On two subsurfaces of a Riemann surface divided by a p-Weil-Petersson curve, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of. By requiring the coincidence of boundary values on, we establish a correspondence between the harmonic functions in these Banach spaces. We analyze the operator arising from this correspondence via the composition operator acting on the Banach space of p-Besov functions on the unit circle.
Katsuhiko Matsuzaki (Tue,) studied this question.