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 Δ _s g = f.