On Carleson Measures of Beltrami Coefficients Being Compatible with Infinitely Generated Fuchsian Groups Related to Denjoy Domian

TL;DR

This paper investigates conditions under which Beltrami coefficients compatible with infinitely generated Fuchsian groups induce Carleson measures, establishing a link between boundary behavior and measure properties in complex analysis.

Contribution

It demonstrates that certain Carleson measure conditions on the boundary imply Carleson measure properties on the disk for Beltrami coefficients compatible with infinitely generated Fuchsian groups, with a contrast for Denjoy domains.

Findings

Carleson measure condition on the boundary implies Carleson measure on the disk for compatible Beltrami coefficients.

The property holds for infinitely generated Fuchsian groups but not for Denjoy domains.

Provides a criterion linking boundary measure conditions to interior measure properties in complex analysis.

Abstract

Let $\Omega$ be a Carleson-Denjoy domain and $G$ be its covering group. Let $\mu$ be a Beltrami coefficient on the unit disk which is compatible with the group $G$. In this paper we show that if $\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain of $G$, then $\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ is a Carleson measure on the unit disk. We also show that the above property does not hold for Denjoy domain.