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In this paper, we give the H\"ormander's $L^2$ theorem for Dirac operatorover an open subset $\Omega\in\R^{n+1}$ with Clifford algebra. Some sufficientcondition on the existence of the weak solutions for Dirac operator has beenfound 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.

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