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Hardy Operators, Function Spaces and Embeddings

  • David E. Edmunds
  • W. Desmond Evans

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages I-XII
  2. David E. Edmunds, W. Desmond Evans
    Pages 1-9
  3. David E. Edmunds, W. Desmond Evans
    Pages 11-61
  4. David E. Edmunds, W. Desmond Evans
    Pages 63-160
  5. David E. Edmunds, W. Desmond Evans
    Pages 161-218
  6. David E. Edmunds, W. Desmond Evans
    Pages 219-273
  7. David E. Edmunds, W. Desmond Evans
    Pages 275-305
  8. Back Matter
    Pages 307-328

About this book

Introduction

Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Of the many developments of the basic theory since its inception, two are of particular interest:

(i) the consequences of working on space domains with irregular boundaries;
(ii) the replacement of Lebesgue spaces by more general Banach function spaces.

Both of these arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.

These aspects of the theory will probably enjoy substantial further growth, but even now a connected account of those parts that have reached a degree of maturity makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.

The significance of generalised ridged domains stems from their ability to 'unidimensionalise' the problems we study, reducing them to associated problems on trees or even on intervals.

This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.

Keywords

Banach function spaces Distribution Hilbert space Sobolev space Sobolev spaces approximation numbers generalised ridged domains integral operators partial differential equation partial differential equations real analysis

Authors and affiliations

  • David E. Edmunds
    • 1
  • W. Desmond Evans
    • 2
  1. 1.Department of MathematicsSussex UniversityBrightonUK
  2. 2.School of MathematicsCardiff UniversityCardiffUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-07731-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-06027-4
  • Online ISBN 978-3-662-07731-3
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site
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