A variant of Hörmanderʼs [formula omitted] existence theorem for the Dirac operator in Clifford analysis

Abstract

In this paper, we give the Hörmanderʼs L2 theorem for the Dirac operator over an open subset Ω∈Rn+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 L2 space with value in Clifford algebra, there exists a weak solution of the Dirac operator such thatD¯u=f with u in the L2 space as well. The method is based on Hörmanderʼs L2 existence theorem in complex analysis and the L2 weighted space is utilized.