Holomorphic Fourier Multipliers on Infinite Lipschitz Surfaces

Tao Qian and Pengtao Li
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Tao Qian: Macau University of Science and Technology, Macau Institute of Systems Engineering
Pengtao Li: Qingdao University, School of Mathematics and Statistics

Chapter Chapter 5 in Singular Integrals and Fourier Theory on Lipschitz Boundaries, 2019, pp 149-167 from Springer

Abstract: Abstract It is well-known that there exists a one-one correspondence between the classical convolution singular integral operators and the Fourier multiplier operators on the Euclidean spaces $$\mathbb {R}^{n}$$ . Because Plancherel’s identity involving the Fourier transform is invalid on Lipschitz surfaces $$\Sigma $$ , the relation between singular Cauchy integral operators and Fourier multipliers on $$\Sigma $$ is an open problem for a long time. In 1994, by the aid of Clifford analysis, Li, McIntosh and Qian [1] introduced a class of holomorphic Fourier multipliers $$H(S^{c}_{\omega ,\pm })$$ on Lipschitz surfaces. In [1], based on the idea of the functional calculus of the Dirac operator, the authors proved the following result: for $$\phi \in K(S_{\omega ,\pm })$$ , there exists a holomorphic function $$b\in H(S^{c}_{\omega ,\pm })$$ such that on the Lipschitz surface, any singular integral operator $$T_{\phi }$$ with the convolution kernel $$\phi $$ corresponds to a Fourier multiplier operator $$M_{b}$$ , where b is the Fourier transform of the kernel $$\phi $$ . In this chapter, we will elaborate on the theory established by the above three authors.

Date: 2019
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DOI: 10.1007/978-981-13-6500-3_5

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