Spherical $$\Pi $$ Π -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.