Holomorphic Fourier Multipliers on Infinite Lipschitz Surfaces

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.