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On the Schwarz reflection principle for monogenic functions

Authors:	Richard Delanghe Frank Sommen

Institution:	(1) Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium

Abstract:	Let ∑ be either an oriented hyperplane or the unit sphere in ${\mathbb{R}}^{m + 1}(m \geq 1)$ , let $\Omega_0 \subset \sum$ be open and connected and let $\tilde{\Omega}$ be an open and connected domain in ${\mathbb{R}}^{m+1}$ such that $\Omega_0\subset \partial \tilde{\Omega}$ . If in $\tilde{\Omega}, \tilde{F}$ is a null solution of the Dirac operator (also called a monogenic function in $\tilde{\Omega}$ ) which is continuously extendable to $\Omega_0$ , then conditions upon $\tilde{F}\|_{\Omega_0}$ are given enabling the monogenic extension of $\tilde{F}$ across $\Omega_0$ . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin $(\frac{1}{2})$ cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin $(\frac{1}{2})$ case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 30G35
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