Zeitschrift für Analysis und ihre Anwendungen 24 (2005), No. 1, 167--178
Copyright Heldermann Verlag 2005
Cauchy Transform and Rectifiability in Clifford Analysis
Juan Bory Reyes
Dept. of Mathematics, University of Oriente, Santiago de Cuba 90500, Cuba
jbory@rect.uo.edu.cu
Ricardo Abreu Blaya
Faculty of Mathematics and Informatics, University of Holguín, Holguín 80100, Cuba
rabreu@facinf.uho.edu.cu
[Abstract-pdf]
\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.
MSC: 30E20; 30E25, 30G35, 45B20
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