Wavelets : Calderon-Zygmund and Multilinear Operators (Cambridge Studies in Advanced Mathematics)

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Wavelets : Calderon-Zygmund and Multilinear Operators (Cambridge Studies in Advanced Mathematics)

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  • 製本 Hardcover:ハードカバー版/ページ数 314 p.
  • 言語 ENG
  • 商品コード 9780521420013
  • DDC分類 515.2433

基本説明

New in paperback. Hardcover is published in 1995. Transl. by David Salinger.

Full Description


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. This 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.

Table of Contents

Translator's note                                  x
Preface to the English edition xi
Introduction xiii
Introduction to Wavelets and Operators xv
The new Calderon-Zygmund operators
Introduction 1 (7)
Definition of Calderon-Zygmund operators 8 (5)
corresponding to singular integrals
Calderon-Zygmund operators and Lp spaces 13 (9)
The conditions T(1) = 0 and tT(1) = 0 for a 22 (2)
Calderon-Zygmund operator
Pointwise estimates for Calderon-Zygmund 24 (6)
operators
Calderon-Zygmund operators and singualr 30 (4)
integrals
A more detailed version of Cotlar's 34 (3)
inequality
The good λ inequalities and the 37 (4)
Muckenhoupt weights
Notes and additional remarks 41 (2)
David and Journe's T(1) theorem
Introduction 43 (2)
Statement of the T(1) theorem 45 (6)
The wavelet proof of the T(1) theorem 51 (3)
Schur's lemma 54 (2)
Wavelets and Vaguelets 56 (1)
Pseudo-products and the rest of the proof 57 (3)
of the T(1) theorem
Cotlar and Stein's lemma and the second 60 (4)
proof of David and Journe's theorem
Other formulations of the T(1) theorem 64 (1)
Banach algebras of Calderon-Zygmund 65 (6)
operators
Banach spaces of Calderon-Zygmund operators 71 (2)
Variations on the pseudo-product 73 (3)
Additional remarks 76 (1)
Examples of Calderon-Zygmund operators
Introduction 77 (2)
Pseudo-differential oeprators and 79 (10)
Calderon-Zygmund operators
Commutators and Calderon's improved 89 (4)
pseudo-differential calculus
The pseudo-differential version of 93 (3)
Leibniz's rule
Higher order commutators 96 (2)
Takafumi Murai's proof that the Cauchy 98 (7)
kernel is L2 continuous
The Calderon-Zygmund method of rotations 105 (6)
Operators corresponding to singular
integrals: their continuity on Holder and
Sobolev spaces
Introduction 111 (1)
Statement of the theorems 112 (2)
Examples 114 (3)
Continuity of T on homogeneous Holder spaces 117 (2)
Continuity of operators in ℒγ 119 (3)
on homogeneous Sobolev spaces
Continuity on ordinary Sobolev spaces 122 (2)
Additional remarks 124 (2)
The T(b) theorem
Introduction 126 (1)
Statement of the fundamental geometric 127 (1)
theorem
Operators and accretive forms (in the 128 (2)
abstract situation)
Construction of bases adapted to a bilinear 130 (2)
form
Tchamitchian's construction 132 (4)
Continuity of T 136 (2)
A special case of the T(b) theorem 138 (3)
An application to the L2 continuity of the 141 (1)
Cauchy kernel
The general case of the T(b) theorem 142 (3)
The space Hb1 145 (4)
The general statement of the T(b) theorem 149 (1)
An application to complex analysis 150 (1)
Algebras of operators associated with the 150 (2)
T(b) theorem
Extensions to the case of vector-valued 152 (1)
functions
Replacing the complex field by a Clifford 153 (2)
algebra
Further remarks 155 (2)
Generalized Hardy spaces
Introduction 157 (1)
The Lipschitz case 158 (5)
Hardy spaces and conformal representations 163 (8)
The operators associated with complex 171 (7)
analysis
The ``shortest'' proof 178 (3)
Statement of David's theorem 181 (4)
Transference 185 (4)
Calderon-Zygmund decomposition of Ahlfors 189 (2)
regular curves
The proof of David's theorem 191 (3)
Further results 194 (1)
Multilinear operators
Introduction 195 (2)
The general theory of multilinear operators 197 (5)
A criterion for the continuity of 202 (5)
multilinear operators
Multilinear operators defined on (BMO)k 207 (3)
The general theory of holomorphic 210 (5)
functionals
Application to Calderon's programme 215 (5)
McIntosh's theory of multilinear operators 220 (6)
Conclusion 226 (1)
Multilinear analysis of square roots of
accretive operators
Introduction 227 (1)
Square roots of operators 228 (4)
Accretive square roots 232 (4)
Accretive sesquilinear forms 236 (2)
Kato's conjecture 238 (1)
The multilinear operators of Kato's 239 (6)
conjecture
Estimates of the kernels of the operators 245 (6)
Lm(2)
The kernels of the operators Lm 251 (3)
Additional remarks 254 (1)
Potential theory in Lipschitz domains
Introduction 255 (1)
Statement of the results 256 (5)
Almost everywhere existence of the 261 (5)
double-layer potential
The single-layer potential and its gradient 266 (4)
The Jerison and Kenig identities 270 (4)
The rest of the proof of Theorems 2 and 3 274 (1)
Appendix 275 (2)
Paradifferential operators
Introduction 277 (1)
A first example of linearization of a 278 (2)
non-linear problem
A second linearization of the non-linear 280 (5)
problem
Paradifferential operators 285 (3)
The symbolic calculus for paradifferential 288 (4)
operators
Application to non-linear partial 292 (2)
differential equations
Paraproducts and wavelets 294 (4)
References and Bibliography 298 (13)
References and Bibliography for the English 311 (2)
edition
Index 313