Singular Integrals and Fourier Theory on Lipschitz Boundaries

Preface
	References
Acknowledgements
Contents
Nomenclature
1 Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves
	1.1 Convolutions and Differentiation on Lipschitz Graphs
	1.2 Quadratic Estimates for Type ω Operators
	1.3 Fourier Transform and the Inverse Fourier Transform on Sectors
	1.4 Convolution Singular Integral Operators  on the Lipschitz Curves
	1.5 Lp-Fourier Multipliers on Lipschitz Curves
	1.6 Remarks
	References
2 Singular Integral Operators on Closed Lipschitz Curves
	2.1 Preliminaries
	2.2 Fourier Transforms Between S0ω and pS0ω(π)
	2.3 Singular Integrals on Starlike Lipschitz Curves
	2.4 Holomorphic Hinfty-Functional Calculus on Starlike Lipschitz Curves
	2.5 Remarks
	References
3 Clifford Analysis, Dirac Operator  and the Fourier Transform
	3.1 Preliminaries on Clifford Analysis
	3.2 Monogenic Functions on Sectors
	3.3 Fourier Transforms on the Sectors
	3.4 Möbius Covariance of Iterated Dirac Operators
	3.5 The Fueter Theorem
	3.6 Remarks
	References
4 Convolution Singular Integral Operators on Lipschitz Surfaces
	4.1 Clifford-Valued Martingales
	4.2 Martingale Type T(b) Theorem
	4.3 Clifford Martingale Φ-Equivalence Between  S(f) and fast
	4.4 Remarks
	References
5 Holomorphic Fourier Multipliers  on Infinite Lipschitz Surfaces
	5.1 Singular Convolution Integrals on Infinite Lipschitz Surfaces
	5.2 Hinfty-Functional Calculus of Functions of n Variables
	5.3 Hinfty-Functional Calculus of Functions of One Variable
	References
6 Bounded Holomorphic Fourier Multipliers on Closed Lipschitz Surfaces
	6.1 Monomial Functions in mathbbRn1
	6.2 Bounded Holomorphic Fourier Multipliers
	6.3 Holomorphic Functional Calculus of the Spherical Dirac Operator
	6.4 The Analogous Theory in mathbbRn
	6.5 Hilbert Transforms on the Sphere and Lipschitz Surfaces
	6.6 Remarks
	References
7 The Fractional Fourier Multipliers  on Lipschitz Curves and Surfaces
	7.1 The Fractional Fourier Multipliers on Lipschitz Curves
	7.2 Fractional Fourier Multipliers on Starlike Lipschitz Surfaces
	7.3 Integral Representation of Sobolev–Fourier Multipliers
	7.4 The Equivalence of Hardy–Sobolev Spaces
	7.5 Remarks
	References
8 Fourier Multipliers and Singular Integrals on mathbbCn
	8.1 A Class of Singular Integral Operators  on the n-Complex Unit Sphere
	8.2 Fractional Multipliers on the Unit Complex Sphere
	8.3 Fourier Multipliers and Sobolev Spaces on Unit Complex Sphere
	References
Bibliography
Index