TRANSMISSION OF HARMONIC FUNCTIONSTHROUGH QUASICIRCLES ON COMPACT RIEMANN SURFACES

摘要

Let R be a compact surface and let be a Jordan curve which separates R into two connected components 1 and 2. A harmonic function h1 on 1 of bounded Dirichlet norm has boundary values H in a certain conformally invariant non-tangential sense on . We show that if is a quasicircle, then there is a unique harmonic function h2 of bounded Dirichlet norm on 2 whose boundary values agree with those of h1. Furthermore, the resulting map from the Dirichlet space of 1 into 2 is bounded with respect to the Dirichlet semi-norm.