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Hardy Operators, Function Spaces and Embeddings

Author: David E Edmunds; W Desmond Evans
Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2004.
Series: Springer monographs in mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Database:WorldCat
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: David E Edmunds; W Desmond Evans
ISBN: 9783662077313 3662077310
OCLC Number: 851392104
Description: 1 online resource (xii, 328 pages).
Contents: Preliminaries --
Hardy-type Operators --
Banach Function Spaces --
Poincaré and Hardy Inequalities --
Generalised Ridged Domains --
Approximation Numbers of Sobolev Embeddings --
References --
Author Index --
Subject Index --
Notation Index.
Series Title: Springer monographs in mathematics.
Responsibility: by David E. Edmunds, W. Desmond Evans.
More information:

Abstract:

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.

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From the reviews: "This interesting monograph is the second joint book by the two authors. Whereas in their first one ... one of the main objects was the study of spectral theory of boundary value Read more...

 
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