Table Of ContentINTEGRAL REPRESENTATIONS
of
FUNCTIONS
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and
¡EDDING THEOREMS
US
Volume I
OLEGV;
VALENTIN P. IL'IN
SERGEI M.NIKOL'SKII
with an i ction by
Mitchell ibleson
dG= U r('
Integral Representations
of Functions and
Imbedding Theorems
Volume I
OLEG V. BESOV,
VALENTIN P. IL’IN,
and SERGEI M. NIKOL SKII
Integral Representations of Func
tions and Imbedding Theorems,
written by three prominent Soviet
mathematicians, is a major work
of lasting significance. It is of
unique value to students, teachers,
and researchers interested in har
monic analysis, functional analysis,
approximation theory, partial dif
ferential equations, singular inte
grals as well as related aspects of
theoretical physics.
The study of spaces of func
tions that satisfy difference condi
tions and functions that satisfy
differentiability conditions devel
oped as a major trend in con
temporary mathematics prior to
World War II. The Russian school
has been a central contributor to
this study, having produced the
pioneering works of S. L. Sobolev
in the 1930s and, more recently,
those of Academician Sergey
Mikhaylovich Nikol’skii who be
came the internationally acknowl
edged leader of the school. Many
of Academician Nikol’skii’s books
and papers, legion in number, have
been translated into foreign lan
guages and published in the U.S.,
Great Britain, and other countries.
The writings of the co-authors,
Oleg Vladimirovich Besov and
Valentin Petrovich Il’in, both se
nior scholars and Nikol’skiPs close
[continued on inside back flap)
INTEGRAL
REPRESENTATIONS
OF FUNCTIONS AND
IMBEDDING THEOREMS
Volume I
SCRIPTA SERIES IN MATHEMATICS
Tikhonov and Arsenin • Solutions of Ill-Posed Problems, 1977
Rozanov • Innovation Processes, 1977
Pogorelov • The Minkowski Multidimensional Problem, 1978
Kolchin, Sevast'yanov, and Chistyakov • Random Allocations, 1978
Boltianskn • Hilbert’s Third Problem, 1978
Besov, ITin, and Nikol'skn • Integral Representations of Functions
and Imbedding Theorems, Volume 1,1978
INTEGRAL
REPRESENTATIONS
OF FUNCTIONS AND
IMBEDDING THEOREMS
Volume I
Oleg V. Besov
Valentin P. D’in
Sergey M. NikoFskii
Steklov Institute of Mathematical Sciences,
Moscow
Edited by
Mitchell H. Taibleson
Washington University
1978
V. H. WINSTON & SONS
Washington, D.C.
A HALSTED PRESS BOOK
JOHN WILEY & SONS
New York Toronto London Sydney
Copyright © 1978, by V. H. Winston & Sons, a Division of
Scripta Technica, Inc.
All rights reserved. Printed in the United States of America.
No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise,
without prior written permission of the publisher.
V. H. Winston & Sons, a Division of Scripta Technica, Inc.,
Publishers
1511 K Street, N.W., Washington, D.C. 20005
Distributed solely by Halsted Press, a Division of John Wiley
& Sons, Inc.
Library of Congress Cataloging in Publication Data
Besov, Oleg Vladimirovich.
Integral representations of functions and imbedding
theorems.
(Scripta series in mathematics)
Translation of IntegraPnye predstavlenifa funkfsii i teoremy
vlozheniia.
Bibliography: p.
Includes index.
1. Functions of several complex variables. 2. Embedding
theorems. I. Il’in, Valentin Petrovich, 1921— joint author.
II. Nikol’skii, Sergei Mikhailovich, joint author. III. Title.
IV. Series.
QA331.B4313 515’.94 78-13337
ISBN 0-470-26540-X
Composition by Isabelle Sneeringer, Scripta Technica, Inc.
CONTENTS
TRANSLATION EDITOR’S PREFACE TO VOLUME I ........... vü
INTRODUCTION ............................................................................. 1
Chapter 1. INTEGRAL INEQUALITIES 5
§1. Lp spaces ....................................................................... 5
§2. The basic integral inequalities .................................... 17
§3. Boundedness of the convolution in Lp ................... 48
§4. Singular integrals in Lp ............................................. 67
Chapter 2. INTEGRAL REPRESENTATIONS OF
DIFFERENTIABLE FUNCTIONS 89
§5. Averaging of functions .............................................. 92
§6. Generalized derivatives .............................................. 96
§7. Integral representations of differentiable functions . 103
§8. The domains of definition of the functions............... 153
v
vi CONTENTS
Chapter 3. ANISOTROPIC SOBOLEV SPACES AND
IMBEDDING THEOREMS 101
§9. Properties of the anisotropic spaces Wlp(G) ..........165
§10. The imbedding of Wlp(G) and Lq(G) in C (G)
and in an Orlicz class. Estimates for the trace of a
function ........................................................................ 180
§11. Coerciveness in the spaceW p(G) .................................. 207
§12. Imbedding of Wlp (G) and when / does not corre
spond to the type of the region G ............................ 224
§13. Inequalities between Z^-norms of mixed derivatives . 242
§14. The behavior of functions in Wlp at <» and the
density of Cq mWlp................................................... 290
§ 15. Multiplicative inequalities for Lp-norms of
derivatives........................................................................ 311
BIBLIOGRAPHY ............................................................................. 331
TRANSLATION EDITOR’S
PREFACE TO VOLUME I
A major trend in contemporary mathematics, for a period exceed
ing 50 years, has been the study of spaces of functions that satisfy
difference conditions (such as the Holder continuity) and functions
that satisfy differentiability conditions, plus the imbedding relations
among and between these various spaces. From the beginning of this
study, the Russian school has been a central contributor and, in
recent years, their acknowledged leader has been Sergey M. Nikol’skn.
In his 1969 book NikoFskii summarized the contributions of his
school, using approximation by entire functions of exponential type
as his main tool. In this book, NikoFskii and his colleagues Valentin
P. Il’in and Oleg V. Besov bring us up to date. Integral representa
tions using kernels that are adapted to the “shape” of the domain of
the function constitute the main tool used in Integral Representations
of Functions and Imbedding Theorems.
The Russian text was written in a somewhat informal style and we
have attempted to preserve the liveliness of the original. As the
translation editor, I should like to add a personal note. A substantial
part of my early mathematical work was built on the studies
described in this book. In particular, the pioneering work of what
vii
viii TRANSLATION EDITOR’S PREFACE TO VOLUME I
most properly are known as Besov spaces was fundamental to my
studies. I hope my efforts on this edition will repay, in part, my debt
to Oleg V. Besov and his co-workers.
There are two volumes in the English-language edition of Integral
Representations of Functions and Imbedding Theorems. The first
three chapters appear in Volume I and the last three in Volume n.
Chapter 1 is concerned with various integral inequalities, and in
particular with a version of the Calderon-Zygmund theory in §4.
Chapter 2 introduces the major idea of the book, integral representa
tions. In Chapter 3 the authors introduce anisotropic Sobolev spaces
on domains that satisfy a horn-condition (a generalization of a
cone-condition) and study imbedding theorems among these spaces.
Mitchell H. Taibleson
Chairman, Dept, of Mathematics
Washington University, Missouri
