lp , 1 < p < 1, and consider the family of Lorentz spaces �Wp ,where 1 < p < 1.

The functions Wp are concave, Wp.0C/ D 0, and W.1/ D 1. Thus, the Lorentzspaces �Wp as well as the spaces Lp are separable. We shall prove in Chapter 12,Example 12.3.3, that �Wp � Lp, 1 < p < 1, and all the inclusions are strict.

Chapter 10Quasiconcave Functions

In this chapter we study quasiconcave functions in conjunction with fundamentalfunctions of symmetric spaces. We show that every concave function is alsoquasiconcave, and conversely, for every quasiconcave function V , there exists its

least concave majorant eV , satisfying inequality1

2eV � V � eV . We also show that

fundamental functions of symmetric spaces are quasiconcave.

10.1 Fundamental Functions and Quasiconcave Functions

The class of quasiconcave functions that we begin to study now will used inChapter 11 for construction of Marcinkiewicz spaces. It contains all Lorentz weightfunctions and consists precisely of all fundamental functions 'X of the symmetricspace X.

Definition 10.1.1. A function V W RC ! RC is called quasiconcave if

1. V.0/ D 0 and V.x/ > 0 for x > 0.2. V.x/ is increasing on .0;1/.

3.V.x/

xis decreasing on .0;1/.

If V is quasiconcave, then the function V� defined by

V�.x/ D x

V.x/; x > 0; (10.1.1)

and extended by zero at x D 0 is also quasiconcave, and .V�/� D V .This follows by interchanging the roles of conditions 2 and 3 for V and V� in the

above definition.


127

128 10 Quasiconcave Functions

0 0x

y y

V

V

x

( )

( )

1

x1

x2 V

V( )

( )

x1

x2

V( )x V( )x

2x xx1 2x

Fig. 10.1 V.x/ is quasiconcave

Fig. 10.2 Convexquasiconcave V.x/

y

x 00 1

V ( )x*

V( )x

1

1

1

x

y

Proposition 10.1.2. If V W RC ! R

C is concave on .0;1/ and satisfiescondition 1 from Definition 10.1.1, then it is quasiconcave.

Proof. Indeed, if a concave function V is defined on Œ0;1/ and satisfies condition 1,

then it is increasing andV.x/

xis decreasing (see Fig. 10.1). Consequently, V is

quasiconcave.ut

10.2 Examples of Quasiconcave Functions

Examples 10.2.1. 1: Let V be defined by

V.x/ D max.1; x/ � 1.0;1/.x/; x � 0:

Then V is quasiconcave, but it is not concave on .0;1/. The correspondingfunction

V�.x/ D min.1; x/ � 1.0;1/.x/; x � 0; (10.2.1)

is concave (Fig. 10.2).2: V.x/ D p

x2 C 1 � 1.0;1/ is quasiconcave, although it is convex (Fig. 10.3).

The corresponding function V�.x/ D xpx2 C 1

is concave.

10.2 Examples of Quasiconcave Functions 129

Fig. 10.3V.x/ D p

x2 C 1 � 1.0;1/ isboth quasiconcave andconvex

0 01

1 1=

y y

x x

V( )x

V*( )x

y x

0

1

( )( )( )

X x1 11 2 3b

1bVVV

y

2b3b

0

1

y y

V

y=x

x( )

=x+1

b b

Fig. 10.4 �0.eV/ is the closed convex hull of �0.V/

Note 10.2.2. In both previous examples, the function

eV.x/ D .x C 1/ � 1.0;1/; x � 0;

is the least concave function that majorizes the function V.x/ on .0;1/.Indeed, let 0 � bn " 1. Then the hypograph

�0.eV/ D f.x; y/ 2 R2 W x � 0; y � x C 1g

of eV contains the hypographs

�0.Vn/ D f.x; y/ 2 R2 W 0 � x � bn; y � x

V.bn/ � 1bn � 1 C 1g

of the restriction Vn D VjŒ0;bn� of V on Œ0; bn�, n D 1; 2; : : :. Since bn " 1,

�0.V1/ � �0.V2/ � : : : �1[

nD1�0.Vn/ � �0.eV/;

and �0.eV/ is the closed convex hull of the hypograph �0.V/ (see Fig. 10.4).

Proposition 10.2.3. Let V1;V2 be two quasiconcave functions. Then

Vmax.x/ D max.V1.x/;V2.x// and Vmin.x/ D min.V1.x/;V2.x//

are quasiconcave functions.

Proof. Clearly, Vmax.0/ D 0, Vmin.0/ D 0 and Vmax.x/ > 0; Vmin.x/ > 0 for x > 0.Let x1; x2 2 R

C, 0 < x1 < x2. Then V1.x1/ � V1.x2/ and V2.x1/ � V2.x2/.Therefore,

Vmax.x1/ D max.V1.x1/;V2.x1// � max.V1.x2/;V2.x2// D Vmax.x2/

and

Vmin.x1/ D min.V1.x1/;V2.x1// � min.V1.x2/;V2.x2// D Vmin.x2/;

i.e., the functions Vmax and Vmin are increasing on .0;1/.On the other hand,

V1.x1/

x1� V1.x2/

x2and

V2.x1/

x1� V2.x2/

x2:

Therefore,

Vmax.x1/

x1D max

�

V1.x1/

x1;

V2.x1/

x1

�

� max

�

V1.x2/

x2;

V2.x2/

x2

�

D Vmax.x2/

x2

and

Vmin.x1/

x1D min

�

V1.x1/

x1;

V2.x1/

x1

�

� min

�

V1.x2/

x2;

V2.x2/

x2

�

D Vmin.x2/

x2:

Thus the functionsVmax.x/

xand

Vmin.x/

xare decreasing on .0;1/, and Vmax.x/ and

Vmin.x/ are quasiconcave. ut

10.3 The Least Concave Majorant

For every quasiconcave function V , there exists eV W RC ! RC with the following

properties:

1. eV is concave on .0;1/.2. V � eV .3. eV is the least function satisfying 1 and 2.

The function eV is called the least concave majorant of V . It can be constructedas follows.

Let �0.V/ be the hypograph of V ,

�0.V/ D f.x; y/ 2 R2; x � 0; V.xC/ � y � 0g;

10.3 The Least Concave Majorant 131

Fig. 10.5 The least concavemajoranteV of V

0 x

y

V x( )

V~

x( )

Fig. 10.6 V.x/ andeV.x/

0 1

1

y

x

y = x

y = x + 1

and let e� 0.V/ be the closed convex hull of �0.V/.Then there is a unique function eV whose hypograph (Fig. 10.5)

� .eV/ D f.x; y/ 2 R2; x � 0; eV.x/ � y � 0g;

coincides with e� .V/.Condition 3 in Definition 10.1.1 provides that eV is finite on Œ0;1/.For example, refer to Note 10.2.2. The function eV.x/ D .x C 1/ � 1.0;1/.x/;

x � 0; is the least concave majorant for two convex quasiconcave functions fromExample 10.2.1 (Fig. 10.6).

Theorem 10.3.1. Let V be a quasiconcave function and eV the least concavemajorant of V. Then

1

2eV.x/ � V.x/ � eV.x/; x � 0: (10.3.1)


Fig. 10.7 A finite “intervalof nonconcavity”

0 x

yV( )b

V( )aA

D

B

C

E

a b

Fig. 10.8 A maximal finite“interval of nonconcavity”

x0

y

E' E

D

D''

D'A

B

C'C

C''

E''

a b

Proof. If eV ¤ V , then there exists an interval Œa; b� on which the graph of V liesbelow the corresponding chord NAB (Fig. 10.7), i.e., Œa; b� is an “interval of concavity”for V .

We may assume that each “interval of nonconcavity” for V is maximal, i.e., a isminimal and b is maximal for the considered part of the graph V .

There are two possibilities:

1: All maximal “intervals of nonconcavity” are finite.2: To the right of all possible finite “intervals of nonconcavity” there is an infinite

such interval Œa;1/; a � 0.

In both cases, it is sufficient to verify the inequality1

2eV � V on each interval of

the mentioned types.Let the interval Œa; b� be a finite “interval of nonconcavity” for V (Fig. 10.8).

By assumption, V increases andV.x/

xdecreases. Therefore, the graph of V is

not lower than the segment NAD, and it is not lower than the segment NDB. Thus it isenough to show that CD � DE.

10.3 The Least Concave Majorant 133

Denote by ˛ and ˇ the measures of angles †CAD and †DOE. Then

AD � OE and ˛ � ˇ

imply

CD D AD � tan˛ � OE � tanˇ D DE;

whence1

2eV � CD � V on Œa; b�.

Recall that the least concave majorant of a quasiconcave function V is the uniquefunction eV on Œ0;1/ such that the hypograph

�0.eV/ D f.x; y/ 2 R2 W x � 0; 0 � y � eV.x/g

coincides with the closed convex hull of the hypograph �0.V/,

�0.V/ D f.x; y/ 2 R2 W x � 0; 0 � y � V.x/g:

Consider the set fV D eVg and set aV D supfV D eVg 2 Œ0;1�.If aV D 1, we choose a sequence fbng such that

bn 2 fV D eVg and 0 � bn " 1

and consider the restrictions

Vn D VjŒ0;bn� and eVn D eVjŒ0;bn�:

Since V.bn/ D eV.bn/, we can apply the first part of the proof to any “interval ofnonconcavity” in Œ0; bn�. Thus, (10.3.1) holds for all x 2 Œ0; bn� and all n, i.e., for allx 2 Œ0;1/.

Turning to the case aV < 1, we have two parallel lines

y D kx and y D V.aV/C k.x � aV/;

where

k D limx!1

V.x/

xD inf

x�aV

V.x/

x:

By construction,

kx � V.x/ � eV.x/ D V.aV/C k.x � aV/; x � aV ;

Fig. 10.9 The case aV < 1

0 a

aV( )

y

x

Fig. 10.10 The case aV < 1

x

y

0 a

aV

E

DA =

C

( )

xV~( )

x

kx

V

y

( )

and hence

eV.x/ � V.x/ � CD D V.aV/C k.x � aV/; x � aV

(see Figs. 10.9 and 10.10).The inequality CD � DE is proved in a similar way as in the first part of the

proof.

Thus1

2eV � V . ut

Note 10.3.2. Figure 10.10 shows that the equality CD D DE is possible only in thecase a D 0. For example, if V.x/ D max.1; x/ � 1.0;1/.x/, then eV.x/ D .x C 1/ �1.0;1/.x/ and

1

2eV.1/ D 1 D V.1/:

10.4 Quasiconcavity of Fundamental Functions 135

Fig. 10.11V.x/ D max.1; x/ � 1.0;1/.x/andeV.x/ D .xC1/ �1.0;1/.x/

0 x1

1

y

C

D

E

xV~( )

xV( )

Thus the inequality1

2eV � V cannot be improved, i.e., the factor

1

2is the best

possible (Fig. 10.11).

10.4 Quasiconcavity of Fundamental Functions

We prove now that the fundamental functions 'X of symmetric spaces are quasicon-cave. To this end, we first show that the fundamental function 'X1 of the associatespace X1 can be easily found in terms of the function 'X.

Proposition 10.4.1. Let 'X D V. Then 'X1 D V�, where V�.x/ D x

V.x/for x > 0

and V�.0/ D 0.

Proof. Using inequalities (5.2.5) from Note 5.2.2, we have

'X1 .x/ D k1Œ0;x�kX1 D supkf kX�1

1Z

0

f � 1Œ0;x�dm

� supkf kX�1

1Z

0

f � � 1Œ0;x�dm D kf � 1Œ0;x�kL1 :

Then

'X1 .x/ � x

V.x/sup

kf kX�1kf � � 1Œ0;x�kX � V�.x/;

since kf � � 1Œ0;x�kX � kf kX. On the other hand, for f D 1Œ0;x�k1Œ0;x�kX

, we have

'X1 .x/ �1Z

0

1Œ0;x�

k1Œ0;x�kX1Œ0;x�dm D x

'X.x/D V�.x/:

Thus,

'X1 .x/ D V�.x/:

utTheorem 10.4.2. The fundamental function 'X of every symmetric space X isquasiconcave.

Proof. The function V.x/ D 'X.x/ D k1Œ0;x�kX is increasing and V.0/ D 0,

since the norm k � kX is monotonic. The function V�.x/ D x

V.x/coincides with

the fundamental function 'X1 of the associate space X1 (Proposition 10.4.1). HenceV� also is increasing. Thus both V and V� are quasiconcave. utNote 10.4.3. We shall prove below (Theorem 11.2.2) that the converse is also true.For every quasiconcave function V, there exists a symmetric space X such that 'X DV. We shall use the Marcinkiewicz space MV�

.

10.5 Quasiconvex Functions

Let V be a concave function on Œ0;1/ and V.0/ D 0. Then the inverse function˚ D V�1 is convex. One should be cautious here, since

V.0C/ > 0 H) ˚.x/ D 0; for 0 � x � V.0C/; (10.5.1)

and

V.1/ < 1 H) ˚.x/ D 1; for x � V.1/ (10.5.2)

(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 thisdefinition.

All properties of quasiconvex functions are easily reduced from the correspond-ing properties of quasiconcave functions.

10.5 Quasiconvex Functions 137

x x

y

0 0

y

0+x x

V

V

V

•

F( )

0+V ( )

( )( )

V •

•

( )

( )

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 existsa greatest convex minorant e˚ such that e˚�1 is the least concave majorant for thequasiconcave function ˚�1, and

e˚� x

2

� ˚.x/ � e˚.x/ x � 0: (10.5.3)

Chapter 11Marcinkiewicz Spaces

In this chapter we study Marcinkiewicz spaces MV constructed by a quasiconcaveweight V . The space MV is equipped with the norm kf kMV D kV�f ��kL1

,

where V�.x/ D x

V.x/, and f ��.x/ D 1

x

Z x

0

f � dm is the maximal Hardy–Littlewood

function of f . We show that .MV ; k � kMV / is a maximal symmetric space. Theassociate space �1

W of every Lorentz space �W with a concave weight W is aMarcinkiewicz space and k � kMV D k � k�1

W. Conversely, the associate space M1

Vcoincides 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 . Thenorm k � kMV in the space MV can be written in the form

kf kMV D kV�f ��kL1; f 2 MV ;

where V�.x/ D x

V.x/, x > 0 and f �� is the maximal Hardy–Littlewood function of

f . For f 2 L1 C L1, we set

f ��.x/ D 1

x

xZ

0

f �dm; x > 0:

First, we describe some elementary properties of f ��.


139

140 11 Marcinkiewicz Spaces

Proposition 11.1.1. Let f 2 L1 C L1 and g 2 L1 C L1. Then

1. f �� is continuous and decreasing on .0;1/, and1

f �� is quasiconcave if f ¤ 0.

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��.

Proof. 1: Since the function f � is nonnegative and decreasing, the function

F.x/ DxZ

0

f �dm

is increasing and concave on .0;1/. Hence f ��.x/ D F.x/

xis decreasing and

continuous, andx

F.x/D 1

f �� is quasiconcave.

Assertions 2 and 3 follow directly from the definition.4: The maximal property (Theorem 4.1.1) implies that

xZ

0

.f C g/�dm D sup

8

<

:

Z

A

jf C gjdm W mA D x

9

=

;

� sup

8

<

:

Z

A

jf jdm W mA D x

9

=

;

C sup

8

<

:

Z

A

jgjdm W mA D x

9

=

;

DxZ

0

f �dm CxZ

0

g�dm:

utAs was mentioned above in Chapter 3, the triangle inequality

.f C g/� � f � C g�

fails in general (Note 3.2.2).On the other hand, the triangle inequality 4 is true and even can be generalized

to the case of infinite sums.

11.1 The Maximal Function f �� 141

Proposition 11.1.2. If fn 2 L1 C L1, then

1X

nD1fn

!��

1X

nD1f ��n :

Proof. Let x > 0. We can clearly assume that1P

nD1f ��n .x/ < 1. Then by

Levi’s theorem, for every measurable set A of measure mA D x, the series1P

nD1fn converges almost everywhere on A. Indeed, using the maximal property of

decreasing rearrangements (Theorem 4.1.1), we have

1

x

Z

A

ˇ

ˇ

ˇ

ˇ

ˇ

1X

nD1fn

ˇ

ˇ

ˇ

ˇ

ˇ

dm � 1

x

1X

nD1

Z

A

jfnjdm � 1

x

1X

nD1

xZ

0

f �n dm D

1X

nD1f ��n .x/:

Hence

supmADx

Z

Aj

1X

nD1fnjdm D

1X

nD1fn

!��.x/ �

1X

nD1f ��n .x/

for all x > 0. utApplying Propositions 11.1.2 and 11.1.3 to symmetric spaces X and nonzero

measurable functions U � 0, we can construct new symmetric spaces XU as follows.For a > 0, define

Ua.x/ D U.x/ � min�

1;a

x

; x � 0:

Proposition 11.1.3. Let X be a symmetric space, and U W RC ! RC a nonzero

measurable function such that Ua 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.

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

1Œ0;a�dm D min

�

a;1

x

�

; x > 0;

we have

k1Œ0;a�kXU D kU1��Œ0;a�kX D kUakX

and Ua 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 ffkg is a sequence in XU with

1X

kD1kfkkXU < 1;

then the series1P

kD1fk converges in norm k � kXU .

We have

1X

kD1kfkkXU D

1X

kD1kU � f ��

k kX < 1:

Since the normed space X is complete, the series1P

kD1U � f ��

k converges in X, i.e.,

1P

kD1U � f ��

k 2 X. By Proposition 11.1.2, we have

U � 1X

kD1fk

!�� U �

1X

kD1f ��k D

1X

kD1U � f ��

k 2 X:

11.2 Definition of Marcinkiewicz Spaces 143

That is,1P

kD1fk 2 XU and

�

�

�

�

�

1X

kD1fk

�

�

�

�

�

XU

D�

�

�

�

�

U � 1X

kD1fk

!��

�

�

�

X

��

�

�

�

�

1X

kD1U � f ��

k

�

�

�

�

�

X

�1X

kD1kfkkXU :

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

11.2 Definition of Marcinkiewicz Spaces

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

V.x/� 1.0;1/.x/.

Using the space XU in the case U D V� and X D L1, we obtain the space

MV D .L1/V�D ff 2 L1 C L1 W V�f �� 2 L1g;

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 < 1gequipped with the norm

kf kMV D kV�f ��kL1

D sup0<x<1

0

@

1

V.x/

xZ

0

f �dm

1

A

is a symmetric space with the fundamental function 'MV D V�.

Proof. By Proposition11.1.3, .MV ; k � kMV / is a symmetric space. The fundamentalfunction of MV is

'MV .x/ D k1Œ0;x�kMV D sup0<u<1

0

@

1

V.u/

uZ

0

1Œ0;x�dm

1

A

D sup0<u<1

�

1

V.u/min.u; x/

�

D max

�

supu�x

u

V.u/; sup

u�x

x

V.u/

�

D x

V.x/D V�.x/;

since both the functions V�.u/ D u

V.u/and V.u/ are increasing. ut

By taking U D V instead of U D V�, we obtain the Marcinkiewicz space MV�

with the fundamental function

'MV�D .V�/� D V:

Thus, for every quasiconcave function V , there exists a symmetric space X D MV�

for which

'X D 'MV�D V:

Invoking Theorem 10.4.2, we can describe the class of all fundamental functions ofsymmetric spaces.

Theorem 11.2.2. A function V is the fundamental function 'X of a symmetric spaceX if and only if V is quasiconcave.

11.3 Duality of Lorentz and Marcinkiewicz Spaces

In this section we describe associate spaces �1W and M1

V of Lorentz spaces �W andthe Marcinkiewicz space MV .

Theorem 11.3.1. Let �W be a Lorentz space with a (concave) weight function W.Then �1

W D MW and k � k�1W

D k � kMW .

Proof. We show that the norm

kgk�1W

D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W f 2 �W ; kf k�W � 1

9

=

;

(11.3.1)

coincides with the norm

kgkMW D sup0<x<1

8

<

:

1

W.x/

xZ

0

g�dm

9

=

;

D kW�g��kL1:

Recall that by Proposition 7.2.1, the supremum in (11.3.1) can be taken over all

f 2 F0, and the integral

1Z

0

fgdm can be replaced by

1Z

0

f �g�dm.

11.3 Duality of Lorentz and Marcinkiewicz Spaces 145

Every function f D f � 2 F0 has the form

f D f � DmX

iD1ci � 1Œ0;bi�; ci > 0; 0 < b1 < b2 � � � < bm:

Therefore,

1Z

0

f �g�dm DmX

iD1ci

biZ

0

g�dm DmX

iD1ciW.bi/ � 1

W.bi/

biZ

0

g�dm

�mX

iD1ciW.bi/ � sup

0<x<1

8

<

:

1

W.x/

xZ

0

g�dm

9

=

;

D kf k�W � kgkMW ;

whence

kgk�1W

D sup

8

<

:

1Z

0

f �g�dm W f 2 F0; kf k�W � 1

9

=

;

� kgkMW

for all g 2 L1 C L1.To verify the reverse inequality, we fix x > 0 and set

fx.u/ D 1

W.x/� 1Œ0;x�.u/:

Then

fx D f �x ; kfxk�W D '�W .x/

W.x/D 1;

and

1Z

0

f �x g�dm D 1

W.x/

xZ

0

g�dm � kgk�1W:

Hence,

kgkMW D sup0<x<1

8

<

:

1

W.x/

xZ

0

g�dm

9

=

;

� kgk�1W:

ut


Consider now the minimal part

�0W D cl�W .F0/ D cl�W .L1 \ L1/

of a Lorentz space �W . By Proposition 7.3.3, we have

.�0W/

1 D MW :

If '�W .0C/ D W.0C/ D 0, then by Theorem 6.5.3, the minimal symmetric space�0

W is separable, and

.MW/ D ..�0W/

1/ D .�0W/

�:

If in addition, �W is separable, then

.MW/ D ��W ;

where as above,

W MW 3 g ! ug 2 ��W ; and ug.f / D

1Z

0

fgdm; f 2 �W :

Corollary 11.3.2. The following are equivalent:

1. W.0C/ D 0 and W.1/ D 1.2. �W is separable.3. .MW/ D ��

W.

We now turn to the associate space .MW/1 D M1

W of a Marcinkiewicz space MW .First, we assume that the weight function W is concave. In this case,

.MW/1 D .�W/

11 D �W :

The latter equality holds because �W is a maximal symmetric space (Proposi-tion 9.2.1). The space �W has property .C/ (Theorem 8.3.3 and Proposition 9.2.1).Therefore, the mapping

�W 3 f ! f 2 .MW/1

is isometric. Thus, we have the following result.

Proposition 11.3.3. If the weight function W is concave, then M1W D �W, and

k � kM1W

D k � k�W .

11.4 Examples of Marcinkiewicz Spaces 147

When the weight function V of MV is not concave, we can use the least concavemajorant W D eV . The previous proposition yields

M1

eVD �

eV and k � kM1

eV

D k � k�eV:

By Theorem 10.3.1,

1

2eV � V � eV;

whence

k � kMeV

� k � kMV � 2k � kMeV

and

1

2k � kM1

eV

� k � kM1V

� k � kM1

eV

:

Thus, by definition of the norms, we have proved the following proposition.

Proposition 11.3.4. Let V be a quasiconcave function and eV the least concavemajorant of V. Then M1

V D �eV and

1

2k � k�

eV� k � kM1

V� k � k�

eV:

11.4 Examples of Marcinkiewicz Spaces

Examples 11.4.1. 1. The space L1. Let V.x/ D 1.0;1/.x/, x � 0 (Fig. 11.1). Then

kf kMV D sup0<x<1

0

@

1

V.x/

xZ

0

f �dm

1

A D1Z

0

f �dm D kf kL1 :

Hence, MV D L1 and k � kMV D k � kL1 .2. The space L1. Let V.x/ D x, x � 0 (Fig. 11.2).

Then

kf kMV D sup0<x<1

0

@

1

x

xZ

0

f �dm

1

A D sup0<x<1

f ��.x/ D kf ��kL1D f ��.0/:


Fig. 11.1 V.x/ for MV D L1

0 x

1

y

xV ( )

Fig. 11.2 V.x/ forMV D L1

0 x

y

=x xV( )

Fig. 11.3 V.x/ forMV D L1 \ L1

0 x1

1

y

xV ( )

Since f �� is continuous and f � is right continuous, we have

f ��.0/ D limx!0

1

x

xZ

0

f �dm D f �.0/ D kf kL1:

Hence, MV D L1 and k � kMV D k � kL1.

3. The space L1 \ L1. Let V.x/ D min.x; 1/, x � 0 (Fig. 11.3). Then

V�.x/ D max.x; 1/ � 1.0;1/.x/; x � 0:

11.4 Examples of Marcinkiewicz Spaces 149

Fig. 11.4 V.x/ and V�.x/ forMV D L1 C L1

0 1 x 0 1 x

y y

1 1xV( ) xV

*( )

For f 2 MV , we have

kf kMV D sup0<x<1

0

@

max.x; 1/

x

xZ

0

f �dm

1

A

D max

8

<

:

sup0<x�1

1

x

xZ

0

f �dm; sup1�x

xZ

0

f �dm

9

=

;

D max

8

<

:

f �.0/;1Z

0

f �dm

9

=

;

D max fkf kL1; kf kL1g D kf kL1\L1

:

Hence, MV D L1 \ L1 and k � kMV D k � kL1\L1.

4. The space L1 C L1. Let V.x/ D max.x; 1/ � 1.0;1/.x/, x � 0 (Fig. 11.4). Then

V�.x/ D min.x; 1/; x � 0:

We have

kf kMV D sup0<x<1

0

@min.x; 1/;1

x

xZ

0

f �dm

1

A

D max

8

<

:

sup0<x�1

xZ

0

f �dm; sup1�x

1

x

xZ

0

f �dm

9

=

;

D max

8

<

:

1Z

0

f �dm; sup1�x

f ��.x/

9

=

;

D1Z

0


Hence, MV D L1 C L1 and k � kMV D k � kL1CL1.

Example 11.4.2. Let 1 < p < 1 and V.x/ D x1p ; x � 0. The norm k � kMV in the

space MV has the form

kf kMV D kV�f ��kL1D sup

0<x<1

�

x� 1p

Z x

0

f �dm

�

;

where V�.x/ D x

x1p

D x1�1p D x

1q for

1

pC 1

qD 1.

In this case, M1V D �V and �1

V D MV , where the Lorentz space �V has the norm

kf k�V D 1

p

Z 1

0

f �.x/x1p �1dx D

Z 1

0

f �.x/dx1p :

The fundamental functions of the spaces 'MV .x/ D V�.x/ D x1q and '�V .x/ D

V.x/ D x1p coincide with the fundamental functions 'Lq D x

1q and 'Lp D x

1p ,

respectively.

Note that .Lp/1 D Lq and .Lq/

1 D Lp since1

pC 1

qD 1. We shall show below

(Examples 12.3.3) that

�V � Lp and Lq � MV ;

where both inclusions are strict.

Chapter 12Embedding �0

eV� X � MV�

In this chapter, we prove the embedding theorem for classes of symmetric spaceshaving the same fundamental functions. The embedding theorem asserts that forevery symmetric space X with a given fundamental function V D 'X, there arecontinuous embeddings �0

eV� X � MV�

. This means that the minimal part �0

eVof

the Lorentz space �eV is the smallest symmetric space whose (concave) fundamental

function eV is equivalent to V , while the Marcinkiewicz space MV�is the largest

symmetric space X with 'X D 'MV�D V . The renorming theorem and other

consequences of the embedding theorem are considered.

12.1 The Embedding Theorem

Let V be a quasiconcave function on RC, and leteV denote the least concave majorant

of V . The quasiconcave function V� is defined by V�.x/ D xV.x/ , x > 0, and

V�.0/ D 0.The Lorentz space �

eV as well as its minimal part �0

eVD cl�

eV.F1/ have

fundamental function '�eV

D eV , where eV and V are equivalent by Theorem 10.3.1.

First, we prove the embedding �0

eV� X for every symmetric space X having the

fundamental function 'X D V .

Proposition 12.1.1. Let X be a symmetric space with the fundamental function'X D V. Then �0

eV� X and kf k�

eV� kf kX for all f 2 �0

eV.

Proof. We prove that

kf k�eV

� kf kX (12.1.1)

for all f 2 F1 � �0

eV.


151

152 12 Embedding �0

eV� X � MV�

0 x

y

1b

1c

2c

2b

3b

nb

Fig. 12.1 f � DmP

iD1

ci � 1Œ0;bi �

Suppose that f D 1A and 0 � mA < 1.In this case, f D 1A 2 �

eV , f � D 1Œ0;x� 2 �eV for x D mA, and

k1Ak�eV

D k1Œ0;x�k�eV

D eV.x/ � V.x/ D k1Œ0;x�kX D k1AkX:

Let f 2 F1. Then f � 2 F0 � �0

eV, and f � can be written in the form (Fig. 12.1)

f � DmX

iD1ci � 1Œ0;bi�; ci > 0; 0 � b1 < b2 < � � � < bm:

For such functions, we have

kf k�eV

D1Z

0

f �deV DmX

iD1cieV.bi/ �

mX

iD1ciV.bi/ D

mX

iD1cik1Œ0;bi�kX

��

�

�

�

�

mX

iD1ci � 1Œ0;bi�

�

�

�

�

�

X

D kf kX:

Thus, the inequality (12.1.1) holds for all f 2 F1.By definition, �0

eVD cl�

eV.F1/. Therefore, for every f 2 �0

eV, one can choose a

sequence fn 2 F1 such that kfn � f k�eV

! 0 as n ! 1.On the other hand, the inequality (12.1.1) provides that fn is a Cauchy sequence

in X, and hence kfn � f0kX ! 0 for some f0 2 X. Since

kfn � f k�eV

! 0 H) kfn � f kL1CL1! 0

12.1 The Embedding Theorem 153

and

kfn � f0kX ! 0 H) kfn � f0kL1CL1! 0;

we have f D f0 2 X. Thus �0

eV� X, and (12.1.1) is valid for all f 2 �0

eV. ut

Further, we show that the Marcinkiewicz space MV�is the largest symmetric

space X having the fundamental function 'X D V .

Proposition 12.1.2. Let X be a symmetric space with the fundamental function'X D V. Then X � MV�

and

kf kMV�� kf kX (12.1.2)

for all f 2 X.

Proof. For every symmetric space X, we have

kf � � 1Œ0;x�kL1 � x

'X.x/kf � � 1Œ0;x�kX; x > 0;

for all f 2 X by Note 5.2.2 and Theorem 5.2.1. This inequality with 'X D V can berewritten as

kf � � 1Œ0;x�kL1

kf � � 1Œ0;x�kX� x

V.x/D V�.x/; x > 0; f 2 X:

Thus,

kf kX D kf �kX � kf � � 1Œ0;x�kX � 1

V�.x/kf � � 1Œ0;x�kL1 D 1

V�.x/

xZ

0

f �dm

and

kf kMV�D sup

0<x<1

8

<

:

1

V�.x/

xZ

0

f �dm

9

=

;

� kf kX:

utIt is worthwhile combining Propositions 12.1.1 and 12.1.2 with Theorem 8.1.1

in the following embedding theorem.

Theorem 12.1.3. Let X be a symmetric space with fundamental function 'X D V.Let eV be the least concave majorant of the quasiconcave function V and V�.x/ D

x

V.x/; x > 0. Then

�0

eV� X0 � X � X11 � MV�

; (12.1.3)


eV� X � MV�

where

kf k�eV

D kf k�0

eV

� kf kX0 D kf kX; f 2 �0

eV

and

kf kX � kf kX11 � kf kMV�; f 2 X:

Proof. From Proposition 12.1.1, we have �0

eV� X, whence

�0

eV� clX.�

0

eV/ D clX.F1/ D X0:

From Proposition 12.1.2, we have X � MV�, whence by the duality relations

between Lorentz and Marcinkiewicz spaces (Section 13.3),

X � MV�H) X1 M1

V�

D �eV�

H) X11 � M11V�

D MV�:

The embedding X � X11 was proved in Section 8.3. All correspondinginequalities for norms also follow. ut

It should be emphasized that all the spaces X0;X;X11, and MV�have the same

quasiconcave fundamental function V , while the spaces �0

eVas well as �

eV have the

fundamental function eV . The concave function eV is equivalent but not necessarilyequal to V . Namely, 1

2eV � V � eV by Theorem 10.3.1.

Note 12.1.4. 1. The Lorentz space �eV is minimal if and only if eV.1/ D 1

(Table 9.1) or, what is equivalent, that V.1/ D 1. Obviously, we have �0

eVD

�eV � X in this case.

Putting X D L01 D L1 \ R0, we obtain V.x/ D 'X.x/ D 1.0;1/, x � 0,and �V D �

eV D L1. Thus �eV is not minimal, �0

eVD L01 D X. We see that

�eV 6� X in this example.

2. Consider the case in which the space X is maximal, i.e., X D X11 as sets. Then

�0

eV� X H) .�0

eV/1 D �1

eV X1 H) �

eV D �11

eV� X11 D X:

Thus �eV � X in this case even if �

eV is not minimal, the inclusion �0

eV� �V is

strict.3. The equality X D X11 does not imply in general the equality of the norms k � kX

and k � kX11 . For example, let X D L1 be equipped with the norm

kf kX D f �.0/C f �.1/; f 2 X D L1:

12.1 The Embedding Theorem 155

We have in this case

kf kX D kf kL1D f �.0/; f 2 L01;

since the seminorm f 7! f �.1/ vanishes identically on L01. Thus we have

V.x/ D 'X.x/ D 1.0;1/.x/; x 2 RC;

and �V D L1 D X, while �0V D L01 6D �V D L1. However, be putting f D

1Œ0;1/, we obtain

k1Œ0;1/kX D 2 > k1Œ0;1/kL1D 1:

Corollary 12.1.5. Let X be a maximal symmetric space with the fundamentalfunction 'X. Then

1. 'X.0C/ > 0 ” X � L1.2. 'X.1/ < 1 ” X L1.3. 'X.0C/ > 0 and 'X.1/ < 1 ” X D L1.

Proof. Let V D 'X. Since X is maximal, we have �eV � X (Note 12.1.4, 2.)

1. If X � L1, then �eV � L1 and V.0C/ > 0 (Table 9.1).

Conversely, the condition V.0C/ > 0 implies that MV�� L1. Indeed,

f 2 MV�” kVkL1

< 1 H) V.0C/f ��.0C/ < 1

H) f ��.0C/ D f �.0/ < 1 ” f 2 L1;

whence

X � MV�� L1:

2. V.1/ < 1 implies 1Œ0;1/ 2 �eV , since

k1Œ0;1/k�eV

D1Z

0

deV D eV.1/ D V.1/ < 1;

whence

1Œ0;1/ 2 �eV � X and X L1:

Conversely,

X L1 H) MV� L1 ” MV�

3 1Œ0;1/:

eV� X � MV�

This means that

k1Œ0;1/kMV�D kV � 1��

Œ0;1/kL1D kVkL1

D V.1/ < 1:

3. Follows from 1 and 2.ut

12.2 The Renorming Theorem

The fundamental functions V D 'X andeV D '�eV

in Theorem 12.1.3 are equivalentbut are not equal if V is not concave. Therefore, it is worthwhile extendingTheorem 12.1.3 with the following renorming theorem.

Theorem 12.2.1. Let X be a symmetric space with the norm k � kX and thefundamental function 'X D V, and let bV be an arbitrary quasiconcave functionthat is equivalent to V. Then there is a new norm k � k0

X on the space X such that

1. .X; k � k0X/ is a symmetric space.

2. k � kX and k � k0X are equivalent.

3. The fundamental function of the space .X; k � k0X/ coincides with bV.

Proof. Let c1V � bV � c2V . We define a new norm on X by setting

kf k1X D maxfc1kf kX; kf kMbV�

g:

Then the space X with the new norm k � k1X is a symmetric space. Also, kf k1X �c1kf kX, and by Theorem 12.1.3,

kf kMbV�

� c2kf kMV�� c2kf kX:

Thus

maxfc1; c2gkf kX � kf k1X � kf kX

and

k1Œ0;x�k1X D maxfc1V.x/;bV.x/g D bV.x/: utWe can put bV D eV in Theorem 12.2.1. Then bV and V are equivalent, and bV is

concave. Thus we obtain the following useful corollary.

12.3 Examples of Lorentz and Marcinkiewicz Spaces 157

Corollary 12.2.2. Let X be a symmetric space and 'X D V its fundamentalfunction. Then there is a new norm k � k0

X on X such that

1. .X; k � k0X/ is a symmetric space.

2. k � kX and k � k0X are equivalent.

3. The fundamental function of the space .X; k � k0X/ coincides with bV, i.e., it is

concave and equivalent to V.

By setting bV D eV , we also have the following corollary.

Corollary 12.2.3. Let X be a symmetric space with the fundamental function V D'X. Then there exists a symmetric norm k � k0

X on X equivalent to k � kX such that thefundamental function of .X; k � k0

X/ is concave.

12.3 Examples of Lorentz and Marcinkiewicz Spaces

Examples 12.3.1. 1. The space L1. Let V.x/ D 'L1.x/ D 1.0;1/.x/, x � 0.

Then V is concave and

kf k�V D V.0C/f �.0/C1Z

0

f �V 0dm D f �.0/ D kf kL1:

Thus for X D L1, we have �V D L1 and k � k�V D k � kL1, while �0

V D X0 DL01 D L1 \ R0 is the minimal part of �V D L1.

We have V�.x/ D x, x � 0. Hence,

kf kMV�D sup

0<x<1

8

<

:

1

x

xZ

0

f �dm

9

=

;

D sup0<x<1

f �� D f ��.0/:

However,

f ��.0/ D limx!0C

1

x

xZ

0

f �dx D f �.0/ D kf kL1;

i.e., MV�D L1 and k � kMV�

D k � kL1.

Thus we have �0V D X0 D L01 � X D L1 D MV�

.2. The space L1. Let V.x/ D 'L1 .x/ D x. Then V� D 1Œ0;1/ and

kf k�V D1Z

0

f �dV D1Z

0

f �dx D kf �kL1 D kf kL1 ;

eV� X � MV�

i.e., for X D L1, we have �V D X D L1. Further,

kf kMV�D sup

0<x<1

8

<

:

xZ

0

f �dm

9

=

;

D1Z

0

f �dm D kf kL1 ;

i.e., MV�D X D L1 and k � kMV�

D k � kL1 .Thus, we have �0

V D X0 D �V D X D MV�.

3. The space L1 \ L1. Let V.x/ D 'L1\L1.x/ D max.x; 1/ � 1.0;1/.x/, x � 0.

Then V is quasiconcave but is not concave. The functioneV.x/ D .xC1/�1.0;1/.x/is the least concave majorant of V . Further,

V�.x/ D min.x; 1/; x � 0; and1

V�.x/D max

�

1

x; 1

�

; x > 0:

Hence,

kf kMV�D sup

0<x<1

8

<

:

1

V�.x/

xZ

0

f �dm

9

=

;

D max

0

@ sup0<x<1

f ��.x/; sup1�x<1

xZ

0

f �dm

1

A D max.kf kL1 ; kf kL1/;

i.e., MV�D L1 \ L1 and k � kMV�

D k � kL1\L1.

Since for X D L1 \ L1,

L1 \ L1 � �eV � X � MV�

D L1 \ L1;

we have

�0

eVD �

eV D X D L1 \ L1 D MV�:

However,

kf k�eV

D f �.0/C1Z

0

f �eV

0dm D kf kL1

C kf kL1 :

Thus for X D L1 \ L1, we have �0

eVD �

eV D X D MV�, k � kX D k � kMV�

,while the norms k � kX and k � k�

eVdo not coincide. Of course, they are equivalent:

1

2.k � kL1

C k � kL1 / � max.k � kL1; k � kL1 / � k � kL1

C k � kL1 ;

i.e.,

1

2k � k�

eV� k � kX � k � kMV�

:

Note that here eV.x/ > V.x/ for all x > 0.

4. The space L1 C L1. Let V.x/ D 'L1CL1.x/ D min.x; 1/, x � 0. Then V is

concave and

kf k�V D1Z

0

f �dV D1Z

0

f �V 0dm D1Z

0

f �dm D kf kL1CL1;

i.e., �V D X D L1 C L1 and k � k�V D k � kX D k � kL1CL1.

Since

�V D X � MV�� L1 C L1;

we have

�V D X D MV�D L1 C L1:

Also,

kf kMV�D sup

0<x�1.V.x/f ��.x// D max

0

@ sup0<x�1

xZ

0

f �dm; supx�1

f ��.x/

1

A

D1Z

0


Thus, all the corresponding norms are equal.

5. The space R0. Let V.x/ D 'R0 .x/ D min.x; 1/. Since the space X D R0 D ff 2L1CL1W f �.1/ D 0g coincides with the minimal part .L1CL1/0 of L1CL1,we can use all the computations from Example 4 above.

So we have �0

eVD X0 D X � �

eV D X11 D MV�, where eV.x/ D V.x/ D

min.x; 1/ and V�.x/ D max.1; x/ � 1.0;1/.x/, x � 0.

Note that all of these examples describe a very special case in which �eV D MV�

.To construct examples with strict inclusions �

eV � X � MV�, consider the case

V D V�.


eV� X � MV�

Fig. 12.2 V.x/ D V�.x/ D px; x � 0

0 x

V(x) = V*(x)

y

Example 12.3.2. Let

V.x/ D V�.x/ D px; x � 0:

The central symmetric space X with the fundamental function V.x/ D 'X.x/ Dpx is the space L2 (Fig. 12.2). Since the function is concave and V.1/ D 1, we

have by Theorem 12.1.3,

�V � L2 � MV�D MV :

We show that the inclusion of �V � L2 is strict. The norm in �V has the form

kf k�V D1Z

0

f �.x/1

2p

xdx:

Consider the function

f .x/ D f �.x/ D min

�

1;1p

x ln.x C e � 1/�

:

Since

.f .x//2 D min

�

1;1

x ln2.x C e � 1/�

;

we have f 2 2 L1 and f 2 L2. However,

kf k�V D1Z

0

f �.x/1

2p

xdx D

1Z

0

1

2p

xdx C

1Z

1

1

2x

1

ln.x C e � 1/dx D 1;

i.e., f 62 �V .

We show now that the embedding L2 � MV is also strict. The norm in MV hasthe form

jf kMV D sup0<x<1

8

<

:

1px

xZ

0

f �dm

9

=

;

:

Consider the function

f .x/ D f �.x/ D 1px:

Then

kf kMV D sup0<x<1

8

<

:

1px

xZ

0

f �dm

9

=

;

D sup0<x<1

8

<

:

1px

xZ

0

1ptdt

9

=

;

D 2;

i.e., f 2 MV . On the other hand,

kf kL2 D1Z

0

f 2dm D1Z

0

1

xdx D 1;

i.e., f 62 L2. Thus, the embedding L2 � MV is also strict.

A similar situation occurs for Lp, 1 < p < 1.

Example 12.3.3. Let V.x/ D x1p , 1 < p < 1. A symmetric space X with 'X.x/ D

x1p is the space Lp. Note that the function V.x/ D x

1p is concave. So V D eV and

�V � Lp � MV�:

Using the functions

f .x/ D f �.x/ D min

1;1

x1p ln.x C e � 1/

!

and

f .x/ D f �.x/ D 1

x1p

;

we verify that both the inclusions �V � Lp and Lp � MV�are strict.

eV� X � MV�

12.4 Comparison of Lorentz and Marcinkiewicz Spaces

Theorem 12.4.1. Let V1 and V2 be two quasiconcave functions, and eV1 and eV2

their least concave majorants. Then the following are equivalent.

1. �eV1

� �eV2

.

2. eV2.x/ � QceV1.x/ for all x � 0 and some Qc > 0.3. V2.x/ � c V1.x/ for all x � 0 and some c > 0.4. M.V1/� � M.V2/� .

Proof. 1 ” 2. Suppose �eV1

� �eV2

. Then by Proposition 6.1.1, there exists aconstant Qc > 0 such that

kf k�eV2

� Qc kf k�eV1:

Hence,

eV2.x/ D '�eV2.x/ D k1Œ0;x�k�

eV2� Qck1Œ0;x�k�

eV1D Qc'�

eV1.x/ D QceV1.x/:

Conversely, suppose that eV2.x/ � QceV1.x/ for all x � 0 and some Qc > 0. Letf 2 �

eV1. Then by (9.1.4),

kf k�eV1

D1Z

0

eV1.�jf j.t//dt < 1;

and hence

kf k�eV2

D1Z

0

eV2.�jf j.t//dt � Qc1Z

0

eV1.�jf j.t//dt < 1;

i.e., f 2 �eV2

. Thus, �eV1

� �eV2

.2 H) 3. Suppose eV2.x/ � QceV1.x/ for all x � 0 and some constant Qc > 0. By

Theorem 10.3.1, we have

V2.x/ � eV2.x/ � QceV1.x/ � 2Qc V1.x/ D cV1.x/

for all x � 0 and c D 2Qc.3 H) 2 is proved in a similar way.3 ” 4. Suppose that V2.x/ � c V1.x/ for all x � 0 and some c > 0 and let

f 2 M.V1/� . Then V1f �� 2 L1 and c V1f �� 2 L1. Since 0 � V2f �� cV1f ��, wehave also V2f �� 2 L1, and therefore, f 2 M.V2/� .

Exercises 163

Conversely, suppose that M.V1/� � M.V2/� . Then by Proposition 6.1.1, thereexists a constant c > 0 such that

kf kM.V2/�� ckf kM.V1/�

:

Hence,

V2.x/ D 'M.V2/�.x/ D k1Œ0;x�kM.V2/�

� ck1Œ0;x�kM.V1/�D c'M.V1/�

.x/ D c V1.x/:

utCorollary 12.4.2. Let V1 and V2 be two quasiconcave functions, and eV1 and eV2

their least concave majorants. Then the following are equivalent:

1. �eV1

D �eV2

.

2. Qc1eV1.x/ � eV2.x/ � Qc2eV1.x/ for all x � 0 and some Qc1; Qc2 > 0.3. c1V1.x/ � V2.x/ � c2 V1.x/ for all x � 0 and some c1; c2 > 0.4. M.V1/� D M.V2/� .

Exercises

17. Concave majorants of nonnegative functions.Let V.x/, x � 0, be a nonnegative function on Œ0;1/. Recall that two such

functions V1 and V2 are said to be equivalent if

c1V1.x/ � V2.x/ � c2V1.x/

for some c1; c2 > 0 and all x � 0.Show that


• V has a finite concave majorant.• V has the least finite concave majorant.• V.x1/ � cV.x2/ for all 0 � x1 < x2 < 1 and some c > 0.

b. The following conditions are equivalent:

• V is equivalent to a positive concave function on Œ0;1/.• The functions V and V� have finite concave majorants.• V.x1/ � cV.x2/ and V�.x1/ � cV�.x2/ for all 0 � x1 < x2 < 1 and some

c > 0.Recall that V�.x/ D x

V.x/ � 1.0;1/.x/, x � 0.

eV� X � MV�

18. Lorentz spaces on Œ0; 1�.Let W be a nonnegative concave function on Œ0; 1�, W.0/ D 0, W.x/ > 0 for

0 < x � 1, and let

�W.0; 1/ D8

<

:

f 2 L0.0; 1/ W kf k�W .0;1/ D1Z

0

f �dW < 19

=

;

be the corresponding Lorentz space.Show that

a. .�W.0; 1/; k�k�W .0;1// is a symmetric space on Œ0; 1� in the sense of Complement2. The fundamental function '�W .0;1/ coincides with W and L1.0; 1/ ��W.0; 1/ � L1.0; 1/.

b. �W.0; 1/ is minimal, i.e., cl�W .0;1/.L1.0; 1// D �W.0; 1/.c. �W.0; 1/ is separable if and only if W.0C/ D 0.d. If W.0C/ > 0, then �W.0; 1/ D L1.0; 1/ as sets, while the norms k � k�W .0;1/

and k � kL1.0;1/ are equivalent.

19. Marcinkiewicz spaces on Œ0; 1�.Let V be a quasiconcave function on Œ0; 1� and let

MV.0; 1/ D ˚

f 2 L0.0; 1/ W kf kMV .0;1/ D kV�f ��kL1.0;1/ < 1�

be the corresponding Marcinkiewicz space on Œ0; 1�.Show that

a. .MV.0; 1/; k � kMV .0;1// is a maximal symmetric space on Œ0; 1� in the senseof Complement 2. The fundamental function 'MV .0;1/ coincides with V� andL1.0; 1/ � MV.0; 1/ � L1.0; 1/.

b. If V�.0C/ > 0, then MV.0; 1/ D L1.0; 1/ as sets, i.e., MV.0; 1/ is minimal andnonseparable.

c. If V 0�.0C/ < 1, then MV.0; 1/ D L1.0; 1/ as sets, i.e., MV.0; 1/ is minimal andseparable.

d. If V�.0C/ D 0 and V 0�.0C/ D 1, then the embeddings L1.0; 1/ � MV.0; 1/ �L1.0; 1/ are strict, and MV.0; 1/ is not minimal and is nonseparable.

Hint: Use the family of functions fcV 0; c > 0g � MV .e. .�1

W ; k � k�1W/ D .MW ; k � kMW / and .M1

W ; k � kM1W/ D .�W ; k � k�W /, where

�W D �W.0; 1/ and MW D MW.0; 1/ are the Lorentz and Marcinkiewicz spaceswith a concave weight W.

20. Spaces �W1 \ �W2 and �W1 C �W2 .Let �W1 and �W2 be two Lorentz spaces with weight functions W1 and W2 on

Œ0;1/.

Exercises 165

Show that

a. The space �W1 \ �W2 is isomorphic to the Lorentz space �W , where W D.max.W1;W2//

� is the least concave majorant of the quasiconcave functionmax.W1;W2/.

b. If the function W1W�12 is decreasing on .0;1/, then the space �W1 C �W2 is

isomorphic to the Lorentz space �W , where W D min.W1;W2/.c. If the function W1W�1

2 is strictly decreasing on .0;1/ and W1.x0/ D W2.x0/ ata point x0 2 .0;1/, then

kf k�W1C�W2D

x0Z

0

f �dW2 C1Z

x0

f �dW1;

and moreover,

.�W1 C �W2 / \ L1 � �W2 and .�W1 C �W2 / \ L1 � �W1 :

21. The space M0V .

Let MV be a Marcinkiewicz space with a quasiconcave function V on Œ0;1/,satisfying conditions V.0C/ D 0 and V.1/ D 1.

Show that

a. M.0/V D ff 2 MV W .V�f ��/.0C/ D .V�f ��/.1/ D 0g is a closed subspace of

MV .b. M.0/

V ¤ f0g ” V�.0C/ D 0.

c. If V�.0C/ D 0, then M.0/V D M0

V , where M0v D clMV .L1 \ L1/ is the minimal

part of MV .

22. Embedding X � X11 and property .C/.Let U and V be two quasiconcave functions such that U � V and

U.0C/ D V.0C/ D 0; U.1/ D V.1/ D 1; U�.0C/ D V�.0C/ D 0;

and X D ff 2 MU W kf k < 1g, where

kf k D kf kMU C limx!0CV�.x/f ��.x/C lim

x!1V�.x/f ��.x/:

Show that

a. .X; k � k/ is a symmetric space.b. MV � X � MU and X11 D MU .c. If the embedding MV � MU is strict, then X fails to have property .C/.d. Moreover, under the same assumptiones, X is not closed in X11 and property .C/

is false in the symmetric space .X; k � k1/, where k � k1 is an arbitrary symmetricnorm that is equivalent to k � kX.

Hint: Note that by the open mapping theorem, X is closed in X11 if and onlyif the canonic embedding X ! X11 is open.

eV� X � MV�

23. Non-embedded pairs of Lorentz spaces.Show that

a. There exist Lorentz spaces �W1 and �W1 such that

�Wi � L1; i D 1; 2; �W1 ª �W2 and �W2 ª �W1 :

b. There exist Lorentz spaces �W1 and �W1 such that

�Wi � L1; i D 1; 2; �W1 ª �W2 and �W2 ª �W1 :

c. Combining parts a and b, construct a pair .�W1 ;�W2 / such that

�W1 \ L1 ª �W2 \ L1; �W2 \ L1 ª �W1 \ L1;

and

�W1 \ L1 ª �W2 \ L1; �W2 \ L1 ª �W1 \ L1:

24. A decreasing family of Lorentz spaces.Let �W0 D �W0 .Œ0; 1�/ and �W1 D �W1 .Œ0; 1�/ be two Lorentz spaces on Œ0; 1�

with weight functions W0 and W1, where W0.x/ < W1.x/ for 0 < x � 1. Let

W˛ D W1�˛0 W˛

1 ; 0 � ˛ � 1;

and let fW˛ be the least concave majorant of W˛ .Show that

a. �W˛ �Wˇfor all 0 � ˛ < ˇ � 1, where the embedding �W˛ � �Wˇ

is strictif �W0 ¤ �W1 .

b. The embeddings �W˛ �Wˇare dense, i.e., cl�W˛

.�Wˇ/ D �W˛ for all 0 �

˛ � ˇ � 1.c. For all 0 � ˛ � 1, the associate space �1

eW˛of �

eW˛coincides with the

Marcinkiewicz space MW˛ .d. If �W0 ¤ �W1 , then all the embeddings MW˛ � MWˇ

are strict for ˛ < ˇ andclMWˇ

.MW˛ / ¤ MWˇ.

Notes

Our presentation of Lorentz and Marcinkiewicz spaces mainly follows [34, Section2.5]. The spaces �W were introduced and studied by Lorentz [38, 40]; see also[20, 26, 39, 41, 82].

Notes 167

Marcinkiewicz spaces MV (also called Lorentz–Marcinkiewicz spaces) appearedfor the first time in [39] as the duals of appropriate Lorentz spaces �W ; see also[37, 38, 45, 70].

The description of quasiconcave functions on RC in Section 12 is close to [34,

Section 2.1], with the exception of construction of the least concave majorants.Our geometric proof of (2.2.4) seems to be new. Another general approach usedin Complement 6 is also taken from [34, Chapter 2]; see also [24].

Quasiconcavity of fundamental functions 'X of symmetric spaces X can beproved by means of dilations �tWRC 3 s ! st 2 R

C. Namely, the functiont ! kf ı �tkX is quasiconcave for every f 2 X, whence 'X.t/ D k1Œ0;1� ı �tkX hasthe property. We do not use such a general approach in the proof of Theorem 10.4.2,since the equality 'X1 D .'X/� suffices.

We use the maximal function f �� in the definition of Marcinkiewicz spaces aswell as in Proposition 11.1.3 and Theorem 11.2.1. The operation f ! f �� wasintroduced by Hardy et al. [27], and it has a great many remarkable applicationsrelated to Hardy and singular Hilbert operators. See [34, Section 2.6], and also[12, 19, 71].

The proofs of the duality �1W D MV�

, M1W�

D �W (Theorems 11.1.3, Proposi-tions 11.3.3 and 11.3.4) and the embedding �0

eV� X � MV�

(Theorem 12.1.13) are

taken from [34, Section 2.5] with replacement of �eV by �0

eV; see also [68, 69].

The description of the minimal part M0V of Marcinkiewicz space MV given in

Complement 7 and Exercise 21 is also due to Semenov [67].Exercise 20 is based on Theorems 5.9 and 5.10 from [34, Section 2.5].The embedded spaces MV � X � MU constructed in Exercise 22 show that

there exist symmetric spaces such that both inclusions X � clX11X � X11 are strict.

Part IVOrlicz Spaces

In this part we study Orlicz spaces L˚ . The class of Orlicz spaces contains basicexamples of symmetric spaces such as Lp, 1 � p � 1, Lp \ Lq, Lp C Lq, 1 �p; q � 1, and Zygmund’s classes Z˛ D L ln˛ L, 0 � ˛ < 1.

We begin with a detailed description of Orlicz functions ˚ . In view of the theoryof symmetric spaces, ˚ is an Orlicz function if ˚�1 is a Lorentz weight function.

The Orlicz spaces arise from Lp spaces as follows.The norm k � kX in any Banach space X can be written as

kf kX D inf

�

c > 0 W�

�

�

�

f

c

�

�

�

�

X� 1

:

In the case of X D Lp with 1 � p < 1, this formula yields

kf kLp D inf

8

<

:

c > 0 W1Z

0

˚

� jf jc

�

dm � 1

9

=

;

;

where ˚.x/ D xp is a nonnegative increasing convex function on Œ0;1/.The required generalization is obtained by replacing the function ˚.x/ D xp by

an arbitrary nonnegative increasing convex function ˚ and defining

L˚ D8

<

:

f 2 L0 W1Z

0

˚

� jf jc

�

dm < 1 for some c > 0

9

=

;

and

kf kL˚ D inf

8

<

:

c > 0 W1Z

0

˚

� jf jc

�

dm � 1

9

=

;

:

170 IV Orlicz Spaces

The Banach space .L˚ ; k � kL˚ / obtained in this way is called an Orlicz space.To include the space L1 into this scheme, one needs to use more general

functions ˚ , each of which is chosen nonnegative, increasing, and convex on someinterval .a; b/, where 0 � a b.

Turning to associate spaces L1˚ of Orlicz spaces L˚ , we shall show that L1˚ D L�as sets for a suitable Orlicz function � , while the norms k � kL1˚

and k � kL� areequivalent.

Every Orlicz space L˚ has property .B/ and also property .C/ with respect tothe norm k � kL1�

. Concerning property .A/, one can observe that there are manynonseparable Orlicz spaces. They do not have property .A/ and are not minimal.

To describe the minimal part L0˚ of the Orlicz space L˚ , it is convenient to use asubset

H˚ D8

<

:

f 2 L0 W1Z

0

˚

� jf jc

�

dm < 1 for all c > 0

9

=

;

called sometimes the heart of the Orlicz space L˚ .When the Orlicz function ˚ is finite, the set H˚ coincides with the minimal part

of the space L˚ . In this case, the equality H˚ D L˚ is equivalent to property .A/,which in turn is equivalent to separability of L˚ . Property .A/ can also be expressedin terms of the function ˚ by so-called .�2/ conditions.

At the end of Part IV we study in detail the spaces Lp \ Lq, Lp C Lq, 1 � p; q �1, and also the Zygmund classes Z˛ , 0 � ˛ < 1.

Chapter 13Definition and Examples of Orlicz Spaces

In this chapter, we begin our study of Orlicz functions ˚ and corresponding Orliczspaces L˚ . We prove that .L˚ ; k � kL˚ / is a maximal symmetric space, find thefundamental functions 'L˚ of L˚ , and consider several simple examples.

13.1 Orlicz Functions

Definition 13.1.1. An Orlicz function is a function ˚ W Œ0; 1/ ! Œ0; 1� suchthat

1. ˚.0/ D 0, ˚.x1/ > 0 for some x1 > 0 and ˚.x2/ < 1 for some x2 > 0.2. ˚ is increasing: x1 � x2 ” ˚.x1/ � ˚.x2/.3. ˚ is convex: ˚.˛x1 C .1 � ˛/x2/ � ˛˚.x1/C .1 � ˛/˚.x2/, 0 � ˛ � 1.4. ˚ 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


171

172 13 Definition and Examples of Orlicz Spaces

˚.x/ DxZ

0

�.u/du; 0 � x � b˚ ;

while �.x/ D 1 for x > b˚ .The convexity of ˚ yields that ˚ is continuous on Œ0; b˚/. Therefore, the left-

continuity 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 arepossible if ˚.b˚/ < 1 (Fig. 13.1).

Since ˚ is increasing on Œ0;1/, it has a left-continuous (generalized) inversefunction ˚�1. This function is finite, concave, continuous on .0;1/, and˚�1.0/ D 0 (Fig. 13.2).

Thus,

a˚ D ˚�1.0C/ D limx!0C˚

�1.x/; b˚ D ˚�1.1/ D limx!1˚�1.x/; (13.1.2)

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.

0

0

0

=

=

0

( )

0 0

a

y y y y

x x b

b

aF

F

F

F

F

F F

F

F

•0>

=

a

bF

F• = •<

aF

bF( )bF

• ••

F F Fbax x

Fig. 13.1 Orlicz functions

0 x 0 x 0 x 0 x

y yy

yF-1 F-1

F-1F-1

bF bFaF

aF

Fig. 13.2 Inverse Orlicz functions

13.2 Orlicz Spaces 173

13.2 Orlicz Spaces

For every Orlicz function ˚ , we define a functional

I˚.f / D1Z

0

˚.jf j/ dm 2 Œ0;1� (13.2.1)

and set

kf kL˚ D inf

�

a > 0 W I˚

�

f

a

�

� 1

2 Œ0;1� (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 anOrlicz space.

We begin with several properties of the functional I˚ , which follow directly fromthe convexity of ˚ .

Proposition 13.2.1. Let f ; g 2 L0 and 0 < ˛ < 1. Then

1. I˚.˛f C .1 � ˛/g/ � ˛I˚.f /C .1 � ˛/I˚.g/.2. I˚.˛f / � ˛I˚.f /.3. The set B˚ D ff 2 L˚ W I˚.f / � 1g is convex.4. 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/ D1Z

0

˚.j˛f C .1 � ˛/gj/ dm

�1Z

0

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

2. Since ˚ is convex and ˚.0/ D 0, we have ˚.˛x/ � ˛˚.x/ and

I˚.˛f / D1Z

0

˚.j˛f j/ dm � ˛

1Z

0

˚.jf j/ dm D ˛I˚.f /

for 0 � ˛ � 1.

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 k�kL˚ are determined by jf j, we may assume that f � 0. If kf kL˚ �

1, there exists a sequence an # 1 such that

1Z

0

˚.f

an/ dm � 1. Since the function

˚ is left-continuous, ˚.f

an/ " ˚.f / almost everywhere. By Levi’s theorem,

1 �1Z

0

˚.f

an/ dm !

1Z

0

˚.f / dm;

i.e., f 2 B˚ .ut

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

h D ˛f

kf kL˚C .1 � ˛/ g

kgkL˚2 B˚ ;

and hence

khkL˚ D�

�

�

�

f C g

kf kL˚ C kgkL˚

�

�

�

�

L˚

D k f C g kL˚

kf kL˚ C kgkL˚� 1:

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 ja

�

� 1

13.2 Orlicz Spaces 175

D j˛j inf

�

a1 > 0 W I˚

� jf ja1

�

� 1

D j˛jkf kL˚ ;

i.e. ˛f 2 L˚ and k˛f kL˚ D j˛jkf kL˚ .

If kf kL˚ D 0, then I˚

�

f

a

�

� 1 for all a > 0. For all b > 0, we obtain

1 � I˚

�

f

a

�

D1Z

0

˚

� jf ja

�

dm � ˚

�

b

a

�

mfjf j � bg:

Then ˚

�

b

a

�

! 1 as a ! 0, and hence mfjf j � bg D 0 for all b > 0, i.e., f D 0.

Thus, k � kL˚ is a norm on L˚ .

Let f 2 L˚ and jgj � jf j. Then I˚

�

f

a

�

< 1 for some a > 0, and

jgj � jf j H) ˚

� jgja

�

� ˚

� jf ja

�

H) I˚

�g

a

� I˚

�

f

a

�

< 1;

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 ffng be a Cauchy sequence in L˚ . Then there exists a subsequence ffnk g for

which

kfnk � fnk�1kL˚ � 1

2k:

We set

g D1X

kD1jfnk � fnk�1 j: (13.2.3)

Since ˚ is convex and left-continuous, we have

˚.g/ D ˚

1X

kD1jfnk � fnk�1 j

!

�1X

kD1

1

2k˚.2kjfnk � fnk�1 j/

and

I˚.g/ D1Z

0

˚.g/ dm �1X

kD1

1

2k

1Z

0

˚.2kjfnk � fnk�1 j/ dm � 1:

Therefore, kgkL˚ � 1, and g 2 L˚ . Consequently, the series (13.2.3) convergesalmost everywhere, and the series

1X

kD1.fnk � fnk�1 /C fn0

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 ffng is a Cauchy sequence and has a convergent subsequence ffnk g, the

sequence ffng 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

I˚.f / D1Z

0

˚.jf j/ dm D1Z

0

�˚ıjf j du D1Z

0

mf˚.jf j/ > xg dx

D1Z

0

mfjf j > ˚�1.x/g dx D1Z

0

mfjf j > yg d˚.y/ D1Z

0

�jf jd˚:

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

I˚.f / D1Z

0

�jf jd˚ D1Z

0

�jgjd˚ D I˚.g/:

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

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˚ , calledYoung 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.

A nice trick with using functions ˚

� jf ja

�

, a > 0, instead of ˚.jf j/, enables us

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

'L˚ .x/ D �

˚�1 �x�1��1 ; x > 0; (13.3.1)

and also

'L˚ .0C/ D b�1˚ D .˚�1.1//�1; 'L˚ .1/ D a�1

˚ D .˚�1.0C//�1: (13.3.2)

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

I˚

�

1

a� 1Œ0;x�

�

D1Z

0

˚

�

1

a� 1Œ0;x�

�

dm DxZ

0

˚

�

1

a

�

dm D x˚

�

1

a

�

:

So

'L˚ .x/ D k1Œ0;x�kL˚ D inf

�

a > 0 W x˚

�

1

a

�

� 1

D inf

(

a > 0 W a ��

˚�1�

1

x

��1)D�

˚�1�

1

x

��1:

This implies (13.3.2), since ˚�1 is increasing and continuous on .0;1/ (Fig. 13.3).ut

The following is a direct consequence of Proposition 13.3.1.

Corollary 13.3.2. Every Orlicz function ˚ is uniquely determined by the funda-mental function 'L˚ of L˚ . Namely,

Fig. 13.3 ˚ and ˚�1

in (13.3.2)

F F

F F0 0x x

y y

-1

a b

F

F

a

b

˚�1.x/ D�

'L˚

�

1

x

��1; x > 0;

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

kf kL˚ D inf

8

<

:

a > 0 W1Z

0

jf jpap

dm � 1

9

=

;

D inf

8

<

:

a > 0 W1Z

0

jf jpdm � ap

9

=

;

D inf

8

ˆ

<

ˆ

:

a > 0 W0

@

1Z

0

jf jpdm

1

A

1p

� a

9

>

=

>

;

D kf kLp ;

i.e., L˚ D Lp, k � kL˚ D k � kLp and 'L˚ .x/ D 'Lp.x/ D ˚�1.x/ D x1p .

Here I˚.f / D kf kL˚ and Y˚ D L˚ D Lp.2. The space L1 (Fig. 13.5). Let

˚.x/ D�

0; 0 � x � 1;

1; x > 1:

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

F F

y y

x x0

1

1

1< < •1

xy

p p

0

=

= xy p=

13.4 Examples of Orlicz Spaces 179

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

F

•

0 1 x

y

Fig. 13.6 Orlicz function forwhich L˚ D L1 \ L1

x

1

1

•

0

y

F

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 � kL1and '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/ D�

x; 0 � x � 1;

1; x > 1:

For every a > 0, the inequality

1Z

0

˚

�

f �

a

�

dm � 1 means that

Z

ff �>ag

f �

adm � 1;

and f � � a. Hence

kf kL˚ D kf �kL˚ D inf

8

<

:

a > 0 W1Z

0

f �dm � a and f � � a

9

=

;

D max

8

<

:

1Z

0

f �dm; f �.0/

9

=

;

D maxfkf kL1 ; kf kL1g D kf kL1\L1

;

i.e., L˚ D L1 \ L1, k � kL˚ D k � kL1\L1and 'L˚ .x/ D 'L1\L1

.x/ D max.x; 1/ �1.0;1/.x/.

Here Y˚ D ff 2 L1 W jf j < 1g 6D L˚ .

4. The space L1 C L1. Let ˚.x/ D .x � 1/ � 1Œ1;1/.x/ (Fig. 13.7).

For every a > 0, we have

˚

�

f �

a

�

D�

f �

a� 1

�

� 1ff �>ag:

Thus (Fig. 13.8)

I˚

�

f �

a

�

DZ

ff �>ag

�

f �

a� 1

�

dm D1Z

a

�f �dm:

Since the function a !1R

a�f �dm is decreasing and continuous, there exists a

unique point a0 for which1R

a0

�f �dm D a0. For such a0, we have

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

x1

x-1

0

y

F

13.4 Examples of Orlicz Spaces 181

Fig. 13.81R

a�f � dm as area

x0

y

a f*

I˚

�

f �

a0

�

D 1 and a0 D kf kL˚ :

By setting

g0 D .f � � a0/ � 1ff �>a0g and h0 D min.f �; a0/;

we obtain a representation of f � of the form f � D g0Ch0, where g0 2 L1, h0 2 L1,and

kg0kL1 D kh0kL1D 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

I˚

�

f

a

�

�1Z

0

˚

� jgja

C jhja

�

dm �1Z

0

˚

� jgja

C 1

�

dm

D1Z

0

jgja

dm � 1:

This means that

kf kL˚ � a D max.kgkL1 ; khkL1//:


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 L1g: (13.4.1)

By comparing (13.4.1) and (4.2.5), we see that the norms k � kL˚ and k � kL1CL1

are equivalent, namely,

kf kL˚ � kf kL1CL1� 2kf kL˚ : (13.4.2)

Therefore, the corresponding fundamental functions are equivalent, and

'L˚ � 'L1CL1� 2'L˚ : (13.4.3)

On the other hand, ˚�1.x/ D .x C 1/ � 1Œ0;1/, whence

'L˚ .x/ D .˚�1.x�1//�1 D x

x C 1; x � 0;

while 'L1CL1D min.x; 1/. We have

x

x C 1� min.x; 1/ � 2x

x C 1; x � 0:

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

I˚.f / DZ 1

0

˚.2 � 1Œ1;1//dm D 1;

i.e., f 62 Y˚ .

Fig. 13.9 'L˚ � 'L1CL1� 2'L˚

Fj

j

x0

y

1

1/2

2L

FL

2

1

Chapter 14Separable Orlicz Spaces

In this chapter, we study conditions of separability for Orlicz spaces L˚ . We con-sider 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 beexpressed in term of an Orlicz function ˚ . The .�2/ condition is described to thisend in detail.

14.1 Young Classes Y˚ and Subspaces H˚

The main purpose of this section is to describe conditions of minimality andseparability for Orlicz spaces L˚ . To this end, we return to our study of Youngclasses Y˚ and corresponding subspaces H˚ .

First, we want to know when the equality H˚ D Y˚ D L˚ holds. Recall that

Y˚ D8

<

:

f 2 L˚ W I˚.f / D1Z

0

˚.jf j/dm < 19

=

;

: (14.1.1)

By Proposition 13.2.1, Y˚ is a convex subset of L˚ that does not necessarilycoincide with L˚ .

To clarify when Y˚ D L˚ , consider the one-parameter family

f˛Y˚ ; 0 < ˛ < 1g:


183

184 14 Separable Orlicz Spaces

For 0 < ˛1 < ˛2 < 1, we have

f 2 ˛1Y˚ ” I˚

�

f

˛1

�

< 1 H) I˚

�

f

˛2

�

< 1 ” f 2 ˛2Y˚ ;

and thus the family is increasing.Consider also the set

H˚ WD�

f 2 L0 W I˚�

f

˛

�

< 1 for all ˛ > 0

D\

0<˛<1˛Y˚

and the Orlicz space

L˚ D�

f 2 L0 W I˚�

f

˛

�

< 1 for some ˛ > 0

D[

0<˛<1˛Y˚

itself. For all 0 < ˛1 < ˛2 < 1, we have

H˚ D\

0<˛<1˛Y˚ � ˛1Y˚ � ˛2Y˚ �

[

0<˛<1˛Y˚ D L˚ : (14.1.2)

Proposition 14.1.1. 1. If

˛1Y˚ D ˛2Y˚

for some 0 < ˛1 < ˛2 < 1, then

H˚ D Y˚ D L˚ :

2. If for some 0 < ˛1 < ˛2 < 1, the inclusion ˛1Y˚ � ˛2Y˚ is strict, then all ofthe inclusions in (14.1.2) are strict.

3. If b˚ < 1, then all inclusions in (14.1.2) are strict.

Proof. 1. Let 0 < ˛1 < ˛2 < 1 and ˛1Y˚ D ˛2Y˚ , i.e.,

k�1Y˚ D Y˚ for k D ˛2

˛1> 1:

Hence

H˚ D\

n�0k�nY˚ D Y˚ and L˚ D

[

n�0knY˚ :

2. Follows in a straightforward way from 1.3. Let f D 1Œ0;1�, f 2 L˚ . Then ˛1 � f 2 Y˚ for some 0 < ˛1 < 1.

If b˚ < 1 and b˚ < ˛2 < 1, we have

14.2 Separability Conditions for Orlicz Spaces 185

I˚.˛2f / D I˚.˛2 � 1Œ0;1�/ � ˚.˛2/ D 1:

That is, ˛2 � 1Œ0;1� 62 Y˚ . Hence, the inclusion ˛1Y˚ � ˛2Y˚ is strict for a pair.˛1; ˛2/, and hence for all pairs 0 < ˛1 < ˛2 < 1. Thus, all the inclusionsin (14.1.2) are strict. utCorollary 14.1.2.

H˚ D L˚ ” H˚ D Y˚ ” Y˚ D 2Y˚ ” Y˚ D L˚ :

Example 14.1.3. Let L˚ D L1, and let the Orlicz function ˚ be defined as inExample 13.4.1, 2), i.e.,

˚.x/ D(

0; 0 � x � 1

1; x > 1:

Then b˚ D 1 < 1 and

I˚.f / DZ 1

0

˚.jf j/dm D(

0; kf kL1� 1

1; kf kL1> 1:

Therefore,

Y˚ D ff 2 L1 W kf kL1� 1g

and

aY˚ D ff 2 L1 W kf kL1� ag; 0 < a < 1;

whence

H˚ D\

0<a<1aY˚ D f0g;

and all inclusions in (14.1.2) are strict.

14.2 Separability Conditions for Orlicz Spaces

Now we show that the equality H˚ D L˚ is equivalent to the separability of L˚ .First, compare the unit ball of Y˚ ,

BY˚ D ff 2 Y˚ W I˚.f / � 1g;

with the unit ball of L˚ ,

BL˚ D ff 2 L˚ W kf kL˚ � 1g:

Proposition 14.2.1. Let L˚ be an Orlicz space.

1. If kf kL˚ D 1, then I˚.f / � kf kL˚ D 1.2. If kf kL˚ < 1, then I˚.f / � kf kL˚ < 1.3. If kf kL˚ > 1, then I˚.f / � kf kL˚ > 1.

Proof. 1. Let a > kf kL˚ D 1. Then I˚

�

f

a

�

� 1 and for every sequence an # 1,

an # kf kL˚ H) jf jan

x

?

jf jkf kL˚

H) ˚

� jf jan

�

x

?˚

� jf jkf kL˚

�

H)

H) I˚

�

f

a

�

x

?I˚

�

f

kf kL˚

�

:

Consequently, I˚

�

f

kf kL˚

�

� 1 and I˚.f / � kf kL˚ .

2. Let 0 < a D kf kL˚ < 1. Then by Proposition 13.2.1,

1

aI˚.f / � I˚

�

f

a

�

< 1;

and hence

I˚.f / � kf kL˚ < 1:

3. Let 1 < a < kf kL˚ . Then by Proposition 13.2.1,

1 < I˚

�

f

a

�

� 1

aI˚.f /:

Therefore, I˚.f / > a. Taking a " kf kL˚ , we get I˚.f / � kf kL˚ > 1. utRecall that a symmetric space X is called minimal if its minimal part

X0 D clX.L1 \ L1/ D clX.F0/

coincides with X. If, in addition, 'X.0C/ D 0, then the space X is separable(Theorem 6.5.3). Our aim now is to study these conditions in the case of Orliczspaces X D L˚ .

We consider two cases: b˚ D 1 and b˚ < 1. According to Corollary 13.3.3,'L˚ .0C/ D 0 in the first case, and 'L˚ .0C/ > 0 in the second case.

Theorem 14.2.2. Let ˚ be an Orlicz function.

1. If b˚ < 1, then H˚ D f0g.2. If b˚ D 1, then H˚ coincides with the minimal part L0˚ of the Orlicz space L˚ .

Proof. 1. If f 2 H˚ , then f 2 aY˚ for all a > 0, i.e.,

I˚

�

f

a

�

D1Z

0

˚

� jf ja

�

dm < 1; a > 0:

If b˚ < 1, then ˚.2b˚/ D 1, and for all f ¤ 0 and a > 0, we have

m

� jf ja

� 2b˚

> 0:

So

I˚

�

f

a

�

D1Z

0

˚

� jf ja

�

dm � ˚.2b˚/ � m

� jf ja

� 2b˚

D 1;

which contradicts the assumption f 2 H˚ � Y˚ .Thus, f D 0 for all f 2 H˚ , i.e., H˚ D f0g.

2. We show that L0˚ � H˚ , provided b˚ D 1.The latter condition means that ˚.x/ < 1 for all x � 0. Let f 2 F0, i.e., f has

the form

f DnX

iD1ci � 1Œai;bi�; 0 � ai < b1 � b < 1:

Then

I˚

�

f

a

�

D1Z

0

˚

� jf ja

�

dm DnX

iD1˚

� jcija

�

.ˇi � ˛1/ < 1

for all a > 0, i.e., f 2 H˚ . So F0 � H˚ .Next we verify that H˚ is a closed linear subspace of L˚ . Let fn 2 H˚ , f 2 L˚ ,

and kf � fnkL˚ ! 0; n ! 1. Then for every a > 0 and sufficiently large n, wehave kf � fnkL˚ < a, and by Proposition 14.2.1,

I˚

�

1

a.f � fn/

�

< 1:

Thus f � fn 2 aY˚ .On the other hand, fn 2 H˚ � aY˚ . By Proposition 13.2.1, the set aY˚ is

convex, and then

1

2f D 1

2.f � fn/C 1

2fn 2 aY˚ ;

i.e., f 2 2aY˚ for all a > 0. Hence, f 2 H˚ . Therefore, H˚ is a closed linearsubspace of L˚ . Since F0 � H˚ , we have also

L0˚ D clL˚ .F0/ � H˚ :

Further, we check the reverse inclusion. Suppose f 2 H˚ . Since

f 2 H˚ ” jf j 2 H˚ ;

we may assume that f � 0. Choose a sequence fn 2 F0 � H˚ such that fn " f . Thenhn D f � fn # 0, and hence hn ! 0 almost everywhere. Since b˚ D 1, the function

˚ is finite on Œ0;1/. Therefore, for every a > 0, the sequence ˚

�

hn

a

�

approaches

zero almost everywhere. Since h1 D f � f1 2 H˚ and hn � h1, we have

I˚

�

h1a

�

D1Z

0

˚

�

h1a

�

dm < 1:

Consequently, the sequence ˚

�

hn

a

�

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

a > 0,

limn!1 I˚

�

hn

a

�

D limn!1

1Z

0

˚

�

hn

a

�

dm D 0:

Hence I˚

�

hn

a

�

< 1 for all sufficiently large n and khnkL˚ < a for all a > 0. Thus,

limn!1 khnkL˚ D lim

n!1 kf � fnkL˚ D 0;

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˚ . ut

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.

Thenjf jan

# 0 and ˚

� jf jan

�

# ˚.jf j/ almost everywhere, since the function ˚ is

finite, increasing, and continuous.

The equality kf kL˚ D 1 implies

�

�

�

�

f

an

�

�

�

�

L˚

> 1, and I˚

�

f

an

�

> 1 by

Proposition 14.2.1, 3.

On the other hand, I˚

�

f

an

�

< 1 for all n, since f 2 H˚ . Therefore, the

sequence of integrable functions ˚

� jf jan

�

is dominated by ˚

�

f

a1

�

and converges

almost everywhere to ˚.jf j/ 2 L1.Consequently,

1 < I˚

�

f

an

�

D1Z

0

˚

� jf jan

�

dm #1Z

0

˚.jf j/dm D I˚.f /;

i.e., I˚.f / � 1. On the other hand,

kf kL˚ D 1 H) I˚.f / � 1;

and thus I˚.f / D 1. utNow 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:

Since

b˚ D 1 ” 'L˚ .0C/ D 0;

the equality L˚ D H˚ means that the symmetric space X D L˚ satisfies condition2 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. L˚ D H˚ .2. L˚ is minimal and 'L˚ .0C/ D 0.3. L˚ is separable.4. L˚ has property .A/.

14.3 The .�2/ Condition

The equivalent conditions of Corollary 14.2.4 can be expressed in terms of Orliczfunctions ˚ .

Definition 14.3.1. An Orlicz function ˚ satisfies the .�2/-condition if a˚ D 0,b˚ D 1, and

sup0<x<1

˚.2x/

˚.x/< 1: (14.3.1)

Conditions a˚ D 0 and b˚ < 1 mean that

0 < ˚.x/ < 1 for x > 0: (14.3.2)

The .�2/-condition can be divided into two separate parts: .�2.0// and.�2.1//, describing the behavior of ˚ at 0 and at 1.

The .�2.0// condition: There exist k > 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

Since the function˚.2x/

˚.x/is continuous on .0;1/, the inequalities (14.3.3),

(14.3.4) imply (14.3.1).

Theorem 14.3.2. Each of the equivalent conditions in Corollary 14.2.4 is alsoequivalent to the .�2/ condition.

Proof. The .�2/ condition implies that ˚.2x/ � k˚.x/ for some k > 0 and allx 2 .0;1/. Hence

I˚.2f / � kI˚.f /

for all f . Suppose that f 2 2Y˚ . Then I˚

�

f

2

�

< 1 and I˚.f / < 1, i.e., f 2 Y˚ .

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

2n˚.xn/; n D 1; 2; : : : :

The function

f D1X

nD1xn � 1An

satisfies

I˚.f / D1Z

0

˚.jf j/ dm D1X

nD1

˚.xn/

2n˚.xn/D

1X

nD1

1

2nD 1 < 1

and

I˚.2f / D1Z

0

˚.2jf j/ dm D1X

nD1

˚.2xn/

2n˚.xn/>

1X

nD1

2n˚.xn/

2n˚.xn/D

1X

nD11 D 1;

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

14.4 Examples of Orlicz Spaces with and Withoutthe .�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

0 < supx2.0;1/

˚.2x/

˚.x/D sup

x2.0;1/

.2x/p

xpD 2p < 1:

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, sincea˚ 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 doesnot 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 anda˚ D 0. Since

limx!1

e2x � 2x � 1ex � x � 1 D 1; lim

x!0Ce2x � 2x � 1ex � x � 1 D 4;

the .�2.0// condition holds, but the .�2.1// condition does not hold. Hence, Y˚ ¤2Y˚ .

We show that for the function

f .x/ D� � 1

2ln x; for 0 < x � 1I

0; for x > 1;

f 2 Y˚ , but 2f 62 Y˚ . Indeed,

I˚.f / D1Z

0

˚.jf j/dm D1Z

0

e� 12 ln x C 1

2ln x � 1

�

dx D 1

2;

while

I˚.2f / D1Z

0

˚.j2f j/dm D1Z

0

�

e� ln x C ln x � 1� dx D C1:

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 15Duality for Orlicz Spaces

In this chapter, we consider associate spaces L1˚ of Orlicz spaces L˚ . Using theLegendre 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� coincideas sets and k � kL� � k � kL1˚

� 2k � kL� . The duality between L˚ and L� is studiedin detail.

15.1 The Legendre Transform

In Sections 15.1 and 15.2, we construct the conjugate Orlicz function � for anOrlicz function ˚ . By definition, ˚ and � are conjugate if they are the Legendretransforms 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 � 0exist almost everywhere and that they are increasing functions.

Let y D y.x/. Then we obtain from (15.1.1), using differentiation with respect tox, that

˚ 0.x/C � 0.y/y0 D y C xy0;


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 theirderivatives ˚ 0 and � 0 are (generalized) inverse functions of each other (Fig. 15.1).

Proposition 15.1.2 (Young Inequality). Let ˚ and � be two conjugate Orliczfunctions; 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 onlyif 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 ofconjugate functions

y=F'(x)

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

aF

aF

aF 0> >

< <bF

bF

bF

•aF 0

bF •

x x

y y

0

15.2 The Geometric Interpretation 197

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

F(x) F(x) F(x)

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

x =Y'(y)

Y(y) Y(y) Y(y)

y

xx

y

0

y

xx

y

0

y

xx

y

0

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 onlyif y D ˚ 0.x/ and x D � 0.y/.

Hence,

˚.x/ D supy

fxy � �.y/g (15.1.6)

and

�.y/ D supx

fxy � ˚.x/g: (15.1.7)

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 thefunction y D ˚.x/. If the derivative ˚ 0.x/ exists at a point x0, the tangent line at thepoint .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/:

Fig. 15.3 Legendre’sparameter m D ˚ 0.x/

y

x0

Φ(x0)

y=Φ(x)

b=–Ψ(m)

y=mx+b

x0

Thus, the family of tangent lines to the graph of y D ˚.x/ can be parametricallyrepresented 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

@mD �x C �.m/ D 0:

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 Legendretransform in the case ofderivative jump

y=m2x+b2

y=F(x)

F(x0)

y=m1x+b1

xx0b1

b2

y

0

Fig. 15.5 Jump of ˚ 0 andlinearity interval of �

y

xy

x0 0x0m

1m

2

m1

m2

x

y

Y

Y

Y

( )

( )

( )

m2

m1 Y

y

( )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 ˚.x2/ � ˚.x1/x2 � x1

and b D ˚.x1/x2 � ˚.x2/x1x2 � 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).


Fig. 15.6 Linearity intervalof ˚

x

y

=y

F(x2)

F( )x

mx b=y +

F(x1)

x1 x20

Fig. 15.7 Constancy intervalof derivative ˚ 0

y

x0

m

x

x1 x2

y

'Y ( )y

'F ( )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 allf 2 L˚ and g 2 L� , and

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� 2kf kL˚ � kgkL� : (15.3.1)

15.3 Duality for Orlicz Spaces 201

Fig. 15.8 Left and righttangent lines at y D m

0 y

x

x =

Y

Y

( )

m

m

( )m

( )y

( )y−mx Y' Y= ( )m+ +

( )m( )y−mx Y' Y= ( )m +

Proof. By (15.1.2), we haveˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

�1Z

0

.˚.jf j/C �.jgj//dm � I˚.f /C I� .g/:

By applying this inequality to the functionsf

kf kL˚and

g

kgkL�, we have

1

kf kL˚ kgkL�

1Z

0

jfgjdm � I˚

�

f

kf kL˚

�

C I�

�

g

kgkL�

�

� 2:

utRecall 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˚D sup

8

<

:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

W f 2 L˚ ; kf kL˚ � 1

9

=

;

:

Using Corollary 15.3.1, we have

L� � L1˚ and kgkL1˚� 2kgkL� ; g 2 L� :

The following theorem states that in fact, L� D L1˚ .


Theorem 15.3.2. Let ˚ and � be conjugate Orlicz functions. Then the symmetricspaces L1˚ and L� coincide as sets, and

kgkL� � kgkL1˚� 2kgkL� ; g 2 L� : (15.3.2)

Proof. We prove that L1˚ � L� and kgkL� � kgkL1˚for every g 2 L1˚ .

Let 0 ¤ g 2 L1˚ and a D kgkL1˚. We show that kgk� � a, i.e.,

1Z

0

�

�

1

ajgj�

dm � 1:

First, suppose that g 2 F1 and the function f D �

�

1

ajgj�

is finite almost

everywhere. By Young’s inequality (15.1.5),

1Z

0

1

ajfgjdm D

1Z

0

�

˚.jf j/C �

�

1

ajgj��

dm D I˚.f /C I�

�

1

ajgj�

:

If I˚.f / � 1, then1R

0

jfgjdm � kgkL1˚D a. Thus,

I�

�

1

ajgj�

� I˚.f /C I�

�

1

ajgj�

D1Z

0

1

ajfgjdm � 1:

On the other hand, if I˚.f / D b > 1, then

I˚

�

1

bjf j�

� 1

bI˚.f /;

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

I˚.f /C I�

�

1

ajgj�

D 1

a

1Z

0

jfgjdm � ab

aD b D I˚.f /:

Hence, I�

�

1

ajgj�

D 0.

Thus in both cases, I�

�

1

ajgj�

� 1, i.e., kgkL� � a D kgkL1˚.

15.3 Duality for Orlicz Spaces 203

Now let g 2 F1 be a function such that �

�

1

ajgj�

is not almost everywhere

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

A � fjgj > ab� g such that 0 < mA < 1. For the function f D 1

b�mA� 1A, we have

I˚.f / D ˚

�

1

b�mA

�

mA � 1;

since kf kL˚ � 1. Consequently,

a D kgkL1˚�

1Z

0

jfgjdm >ab�

b�mAmA D a:

This contradiction shows that jgj � ab� almost everywhere.

Taking an " 1 with an < 1, we havean

ajgj � anb� < b� , whence �

�an

ajgj

is

finite.In this case, we can prove as above that

I�

�ang

a

� an < 1:

Since an " 1 and � is left continuous, we obtain I�� g

a

� � 1.For every g 2 L� , we choose a sequence gn in F1 such that gn " jgj. Since

I�

gn

kgnkL1˚

!

� 1 and kgnkL1˚� kgkL1˚

D a;

we have I�

�

1

agn

�

� 1 and hence I�

�

1

ag

�

� 1. Thus, kgkL� � a D kgkL1˚. ut

Replacing ˚ and � in Theorem 15.3.2, we have L˚ D L1� . Thus every Orliczspace 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˚� 1C I� .g/; g 2 L� .

Proof.

kgkL1˚D sup

kf kL˚ �1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

� supkf kL˚ �1

1Z

0

.˚.jf j/C �.jgj//dm

D supkf kL˚ �1

.I˚.f /C I� .g// � 1C I� .g/:

utNote 15.3.5. Replacing ˚ and � in Proposition 15.3.4, we have

kf kL1�� 1C I˚.f /; f 2 L˚ ;

and takingf

cinstead of f , we have

kf kL1�� c

�

1C I˚

�

f

c

��

for all c > 0:

Moreover, it can be proved that

kf kL1�D inf

c>0c

�

1C I˚

�

f

c

��

:

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 allx � 0 (Fig. 15.9).

Fig. 15.9 ˚ 0 and � 0 whena˚ > 0 and b˚ < 1

F

F'

x y

y x

0 0

= ( )

aFb

y x

Y'= ( )x y

F'= ( )y x

Y'= ( )x 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 b�1

˚ .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 thefollowing 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 14.3.2and 15.3.2 the following result.

Theorem 15.4.1. The following are equivalent:

1. ˚ satisfies the .�2/ condition.2. L˚ is separable.3. L˚ D H˚ .4. L� D .L� /, where � is the conjugate function for ˚ .

By combining this theorem with the results of Section 8.4, we obtain the followingcorollary.

Corollary 15.4.2. Let˚ be � be two conjugate Orlicz functions. The following areequivalent:

1. Both ˚ and � satisfy the .�2/ condition.2. Both L˚ and L� are separable.3. .L˚/ D L�

� ; .L� / D L� ;4. L˚ is reflexive.5. L� is reflexive.

Chapter 16Comparison of Orlicz Spaces

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

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 suchthat

˚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.a1x/; 0 � x � x1 forsome a1 > 0 and x1 > 0.

2. ˚1 �1 ˚2 if and only if ˚2.x/ � ˚1.a2x/; x � x2 for some a2 > 0 and x2 > 0.


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 x

b, we have

˚2.y/ D ˚2

� x

b

� 1

b˚2.x/ � 1

bb˚1.ax/ D ˚1.aby/

for all 0 � x � x0. That is,

˚2.y/ � ˚1.a1y/; 0 � y � y1;

where a1 D ab and y1 D x0b

, i.e., ˚1 �0 ˚2.

The converse is obvious.2. The proof is similar to that of 1.

utProposition 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.a1x/; 0 � x � x1;

and

˚2.x/ � ˚1.a2x/; x � x2;

for some positive a1, a2, x1, and x2.We may assume without loss of generality that ˚2.x2/ < 1. Otherwise, we

choose x2 such that 0 � x2 � x2 and ˚2.x2/ < 1. Then by setting a2 D a2 � x2x2

, we

have

˚2.x/ � ˚2.x � x2x2/ � ˚1.a2x � x2

x2/ D ˚1.a2x/

for all x � x2.Similarly, we may assume that ˚1.a1x1/ > 0.

16.2 The Embedding Theorem for Orlicz Spaces 209

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

b D8

<

:

1; for x2 � x1;

maxx1�x�x2

�

˚2.x/

˚1.ax/

�

; for x1 < x2:

Then ˚2.x/ � b˚1.ax/ for all x � 0. ut

16.2 The Embedding Theorem for Orlicz Spaces

Let L˚1 and L˚2 be two Orlicz spaces. Our aim is to characterize the embeddingL˚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 thecorresponding fundamental functions of Orlicz spaces L˚1 and L˚2 . Then thefollowing are equivalent:

1. ˚1 � ˚2;2. L˚1 � L˚2 ;3. kf kL˚2 � akf kL˚1 for some a > 0;4. 'L˚2 � a'L˚1 for some a > 0;5. ˚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,

I˚1

�

f

c

�

D1Z

0

˚1

� jf jc

�

dm < 1:

Consequently,

I˚2

�

f

ac

�

D1Z

0

˚2

� jf jac

�

dm � b

1Z

0

˚1

� jf jc

�

dm D bI˚1

�

f

c

�

< 1;

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

�

˚�12

�

x�1��1 � a�

˚�11

�

x�1��1 ; x > 0:

By taking x�1 D ˚2.y/, we obtain

˚�11 .˚2.y// � a˚�1

2 .˚2.y// D ay; y > 0;

or

˚2.y/ � ˚1.ay/; y > 0:

5 H) 1 is obvious. utNote 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

�

c2c1

x

�

; x � 0;

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 � cI1; x > c:

16.2 The Embedding Theorem for Orlicz Spaces 211

Fig. 16.1 Functions˚c.x/ D ecx � 1; c � 0

x0

y

ec2x–1 ec1x–1

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

x0 x

yy

01

F

• •

c

1c F

1c

1c 2c

For 0 < c1 < c2 < 1, we have

˚c2 .x/ � ˚c1 .x/ D ˚c2

�

c1c2

x

�

; x � 0;

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 comparisontheorem for Lorentz and Marcinkiewicz spaces (Theorem 12.4.1), the above Theo-rem 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 thefundamental functions of Orlicz spaces L˚1 and L˚2 .

Since all Orlicz spaces are maximal (Theorem 8.2.2), Theorem 12.1.3 implies

�eV1

� L˚1 � MV1�; �eV2

� L˚2 � MV2�;

whereeVi is the least concave majorant of Vi, and Vi�.x/ D x

Vi.x/for x > 0, i D 1; 2.

We claim that

�eV1

� �eV2

” L˚1 � L˚2 ” MV1�� MV2�

:

Indeed, each of the embeddings holds if and only if V2 � aV1 for some a > 0. (Thefirst 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 Orliczspaces 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 ” ˚ 02.1/ < 1:

2. Let L˚2 D L1, ˚2.x/ D x, x � 0. In this case,

L˚1 � L1 ” ˚ 01.0C/ > 0:

3. Let L˚1 D L1, ˚1.x/ D 1.0;1/.x/. In this case,

L1 � L˚2 ” a˚2 > 0:


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 offunctions ˚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 areequivalent:

1. ˚1 � ˚2.2. L˚1 D L˚2 as sets.3. k � kL˚1 and k � kL˚2 are equivalent, i.e.,

a1kf kL˚1 � kf kL˚2 � a2kf 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.a1x/ � ˚2.x/ � ˚1.a2x/; 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 thesame. For example, condition 5 is equivalent to

1

a2˚�11 .y/ � ˚�1

2 .y/ � 1

a1˚�11 .y/; y > 0;

where y D ˚2.x/ . Hence,

a1

�

˚�11

�

1

x

��1��

˚�12

�

1

x

��1� a2

�

˚�11

�

1

x

��1; x > 0;

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,

�eV1

D �eV2

” 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 ” ˚ 02.0C/ > 0 and ˚ 0

2.1/ < 1:


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

˚˛.x/ D8

<

:

0; 0 � x � 1;xR

1

.ln u/˛du; x > 1:

Since the derivatives

˚ 0 .x/ D ln˛ x � 1Œ1;1/.x/; x � 0;

0 0

y y y

x x1 1 0 x1

1

F'0

0 11

1

1

>( )

< <

F'a

aaa

a

e

F'a

a( )

Fig. 16.3 Derivatives of ˚˛.x/

Fig. 16.4 Functions˚˛.x/; 0 � ˛ < 1

y y

0 1 x 0 1 x

Fa F0

a>0

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 (seeCorollary 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:

16.4 Zygmund Classes 215

Since for ˛ � 1,

˚0˛.x/ D ˚ 0 .x/C ˛˚ 0 �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 itsgreatest convex minorant .˚˛/

Ï. For example, if ˛ D 0, then the function ˚0 is thegreatest 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

� 0 .x/ D ex1=˛ ;

where

�˛.x/ DxZ

0

eu1=˛du; x � 0;

is the conjugate of the Orlicz function ˚˛ .

Fig. 16.5 ˚0 and its greatestconvex minorant

10 x 10

1

y

x

y

1

F0

F0

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˛ andZ1˛ .

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˛ hasthe form

Z0˛ D L0˚˛ D H˚˛ D Z˛ \ R0:

Chapter 17Intersections and Sums of Orlicz Spaces

In this chapter, we study the intersections and the sums of Orlicz spaces. Inparticular, for every pair of mutually conjugate Orlicz functions ˚ and � , the Orliczspaces .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 correspondingOrlicz 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˚2g

are equipped with the norms

kf kL˚1\L˚2 D max.kf kL˚1 ; kf kL˚2 / (17.1.1)

and

kf kL˚1CL˚2 D inffkgkL˚1 C khkL˚2 ; f D g C h; g 2 L˚1; h 2 L˚2g; (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/:


217

218 17 Intersections and Sums of Orlicz Spaces

0

yyy

0 0x x x

~( )

F1

F2

F2

F1F1

x x1

y1

y2

2

F1F2

F2F2

F1F2F1

Ÿ Ÿ Ÿ

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 notnecessarily a convex function, i.e., it is not an Orlicz function in general (Fig. 17.1).

Since the inverse functions ˚�11 and ˚�1

2 are concave, the function ˚�11 _ ˚�1

2

is quasiconcave (Proposition 10.2.3). Its least concave majorant .˚�11 _ ˚�1

2 /Ï isthe 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

�

< 1

for some a1 > 0 and a2 > 0, whence

I˚1_˚2�

f

a

�

� I˚1

�

f

a1

�

C I˚2

�

f

a2

�

< 1

for a D max.a1; a2/. Thus L˚1 \ L˚2 � L˚1_˚2 , and hence

L˚1_˚2 D L˚1 \ L˚2 :

17.1 The Intersection and the Sum of Orlicz Spaces 219

2. Let .˚1 ^ ˚2/Ï be the greatest convex minorant of ˚1 ^ ˚2. Then

˚i � ˚1 ^ ˚2 � .˚1 ^ ˚2/Ï; i D 1; 2:

Hence

L˚1 � L.˚1^˚2/Ï ; L˚2 � L.˚1^˚2/Ï

and

L˚1 C L˚2 � L.˚1^˚2/Ï :

On the other hand, let f 2 L.˚1^˚2/Ï . We may assume without loss of generalitythat f � 0. Then for some a > 0,

I.˚1^˚2/Ï�

f

a

�

< 1;

and by (17.1.3),

1Z

0

.˚1 ^ ˚2/�

f

2a

�

dm D1Z

0

˚1

�

f

2a� 1f˚1�˚2g

�

dm

C1Z

0

˚2

�

f

2a� 1f˚2<˚1g

�

dm < 1:

Then

g D f

2a� 1f˚1�˚2g 2 L˚1 ; h D f

2a� 1f˚2<˚1g 2 L˚2;

and hence f D g C h 2 L˚1 C L˚2 . Thus,

L.˚1^˚2/Ï � L˚1 C L˚2

and

L.˚1^˚2/Ï D L˚1 C L˚2 :

This completes the proof. utConsider now the associate spaces of sums and intersections of Orlicz spaces.

Using the Legendre transform

˚L.x/ D supy.xy � ˚.y//

of ˚ , we obtain

˚L1 _ ˚L

2 D .˚1 ^ ˚2/L;.˚1 _ ˚2/L.x/ � .˚L

1 ^ ˚L2 /.x/ � .˚1 _ ˚2/L.2x/;

.˚1 ^ ˚2/LL.x/ � .˚1 ^ ˚2/.x/ � .˚1 ^ ˚2/LL.2x/: (17.1.4)

In these relations, the function ˚1 _ ˚2 is convex, and the function .˚1 ^ ˚2/LL

coincides with the greatest convex minorant of the quasiconvex function ˚1 ^ ˚2.The inequalities (17.1.4) follow from Theorem 10.5.2.

By inequalities (17.1.4) and Proposition 17.1.1, we also have the following.

Proposition 17.1.2. 1. .L˚1 \ L˚2/1 D L˚1 C L˚2 .

2. .L˚1 C L˚2/1 D L˚1 \ L˚2 .

17.2 The Spaces L˚ C L1 and L� \ L1

As a particular case, we consider the spaces L˚ C L1, L˚ \ L1, L� C L1, andL� \ L1, where ˚ and � are two conjugate Orlicz functions. Since

a˚ > 0 H) L˚ L1 H) L˚ C L1 D L˚

and

b˚ < 1 H) L˚ � L1 H) L˚ C L1 D L1;

we may assume without loss of generality that a˚ D 0 and b˚ D 1.By Proposition 17.1.1, we have L˚ C L1 D L.˚^˚1/Ï , where

˚1 D�

0; 0 � x � 1;

1; x > 1;

and .˚ ^ ˚1/Ï is the greatest convex minorant of ˚ ^ ˚1 D min.˚;˚1/. Thefunction .˚ ^ ˚1/Ï has the form

.˚ ^ ˚1/Ï.x/ D

8

ˆ

<

ˆ

:

0; 0 � x � 1;

˚.x0/x�1x0�1 ; 1 < x � x0;

˚.x/; x > x0:

Here x0 is chosen in such a way that the supporting line

y D ˚.x0/x � 1x0 � 1

17.2 The Spaces L˚ C L1 and L� \ L1 221

•

1

1( )

x0110 0 0 0

y y y y

xxx

V ( )

x

F F•

•

F

F

F

V

•F F ˜

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

0

y x

0 1

1

2

1

0

0

2

-1

( )x yF

=y x( )F = =y xx( )F

1( )F

y( )F

-1=x y( )F

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/ � ˚�10 .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 L1g:


Fig. 17.4 Functions � and� _ ˚1

0

y y

x 0

V

xx

FY Y

1

0 x0

F1

It is equivalent but is not equal to the natural norm of L˚ C L1 defined by

kf kL˚CL1D inffkgkL˚ C khkL1

W f D g C h; g 2 L˚ ; h 2 L1g:

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 spaceL�_˚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

˚L1 D ˚1 and ˚L1 D ˚1 and relations (17.1.4), we have

.˚ ^ ˚1/L D � _ ˚1and

.� _ ˚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

.˚ ^ ˚1/Ï.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)

.˚ ^ ˚1/Ï.x/ D�

˚.x/; 0 � x � 1;

x C ˚.1/ � 1; x > 1:

0 x 0

V

x 0

1x

1x x

y y y

~F F

1F

1F

V ( )

( )

F

F

1F

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

Fig. 17.6 Function.˚ ^ ˚1/

Ï

0 1 x

y

y= F(x)

F'(1)=1

( )y x F= _+ 1( )F 1

1

Fig. 17.7 Functions �;˚1,and � _ ˚1

0

y y

V

1 x 0 1 x

F•

F•

• •Y Y

For every Orlicz function � , the space L� \ L1 coincides with the Orlicz spaceL�_˚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 spacewith the Orlicz function ˚.x/ D xp. In this case, Lp D L˚ D H˚ and k � kLp Dk � kL˚ .

It will be convenient to use additional Orlicz functions

˚p.x/ D xp

p; x � 0:

17.4 The Spaces Lp \ Lq and Lp C Lq, 1 � p � q � 1 225

We have

kf kL˚pD inf

8

<

:

a > 0 W1Z

0

jf jpp ap

dm � 1

9

=

;

D inf

8

<

:

a > 0 W1Z

0

� jf jp1=p

�p

dm � ap

9

=

;

D inf

8

ˆ

<

ˆ

:

a > 0 W0

@

1Z

0

� jf jp1=p

�p

dm

1

A

1p

� a

9

>

=

>

;

D 1

p1=pkf kLp :

Thus, L˚p D L˚ D Lp and k � kL˚pD 1

p1=pk � kL˚ D 1

p1=pk � kLp .

The function ˚p corresponds to the normalizing condition ˚ 0p.1/ D 1, instead

of ˚.1/ D 1. The advantage of such normalization is that the conjugate powerfunctions

y D ˚ 0p.x/ D xp�1 and x D ˚ 0

p0.y/ D yp0�1;1

pC 1

p0 D 1;

are mutually inverse (.p � 1/.p0 � 1/ D 1), i.e., the Orlicz functions ˚p and ˚p0 areconjugates of each other (Fig. 17.8).

In the case p D 1, we have

˚1.x/ D limp!1˚p.x/ D lim

p!1xp

pD�

0; 0 � x � 1;

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 andLp 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

x

y

0

(y)p'

y

x

F

( )p xF

( )( ) =1p p'1 1_ _

=

=

1

1

y

x

x

y

p

p' _

_

For 1 � p � q < 1, we have

.˚p _ ˚q/.x/ D max.˚p.x/; ˚q.x// D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

xp

p; if 0 �

�

q

p

� 1q�p

;

xq

q; if x >

�

q

p

� 1q�p

;

and

.˚p ^ ˚q/.x/ D min.˚p.x/; ˚q.x// D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

xq

q; if 0 �

�

q

p

� 1q�p

;

xp

p; if x >

�

q

p

� 1q�p

:

The greatest convex minorant .˚p ^ ˚q/Ï.x/ of .˚p ^ ˚q/.x/ has the form

(Fig. 17.9)

.˚p ^ ˚q/Ï.x/ D

8

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

:

xq

q; 0 � x � x1

xq�11 x � q � 1

qxq1; x1 < x � x2;

xp

p; x > x2;

where

x1 D

p.q � 1/q.p � 1/

�p�1q�p

; x2 D

p.q � 1/q.p � 1/

�q�1q�p

:

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

y D xq�11 x � q � 1

qxq1

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

0

yyy

0 0x x x

V ~( )

FF

qFp

q

FppFp

x x1

y1

y2

2

F

V

qF

qFqF

pFVqFpF

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

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:

˚p;q.x/ D

8

ˆ

<

ˆ

:

xp

p; 0 � x � 1I

xq

qC 1

p� 1

q; x > 1;

.p; q < 1/;

˚p;1.x/ D8

<

:

xp

p; 0 � x � 1I

1; x > 1;.q D 1/;

˚1;q.x/ D8

<

:

0; 0 � x � 1Ixq

q� 1

q; x > 1:

.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, thereexists ˚ 0

p;q.1/ D 1 (Fig. 17.10).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 :

x0 11 q/1 p

qxq

p,q<• q<•p<•

p,•

p q

p,q•

1+ 1

px px/ 1 p/ 1 p/

x0 1 x0 1 x0 1

y y y yF F •,q

F

qxq

pxp

q1

qxq

pp

Fig. 17.10 Functions ˚p;q; ˚p;1; ˚1;q

The conjugate functions�p;q of Orlicz functions˚p;q can be easily found. Indeed,for 1 < p; q < 1,

˚ 0p;q.x/ D

�

xp�1; 0 < x � 1Ixq�1; x > 1:

Hence

� 0p;q.x/ D

(

x1

p�1 ; 0 < x � 1Ix

1q�1 ; x > 1;

and

�p;q D

8

ˆ

<

ˆ

:

p � 1p

xp

p�1 ; 0 < x � 1Iq � 1

qx

qq�1 C p � 1

p� q � 1

q; x > 1:

Using the conjugate exponents .p0

; q0

/,

1

qC 1

p0D 1;

1

pC 1

q0D 1;

we obtain

p � 1p

D 1

q0;

q � 1q

D 1

p0;

and hence

�p;q D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

xq0

q0; 0 < x � 1I

xp0

p0C 1

q0� 1

p0; x > 1

9

>

>

>

=

>

>

>

;

D ˚q0

;p0 :

If 1 � p � q � 1 (and hence 1 � p0 � q

0 � 1), then

.Lp \ Lq/1 D L1˚p;q

D L�p;q D L˚q0;p

0D Lp0 C Lq0 ;

and

.Lp C Lq/1 D L1˚q;p

D L�q;p D L˚p0;q

0D Lp0 \ Lq0 :

Exercises 229

Exercises

25. Orlicz spaces on general measure spaces. Let .˝ ; / be a measure spacewith a � -finite (finite or infinite) measure and let ˚ be an Orlicz function onŒ0;1/. Let L˚ D L˚.˝;/ be the corresponding Orlicz space on .˝ ; /, that is,

L˚.˝;/ D ff 2 L0.˝;/ W kf kL˚ .˝ ;/ < 1g;where

kf kL˚ .˝;/ D inf

�

c > 0 W I˚

�

f

c

�

� 1

and I˚.f / DZ

˝

˚.jf j/d:

Show that

a. .L˚.˝;/; k � kL˚ .˝ ;// is a symmetric space on .˝ ; / in the sense ofComplement 1.

b. Let the measure be nonatomic and infinite. Then the standard space corre-sponding to the symmetric space L˚.˝ ; / coincides with the Orlicz spaceL˚.RC;m/ on Œ0;1/, i.e.,

L˚.RC;m/ D ff 2 L0.˝;/ W f � D g� for some g 2 L˚.˝;/g;

andZ

˝

˚.jf j/d DZ

RC

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

c. Let the Orlicz function ˚ be finite on .0;1/ (b˚ D 1) and let

H˚.˝;/ D8

<

:

f 2 L0.˝;/ WZ

˝

˚

� jf jc

�

d < 1 for all c > 0

9

=

;

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˚ DclL˚ .F1/, where F1 D F1.˝;/ denotes the set of all simple -integrablefunctions 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 correspondingassociate space L1˚ D .L˚.˝;//1 is defined by

L1˚ D fg 2 L0.˝;/ W ug 2 L� g;


where

kgkL1˚D kugkL�

˚and ug.f / D

Z

˝

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 theOrlicz 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� DH� , 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:

Exercises 231

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 thatH˚.˝;/ 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, thenH˚.˝;/ 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 ffng in L˚ is said to be

˚-mean convergent to f 2 L˚ if limn!1 I˚.fn � f / D 0 (Here I˚.f / D R

˝

˚.jf j/d).

Show that

a. limn!1 kfn � f kL˚ D 0 ” lim

n!1 I˚.k.fn � f // D 0 for all k > 0.

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.

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˚ andH˚ ¤ L˚ .

b. If fn.x/ D � ln x � 1Œ0; 1n �.x/, then fn # 0, but kfnkL˚ � 1 for all n, i.e., L˚ fails tohave property .A/.

c. If fn.x/ D � 12

ln x � 1Œ0; 1n �.x/, then I˚.fn/ ! 0, but kfnkL˚ � 12, i.e., ˚-mean

convergence does not imply convergence in norm k � kL˚ .d. If fn.x/ D � ln x�1Œ 1n ;1�.x/, then fn 2 H˚ , fn " f 62 H˚ , although sup

nkfnkL˚ < 1,

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

'L˚ .x/ D �

˚�1 �x�1��1 ; x > 0:

Show that

a. Let ˚1 and ˚2 be two Orlicz functions. Then the following conditions areequivalent:

• L˚1 D L˚2 as sets.• ˚1 � ˚2 in the sense of Definition 16.3.1.• The functions ˚�1

1 and ˚�12 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 concavemajorant 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 L�1 DL1˚ as sets, and the Orlicz functions �1 and � are equivalent in the sense ofDefinition 16.3.1.

c. X D L2 is the only Orlicz space for which X D X1.

Exercises 233

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

Lp;q D

8

ˆ

<

ˆ

:

f 2 L0 W kf kLp;q D0

@

1Z

0

Œf �.x/�q d.xqp /

1

A

1q

< 1

9

>

=

>

;

;

for 1 � p < 1, 1 � q < 1, and

Lp;1 D�

f 2 L0 W kf kLp;1 D supx>0

x1p f �.x/ < 1

;

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 � k0

Lp;q, where

kf k0Lp;q

D kf ��kLp;q f 2 Lp;q; f ��.x/ D 1

x

Z x

0

f �dm:

In this case, .Lp;q; k � k0Lp;q/ is a symmetric space.

Hint: Apply Jensen’s inequality ˚�R

fdm� � R

˚.f /d with the convexfunction '.x/ D xq and the probability measure

d.u/ D t.1p �1/d.u.1�

1p //

on the interval Œ0; t�. Check that

0

@

tZ

0

f .u/du

1

A

q

��

p

p � 1�q�1

t.q�1/.p�1/

p

tZ

0

.f �.u//quq�1

p du

and that the inequality implies

kf ��kLp;q � p

p � 1kf kLp;q

for all 1 < p < 1 and 1 � q < 1.


Notes

The basic material on Orlicz spaces presented in Part IV is taken from [18, Chapter2]. These authors deal with Orlicz spaces L˚.˝;/ on general � -finite measurespaces .˝;/. Our presentation is adapted mainly to the symmetric spaces on.RC;m/, while most of definitions and results related to the spaces L˚.˝;/ areformulated 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, andlater refined in [33, 54, 81].

The Young classes were introduced in [77]. The subspaces H˚ called the heartsof L˚ originated in [42] and [49].

W. Orlicz introduced the spaces L˚ in [54, 55] by means of the conjugatefunction � 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 calledthe Orlicz norm on L˚ .

Inequality (15.1.5) in Property 15.1.2 is due to Young [77]; see also [27, Section4.8]. A more general form of Young’s inequality linked to the Legendre transformcan be found in [6, 21] and [61].

The proof of Theorem 15.3.2 and other related results in Section 15.3 are takenfrom [18], Section 2.2. Proposition 15.3.4 can be refined as

kgkL1˚D inf

k>0

1

k.1C I˚.kg//; g 2 L1˚ I

see Note 15.3.5. The above equality was proved in [33] in the case of finitemeasure .

The .�2/-condition was used already in [54]. It yields the separability of theOrlicz space L˚.˝;/ on .˝;/, provided that the measure space is separable;see Complement 1. In this case, one can get many equivalents of separabilityby combining the corresponding results of Chapters 14 and 15 with property .A/and other separability conditions obtained earlier for general symmetric spaces inChapters 6 and 7.

Zygmund’s classes Z˛ arise in the study of Hardy–Littlewood maximal func-tions. 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, Section2.2]. Edgar and Sucheston [18, Chapter 2] identified the spaces as the hearts ofZygmund classes Zk D L logk L, constructed as Orlicz spaces L˚ with suitable ˚ .

Notes 235

Note that one-parameter families of symmetric spaces fLpg, fZ˛g, fR˛g as wellas the Lorentz spaces f�

eW˛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 spacesand 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 . Theupper 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 limx!C1 �jf j.x/ D 0, then there is a unique decreasing right-

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 � aredefined on R

C, 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) normk � 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 asymmetric space on .˝;/. It is convenient to consider, together with X.˝;/, thespace X.RC;m/. The space consists of all h 2 L0.RC;m/ such that f � D h� forsome 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 symmetricspace on .RC;m/. For example, if there exists a measure-preserving isomorphism

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

237

238 Complements

� W .˝;/ ! .RC;m/, it induces the isometric isomorphism f ! f ı ��1 betweenX.˝;/ and X.RC;m/.

Next assume that the measure is finite and nonatomic, let ˝ D a < 1. Forevery 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 ofL0.Œ0; a�;ma/ for all f 2 L0.˝;/, where ma D mjŒ0;a� is the usual Lebesguemeasure on Œ0; a�.

If there exists a measure-preserving isomorphism � W .˝;/ ! .Œ0; a�;ma/, thenthe 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 isomorphismbetween 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. Thusin this case, every symmetric space X.˝;/ on .˝;/ can be identified with thecorresponding 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 asymmetric space on .N; ]/, where ] is the counting measure on N.

Recall that a measure space .˝;/ is said to be separable if the � -algebra F ofall -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 standardspace. To see this, consider the (uncountable) direct product

.˝;/ DY

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

The measure space is nonseparable, and for 1 � p < 1, the space X.˝;/ DLp.˝;/ is nonseparable, while the corresponding standard space X.Œ0; 1�;m1/ DLp.Œ0; 1�;m1/ is separable.

The conditions for separability of symmetric spaces on general measure spacescan be described as follows.

• Let X.˝;/ be a symmetric space on .˝;/, where is a nonatomic infinitemeasure, and let X.RC;m/ be the corresponding standard space. Then thefollowing conditions are equivalent:

1. X.˝;/ is separable.2. X.RC;m/ is separable and .˝;/ is separable.3. X.˝;/ has property .A/ and .˝;/ is separable.

Complements 239

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 consideredas 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 correspond-ing 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 all1 � p � q � 1,

L1.0; 1/ � Lq.0; 1/ � Lp.0; 1/ � L1.0:1/

and

k � kL1� k � kLq � k � kLp � k � kL1 :

240 Complements

Minimality and Separability

A symmetric space X D X.0; 1/ is called minimal if L1 is dense in X, i.e., Xcoincides 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. X is separable.2. X is minimal and X ¤ L1.3. X has property .A0/ (see. Proposition 6.5.2).4. .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

X1 D8

<

:

f 2 L1 W kf kX1 D supkgkX�1

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1Z

0

fgdm

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

< 19

=

;

:

• .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 haveproperty .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).

Complements 241

• The following conditions are equivalent.

1. X D X11 and k � kX D k � kX11 .2. X has the property

.BC/ffng � X; 0 � fn " and sup

nkfnkX < 1 H) fn " f and

jfnkX " kf kX for some f 2 X:

3. If fn.st/�! f in L0 and sup

nkfnkX � 1, then kf kX � 1 (cf. Theorem 8.3.5).

Property .BC/ means that X has both properties .B/ and .C/. It is equivalent tothe 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/g1nD1 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/g1nD1 of the sequence jf j can be written as

f �.n/ D inffx 2 RC W j�jf 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/g1nD1 2 l0 W kf kl1 D sup

nf .n/ < 1g

is the space of all bounded sequences. Here, either f and f � belong to the space

c0 D fg D fg.n/g1nD1 2 l1 W lim

n!1 g.n/ D 0g;

or

limn

f .n/ D infn

f �.n/ D lim f �.n/ > 0:

242 Complements

Note that in the first case, there exists a permutation � W N ! N such thatf � D jf j ı � .

A symmetric sequence space X.N/ is a nonzero linear ideal Banach sublattice ofthe 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/g1nD1 W kf klp < 1g;

with

kf klp WD 1X

nD1jf .n/jp

! 1p

; 1 � p < 1;

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.

Complements 243

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 equiva-lent:

1. X.N/ is minimal.2. X.N/ is separable.3. X.N/ has the following property .A/:

.A/ ffkg � X; 0 � fk # 0 ) kfkkX.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 � � f �

N kX.N/ ! 0, N ! 1,where

f �N .n/ D

�

f �.n/; n � NI0; n > N:

The associate space X1.N/ has the form

X1 D(

g D fg.n/g1nD1 2 l1 W

ˇ

ˇ

ˇ

ˇ

ˇ

1X

nD1f .n/g.n/

ˇ

ˇ

ˇ

ˇ

ˇ

< 1 for all f 2 X

)

and

kgkX1 D sup

( ˇ

ˇ

ˇ

ˇ

ˇ

1X

nD1f .n/g.n/

ˇ

ˇ

ˇ

ˇ

ˇ

W kf kX � 1

)

:

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

4 The Spaces Lp; 0 

=

>

;

:

244 Complements

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;

where C D 21p �1

> 1 is a constant that cannot be reduced to 1.Thus, k�kLp is a quasinorm and .Lp; k�kLp/ 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 k�kLp . 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

np.f / D1Z

0

jf jpdm D kf kpLp;

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 jcjpnp.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 scalarmultiplication are continuous in the metric dp.

Thus for 0 < p < 1, the space Lp is an F-space, i.e., a complete linear topologicalspace Lp with translation-invariant metric dp, in which the operations of addition andmultiplication by a scalar are continuous.

Complements 245

Recall that the space L0 with the metric d0 described in Section 6.3 has analogousproperties. Like the space L0, the linear topological spaces Lp, 0 < p < 1 are notnormable, 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 issequentially 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

n!1 u.fn/, then limn!1 u.fn/ D

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 weaktopology �.X;X�/. But it does not imply in general sequential completeness in theweak 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 weaklysequentially complete. The space L1 has the predual space L1, but L1 is notweakly sequentially complete. The space L1 has no Banach predual space, but itis weakly sequentially complete.

In the case that X is a symmetric space, properties .A/ and .B/ give an effectivecriterion 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

nkfnkX < 1, then there

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

l1 D ff D ff .n/g1nD1 2 R

N W kf kl1 D supn

f .n/ < 1g

and its closed subspace

c0 D ff D ff .n/g1nD1 2 l1 W lim

n!1 f .n/ D 0g:

246 Complements

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

6 The Least Concave Majorant

Let V W RC ! R

C be an arbitrary nonnegative function. To construct its leastconcave majorant eV , we consider the subgraph

�0.V/ D f.x; y/ 2 R2 W x � 0; 0 � y � V.x/g:

We denote by A�0.V/ the closed convex hull of the set �0.V/. Let eV W RC !

Œ0;C1� be a function on RC such that �0.eV/ D A�0.V/.

Only the two following cases are possible: either eV.x/ � C1 or eV.x/ < 1 forall x 2 R

C . In the second case, eV is just the least concave majorant for V . We have

eV.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, andm1 C m2 C � � � C mn D 1.

Denote by

x D m1x1 C m2x2 C � � � C mnxn

the center of mass of the system and set

V.x/ D sup.m1V.x1/C m2V.x2/C � � � C mnV.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 eV is a concave function, we have

eV.x/ �nX

iD1m1eV.x1/;

and hence eV.x/ � V.x/ for all x 2 RC.

Complements 247

On the other hand, it is easy to verify that the function V.x/ itself is concave.Therefore, eV � V � V implies that eV D V .

For the quasiconcave function V , its least concave majorant eV exists and isequivalent to it, namely 1

2eV � V � eV (see Theorem 10.3.1). For an arbitrary positive

function V we can find sufficient conditions under which V and eV are equivalent.The conditions can be formulated by means of dilation indices of the function V .Let

p.V/ D limx!C1

ln x

lnbV.x/D sup

x>1

ln x

lnbV.x/

and

q.V/ D limx!C0

ln x

lnbV.x/D inf

0<x<1

ln x

lnbV.x/:

Here, the dilation function

bV.x/ D sup0<t<1

V.tx/

V.t/; x 2 .0;1/

is semimultiplicative, which provides the existence of the above limits.Clearly, p.V/ � q.V/.

• If 1 < p.V/ � q.V/ < 1, then the function V has the least concave majorant eV ,which is equivalent to V .

When the function V is quasiconcave,

1 � p.V/ � q.V/ � 1;

and both equalities 1 D p.V/ and q.V/ D 1 are possible.On the other hand, for the function V.x/ D max.x; x.1C ln x//, we have bV D V

and 1 D p.V/ D q.V/ < 1. However, V clearly has no concave majorants.

7 The Minimal Part M0V of the Marcinkiewicz Space MV

Let V be a quasiconcave function on RC, and let

MV D ff 2 L1 C L1 W kf kMV D supx.V�.x/f ��.x// < 1g

248 Complements

be the corresponding Marcinkiewicz space. Here V�.x/ D x

V.x/� 1.0;1/.x/ and

f ��.x/ D 1x

xR

0

f �dm; x > 0 (see Sections 12.1 and 13.1).

Consider the minimal part M0V D clMV .L1 \ L1/ of MV .

• If V.0C/ D 0; V.1/ D 1 and V�.0C/ D 0, then the minimal part M0V of the

space MV has the form

M0V D M.0/

V D ff 2 MV W limx!C0V�.x/f ��.x/ D lim

x!1 V�.x/f ��.x/ D 0g:

Recall that the fundamental function 'MV of the space MV coincides with thefunction V�. Therefore, the condition

'MV .0C/ D V�.0C/ D 0

implies separability of the minimal space M0V (see Section 13.2).

If V�.0C/ > 0, then M.0/V D f0g ¤ M0

V .If V.0C/ > 0, the corresponding Lorentz space �

eV is contained in L1.Therefore, its associate space �1

eVD MV contains L1.

Moreover, under the condition MV L1, one can use the space

ff 2 MV W limx!1 V�.x/f ��.x/ D 0g

instead of the space M.0/V .

If V.1/ < 1, then �eV L1 and MV � L1. Thus one can use the space

ff 2 MV W limx!0C V�.x/f ��.x/ D 0g

instead of M.0/V .

If V.1/ D 1 and V.0C/ > 0, then �eV 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 D

8

ˆ

<

ˆ

:

f 2 L0 W kf kLp;q D0

@

1Z

0

Œf �.x/�q d.xqp /

1

A

1q

< 1

9

>

=

>

;

;

Complements 249

for 1 � p < 1, 1 � q < 1, and

Lp;1 D�

f 2 L0 W kf kLp;1 D supx>0

x1p f �.x/ < 1

;

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

'Lp;q D k1Œ0; x�kLp;q D x1p D 'Lp.x/; x � 0:

Therefore, just the index p is regarded as the main index, and the index q

determines the appropriate weight function W.x/ D xqp .

• The family Lp;q increases with q, that is, Lp;q1 � Lp;q2 and kf kLp;q1� kf kLp;q2

for1 � q1 � q2 � 1.

Thus for each fixed p, the space Lp;1 is the least space in the family, and it is the

Lorentz space �W with the weight function W D x1p .

If 1 � q � p < 1, then the weight function W.x/ D xqp is concave, and it can

be verified that

• .Lp;q; k � kLp;q/ is a symmetric space for all 1 � q � p < 1.

In the case 1 � p < q < 1, the weight function W.x/ D xqp is convex, and the

triangle inequality fails for k�kLp;q . However, k�kLp;q is a quasinorm and .Lp;q; k�kLp;q/

is a quasi-Banach space.Moreover, if 1 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

p � 1kf kLp;q

(see Exercise 32).

Note that for q D 1 and p > 1, the symmetric space .Lp;1; k � k�Lp;1

/ is just the

Marcinkiewicz space MV with V.x/ D x1�1p , x � 0, where V�.x/ D x

1p 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 R

C. Let

L˚;W D ff 2 L0 W kf kL˚;W < 1g;

250 Complements

where

kf kL˚;W D sup

8

<

:

c > 0 W1Z

0

˚

� jf �jc

�

dW � 1

9

=

;

:

• 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.

The space Lp;q is obtained with ˚.x/ D xq and W.x/ D xqp .

In fact, k � kL˚;W is a norm or a quasinorm that admits an equivalent symmetric normfor a very wide class of pairs .˚;W/. These spaces are called Orlicz–Lorentz spaces.

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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. Prince-

ton 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 21566. 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.

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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

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(1929)74. Titchmarsh, E.: On conjugate functions. Proc. Lond. Math. Soc. 2(1), 49–80 (1929)

254 References

75. Vulikh, B.Z.: Introduction to the Theory of Partially Ordered Spaces, xv, 387 p. Wolters-Noordhoff Scientific Publications Ltd., Groningen (1967). In Russian: Gosizdat phis-math lit.,Moscow, (1961); Translated from the Russian by Leo F. Boron, with the editorial collaborationof Adrian C. Zaanen and Kiyoshi Iséki (English)

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Index

Symbols˚1 � ˚2, 212˚1 ˚2, 207˚1 0 ˚2, 207˚1 1 ˚2, 207�-algebra

B, 5B.RC/, 5Fm, 5Fm.0; n/, 25Fm.R

C/, 5a˚ , 171b˚ , 171b , 196

fn.st/�! f , 70

gn # g, 21gn " g, 21

BBanach

lattice, 14ideal , 14

CClass

Young Y˚ , 176, 183Zygmund Z˛ , 214

Condition.�2.0//, 190.�2.1//, 190.�2/, 190

Convergence˚-mean, 231almost everywhere, 54essentially uniform, 22in measure, 54order, 108stochastic, 70

Cutoff functionright f � � 1Œ0;n�, 81upper min.f �; n/, 81

Ddilation indices p.V/, q.V/, 247

EEmbedding, see also Theorem embedding

�eV1

� �eV2

, 162 W X ! X��, 103L1 C L1 � .L1 \ L1/1, 91L1 \ L1 � .L1 C L1/1, 91L˚1 � L˚2 , 209M.V1/� � M.V2/� , 162X � X11, 165X1 � 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/�, 1041 W X11 ! .X1/�, 104i W X ! X11, 104


255

256 Index

FFactor space

.X1 � X2/=L, 74Function

.˚ ^ ˚1/Ï, 220

.˚1 ^ ˚2/Ï, 218

.˚p ^ ˚q/Ï, 226

1A, 6F.x; y;m/, 198Ua, 141V , 127V�, 127Vmax, 129Vmin, 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, 228min.f ; n/, 50min.f �; n/, 50˚˛ , 214�, 171 , 196eV , 129, 130, 246e˚ , 136f �, 11f ��, 139f �1 , 9f", 33W, 115Lorentz, 115weight, 115characteristic, 6concave, 115decreasing rearrangement, 11distribution �f , 5epigraph � 0.g/, 10finite, 23, 38fundamental, 60

'�W , 119'L1

, 61

'Lp ; 1 � p < 1, 60'L˚ , 177'Lp\L1

; 1 � p < 1, 77'MV , 143'MV�

, 143'R0 , 77'X1 , 135'X, 60'L1CL1

, 61'L1\L1

, 61generalized inverse, 10graph � .g/, 10hypograph �0.g/, 10least concave majorant, 130locally integrable, 49maximal, 139Orlicz, 171

conjugate, 196quasiconcave, 127quasiconvex, 136semiadditive, 66separating, 101simple

integrable, 23, 38step, 33, 36

FunctionalI˚ W L0 ! R

C, 173p0 W L0 ! R

C, 66u W X ! R, 24, 83ug W L1 ! R, 84ug W L1 ! R, 84

Functionsequimeasurable, 7

GGroup A.m/, 35

IIdeal lattice

symmetric, 109Inequality

Hölder, 18Minkowski, 18Young, 17, 196

MMaximal property

f �g�, 86f �, 41

Measureu, 84

Index 257

m, 5m ı ��1, 35W , 116

Metricd0.f ; g/, 67translation-invariant, 67

NNorm

k.f1; f2/kX1�X2 , 46kf k0, 102kf k�W

, 116kf kL01 , 31kf kL1˚

, 201kf kL1CL1

, 46, 182kf kL1\L1

, 29, 180kf kL˚ , 173kf kL1

, 14kf kLp

, 178kf kLp

; 1 � p < 1, 14kf kL˚0

, 221kf kL˚1CL˚2

, 217kf kL˚1\L˚2

, 217kf kL˚p

, 225

kf kL1

, 179kf kLL˚CL1

, 222kf kMV

, 143kf kX1CX2 , 46kf kX1\X2 , 29kgk�1

W, 144

kgkMW, 144

kgkX1 , 85khkX11 , 95kukX� , 24, 83�

�ug

�

�

L�

1, 84

�

�ug

�

�

L�

1

, 84Lorentz, 115monotonic, 14monotonically complete, 97order complete, 108order continuous, 79order semicontinuous, 99

PProperty

.A/, 79on .0; 1/, 106on N, 107, 243

.AB/, 103, 245

.A0/, 79on .0; 1/, 240

.B/, 97on .0; 1/, 240

.BC/, 102on .0; 1/, 240

.C/, 99

.F/, 102Fatou, 102maximality

on .0; 1/, 240minimality

on .0; 1/, 106, 240on N, 107, 243

separabilityon .0; 1/, 106, 240on N, 107, 243

weakly sequentially complete, 245

SSet

B˚ , 173BL˚ , 186BY˚ , 186Va, 104� , 10� .eV/, 131� .g/, 10� 0.g/, 10�0.V/, 129�0.Vn/, 129�0.eV/, 129�0.g/, 10R

C, 5F0, 152F1, 152F.0/, 24F0, 23, 38, 121F1, 23, 38Lp; 1 � p � 1, 14R0, 50X0, 75Y˚ , 176, 183clX.L1 \ L1/, 75clX.Y/, 72indicator, 6

Setsupper Lebesgue, 5

SpaceM0

V , 247X1.N/, 243..L1 C L1/1; k�k.L1CL1/1 /, 91..L1 \ L1/1; k�k.L1\L1/1 /, 91

258 Index

Space (cont.).�W ; k�k�W

/, 116.RC;m/, 13.L0

1; k�kL01 /, 31

.L0; d0/, 67

.L1 C L1/0, 76, 97

.L1 C L1/1, 91

.L1 C L1/11, 96

.L1 C L1; k�kL1CL1/, 46

.L1 \ L1/0, 76

.L1 \ L1/1, 91

.L1 \ L1/11, 96

.L˚ ; k�kL˚ /, 173

.Lp C Lq/1, 228

.Lp; k�kLp/; 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 ; k�kMV/, 143

.X0/1, 91

.X0; k�kX0 /, 75

.X1; k�kX1 /, 89

.X11/0, 97

.X1 C X2; k�kX1CX2 /, 74

.X1 \ X2; k�kX1\X2 /, 73

.XU ; k�kXU/, 141

F-space, 54, 68, 244�0

W , 121, 146�1

W , 119�11

W , 119�1

W , 144�W.0; 1/, 164�W1 C �W2 , 164�W1 \ �W2 , 164�eV , 147

H˚ , 187H˚˛ , 216H˚.˝ ; /, 229L ln˛ L, 215L�

1 , 84L�

1, 85

L0˚ , 187L0

1, 76, 96

L0˚˛ , 216L1˚ , 195L1˚ , 230L11

1, 96

L11p ; 1 � p < 1, 96L0.0; 1/, 239

L0, 66L˚ C L1, 223L˚ C L1, 220L� , 195L� \ L1, 220L� \ L1, 223Lp.0; 1/, 239Lp.0; 1/, 51Lp C L1, 53Lp C L1, 53Lp C Lq, 224, 225Lp; 0 < p < 1, 243Lp; 0 < p < 1, 54Lp \ L1, 53Lp \ L1, 53Lp \ Lq, 224, 225L0p; 1 � p < 1, 76L.˚1^˚2/Ï , 218L1 \ L1, 29L˚;W , 250L˚1_˚2 , 218L˚1 C L˚2 , 217L˚1 \ L˚2 , 217L˚˛ ; 0 � ˛ < 1, 214L˚p , 225L˚.˝ ; /, 229L˚˛ , 215Lp;q, 233Lp.˝;/, 237M0

V , 165M1

W , 119, 146M1

eV, 147

MV .0; 1/, 164MV�

, 143R0, 50, 76, 97X.˝;/ , 237X.0; 1/, 239X.N/, 241X�, 83X1, 85X11, 95X1 C X2, 46X1 � X2, 46X1 \ X2, 29Z1˛ , 215Z0, 214Z˛; 0 � ˛ < 1, 214Z0˛ , 216c0, 241l0, 241lp, 52, 242.L1/, 85.L˚/, 205

Index 259

.L� /, 205.L1/, 84.Lq/, 90.MW/, 146.X1/, 89cl�W .F0/, 146cl�W .L1 \ L1/, 146clL1CL1

.F0/, 50clL1\L1

.F0/, 39clL1 .F0/, 39clL1

.F0/, 39clX.L1 \ L1/, 75clX.Y/, 39, 72L1˚ , 201Lp;q, 248dual, 24, 83

L�

p ; 1 � p � 1, 25X�, 24

Lorentzon .0; 1/, 164

Lorentz �W , 116Lorentz Lp;q, 248Lorentz–Orlicz L˚;W , 250Marcinkiewicz

on .0; 1/, 164Marcinkiewicz MV , 143Orlicz

general measure space, 229Orlicz L˚ , 173quasi-Banach symmetric, 54rearrangement invariant, 14

symmetric, 14general measure space, 237on Œ0; 1�, 239

symmetric sequence, 241weakly sequentially complete, 245

Subspacenorming, 99

Symmetric space.X; k�kX/, 13associate, 90general measure space, 237maximal, 96minimal, 76minimal part, 76on Œ0; 1�, 239reflexive, 103second associate, 95separable, 77

TTheorem embedding

�0

eV� X � MV�

, 153L1 \ L1 � X � L1 C L1, 61X0 � X � X11, 96on Œ0; 1�, 106, 239on N, 106, 242

Transform Legendre, 195Transformation

measure-preserving, 35invertible, 35

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

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