On Carleson Measures of Beltrami Coefficients Being Compatible with Infinitely Generated Fuchsian Groups Related to Denjoy Domian

Let $\mathrm{\Omega }$ be a Carleson-Denjoy domain and $G$ be its covering group. Let $\mu$ be a Beltrami coefficient on the unit disk which is compatible with the group $G$. In this paper we show that if $\frac{\mathrm{\mid }\mu {\mathrm{\mid }}^2}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}dxdy$ satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain of $G$, then $\frac{\mathrm{\mid }\mu {\mathrm{\mid }}^2}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}dxdy$ is a Carleson measure on the unit disk. We also show that the above property does not hold for Denjoy domain.