Bounded and Compact Integral Operators / Edition 1

by David E. Edmunds, V.M Kokilashvili, Alexander Meskhi

Overview

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. Itfocuses onintegral operators naturally arising in boundary value problems for PDE, the spectral theory of differential operators, continuum and quantum mechanics, shastic 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, togive var­ ious 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 theinvestigation ofclassical 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 (com­ pactly). When a given operator is not bounded (compact), for example in some Lebesgue space, we look for weighted spaces where boundedness (compact­ ness) holds. We develop the ideas and the techniques for the derivation of appropriate conditions, in terms of weights, which are equivalent to bounded­ ness (compactness).

Product Details

ISBN-13:	9781402006197
Publisher:	Springer Netherlands
Publication date:	05/31/2002
Series:	Mathematics and Its Applications , #543
Edition description:	2002
Pages:	643
Product dimensions:	6.14(w) x 9.21(h) x 0.36(d)

Table of Contents

1. Hardy-Type Operators.- 2. Fractional Integrals on the Line.- 3. One-Sided Maximal Functions.- 4. Ball Fractional Integrals.- 5. Potentials on RN.- 6. Fractional Integrals on Measure Spaces.- 7. Singular Numbers.- 8. Singular Integrals.- 9. Multipliers of Fourier Transforms.- 10. Problems.- References.