The Π-operator on Some Conformally Flat Manifolds and the Upper Half Space

2020/06/28 by Ryan, Wanqing Cheng John
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2006.15676

Abstract

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 L² isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the L² 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 U/Γ, where U is a simply connected subdomain of either ℝⁿ or \mathbbSⁿ, and Γ is a Kleinian group acting discontinuously on U. The Π-operators on those manifolds also preserve the isometry property in certain L² spaces, and their L^p norms are bounded by the L^p norms of the Π-operators on ℝⁿ or \mathbbSⁿ, depending on where U 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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