The Fractional Fourier Multipliers on Lipschitz Curves and Surfaces

The main contents of this chapter are based on some new developments on the holomorphic Fourier multipliers which are obtained by the two authors in recent years, see the author’s paper joint with Leong [1] and the joint work [2]. In the above chapters, we state the convolution singular integral operators and the related bounded holomorphic Fourier multipliers on the finite and infinite Lipschitz curves and surfaces. Let \(S^{c}_{\mu ,\pm }\) and \(S^{c}_{\mu }\) be the regions defined in Sect. 1.1. The multiplier b belongs to the class \(H^{\infty }(S^{c}_{\mu ,\pm })\) defined as $$H^{\infty }(S^{c}_{\mu })=\Big \{b:\ S^{c}_{\mu }\rightarrow \mathbb {C}:\ b_{\pm }=b\chi _{\{z\in \mathbb {C}:\ \pm \text {Re}z>0\}} \in H^{\infty }(S^{c}_{\mu ,\pm })\Big \},$$ where \(H^{\infty }(S^{c}_{\mu ,\pm })\) is defined as the set of all holomorphic function b satisfying \(|b(z)|\leqslant C_{\nu }\) in any \(S^{c}_{\nu ,\pm },\ 0<\nu <\mu \).