Plemelj\u2013Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

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 Calderón–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 for k=1,2, which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Gamma , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L^2 anti-holomorphic one-forms on R with respect to the inner product on Sigma _1. We show that the restriction of the Schiffer operator T(Sigma _1,Sigma _2) to V is an isomorphism onto the set of exact holomorphic one-forms on Sigma _2. Using the relation between this 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.