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

Abstract

Let SHf be the Schwarzian derivative of a univalent harmonic function f in the unit disk D, compatible with a finitely generated Fuchsian group G of the second kind. We show that if SHf21−z23dxdy satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G, then SHf21−z23dxdy is a Carleson measure in D.