Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

Abstract

Let $S_{\text{H}}\left(f\right)$ be the Schwarzian derivative of a univalent harmonic function $f$ in the unit disk $\mathbb{D}$ , compatible with a finitely generated Fuchsian group $G$ of the second kind. We show that if ${\left|S_{\text{H}}\left(f\right)\right|}^2{\left(1-{\left|z\right|}^2\right)}^{3}dxdy$ satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain $\mathcal{F}$ of $G$ , then ${\left|S_{\text{H}}\left(f\right)\right|}^2{\left(1-{\left|z\right|}^2\right)}^{3}dxdy$ is a Carleson measure in $\mathbb{D}$ .