Lipschitz边界上的奇异积分与Fourier理论

在第一章中介绍Lipschitz曲线上的Fourier乘子理论，主要介绍一维无穷曲线上的Fourier乘子、奇异积分和泛函演算理论；第二章主要介绍单位圆的Lipschitz扰动上Fourier乘子理论以及相关问题的研究。第三章主要介绍用Clifford分析的背景知识。第四章和第五章则主要着眼于阐述利用Clifford分析的手段处理Lipschitz曲面上的全纯Fourier乘子和相应的奇异积分，包括Futuer定理，Clifford鞅等内容。第六、七、八章。分别介绍星形Lipschitz曲面上的的Fourier乘子理论，包括星形Lipschitz曲面上有界和无界Fourier乘子的核函数估计、奇异积分表示以及在高维复球面上的推广等内容。

1 Singular Integrals and Fourier Multipliers on Infinite Lipschitz Curves
1.1 Convolutions and Differentiation on Lipschitz Graphs
1.2 Quadratic Estimates for Type co 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 S and PS(π)
2.3 Singular Integrals on Starlike Lipschitz Curves
2.4 Holomorphic H∞-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 Mobius 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 f*
4.4 Remarks
References
5 Holomorphic Fourier Multipliers on Infinite Lipschitz Surfaces
5.1 Singular Convolution Integrals on Infinite Lipschitz Surfaces
5.2 H∞-Functional Calculus of Functions of n Variables
5.3 H∞-Functional Calculus of Functions of One Variable
References
6 Bounded Holomorphic Fourier Multipliers on Closed Lipschitz Surfaces
6.1 Monomial Functions in Rn
6.2 Bounded Holomorphic Fourier Multipliers
6.3 Holomorphic Functional Calculus of the Spherical Dirac Operator
6.4 The Analogous Theory in Rn
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 Cn
8.1 A Class of Singular Integral Operators on the n-ComplexUnit Sphere
8.2 Fractional Multipliers on the Unit Complex Sphere
8.3 Fourier Multipliers and Sobolev Spaces on Unit Complex Sphere
References
Bibliography
Index