Wavelets: Calderon-Zygmund and Multilinear Operators

Wavelets: Calderon-Zygmund and Multilinear Operators

by Yves Meyer, Ronald Coifman
     
 

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ISBN-10: 0521420016

ISBN-13: 9780521420013

Pub. Date: 05/29/1997

Publisher: Cambridge University Press

Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. In this volume the theory of paradifferential operators and the Cauchy kernel on Lipschitz curves are discussed with the emphasis firmly on their connection with wavelet bases. Sparse matrix representations of these operators can be given

Overview

Now in paperback, this remains one of the classic expositions of the theory of wavelets from two of the subject's leading experts. In this volume the theory of paradifferential operators and the Cauchy kernel on Lipschitz curves are discussed with the emphasis firmly on their connection with wavelet bases. Sparse matrix representations of these operators can be given in terms of wavelet bases which have important applications in image processing and numerical analysis. The method is now widely studied and can be used to tackle a wide variety of problems arising in science and engineering. Put simply, this is an essential purchase for anyone researching the theory of wavelets.

Product Details

ISBN-13:
9780521420013
Publisher:
Cambridge University Press
Publication date:
05/29/1997
Series:
Cambridge Studies in Advanced Mathematics Series, #48
Pages:
336
Product dimensions:
5.98(w) x 8.98(h) x 0.87(d)

Table of Contents

Translator's notex
Preface to the English editionxi
Introductionxiii
Introduction to Wavelets and Operatorsxv
7The new Calderon-Zygmund operators
1Introduction1
2Definition of Calderon-Zygmund operators corresponding to singular integrals8
3Calderon-Zygmund operators and L[superscript p] spaces13
4The conditions T(1) = 0 and [superscript t]T(1) = 0 for a Calderon-Zygmund operator22
5Pointwise estimates for Calderon-Zygmund operators24
6Calderon-Zygmund operators and singular integrals30
7A more detailed version of Cotlar's inequality34
8The good [lambda] inequalities and the Muckenhoupt weights37
9Notes and additional remarks41
8David and Journe's T(1) theorem
1Introduction43
2Statement of the T(1) theorem45
3The wavelet proof of the T(1) theorem51
4Schur's lemma54
5Wavelets and Vaguelets56
6Pseudo-products and the rest of the proof of the T(1) theorem57
7Cotlar and Stein's lemma and the second proof of David and Journe's theorem60
8Other formulations of the T(1) theorem64
9Banach algebras of Calderon-Zygmund operators65
10Banach spaces of Calderon-Zygmund operators71
11Variations on the pseudo-product73
12Additional remarks76
9Examples of Calderon-Zygmund operators
1Introduction77
2Pseudo-differential operators and Calderon-Zygmund operators79
3Commutators and Calderon's improved pseudo-differential calculus89
4The pseudo-differential version of Leibniz's rule93
5Higher order commutators96
6Takafumi Murai's proof that the Cauchy kernel is L[superscript 2] continuous98
7The Calderon-Zygmund method of rotations105
10Operators corresponding to singular integrals: their continuity on Holder and Sobolev spaces
1Introduction111
2Statement of the theorems112
3Examples114
4Continuity of T on homogeneous Holder spaces117
5Continuity of operators in L[subscript gamma] on homogeneous Sobolev spaces119
6Continuity on ordinary Sobolev spaces122
7Additional remarks124
11The T(b) theorem
1Introduction126
2Statement of the fundamental geometric theorem127
3Operators and accretive forms (in the abstract situation)128
4Construction of bases adapted to a bilinear form130
5Tchamitchian's construction132
6Continuity of T136
7A special case of the T(b) theorem138
8An application to the L[superscript 2] continuity of the Cauchy kernel141
9The general case of the T(b) theorem142
10The space H[superscript 1 subscript b]145
11The general statement of the T(b) theorem149
12An application to complex analysis150
13Algebras of operators associated with the T(b) theorem150
14Extensions to the case of vector-valued functions152
15Replacing the complex filed by a Clifford algebra153
16Further remarks155
12Generalized Hardy spaces
1Introduction157
2The Lipschitz case158
3Hardy spaces and conformal representations163
4The operators associated with complex analysis171
5The "shortest" proof178
6Statement of David's theorem181
7Transference185
8Calderon-Zygmund decomposition of Ahlfors regular curves189
9The proof of David's theorem191
10Further results194
13Multilinear operators
1Introduction195
2The general theory of multilinear operators197
3A criterion for the continuity of multilinear operators202
4Multilinear operators defined on (BMO)[superscript k]207
5The general theory of holomorphic functionals210
6Application to Calderon's programme215
7McIntosh's theory of multilinear operators220
8Conclusion226
14Multilinear analysis of square roots of accretive operators
1Introduction227
2Square roots of operators228
3Accretive square roots232
4Accretive sesquilinear forms236
5Kato's conjecture238
6The multilinear operators of Kato's conjecture239
7Estimates of the kernels of the operators L[superscript (2) subscript m]245
8The kernels of the operators L[subscript m]251
9Additional remarks254
15Potential theory in Lipschitz domains
1Introduction255
2Statement of the results256
3Almost everywhere existence of the double-layer potential261
4The single-layer potential and its gradient266
5The Jerison and Kenig identities270
6The rest of the proof of Theorems 2 and 3274
7Appendix275
16Paradifferential operators
1Introduction277
2A first example of linearization of a non-linear problem278
3A second linearization of the non-linear problem280
4Paradifferential operators285
5The symbolic calculus for paradifferential operators288
6Application to non-linear partial differential equations292
7Paraproducts and wavelets294
References and Bibliography298
References and Bibliography for the English edition311
Index313

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