A variant of Hörmanderʼs existence theorem for the Dirac operator in Clifford analysis

In this paper, we give the Hörmanderʼs L 2 theorem for the Dirac operator over an open subset Ω ∈ R n + 1 with Clifford algebra. Some sufficient condition on the existence of the weak solutions for the Dirac operator has been obtained 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 the Dirac operator such that D ¯ u = f with u in the L 2 space as well. The method is based on Hörmanderʼs L 2 existence theorem in complex analysis and the L 2 weighted space is utilized.