JournalszaaVol. 24 , No. 1DOI 10.4171/zaa/1235

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

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.