# On Carleson measures induced by Beltrami coefficients being compatible with Fuchsian groups

## TL;DR

This paper investigates conditions under which Beltrami coefficients compatible with convex co-compact Fuchsian groups induce Carleson measures, establishing a link between boundary conditions and measure properties in the unit disk.

## Contribution

It proves that certain boundary Carleson conditions on Beltrami coefficients imply the measure is Carleson in the disk, extending understanding of their geometric and analytic properties.

## Key findings

- Boundary Carleson condition implies measure is Carleson in the disk
- Compatibility with Fuchsian groups influences measure properties
- Results connect boundary behavior with interior measure conditions

## Abstract

Suppose $\mu$ be a Beltrami coefficient on the unit disk, which is compatible with a convex co-compact Fuchsian group $G$ of the second kind. In this paper we show that if $\displaystyle\frac{|\mu|^{2}}{1-|z|^{2}}dxdy $ satisfies the Carleson condition on the infinite boundary boundary of the Dirichlet domain of $G$, then $\displaystyle\frac{|\mu|^{2}}{1-|z|^{2}}dxdy$ is a Carleson measure on the unit disk.