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

The Π-operator, also known as Ahlfors-Beurling transform, plays an important role in solving the existence of locally quasiconformal solutions of Beltrami equations.In this paper, we first construct the Π-operator on a general Clifford-Hilbert module. This Π-operator is also an 2 isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the 2 space of a measure space. Then, we show that this technique can be applied to construct the classical Π-operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by ∕Γ, where is a simply connected subdomain of either ℝ +1 or , and Γ is a Kleinian group acting discontinuously on . The Π-operators on those manifolds also preserve the isometry property in certain 2 spaces, and their norms are bounded by the norms of the Π-operators on ℝ +1 or , depending on where lies. The applications of the Π-operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the Π-operator theory in the upper-half space with the hyperbolic metric.

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