Transmission of harmonic functions through quasicircles on compact riemann surfaces

Abstract

Let R be a compact surface and let Γ be a Jordan curve which separates R into two connected components σ1 and σ2. A harmonic function h1 on σ1 of bounded Dirichlet norm has boundary values H in a certain conformally invariant non-tangential sense on Γ. We show that if Γ is a quasicircle, then there is a unique harmonic function h2 of bounded Dirichlet norm on σ2 whose boundary values agree with those of h1. Furthermore, the resulting map from the Dirichlet space of σ1 into σ2 is bounded with respect to the Dirichlet semi-norm.