Riemann Boundary Value Problem on Quasidisks, Faber Isomorphism and Grunsky Operator

Abstract

Let $\Gamma$ be a bounded Jordan curve with complementary components $\Omega ^{\pm }$. We show that the jump decomposition is an isomorphism if and only if $\Gamma$ is a quasicircle. We also show that the Bergman space of $L^{2}$ harmonic one-forms on $\Omega ^{+}$ is isomorphic to the direct sum of the holomorphic Bergman spaces on $\Omega ^{+}$ and $\Omega ^{-}$ if and only if $\Gamma$ is a quasicircle. This allows us to derive various relations between a reflection of harmonic functions in quasicircles and the jump decomposition on the one hand, and the Grunsky operator, Faber series and kernel functions of Schiffer on the other hand. It also leads to new interpretations of the Grunsky and Schiffer operators. We show throughout that the most general setting for these relations is quasidisks.