Hardy Operators, Function Spaces and Embeddings
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 variatio
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David E. Edmunds W. Desmond Evans
Hardy Operators, Funetion Spaees and Embeddings With 6 Figures
~ Springer
David E. Edmunds Department of Mathematics Sussex University Brighton BN1 9RF, United Kingdom e-mail: [email protected] W. Desmond Evans School of Mathematics Cardiff Universi ty Cardiff CF24 4YH, Uni ted Kingdom e-mail: [email protected]. uk
Library of Congress Control Number: 2004108695
The cover figure is taken from a paper by L.E. Fraenkel and is Fig. 5.5 on page 232 of the text.
Mathematics Subject Classification (2000): 26DlO, 26D15, 34L20, 35J05, 35P20, 45D05, 45P05, 46B50, 46E35, 47B06, 47BlO ISSN 1439-7382 ISBN 978-3-642-06027-4
ISBN 978-3-662-07731-3 (eBook)
DOI 10.1007/978-3-662-07731-3
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Preface
Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have far many years proved to be absolutely indispensable in the study of partial differential equations and variational problems. The embedding theorems and inequalities which feature so large in first courses on function spaces are key ingredients of the proofs of existence and regularity for elliptic boundary-value problems. There have been many developments of the basic theory since its inception, and of these we distinguish two which seem to us to be 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 demands imposed by concrete problems. For example, the ubiquitous nature of sets with fractal boundaries make::> it unnecessary to give an extended justification of (i), while (ii) is very natural when face
