Eigenvalues, Embeddings and Generalised Trigonometric Functions
The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone de
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For further volumes: http://www.springer.com/series/304
2016
•
Jan Lang
•
David Edmunds
Eigenvalues, Embeddings and Generalised Trigonometric Functions
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Prof. Jan Lang Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, Ohio 43210 USA [email protected]
Prof. David Edmunds University of Sussex Department of Mathematics Pevensey 2, North-South Road Brighton BN1 9QH United Kingdom [email protected]
ISBN 978-3-642-18267-9 e-ISBN 978-3-642-18429-1 DOI 10.1007/978-3-642-18429-1 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011924532 Mathematics Subject Classification (2011): 41A35, 41A46, 47B06, 33E30, 47G10, 35P05, 47A75, 35P15, 46E35, 47B05 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The main theme of these notes is the study, from the standpoint of s-numbers, of operators of Hardy type and related Sobolev embeddings. More precisely, let p, q ∈ (1, ∞) and suppose that I is the interval (a, b), where −∞ < a < b < ∞. Maps T : L p (I) → Lq (I) of the form (T f )(x) = v(x)
x a
u(t) f (t)dt,
(1)
where u and v are prescribed functions satisfying some integrability conditions, are said to be of Hardy type. They are of importance in connection with ‘small ball’ problems in probability theory [87] and also in the theory of embeddings of Sobolev spaces when the underlying subset Ω of Rn is a generalised ridged domain, which means crudely that Ω has a central axis (the generalised ridge) that is the image of a tree under a Lipschitz map [42]. In addition, the literature on such maps T has grown to such an extent that the topic has acquired an independent life. Our object is, so far as we are able, to give an account of the present state of knowledge in this area in the hope that it will stimulate further work. In addition to the main theme, topics that arise naturally include the geometry of Banach spaces, generalised trigonometric functions and the p-Laplacian, and we hav
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