The Hilbert Transform on a Smooth Closed Hypersurface

Abstract

In this paper a condensed account is given of results connected to the Hilbert transform on the smooth boundary of a bounded domain in Euclidean space and some of its related concepts, such as Hardy spaces and the Cauchy integral, in a Clifford analysis context. Clifford analysis, also known as the theory of monogenic functions, is a multidimensional function theory, which is at the same time a generalization of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. It offers a framework which is particularly suited for the integrated treatment of higher dimensional phenomena, without having to rely on tensorial approaches.