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

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.