Clifford Wavelets, Singular Integrals, and Hardy Spaces
The book discusses the extensions of basic Fourier Analysis techniques to the Clifford algebra framework. Topics covered: construction of Clifford-valued wavelets, Calderon-Zygmund theory for Clifford valued singular integral operators on Lipschitz hyper-
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Marius Mitrea
Clifford Wavelets, Singular Integrals, and Hardy Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Marius Mitrea Institute of Mathematics of the Romanian Academy P. O. Box 1-764 RO-70700 Bucharest, Romania and Department of Mathematics University of South Carolina Columbia, SC 29208, USA
Mathematics Subject Classification (1991): 30035, 42B20, 42B30, 31B25
ISBN 3-540-57884-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57884-6 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 SPIN: 10130077
46/3140-543210 - Printed on acid-free paper
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Table of Contents Page Introduction
IX
Chapter 1: Clifford Algebras
1
§1.1 Real and complex Clifford algebras
1
§1.2 Elements of Clifford Analysis
5
§1.3 Clifford modules
.....
11
16
Chapter 2: Constructions of Clifford Wavelets §2.1 Accretive forms and accretive operators
17
§2.2 Clifford Multiresolution Analysis. The abstract setting
18
§2.3 Bases in the wavelet spaces . . . . . . . . . .
23
ern)
26
. . . . . . . . . . . .
30
§2.4 Clifford Multiresolution Analyses of L 2 (lR.m ) 0 §2.5 Haar Clifford wavelets
Chapter 3: The £2 Boundedness of Clifford Algebra Valued Singular Integral Operators
.
42
§3.1 The higher dimensional Cauchy integral
43
§3.2 The Clifford algebra version of the T(b) theorem
53
Chapter 4: Hardy Spaces of Monogenic Functions
60
§4.1 Maximal function characterizations
61
§4.2 Boundary behavior
70
§4.3 Square function characterizations
73
§4.4 The regularity of the Cauchy operator
82
VII
Chapter 5: Applications to the Theory of Harmonic Functions
87
§5.1 Potentials of single and double layers.
87
§5.2 L 2-estimates at the boundary
90
§5.3 Boundary value problems for the Laplace operator mains
in Lipschitz do-
93
.
§5.4 A Burkholder-Gundy-Silverstein type theorem for monogenic functions and applications References
98
. . .
106
Notational Index
113
Subject Index. .
114
VIII
Introduction As the seminal work of Zygmund [Zyj describes the state of the art in the mid 30's, much of classical Fourier Analysis, dealing with the boundary behavior of harmonic functions in the unit disc or the upper-half plane, has initially been developed with the aid of complex-variable methods. The success of extending these results to higher dimensions, the crowning
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