Singular Integrals and Fourier Theory on Lipschitz Boundaries

The main purpose of this book is to provide a detailed and comprehensive survey of the theory of singular integrals and Fourier multipliers on Lipschitz curves and surfaces, an area that has been developed since the 1980s. The subject of singular integrals and the related Fourier multipliers on Lipschitz curves and surfaces has an extensive background in harmonic analysis and partial differential equations. The book elaborates on the basic framework, the Fourier methodology, and the main results in various contexts, especially addressing the following topics: singular integral operators with holomorphic kernels, fractional integral and differential operators with holomorphic kernels, holomorphic and monogenic Fourier multipliers, and Cauchy-Dunford functional calculi of the Dirac operators on Lipschitz curves and surfaces, and the high-dimensional Fueter mapping theorem with applications. The book offers a valuable resource for all graduate students and researchers interested in singular integrals and Fourier multipliers.

States systemically the theory of singular integrals and Fourier multipliers on the Lipschitz graphs and surfaces Elaborates the basic framework, essential thoughts and main results Reveals the equivalence between the operator algebra of the singular integrals, Fourier multiplier Operators and the Cauchy-Dunford functional calculus of the Dirac operators

Inhalt
Singular integrals and Fourier multipliers on infinite Lipschitz curves.- Singular integral operators on closed Lipschitz curves.- Clifford analysis, Dirac operator and the Fourier transform.- Convolution singular integral operators on Lipschitz surfaces.- Holomorphic Fourier multipliers on infinite Lipschitz surfaces.- Bounded holomorphic Fourier multipliers on closed Lipschitz surfaces.- The fractional Fourier multipliers on Lipschitz curves and surfaces.- Fourier multipliers and singular integrals on Cn