Bounded and Compact Integral Operators

Transcription

1 Bounded and Compact Integral Operators

2 Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre f or Mathematics and Computer Science. Amsterdam, The Netherlands Volume 543

3 Bounded and Compact Integral Operators by David E. Edmunds Centre for Mathematical Analysis and its Application. University of Sussex. Sussex. United Kingdom Vakhtang Kokilashvili A. Razmadze Mathematicallnstitute. Georgian Academy of Sciences. Tbilisi. Georgia and Alexander Meskhi A. Razmadze Mathematicallnstitute. Georgian Academy of Sciences. Tbilisi. Georgia lit... " Springer-Science+Business Media, B.Y.

4 A c.i.p. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI DOl / !O.l007/ Printed on acid-free paper All Rights Reserved SpringerScience+Business Media Dordrecht2002 Originallypublishedby KluwerAcademicPublishersin Softcoverreprintof the hardcover Ist edition2002 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

5 Contents Preface Acknowledgments Basic notation IX Xlll xv 1. HARDY-TYPE OPERATORS Boundedness and compactness in BFS An extension of the Hardy transform Estimates for approximation numbers Norms of positive operators Notes and comm ents on Chapter FRACTIONAL INTEGRALS ON THE LINE Fractional integrals Two-weight problems Trace inequalities ("Diagonal case"). Examples Weak-type inequalities. Examples Integral transforms with power-logarithmic kernels Erdelyi-K6ber operators Integral transforms with positive kernels Extended Erdelyi-Kober operators Generalized one-sided potentials One-sided potentials on the half-space Weighted criteria in Lorentz spaces Applications to Abel's integral equations On some Volterra-type integral equations Application to the existence of positive solutions of nonlinear integral equations Notes and comments on Chapter v

6 VI BOUNDED & COMPACT INTEGRAL OPERATORS 3. ONE-SIDED MAXIMAL FUNCTIONS One-sided maximal functions Mapping properties of potentials on the line Potentials T, on the line (the case 0 <, < ~) Weak-type inequalities Potentials on bounded intervals Two-weight criteria for fractional maximal functions Generalized one-sided maximal functions Potentials with power-logarithmic kernels Multiple potentials One-sided Hormander-type maximal functions Notes and comments on Chapter BALL FRACTIONAL INTEGRALS Boundedness criteria Compactness criteria The measure of non-compactness Mapping properties in Lorentz spaces Notes and comments on Chapter POTENTIALS ON R N Truncated potentials Two-weight compactness conditions for Riesz potentials Integral transforms with radial kernels Notes and comments on Chapter FRACTIONAL INTEGRALS ON MEASURE SPACES Integral transforms on nonhomogeneous spaces Theorems of Adams type Truncated potentials on SHT Riesz potentials in the half-space Truncated potentials in the half-space Two-weight (p,p) type inequalities Theorems of Koosis type Fractional maximal functions on SHT Weighted estimates in Lorentz spaces Notes and comments on Chapter SINGULAR NUMBERS Volterra integral operators Riemann-Liouville-type operators 461

7 Contents VB 7.3 Potential-type operators Hardy-type operators Asymptotic behaviour of singular and entropy numbers Notes and comments on Chapter SINGULAR INTEGRALS 8.1 Two-weight strong-type estimates 8.2 Weak-type estimates 8.3 Singular integrals on nonhomogeneous spaces 8.4 Two-weight inequalities for the Hilbert transform 8.5 Estimates in Lorentz spaces 8.6 Cauchy-Szego projection 8.7 Singular integrals via Clifford analysis 8.8 Theorems of Koosis type 8.9 Notes and comments on Chapter 8 9. MULTIPLIERS OF FOURIER TRANSFORMS 9.1 Weighted Triebel-Lizorkin spaces 9.2 Two-weight multipliers in Triebel-Lizorkin spaces 9.3 (FJ:~, F5 :~) multipliers. The case 1 < q ~ p < Multipliers in weighted spaces with mixed norms 9.5 Examples 9.6 Notes and comments on Chapter PROBLEMS REFERENCES INDEX

8 Preface The monograph presents some of the authors' recent and original results concerning boundedness and compactness problems in Banach function spaces both for classical operators and integral transforms defined, generally speaking, on nonhomogeneous spaces. It focuses on integral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, stochastic processes etc. The book may be considered as a systematic and detailed analysis of a large class of specific integral operators from the boundedness and compactness point of view. A characteristic feature of the monograph is that most of the statements proved here have the form of criteria. These criteria enable us, for example, to give various explicit examples of pairs of weighted Banach function spaces governing boundedness/compactness of a wide class of integral operators. The book has two main parts. The first part, consisting of Chapters 1-5, covers the investigation of classical operators: Hardy-type transforms, fractional integrals, potentials and maximal functions. Our main goal is to give a complete description of those Banach function spaces in which the above-mentioned operators act boundedly (compactly). When a given operator is not bounded (compact), for example in some Lebesgue space, we look for weighted spaces where boundedness (compactness) holds. We develop the ideas and the techniques for the derivation of appropriate conditions, in terms of weights, which are equivalent to boundedness (compactness). It should be stressed that there is a wide range of problems of Mathematical Physics whose solutions are closely connected to the subject matter of the book. We emphasize the very profound impact of trace inequalities on spectral problems of differential operators, and in particular on eigenvalue estimates for Schrodinger operators (see [87]); and the close connection with the solubility of certain semilinear differential operators with minimal restrictions on the regularity of the coefficients and data. In fact, the existence of positive solutions of certain nonlineardifferential equations is equivalent to the IX

9 x BOUNDED & COMPACT INTEGRAL OPERATORS validity of a certain two-weighted inequality for a potential-type operator, in which the weights are expressed in terms of coefficients and data (Cf. [120], [13], [3], [197], [303J etc). One of the most challenging problems of the spectral theory of differential operators is the derivation of eigenvalue and singular value estimates of integral operators in terms of their kernels. The works [34], [235], [164J mark an important stage in the development of this theory ; see also [66], [79J. Until recently the list of non-trivial cases in which sharp two-sided estimates are available was rather short. Here we present two-sided estimates of the singular numbers for a large class of operators. The subjects of our investigation (fractional integrals, potentials, maximal functions, singular integrals) are important tools for solving a variety ofprob lems in several areas of mathematics and its applications. Some examples are worth mentioning here. The Hardy-type transforms are closely related to the solubility problems of nonlinear PDE (see, e.g., [63]). They are of considerable importance in the prediction of stock price futures in an equity market solely on the base of past performance of the stock price or market evaluation respectively. Integral equations involving generalized Hardy operators arise in the theory of automatic complex control systems (see Notes to Chapter 1). Concerning frac-, tional integrals, from the historical point of view we recall the then completely new method of M. Riesz for the solution of the Cauchy problem for PDE of hyperbolic type by means of a semi-group of operators corresponding to the Riemann-Liouville transform. We also mention the close connection between potentials and integral representations for the solution of differential equations; potential and singular integral methods in Mathematical Physics, etc. The considerable interest in fractional calculus in recent years has been stimulated by applications in different fields of science, including Stochastic Analysis of Long Memory Processes, Numerical Analysis, Physics, Chemistry, Engineering, Biology, Economics and Finance. For the theory of fractional integration and differentiation we refer to the well-known monograph [255]. For multidimensional fractional integrals see [247]. For a wide range of applications of modem Fractional Calculus see the references from [107J, [248]. In the second part of the book our attention is concentrated on the investigation of integral transforms on general measure spaces from the boundedness and compactness viewpoint. Topological spaces endowed with a locally finite complete measure and quasi-metric are considered. By a nonhomogeneous space we mean a measure space, with a quasi-metric, in which the doubling condition is not assumed and so may fail. Ifthe measure satisfies the doubling condition, then the measure space is said to be of homogeneous type (SHT). For these spaces we refer to the well-known monographs [56], [286]. SHT in all their generalities naturally arise when studying boundary value problems for PDE with variable coefficients, for instance, when the quasi-metric might be

10 PREFACE Xl induced by a differential operator, or tailored to fit kernels of integral operators. A weight theory for a wide class of integral transforms on SHT was developed in the monograph [l00]. We observe that the general setting of the problem enabled new results to be obtained even in some classical cases. Moreover, it should be emphasized that in that monograph a novel concept was developed, namely a two-weight inequality for integral transforms with arbitrary positive kernels in nonhomogeneous spaces, that is, in spaces where the doubling condition need not hold. In the late 1990s, in the papers [ ], [ ], it was shown that for the validity of almost all statements concerning Calderon-Zygmund singular integrals on an arbitrary metric space with nonatomic measure the doubling condition is not needed. In the present book we are interested in the development of this approach to integral transforms with positive kernels and singular integrals in the framework of the theory of two-weighted estimates. Now we shall briefly describe the contents of the book. Chapter 1 deals with Hardy transforms on measure spaces: two-weighted boundedness and compactness criteria are established. Here the exact values of the norms of integral transforms with positive kernels on certain cones of functions are explored. Results relating to the measure of noncompactness and two-sided estimates for the approximation numbers of integral transforms are distributed in various chapters (Chapters 1,2,4,5), corresponding to the study of particular operators. Chapters 2 and 4 contain an exposition of authors' latest results on the boundedness and compactness of fractional integrals on the line and also in the multidimensional case. Our approach is to do with criteria of boundedness and compactness from Lfn to L~ (1 < p ::; q < (0). For the Riemann-Liouville and Weyl operators criteria the trace inequalities are established. The "diagonal case" p = q is essentially treated. The latter enables us to give a new criteria for the trace inequality for potentials as well. The conditions which emerge are transparent and easy to verify. Experience of dealing with fractional integrals with specific kernels enables us to extend these results to a certain class ofintegral operators. Applications to the solvability problems of Abel's integral equation and certain superlinear inhomogeneous integral equations are presented. Chapter 3 is devoted to fractional maximal functions, one-sided maximal functions and potentials on the line. In Chapter 5, our attention is concentrated on boundedness and compactness problems for Riesz potentials and truncated potentials in a half-space. We are concerned with necessary and sufficient conditions for boundedness (compactness) in weighted Lebesgue spaces. In Chapter 6, we develop the study of integral transforms with positive kernels on nonhomogeneous spaces. We present a complete description of those

11 XlI BOUNDED & COMPACT INTEGRAL OPERATORS measures on which potential-type integral transforms generate bounded operators in Lebesgue spaces. Here theorems of Sobolev and Adams type in nonhomogeneous spaces are proved. In Chapter 7, on the basis of the results of the previous chapters, two-sided estimates are derived for the singular numbers of operators of Volterra and potential type. We establish necessary and sufficient conditions for these operators to belong to Schatten-von Neumann ideals. Asymptotic formulae for singular and entropy numbers of some Volterra-type operators are presented. Chapter 8 deals with two-weighted estimates for singular integrals in nonhomogeneous spaces. We provide a special treatment of the problem in SHT. In another section focussing on Heisenberg groups, two-weighted estimates for Cauchy-Szego projections are derived. For the Hilbert transform, twoweighted criteria are given for a certain class of pairs of weights. In the same chapter two-weight estimates for higher-dimensional singular integrals defined on Lipschitz surfaces in Clifford algebras are presented. Chapter 9 is devoted to two-weight problems for Fourier multipliers. Concerning one-weight estimates with Ap-weights we recall [169], [268], [140], and with exponential weights- [267], [170], [42]. A feature to which we draw attention is that estimates for Fourier multipliers with different weights on the two sides are presented in this book for the first time. We prove twoweighted inequalities for multipliers in Triebel-Lizorkin spaces. The setting of the problem in the framework of two-weight theory enables us to determine new classes of multipliers that it was known in unweighted case. The results of this direction are heavily based on the criteria of boundedness from Lfn to L~ (1 < p ::; q < (0) for fractional and singular integrals derived in previous chapters. Finally we provide a list of problems which were open at the time of completion of the book. We hope that this list will be useful in stimulating further research. A few words about the organization of the book are necessary. The enumeration of theorems, lemmas, formulas, etc. follows the natural three-digit system, where after the chapter and the section a consecutive number within the section is used. There are three categories for numbering: theorems, lemmas, propositions and remarks, and then come the formulae. The bibliography contains about 310 titles; we have collected the most relevent ones related to the topics presented here and it is by no means complete; our concern was to provide a basic orientation rather than a database. The book is aimed at a rather wide audience, ranging from researchers in Functional and Harmonic Analysis to experts in Applied Mathematics and prospective students.

12 Acknowledgments We wish to express our sincere gratitude to Dr. Maia Kvinikadze, Dr. Lida Gogolauri and Dr. Inga Gabisonija from A. Razmadze Mathematical Institute of Georgian Academy of Sciences for the energy and patience they devoted to the production of the TEX master of this book. We would also like to thank Mr. Tom Armour from the University of Sussex for helping us with solving various technical problems. The work on this book started in 1997, when the second author visited the Centre for Mathematical Analysis and its Applications, University of Sussex. The collaboration was developed during the visits of the second and third authors to Sussex. They express their deep gratitude to the Centre for support and warm hospitality. Partial support of Grant No GRIN33034/01 of EPSRC, Grant of the Royal Society, an INTAS Fellowship Grant for Young Scientists (Fellowship Reference No YSF 01/1-8) and Grant No. 1.7 of the Georgian Academy of Sciences is gratefully acknowledged. XIII

13 Basic notation R : real line R+ = [0, 00) H" : n-dimensional Euclidean space N : set of all positive integers Z+ = N U {O} Z: set of all integers n! = 1 2 n X = X x R+ R ~+ l = R n x R+ - 2 R+ = R+ x R+ dist (a, A) : distance from a to A. B (x, r) : open ball in R" with centre x and radius r B (x,r): closed ball in R" with centre x and radius r e n: n- dimensional complex plane n: domain in R n an t a : an is increasing and lim an = a n-too an +a : an is decreasing and lim an = a n-too I f h. b II R". 1r m / 2 am: vo ume 0 t e unit a III, r.e. am = f (l+m j 2) f ;:;;; g: there exist positive constants Cl and C2 such that c. I ::; 9 ::; c2f an ;:;;; b n : there exist positive constants Cl and C2 such that Cl an ::; b., ::; C2an for all n E N f.l x u: product measure [r]: integer part of the constant r C k (n ): set of k times differentiable functions <jj, F (ep ): Fourier transform rp, F - 1 (ep) : inverse Fourier transform SHT: space of homogemeous type xv

14 XVI BOUNDED & COMPACT INTEGRAL OPERATORS BPS: Banach function space RD : reverse doubling condition P DE: partial differential equatins o denotes the end of the proof