A variant of Hörmander's $L^2$ existence theorem for Dirac operator in Clifford analysis

Abstract

In this paper, we give the Hörmander's $L^2$ theorem for Dirac operator over an open subset $Ω\in\R^{n+1}$ with Clifford algebra. Some sufficient condition on the existence of the weak solutions for Dirac operator has been found in the sense of Clifford analysis. In particular, if $Ω$ is bounded, then we prove that for any $f$ in $L^2$ space with value in Clifford algebra, there exists a weak solution of Dirac operator such that $$\bar{D}u=f$$ with $u$ in the $L^2$ space as well. The method is based on Hörmander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted space is utilised.