Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators.

Let R be a compact Riemann surface and Γ be a Jordan curve separating R into connected components Σ 1 and Σ 2 . We consider Calderón–Zygmund type operators T (Σ 1 , Σ k) taking the space of L 2 anti-holomorphic one-forms on Σ 1 to the space of L 2 holomorphic one-forms on Σ 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 Γ , 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 Σ 1 . We show that the restriction of the Schiffer operator T (Σ 1 , Σ 2) to V is an isomorphism onto the set of exact holomorphic one-forms on Σ 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. [ABSTRACT FROM AUTHOR]