A variant of H\"ormander's $L^2$ theorem for Dirac operator in Clifford analysis

TL;DR

This paper extends H"ormander's $L^2$ theorem to the Dirac operator within Clifford analysis, establishing conditions for the existence of weak solutions in bounded domains using weighted $L^2$ spaces.

Contribution

It introduces a new $L^2$ existence theorem for the Dirac operator in Clifford analysis, adapting H"ormander's method to this setting.

Findings

Existence of weak solutions for Dirac operator in bounded domains.

Sufficient conditions for solutions in weighted $L^2$ spaces.

Application of complex analysis techniques to Clifford analysis.

Abstract

In this paper, we give the H\"ormander's $L^2$ theorem for Dirac operator over an open subset $\Omega\in\R^{n+1}$ 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 $\Omega$ 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 H\"ormander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted space is utilised.