Spherical $\Pi$-type Operators in Clifford Analysis and Applications

TL;DR

This paper introduces two spherical $ ext{Pi}$-operators in Clifford analysis, demonstrating their isometric properties and applying them to solve spherical Beltrami equations, advancing the mathematical tools for analysis on spheres.

Contribution

The paper defines two new spherical $ ext{Pi}$-operators, proves their isometric properties, and applies them to solve spherical Beltrami equations, extending Clifford analysis techniques.

Findings

First $ ext{Pi}$-operator is an $L^2$ isometry up to isomorphism.

Second $ ext{Pi}$-operator is constructed as an isometric $L^2$ operator.

Both operators are applied to solve spherical Beltrami equations.

Abstract

The $\Pi$-operator (Ahlfors-Beurling transform) plays an important role in solving the Beltrami equation. In this paper we define two $\Pi$-operators on the n-sphere. The first spherical $\Pi$-operator is shown to be an $L^2$ isometry up to isomorphism. To improve this, with the help of the spectrum of the spherical Dirac operator, the second spherical $\Pi$ operator is constructed as an isometric $L^2$ operator over the sphere. Some analogous properties for both $\Pi$-operators are also developed. We also study the applications of both spherical $\Pi$-operators to the solution of the spherical Beltrami equations.