Generalized Π-operator in the theory of slice monogenic functions and applications

Abstract

The Π-operator plays an important role in complex analysis, especially in the theory of generalized analytic functions in the sense of Vekua. In this paper, we introduce a generalized Π-operator in the theory of slice monogenic functions, and some mapping properties of the generalized Π-operator are also introduced. Further, a left and right inverse and the adjoint operator of the generalized Π-operator are given. As an application, we introduce a slice Beltrami equation, which reduces to the classical complex Beltrami equation when the dimension is 2. We show details that the norm estimate of the generalized Π-operator can determine the existence of solutions of the slice Beltrami equation.