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

TL;DR

This paper extends Hörmander's $L^2$ existence theorem to the Dirac operator within Clifford analysis, providing conditions for 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 Hörmander's method to this setting.

Findings

Existence of weak solutions for Dirac operator in bounded domains.

Conditions for solutions in $L^2$ spaces with Clifford algebra values.

Application of weighted $L^2$ spaces in 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.