Weighted Littlewood-Paley Theory and Exponential-Square Integrability

Littlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense.

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1924

Michael Wilson

Weighted Littlewood-Paley Theory and Exponential-Square Integrability

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Author Michael Wilson Department of Mathematics University of Vermont Burlington, Vermont 05405 USA e-mail: [email protected]

Library of Congress Control Number: 2007934022 Mathematics Subject Classification (2000): 42B25, 42B20, 42B15, 35J10 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-74582-2 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-74587-7 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008  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. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 12114160

41/SPi

543210

I dedicate this book to my parents, James and Joyce Wilson.

Preface

Littlewood-Paley theory can be thought of as a profound generalization of the Pythagorean theorem. If x ∈ Rd —say, x = (x1 , x2 , . . . , xd )—then we define d x’s norm, x, to be ( 1 x2n )1/2 . This norm has the good property that, if y = (y1 , y2 , . . . , yd ) is any other vector in Rd , and |yn | ≤ |xn | for each n, then y ≤ x. In other words, the size of x, as measured by the norm function, is determined entirely by the sizes of x’s components. This remains true if we let the dimension d increase to infinity, and define the norm ∞of a vector (actually, an infinite sequence) x = (x1 , x2 , . . .) to be x ≡ ( 1 x2n )1/2 . In analysis it is often convenient (and indispensable) to decompose functions f into infinite series,  (0.1) f (x) = λn φn (x), where the functions φn come from some standard family (such as the Fourier system) and the λn ’s are complex numbers. (For the time being we will not specify how the series 0.1 is supposed to converge.) Typically the coefficients λn are defined by integrals of f against some other functions ψn . If we are interested about convergence in the sense of L2 (or “mean-square”), and if the φn ’s comprise a complete orthonormal family, then each ψn can be taken to be φ¯n , the complex conjugate of φn ; i.e.,  λn = f (x) φ¯n (x) dx, 

and we have

2

|f (x)| dx =



2

|λn | .

(For the time b