Integral Representations of Functions and Imbedding Theorems

323 Pages·1979·11.785 MB·English

by  Oleg V. Besov, Valentin P. Il’in, Sergey M. Nikol’skii

Integral Representations
of Functions and
Imbedding Theorems
Volume II
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’skifs 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 IPin, both se­
nior scholars and Nikol’skii’s close
(continued on inside back flap)
INTEGRAL
REPRESENTATIONS
OF FUNCTIONS AND
IMBEDDING THEOREMS
Volume II
SCRIPTA SERRES 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
Boltianskiï • Hilbert’s Third Problem, 1978
Besov, I Pin, and Nikol'skii* * Integral Representations of Functions
and Imbedding Theorems, Volume 1,1978
Besov, I Tin, and Nikol'skM • Integral Representations of Functions
and Imbedding Theorems, Volume II, 1978
Sprindfuk • The Metric Theory of Diophantine Approximations, 1979
INTEGRAL
REPRESENTATIONS
OF FUNCTIONS AND
IMBEDDING THEOREMS
Volume II
Oleg V. Besov
Valentin P. D’in
Sergey M. Nikol’skn
Steldov Institute of Mathematical Sciences,
Moscow
Edited by
Mitchell H. Taibleson
Washington University
1979
V. H. WINSTON & SONS
Washington, D.C.
A HALSTED PRESS BOOK
JOHN WILEY & SONS
New York Toronto London Sydney
Copyright © 1979, 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
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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. Vol. II.
(Scripta series in mathematics)
Translation of Integral’nye predstavleniia funktsii i teoremy
vlozhenifa.
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 0-470-26593-0
Composition by Isabelle Sneeringer, Scripta Technica, Inc.
CONTENTS
TRANSLATION EDITOR’S PREFACE TO VOLUME I I .......... vii
Chapter 4. SPACES OF FUNCTIONS WITH
DIFFERENTIAL-DIFFERENCE
CHARACTERISTICS............................................... 1
§ 16. Estimates of the moduli of continuity in
Lp-norms......................................................................... 2
§17. Imbedding and extension theorems for
generalized Holder spaces............................................... 43
§18. The spaces BpQ and their connection with
the spaces Wlp .............................................................. 58
§19. The density of the smooth functions in
Wlp(G) and B'.e(G).................................................... 86
V
vi CONTENTS
Chapter 5. TRACES OF FUNCTIONS IN ISOTROPIC
SPACES ON MANIFOLDS.................................... 103
§20. Traces of functions in the spaces on a
Lipschitzian surface......................................................... 103
§21. A lemma on the invariance of classes under
change of variables......................................................... 134
§22. Differentiable manifolds.............................................. 160
§23. Classes of functions on a differentiable manifold. . . 171
§24. The trace of a function on a differentiable
manifold. The direct imbedding theorem for traces . 177
§25. Inverse theorems on traces ......................................... 199
Chapter 6. SOME ADDITIONAL RESULTS............................... 211
§26. Compactness of sets in spaces of differentiable
functions ........................................................................ 211
§27. The function spaces Wlp a x(G) ............................... 230
§28. The function spaces &*[p'tQS(G)............................... 253
§29. A multiplicative estimate of the mixed
modulus of smoothness .............................................. 287
BIBLIOGRAPHY ............................................................................. 297
TRANSLATION EDITOR’S
PREFACE TO VOLUME II
A major trend in contemporary mathematics, for a period that
now exceeds 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 begin­
ning of this study, the Russian school has been a central contributor
and, in recent years, their acknowledged leader has been Sergey M.
Nikol’skii. In his 1969 book Nikol’skii summarized the contributions
of his school, using approximation by entire functions of exponential
type as his main tool. In this book, Nikol’skii and his colleagues
Valentin P. Il’in and Oleg V. Besov brings us up to date. Integral
representations 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 PREFACE
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.
The English-language edition of Integral Representations of Func­
tions and Imbedding Theorems comprises two volumes. The first
volume presented various integral inequalities (in the first chapter),
integral representations (in the second), and anisotropic Sobolev
spaces on domains that satisfy a horn-condition (in the third).
The fourth chapter, first in the second volume, treats anisotropic
spaces of Nikol’skii and Besov on domains that satisfy a “shape”-
condition. These are spaces of functions whose differences satisfy
Holder conditions. It then discusses imbedding theorems both among
these spaces and between them and the Sobolev spaces. The fifth
chapter extends many of the earlier results to traces of functions on
smooth manifolds. The final chapter is concerned with several topics.
Principally, it discusses compactness results for the spaces studied and
presents some generalizations of the notion of a Sobolev space
involving spaces of Morrey and Campanato.
Mitchell H. Taibleson
Chairman, Dept, of Mathematics
Washington University, Missouri