Table Of ContentDevelopments in Mathematics
Ben-Zion A. Rubshtein
Genady Ya. Grabarnik
Mustafa A. Muratov
Yulia S. Pashkova
Foundations of
Symmetric Spaces
of Measurable
Functions
Lorentz, Marcinkiewicz and Orlicz Spaces
Developments in Mathematics
VOLUME 45
Series Editors:
Krishnaswami Alladi, University of Florida, Gainesville, FL, USA
Hershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel
More information about this series at http://www.springer.com/series/5834
Ben-Zion A. Rubshtein • Genady Ya. Grabarnik
Mustafa A. Muratov • Yulia S. Pashkova
Foundations of Symmetric
Spaces of Measurable
Functions
Lorentz, Marcinkiewicz and Orlicz Spaces
123
Ben-Zion A. Rubshtein Genady Ya. Grabarnik
Mathematics Mathematics and Computer Science
Ben Gurion University of the Negev St. John’s University
Be’er Sheva, Israel New York, NY, USA
Mustafa A. Muratov Yulia S. Pashkova
Mathematics and Computer Sciences Mathematics and Computer Sciences
V.I. Vernadsky Crimean Federal University V.I. Vernadsky Crimean Federal University
Simferopol, Russian Federation Simferopol, Russian Federation
ISSN 1389-2177 ISSN 2197-795X (electronic)
Developments in Mathematics
ISBN 978-3-319-42756-0 ISBN 978-3-319-42758-4 (eBook)
DOI 10.1007/978-3-319-42758-4
Library of Congress Control Number: 2016953731
Mathematics Subject Classification (2010): 46E30, 46E35, 26D10, 26D15, 46B70, 46B42, 46B10,
47G10
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To
Inna, Tanya, Andrey
Fany, Yaacob, Laura, Golda
Ajshe, Elvira, Enver, Lenur
Anna, Ludmila, Sergey
Foreword
This book is the first part of the textbook Symmetric Spaces of Measurable
Functions. It contains the main definitions and results of the theory of symmetric
(rearrangement invariant) spaces. Special attention is paid to the classical spaces
Lp, Lorentz, Marcinkiewicz, and Orlicz spaces.
The book is intended for master’s and doctoral students, researchers in mathe-
matics and physics departments, and as a general manual for scientists and others
who use the methods of the theory of functions and functional analysis.
vii
Preface
This book is the first, basic, part of a more advanced textbook Symmetric Spaces of
Measurable Functions. It contains an introduction to the theory, including a detailed
study of Lorentz, Marcinkiewicz, and Orlicz spaces.
The theory of symmetric (rearrangement invariant) function spaces goes back
to the classical spaces Lp, 1 p 1. The theory was intensively developed
during the last century, mainly in the context of general Banach lattices. It presents
many interesting and deep results having important applications in various areas of
function theory and functional analysis. The theory has a great many applications
in interpolation of linear operators, ergodic theory, harmonic analysis, and various
areas of mathematical physics.
The authors of this book (at different years and in different countries) have
studied and taught the theory of symmetric spaces. They discovered independently
the following surprising fact: despite the abundance of monographs, there was no
book suitable for our purposes either in the Russian mathematical literature or in the
mathematical literature of the rest of the world.
In fact, we wished to have a book with a relatively small volume that met the
following criteria:
1. The book should contain basic concepts and results of the general theory of
symmetric spaces with the main focus on a detailed exposition of classical spaces
Lp; 1 p 1, and Lorentz, Marcinkiewicz, and Orlicz spaces, as well.
2. The book should be accessible to master’s students, doctoral students, and
researchers in mathematics and physics departments who are familiar with the
basics of the measure theory and functional analysis in the framework of standard
university courses.
3. The material of the book should correspond to a one-semester special course of
lectures (about 4 months or 17–18 weeks).
4. The presentation should not require any additional source except standard
references on basic concepts and theorems of measure theory and functional
analysis.
ix
x Preface
In our opinion, this book, offered now to the reader, completely meets the above
requirements.
We can point out three main sources from which the material of the book was
adopted.
First is a monograph by S. G. Krein, J. I. Petunin, E. M. Semenov, Interpolation
of Linear Operators.
The second source is two volumes of J. Lindenstrauss, L. Tzafriri, Classical
Banach Spaces I. Sequence Spaces and Classical Banach Spaces II. Function
Spaces.
Third, the part devoted to Orlicz spaces is based on a nice exposition of this theme
in the book by G. A. Edgar, L. Sucheston, Stopping Times and Directed Processes.
Our book includes four parts comprising seventeen chapters. This allows us to
divide the corresponding one-semester lecture course into 4 months or 17 weeks,
and rigorously restricts, in turn, the volume of material.
As a result a great many important related topics have not been included in the
main part of the book. The reader can find this additional material in the exercises
at the end of each part and in the section called “Complements” at the end of the
book. Throughout the main exposition, we deal only with symmetric spaces on the
C
half-line R D Œ0;1/, while the symmetric spaces on the interval Œ0; 1 and the
symmetric sequence spaces are considered in the exercises and complements.
Each of the four parts begins with an overview and then is divided into chapters.
Each part concludes with exercises and notes. Complements are located at the end
of the book together with references and an index.
Complements and exercises are intended for independent study.
The list of references contains some historical material, the books and articles
from which we took terminology, results, and their proofs, and also a bibliography
for further rending. The list of references is not, of course, comprehensive, but it
points out, we hope, the most of important directions of the theory.
Be’er Sheva, Israel Ben-Zion A. Rubshtein
New York, NY, USA Genady Ya. Grabarnik
Simferopol, Russia Mustafa A. Muratov
Simferopol, Russia Yulia S. Pashkova
Contents
Part I Symmetric Spaces. The Spaces Lp, L1 \ L1, L1 C L1
1 Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Distribution Functions, Equimeasurable Functions. . . . . . . . . . . . . . . . . 5
1.2 Generalized Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Decreasing Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Integrals of Equimeasurable Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Definition of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Example. Lp, 1 p 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Spaces Lp; 1 p 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Hölder’s and Minkowski’s Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Completeness of Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Separability of Lp, 1 p < 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 The Space L1 \ L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1 The Intersection of the Spaces L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 29
0
3.2 The Space L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Approximation by Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Measure-Preserving Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Approximation by Simple Integrable Functions . . . . . . . . . . . . . . . . . . . . 38
4 The Space L1 C L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 The Maximal Property of Decreasing Rearrangements . . . . . . . . . . . . 41
4.2 The Sum of L1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Embeddings L1 L1 C L1 and L1 L1 C L1.
The Space R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
xi
xii Contents
Part II Symmetric Spaces. The Embedding Theorem.
Properties .A/; .B/; .C/
5 Embeddings L1 \ L1 X L1 C L1 L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 The Embedding Theorem L1 \ L1 X L1 C L1 . . . . . . . . . . . . . 61
5.3 The Space L0 and the Embedding L1 C L1 L0 . . . . . . . . . . . . . . . . . 66
6 Embeddings. Minimality and Separability. Property .A/ . . . . . . . . . . . . . . 71
6.1 Embedded Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 The Intersection and the Sum of Two Symmetric Spaces . . . . . . . . . . 73
6.3 Minimal Symmetric Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.4 Minimality and Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Separability and Property .A/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.1 Dual and Associate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 The Maximal Property of Products f g . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Examples of Associate Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1
7.4 Comparison of X and X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8 Maximality. Properties (B) and (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.1 The Second Associate Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2 Maximality and Property .B/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11
8.3 Embedding X X and Property .C/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.4 Property .AB/. Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Part III Lorentz and Marcinkiewicz Spaces
9 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.1 Definition of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.2 Maximality. Fundamental Functions of Lorentz Spaces . . . . . . . . . . . 119
9.3 Minimal and Separable Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.4 Four Types of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
10 Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.1 Fundamental Functions and Quasiconcave Functions . . . . . . . . . . . . . . 127
10.2 Examples of Quasiconcave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.3 The Least Concave Majorant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.4 Quasiconcavity of Fundamental Functions . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.5 Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11 Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.1 The Maximal Function f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.2 Definition of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.3 Duality of Lorentz and Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . 144
11.4 Examples of Marcinkiewicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
