Contents
1 Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves 1
1.1 Convolutions and Differentiation on Lipschitz Graphs 2
1.2 Quadratic Estimates for Type co Operators 6
1.3 Fourier Transform and the Inverse Fourier Transform on Sectors 6
1.4 Convolution Singular Integral Operators on the Lipschitz Curves 22
1.5 Lp-Fourier Multipliers on Lipschitz Curves 29
1.6 Remarks 41
References 42
2 Singular Integral Operators on Closed Lipschitz Curves 43
2.1 Preliminaries 44
2.2 Fourier Transforms Between S and PS(π) 48
2.3 Singular Integrals on Starlike Lipschitz Curves 54
2.4 Holomorphic H*-Functional Calculus on Starlike Lipschitz Curves 61
2.5 Remarks 65
References 65
3 Clifford Analysis, Dirac Operator and the Fourier Transform 67
3.1 Preliminaries on Clifford Analysis 67
3.2 Monogenic Functions on Sectors 74
3.3 Fourier Transforms on the Sectors 79
3.4 Mobius Covariance of Iterated Dirac Operators 94
3.5 The Fueter Theorem 100
3.6 Remarks 114
References 115
4 Convolution Singular Integral Operators on Lipschitz Surfaces 117
4.1 Clifford-Valued Martingales 117
4.2 Martingale Type T(b) Theorem 125
4.3 Clifford Martingale O-Equivalence Between S(f) and f* 140
4.4 Remarks 147
References 147
5 Holomorphic Fourier Multipliers on Infinite Lipschitz Surfaces 149
5.1 Singular Convolution Integrals on Infinite Lipschitz Surfaces 149
5.2 H*-Functional Calculus of Functions of n Variables 156
5.3 H*-Functional Calculus of Functions of One Variable 162
References 166
6 Bounded Holomorphic Fourier Multipliers on Closed Lipschitz Surfaces 169
6.1 Monomial Functions in Rn 169
6.2 Bounded Holomorphic Fourier Multipliers 186
6.3 Holomorphic Functional Calculus of the Spherical Dirac Operator 200
6.4 The Analogous Theory in Rn 203
6.5 Hilbert Transforms on the Sphere and Lipschitz Surfaces 206
6.6 Remarks 219
References 219
7 The Fractional Fourier Multipliers on Lipschitz Curves and Surfaces 221
7.1 The Fractional Fourier Multipliers on Lipschitz Curves 224
7.2 Fractional Fourier Multipliers on Starlike Lipschitz Surfaces 239
7.3 Integral Representation of Sobolev-Fourier Multipliers 254
7.4 The Equivalence of Hardy-Sobolev Spaces 270
7.5 Remarks 272
References 273
8 Fourier Multipliers and Singular Integrals on Cn 275
8.1 A Class of Singular Integral Operators on the n-ComplexUnit Sphere 275
8.2 Fractional Multipliers on the Unit Complex Sphere 289
8.3 Fourier Multipliers and Sobolev Spaces on Unit Complex Sphere 298
References 300
Bibliography 303
Index 305