Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Let $R$ be a compact surface and let $\Gamma$ be a Jordan curve which separates $R$ into two connected components $\Sigma_1$ and $\Sigma_2$. A harmonic function $h_1$ on $\Sigma_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $\Gamma$. We show that if $\Gamma$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $\Sigma_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $\Sigma_1$ into $\Sigma_2$ is bounded with respect to the Dirichlet semi-norm.