A Calculus Scheme for Clifford Distributions

Abstract

The aim of the paper is to construct the fundamental solution of an arbitrary complex power of the Dirac operator, these powers being defined as convolution operators with a kernel expressed in terms of specific distributions in Euclidean space. The desired fundamental solution is found, at least formally, in terms of the same families of distributions as those arising in the kernel of the corresponding operator. Clearly, in order to prove these results in a rigorous way, we first have to investigate the definition and properties of both the convolution and the product of arbitrary elements of the families of distributions under consideration, leading to a very attractive pattern of mutual relations between them.