Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions defined on an open set of the Euclidean space and with values in the Clifford Algebra . Slice monogenic functions belong to the kernel of the global differential operator with non constant coefficients given by Since the operator is not elliptic and there is a degeneracy in , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation and some of its variations. In particular, we focus our attention on the solutions of the differential equation where is the Euler operator, from which we deduce properties of the solutions of the equation GF (x(0),(x) under bar) = G(x(0),(x) under bar)