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
Cauchy Transform and Rectifiability in Clifford Analysis cover
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Abstract

\def\R{\mathbb R} Let Γ\Gamma be an nn-dimensional rectifiable Ahlfors-David regular surface in Rn+1\R^{n+1}. Let uu be a continuous R0,n\R_{0,n}-valued function on Γ\Gamma, where R0,n\R_{0,n} is the Clifford algebra associated with Rn\R^n. Then we prove that the Cliffordian Cauchy transform

(CΓu)(x):=Γ yxAn+1yxn+1n(y)u(y)dHn(y),xΓ,({\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 Γ\Gamma if and only if the truncated integrals

SΓ,ϵu(z):=Γ{yzϵ} yzAn+1yzn+1n(y)(u(y)u(z))dHn(y){\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 Γ\Gamma as ϵ0\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