The Spherical $\pi$-Operator

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 ${\mathrm{\Gamma }}_{\alpha }$ as an $\alpha$- shift of of ${\mathrm{\Gamma }}_{o}mega$, where ${\mathrm{\Gamma }}_{o}mega$ 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 ${\mathrm{\Gamma }}_{a}lpha$.