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

Abstract

Let $\mu$ be a Beltrami coefficient on the unit disk, which is compatible with a finitely generated Fuchsian group $G$ of the second kind. 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.