Weil-Petersson curves and Dirichlet finite harmonic functions on Riemann surfaces

On two subsurfaces of a Riemann surface divided by a p-Weil-Petersson curve γ, we consider the spaces of harmonic functions whose p-Dirichlet integrals are finite in the complementary domains of γ. By requiring the coincidence of boundary values on γ, 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.