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

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.