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

The $\mathrm{\Pi }$-operator (Ahlfors-Beurling transform) plays an important role in solving the Beltrami equation. In this paper we define two $\mathrm{\Pi }$-operators on the n-sphere. The first spherical $\mathrm{\Pi }$-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 $\mathrm{\Pi }$ operator is constructed as an isometric $L^2$ operator over the sphere. Some analogous properties for both $\mathrm{\Pi }$-operators are also developed. We also study the applications of both spherical $\mathrm{\Pi }$-operators to the solution of the spherical Beltrami equations.