Wavelets: CalderÃ³n-Zygmund and Multilinear

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

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 corresponding to singular integrals

8

(5)

Calderon-Zygmund operators and Lp spaces

13

(9)

The conditions T(1) = 0 and tT(1) = 0 for a Calderon-Zygmund operator

22

(2)

Pointwise estimates for Calderon-Zygmund operators

24

(6)

Calderon-Zygmund operators and singualr integrals

30

(4)

A more detailed version of Cotlar's inequality

34

(3)

The good λ inequalities and the Muckenhoupt weights

37

(4)

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 of the T(1) theorem

57

(3)

Cotlar and Stein's lemma and the second proof of David and Journe's theorem

60

(4)

Other formulations of the T(1) theorem

64

(1)

Banach algebras of Calderon-Zygmund operators

65

(6)

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 Calderon-Zygmund operators

79

(10)

Commutators and Calderon's improved pseudo-differential calculus

89

(4)

The pseudo-differential version of Leibniz's rule

93

(3)

Higher order commutators

96

(2)

Takafumi Murai's proof that the Cauchy kernel is L2 continuous

98

(7)

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 on homogeneous Sobolev spaces

119

(3)

Continuity on ordinary Sobolev spaces

122

(2)

Additional remarks

124

(2)

The T(b) theorem

Introduction

126

(1)

Statement of the fundamental geometric theorem

127

(1)

Operators and accretive forms (in the abstract situation)

128

(2)

Construction of bases adapted to a bilinear form

130

(2)

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 Cauchy kernel

141

(1)

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 T(b) theorem

150

(2)

Extensions to the case of vector-valued functions

152

(1)

Replacing the complex field by a Clifford algebra

153

(2)

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 analysis

171

(7)

The ``shortest'' proof

178

(3)

Statement of David's theorem

181

(4)

Transference

185

(4)

Calderon-Zygmund decomposition of Ahlfors regular curves

189

(2)

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 multilinear operators

202

(5)

Multilinear operators defined on (BMO)k

207

(3)

The general theory of holomorphic functionals

210

(5)

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 conjecture

239

(6)

Estimates of the kernels of the operators Lm(2)

245

(6)

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 double-layer potential

261

(5)

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 non-linear problem

278

(2)

A second linearization of the non-linear problem

280

(5)

Paradifferential operators

285

(3)

The symbolic calculus for paradifferential operators

288

(4)

Application to non-linear partial differential equations

292

(2)

Paraproducts and wavelets

294

(4)

References and Bibliography

298

(13)

References and Bibliography for the English edition

311

(2)

Index

313

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Wavelets: CalderÃ³n-Zygmund and Multilinear Operators

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