Holomorphic Fourier Multipliers on Infinite Lipschitz Surfaces

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.