Cauchy Transform and Rectifiability in Clifford Analysis

Abstract

\def\R{\mathbb R} Let $\Gamma$ be an $n$-dimensional rectifiable Ahlfors-David regular surface in ${\mathbb{R}}^{n+1}$. Let $u$ be a continuous ${\mathbb{R}}_{0,n}$-valued function on $\Gamma$, where ${\mathbb{R}}_{0,n}$ is the Clifford algebra associated with ${\mathbb{R}}^n$. Then we prove that the Cliffordian Cauchy transform

has continuous limit values on $\Gamma$ if and only if the truncated integrals

converge uniformly on $\Gamma$ as $ϵ\to 0$.