The π-operator on some conformally flat manifolds and hyperbolic half space

The pi-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 pi-operator on a general Clifford-Hilbert space. This pi-operator is also an L-2 isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the L-2 space of a measure space. Then, we show that this technique can be applied to construct the classical pi-operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by U/Gamma, where U is a simply connected subdomain of either Double-struck capital Rn+1 or n-1, and Gamma is a Kleinian group acting discontinuously on U. The pi-operators on those manifolds also preserve the isometry property in certain L-2 spaces, and their L-p norms are bounded by the L-p norms of the pi-operators on Double-struck capital Rn+1 or n-1, depending on where U lies. The applications of the pi-operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the pi-operator theory in the upper-half space with the hyperbolic metric.