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Zeitschrift für Analysis und ihre Anwendungen


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Volume 24, Issue 1, 2005, pp. 167–178
DOI: 10.4171/ZAA/1235

Published online: 2005-03-31

Cauchy Transform and Rectifiability in Clifford Analysis

Juan Bory Reyes[1] and Ricardo Abreu Blaya[2]

(1) University of Oriente, Santiago De Cuba, Cuba
(2) University of Holguín, Cuba

\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 $\epsilon\to 0$.

Keywords: Clifford analysis, Cauchy transform, rectifiability

Bory Reyes Juan, Abreu Blaya Ricardo: Cauchy Transform and Rectifiability in Clifford Analysis. Z. Anal. Anwend. 24 (2005), 167-178. doi: 10.4171/ZAA/1235