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

$$
({\cal C}_{\Gamma}u)(x):= \int_{\Gamma}\ \frac{\overline{y-x}}{A_{n+1}|y-x|^{n+1}}n(y)u(y) \,d{\cal H}^{n}(y),\quad x\notin\Gamma,
$$

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

$$
{\cal S}_{\Gamma,\,\epsilon}u(z):=\int_{\Gamma\setminus\{|y-z|\le\epsilon\}} \ \frac{\overline{y-z}}{A_{n+1}|y-z|^{n+1}}n(y)(u(y)-u(z))\,d{\cal H}^{n}(y)
$$

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

Cauchy Transform and Rectifiability in Clifford Analysis

Juan Bory Reyes

Ricardo Abreu Blaya

Abstract

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