Regular functions of several quaternionic variables and the Cauchy-Fueter complex

Abstract

We employ a classical idea of Ehrenpreis, together with a new algebraic result, to give a new proof that regular functions of several quaternionic variables cannot have compact singularities. As a byproduct we characterize those inhomogeneous Cauchy-Fueter systems which admit solutions on convex sets. Our method readily extends to the case of monogenic functions on Clifford Algebras. We finally study a free resolution of the Cauchy-Fueter complex of differential operators and we obtain some new duality theorems which hint at a hyperfunction theory of several quaternionic variables.