Plemelj-Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operator

Abstract

Let $R$ be a compact Riemann surface and $\Gamma$ be a Jordan curve separating $R$ into connected components $\Sigma_1$ and $\Sigma_2$. We consider Calder\'on-Zygmund type operators $T(\Sigma_1,\Sigma_k)$ taking the space of $L^2$ anti-holomorphic one-forms on $\Sigma_1$ to the space of $L^2$ holomorphic one-forms on $\Sigma_k$, which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves $\Gamma$, to general quasicircles in a characterizing manner, and prove new identities for adjoints of the Schiffer operators. Furthermore, we show that if $V$ is the space of anti-holomorphic one-forms orthogonal to $L^2$ forms on $R$ with respect to the inner product on $\Sigma_1$, then the Schiffer operator $T(\Sigma_1,\Sigma_2)$ is an isomorphism onto the set of exact one-forms on $\Sigma_2$. Using the relation between the Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.