Generalized {\Pi}-operator in the theory of slice monogenic functions and applications

TL;DR

This paper introduces a generalized {\Pi}-operator within slice monogenic functions, explores its properties, and applies it to solve a slice Beltrami equation, extending complex analysis concepts to higher dimensions.

Contribution

It defines a new generalized {\Pi}-operator in slice monogenic functions and investigates its properties and applications, including solving a higher-dimensional Beltrami equation.

Findings

The generalized {\Pi}-operator has specific mapping properties.

Inverse and adjoint operators of the generalized {\Pi}-operator are established.

Norm estimates of the operator relate to the existence of solutions.

Abstract

The {\Pi}-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 {\Pi}-operator in the theory of slice monogenic functions, and some mapping properties of the generalized {\Pi}-operator are also introduced. Further, a left and right inverse and the adjoint operator of the generalized {\Pi}-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 {\Pi}-operator can determine the existence of solutions of the slice Beltrami equation.