The Spherical $\pi$-Operator

TL;DR

This paper introduces the spherical π-operator on domains of the unit sphere in R^n, explores its properties, and develops related operators like the spherical Dirac operator using Gegenbauer polynomials.

Contribution

It defines the spherical π-operator and the spherical Dirac operator, providing new results and mapping properties in the context of spherical analysis.

Findings

Defined the spherical π-operator on the unit sphere.

Developed properties and mapping results for the operator.

Introduced the spherical Dirac operator using Gegenbauer polynomials.

Abstract

In this article, we define the spherical $\pi$-operator over domains in the $(n-1)$-D unit sphere $S^n$ of $R^n$ and develop new and analogous results on the operator it self and its mapping properties. We introduce the spherical Dirac operator $\Gamma_\alpha$ as an $\alpha$- shift of of $\Gamma_omega$, where $\Gamma_omega$ is the negative of the wedge (or Grassmann) product of $\omega$ with that of the Dirac operator $D_\omega$. A gegenbauer polynomial is used as a Cauchy kernel for the spherical Dirac operator $\Gamma_alpha$.