Weighted inequalities in Lorentz and Orlicz spaces

WEIGHTED INEQUALITIES IN LORENTZ AND ORLICZ SPACES

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WEIGHTED INEQUALITIES IN LORENTZ AND ORLICZ SPACES

Vakhtang Kokilashvili Institute of Mathematics Georgian Acad. Sci., Tbilisi

Miroslav Krbec Institute of Mathematics Czechoslovak Acad. Sci., Prague

World Scientific Singapore NewJersey Jersey• L• London • Hong Kong Singapore ••New

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main St., River Edge, NJ 07661 UK office: 73 Lynton Mead, Totleridge, London N20 8DH

Library of Congress Cataloging-in-Publication data is available.

WEIGHTED INEQUALITIES IN LORENTZ AND ORLICZ SPACES Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0612-7

Printed in Singapore by Utopia Press.

To our wives for their constant patience and support

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FOREWORD

Generally, this book is devoted to the problem of a characterization of weighted function spaces for which a given integral operator is bounded from one weighted space into another. According to coincidence and non-coincidence of the weights involved, these problems are usually called one weight and two weight problems, respectively. The weighted norm inequalities have become one of the most dynamically developing parts of harmonic analysis since the early 70's and the pioneering result by B. Muckenhoupt [1]. Solutions of many important problems have been closely linked with weight problems. The mentioned paper by B. Muckenhoupt triggerred a flood of results on weighted inequalities and related topics; in this paper it was shown among others that the one weight norm inequality for the maximal operator is true iff the weight satisfies the so called Ap condition. The Ap weights provide an extraordinary beautiful answer to a number of challenging problems which had arisen already in the 30's in connection with fundamental results due to G. H. Hardy and J. E. Littlewood [1], [2]. Theorems on boundedness of the Hilbert transform, fractional order maximal functions, fractional integrals followed very soon; see R. A. Hunt, B. Muckenhoupt and R. L. Wheeden [1], B. Muckenhoupt and R. L. Wheeden [1], R. R. Coifman and C. Fefferman [1]. Observe that two weight problems turned out to be of somewhat other nature. E. Sawyer [1] was the first who solved this problem for maximal func­ tions and later in [5] for fractional integrals. A solution of the two weight prob­ lem of weak type for potentials is due to E. Sawyer [2] and independently to M. Gabidzashvili [1], [4]. The latter author in [1] has found a more easily verifi­ able necessary and sufficient condition for validity of the two weight inequality of weak type for fractional integrals. It is perhaps impossible to give a full account of related problems and to trace back the (by no means straightforward) way to the first celebrated papers on weighted inequalitites in the early 70's. Several branches of analysis meet here, influencing each other, and one could mention a very large number of papers and still the list would be far from being complete. Let us recall the recent monographs by J. Garcia-Cuerva and J. Rubio de Francia [1], V. Kokilashvili [5], and A. Torchinsky [2], further the survey vn

viii Foreword papers by B. Muckenhoupt [2], E. Dynkin and B. Osilenker [1], V. Kokilashvili [4]. A large bibliography can be found in them. The importance of the weight problems stems not only from the theory of functions itself, but is also clear from the numerous applications to imbedding theorems, theory of basis, spectral theory of differential operators, boundary value problems, in particular, regularity, degenerate p.d.e.'s, singular integral equations, theory of analytic functions etc. We refer to B. Khvedelidze [1], K. T. Mynbaev and M. 0 . Otelbaev [1], D. E. Edmunds and W. D. Evans [1], L. D. Kudryavcev and S. M. Nikolskii [1], S. M. Nikolskii, P. I. Lizorkin and N. V. Miroshin [1], E. Stredulinsky [1], B. Opic and A. Kufner [1] to recall at least some of the monographs and/or survey papers dealing with the applica­ tions. This book is intended as a survey, (uncomplete as it may be) of weighted norm inequalities in Lorentz and Orlicz spaces and Zygmund classes, which have become one of the well developed offshoots of the theory during the last several years. The results are mostly scattered in various journals and a substantial part of the book consists of the latest results, some of them still unpublished. Many of the presented theorems and also the last but one cited monograph have been published in Russian so that this book might also be considered as a break through the language and information barrier. Now a few words about the contents. The book is divided into six chapters. The first one deals with general (nonweighted) inequalities for the maximal function and Riesz transforms in the <j>{L) classes under rather general assumptions on the function $ which need not be necessarily a Young func­ tion. Chapter 2 surveys the results starting with the paper by R. Kerman and A. Torchinsky [1] on the index characterization of weights in the reflexive Orlicz spaces, there is a complete characterization of weights in two weight weak type inequalities and other relevant results on fractional order maximal functions, anisotropic Riesz potentials etc. The following Chapter 3 contains theorems about strong and weak type inequalities for the Riesz and Hilbert transform. Of particular interest in harmonic analysis have always been the Zygmund classes Llog L and their generalizations. Chapter 4 is devoted to strong and weak type inequalities for the maximal function, the strong maximal function and the Riesz transform in them. The last two chapters present recent results on fractional order maximal functions and potentials in weighted Lorentz spaces: a characterization of triples of weights is given which

Foreword ix guarantees validity of inequalities of weak type for fractional maximal functions and potentials, two weight problem for potentials is solved in Lebesgue spaces; the condition obtained here is rather simple and easy to verify in comparison with those previously known, further, the one weight problem for fractional maximal functions integrals and Riesz transforms are solved. The references include papers and monographs having an immediate con­ nection with the theory presented and some of those dealing with applications. The list is by no means complete. The book is not an encyclopedia, however, it is self-contained if the reader is familiar (apart from fundaments of real and functional analysis) with the basic techniques used in the real variable methods of the harmonic analysis: covering theorems, decompositions, interpolation. A knowledge of introduc­ tory parts of de Guzman [1], Bergh and Lofstrom [1] or the part on classical interpolation theorems in Triebel [1], and of the chapters devoted to Ap weights in Garcia-Cuerva and Rubio de Francia [1], Torchinsky [2] will be useful. Ref­ erences are made when it comes to use of something special. We will be grateful for any comments on relevant results and for pointig out imperfections or mistakes of any kind which might occur in the text. Let us point out solutions to several important problems which are pre­ sented in this book: two weight problems for fractional maximal functions and potentials in Lorentz spaces, two weight problems for the maximal function and the Riesz transform in Orlicz spaces and Zygmund classes, another de­ scription of the Aoo class, one weight problem for the Riesz transform and one weight problem of weak type for the strong maximal function in the Zygmund class. It is the authors' intention that this book not only surveys recent results, but will also stimulate further research in this challenging area. Some of the open problems are listed in a special section. This book was typeset in .4A/(S-TEX, Version 2.0 and camera-ready output is from Hewlett-Packard Laser Jet II. Special thanks goes to Dr. S. Hojek from the Institute of Mathematics of Czech. Acad. Sci. in Prague for his kind TEXnical assistance.

x Foreword We thank to the staff of the World Scientific, particularly, to Prof. J. G. Xu, for their patience during production and the extraordinary prompt publishing. Our thanks also goes to Institutes of Mathematics in Tbilisi and Prague for giving us the opportunity to spend a couple of weeks together in both places last and this year. Last but not least we wish to thank to our colleagues both in Tbilisi and Prague, in particular to M. Gabidzashvili, I. Genebashvili, A. Gogatishvili, and L. Pick for their cooperation and many valuable discussions, while we were preparing the manuscript.

Tbilisi and Prague, October 1990 and July 1991

The authors

CONTENTS

Foreword Chapter 1

vii Integral o p e r a t o r s in nonweighted Orlicz classes

1.1 Preliminaries, Orlicz Classes and Spaces, Quasiconvex functions 1.2 Maximal functions in $(L) classes 1.3 Vector-valued maximal functions in 0n -

z£R

1

J/

Orlicz-Moirey

$(f(y))dy < oo}.

B(z,r)

Clearly, $0(L) = $(L). If only 0 G $, we can define the Orlicz space L$ as the linear hull of 0;y#(/(y)/A)(fy [0,oo) is said to be a weight function. Starting with Chapter 2 we will work with weighted Orlicz classes and weighted Orlicz spaces: Let Q be a weight on Rn and 0 G $■ Then the weighted Orlicz class is the set of all functions / for which / $(f(x))e(x)

dx < oo

and the weighted Orlicz space is its linear hull. We will adopt the notation L$(Q) for the latter set. If # is a Young function, one can define the weighted Luxemburg norm and the weighted Orlicz norm in analogy with the above nonweighted norms. The following concept is one of the most frequently occuring in the theory of the Orlicz spaces and classes. Definition 1.1.4. A function 4> G $ satisfies the (global) A2 condition if there exists c > 0 such that $(2t) < c$(t), t > 0. If 0 such that (1.1.1)

u(t) is quasiconvex on [0, oo); (ii) the inequality 0(txi + (1 - t)x2) < CI(