On the Schwarz reflection principle for monogenic functions

摘要

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_0subset 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.