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