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 $\frac{|\mu {|}^2}{1-|z{|}^2}dxdy$ satisfies the Carleson condition on the infinite boundary boundary of the Dirichlet domain of $G$, then $\frac{|\mu {|}^2}{1-|z{|}^2}dxdy$ is a Carleson measure on the unit disk.

10 pages