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Singular Integrals and Fourier Theory on Lipschitz Boundaries

Qian Tao and Pengtao Li
Publisher: 
Springer
Publication Date: 
2019
Number of Pages: 
306
Format: 
Hardcover
Price: 
139.99
ISBN: 
978-981-13-6499-0
Category: 
Monograph
[Reviewed by
Eric Stachura
, on
12/22/2019
]
This book is largely a research monograph detailing the work of Alan McIntosh and collaborators on singular integrals and Fourier multipliers on Lipschitz surfaces. There are three main sections of this book: singular integrals and Fourier multipliers on Lipschitz curves in one complex variable; on graph type Lipschitz surfaces; and on starlike Lipschitz surfaces. 
 
The first two chapters are devoted to the theory of singular integrals and Fourier multipliers on Lipschitz curves, and includes results such as:
 
  • The \( L^p \) boundedness of singular convolution integral operators on such curves; 
  • The \(H^{\infty} \) functional calculus of the Fourier multipliers
In chapters 3-5, singular integrals and Fourier multipliers are treated systematically on Lipschitz surfaces using Clifford analysis. These chapters include results such as:
 
  • A generalized Fueter Theorem in the setting of Clifford algebras; 
  • A Clifford martingale \(T(b) \) theorem, implying boundedness of Cauchy-type singular integral operators;
  • The correspondence between \( H^\infty \) Fourier multipliers, singular integrals of monogenic kernels on Lipschitz surfaces, and the \( H^{\infty} \) functional calculus of the spherical Dirac operator
The last three chapters are devoted to the theory of holomorphic Fourier multipliers on starlike Lipschitz surfaces. 
 
The main audience for this book would be those interested in the importance of Fourier multipliers in Harmonic Analysis. In particular, the big questions addressed in this book arise from the main question of \( L^p \) boundedness of singular integral operators on Lipschitz surfaces. Each chapter has a nice introduction outlining the main results. Additionally, there is a short bibliography at the end of each chapter, illuminating which articles the material from the chapter comes from. 
 
Overall this book would serve as a nice reference on recent developments on singular integrals and Fourier multipliers on various Lipschitz surfaces. 

 

Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations. 

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