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.

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