Bounded Holomorphic Fourier Multipliers on Closed Lipschitz Surfaces

Abstract

On the infinite Lipschitz graph, the theory of singular integrals has been established in [1,2,3,4,5,6]. In [7, 8], the authors discussed the singular integrals and Fourier multipliers for the case of starlike Lipschitz curves on the complex plane. The cases of $n-$tours and their Lipschitz disturbance are studied in [9, 10]. In 1998 and 2001, by a generalization of Fueter’s theorem, T. Qian established the theory of bounded holomorphic Fourier multipliers and the relation with singular integrals on Lipschitz surfaces in the setting of quaternionic space and Clifford algebras with general dimension, respectively. Fueter’s theorem and its generalizations seem to be the unique method to deal with singular integral operator algebras in the sphere contexts. In this chapter, we systematically elucidate the results obtained by Qian [11,12,13]. Denote by $\mathbb {R}^{n}_{1}$ and $\mathbb {R}^{n}$ the linear subspaces of $\mathbb {R}_{(n)}$ spanned by $\{e_{0}, e_{1}, \ldots , e_{n}\}$ and by $\{e_{1},e_{2},\ldots , e_{n}\}$, respectively.