On a Spatial Generalization of the Complex $\Pi$-Operator

Abstract

The $\Pi$-operator plays a mayor role in complex analysis, especially in the theory of generalized analytic functions in the sense of Vekua. The present paper deals with a hyper-complex generalization of the complex $\Pi$-operator which turns out to have most of the useful properties of its complex origin such as mapping properties and invertibility. At the end an application of the generalized $\Pi$-operator to the solution of a hypercomplex Beltrami equation will be studied.