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