Extrapolation and Optimal Decompositions with Applications to Analys
This book develops a theory of extrapolation spaces with applications to classical and modern analysis. Extrapolation theory aims to provide a general framework to study limiting estimates in analysis. The book also considers the role that optimal decompo
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Mario Milman
Extrapolation and Optimal Decompositions with Applications to Analysis
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Author Mario Milman Department of Mathematics Florida Atlantic University Boca Raton, FL 33431, USA E-mail: Milman @acc.fau.edu
Mathematics Subject Classification (1991): Primary: 46M35, 46E35, 42B20 Secondary: 35R15, 58D25
ISBN 3-540-58081-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-58081-6 Springer-Verlag New York Berlin Heidelberg
CIP-Data applied for 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, re-use of illustrations, recitation, broadcasting, reproduction on microfilms 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- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera-ready by author SPIN: 10130043 46/3140-543210 - Printed on acid-free paper
To Vanda
Preface In these notes we continue the development of a theory of extrapolation spaces initiated in [57]. One of our main concerns has been to connect the fundamental processes associated with the construction of interpolation and extrapolation spaces "optimal decompositions" with a number of problems in analysis. In particular we study extrapolation of inequalities related to Sobolev embedding theorems, higher order logarithmic Sobolev inequalities, a.e.convergence of Fourier series, bilinear extrapolation of estimates in different settings with applications to PDE's, apriori estimates for abstract parabolic equations, commutator inequalities with applications to compensated compactness, a functional calculus associated with positive operators on a Banach space, and the iteration method of Nash/Moser to solve nonlinear equations. Many of the results presented in these notes are new and appear here for the first time. We have also included a number of open problems throughout the text. While we hope that these features could make our work attractive to specialists in the field of function spaces it is also hoped that the central role that the applications play in our development could also make it of interest to classical analysts working in other areas. In order to facilitate the task of these prospective readers we have tried to provide sufficient background information with complete references and included a brief guide to the literature on interpolation theory. We have also tried to make the contents of different chapters as independent of each other as possible while at the same time avoiding too much repetition. Finally we have also included a subject index and notation index. It is a pleasure to record here my gr
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