On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

Abstract

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator \({\mathcal{D}}^{(k)}_{s}\) on locally defined polynomial forms on the unit sphere of \({\mathbb{R}}^{n}\); functions valued in the universal Clifford algebra \({\mathbb{C}}(V_{n,n})\), here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations \({\mathcal{D}}^{(k)}_{s} g = f\). We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δsg = f.