The Beurling operator for the hyperbolic plane

TL;DR

This paper introduces a Beurling operator tailored for the hyperbolic plane, establishing its $L^2$ norm identity and deriving $L^p$ estimates, advancing the understanding of harmonic analysis in hyperbolic geometry.

Contribution

It constructs a new Beurling operator for the hyperbolic plane and provides fundamental $L^p$ bounds and identities for it.

Findings

Established an $L^2$ norm identity for the hyperbolic Beurling operator

Derived $L^p$ estimates for the operator

Extended harmonic analysis tools to hyperbolic geometry

Abstract

We find a Beurling operator for the hyperbolic plane, and obtain an $L^2$ norm identity for it, as well as $L^p$ estimates.