Cauchy Transform and Rectifiability in Clifford Analysis

Juan Bory Reyes; Ricardo Abreu Blaya

doi:10.4171/zaa/1235

JournalszaaVol. 24, No. 1pp. 167–178

Cauchy Transform and Rectifiability in Clifford Analysis

Juan Bory Reyes
University of Oriente, Santiago De Cuba, Cuba
Ricardo Abreu Blaya
University of Holguín, Cuba

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Abstract

\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 $\epsilon\to 0$ .

Cite this article

Juan Bory Reyes, Ricardo Abreu Blaya, Cauchy Transform and Rectifiability in Clifford Analysis. Z. Anal. Anwend. 24 (2005), no. 1, pp. 167–178

DOI 10.4171/ZAA/1235