Spherical Pi-type operators in Clifford analysis and applications

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.