Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

TL;DR

This paper studies Schiffer operators on Riemann surfaces with quasicircle boundaries, proving isomorphisms and approximation results for Bergman and Dirichlet spaces, extending classical complex analysis tools to more general surfaces.

Contribution

It establishes the isomorphism property of Schiffer integral operators on Riemann surfaces with quasicircle boundaries and derives approximation theorems for function spaces.

Findings

Schiffer integral operator is an isomorphism between specific Bergman spaces.

Versions of Plemelj-Sokhotski isomorphism and jump decomposition are proved.

Approximation theorems for Bergman and Dirichlet spaces on Riemann surfaces are established.

Abstract

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $\Sigma$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $\Sigma$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $\Sigma$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $\Sigma$.