Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral
Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral oper
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Herv´e Pajot
Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral
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Author Herv´e Pajot Department of Mathematics University of Cergy-Pontoise 2 Avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France E-mail: [email protected]
Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
Mathematics Subject Classification (2000): 28A75, 30C85, 42B20 ISSN 0075-8434 ISBN 3-540-00001-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de c Springer-Verlag Berlin Heidelberg 2002 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10891746
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Introduction These notes deal with complex analysis, harmonic analysis and geometric measure theory. My main motivation is to explain recent progress on the Painlev´e Problem and to describe their connections with the study of the L2 -boundedness of the Cauchy singular integral operator on Ahlfors-regular sets and the quantitative theory of rectifiability. Let E ⊂ C be a compact set. We say that E is removable for bounded analytic functions if, for any open set U ⊃ E, any bounded analytic function f : U \ E → C has an analytic extension to the whole U . The Painlev´ e problem can be stated as follows: Find a geometric/metric characterization of such removable sets. In 1947, L. Ahlfors [1] introduced the notion of analytic capacity of a compact set E: γ(E) = sup{|f (∞)|, f : C \ E → C is analytic bounded with ||f ||∞ ≤ 1} and proved that E is removable if and only if γ(E) = 0. But, as wrote Ahlfors himself (in this quotation, M (G) is the analytic capacity of the boundary of G where G is a complex domain of finite connectivity), “Of course our theorem is only a
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