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 onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. 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 \(\Sigma \).

我们考虑任意属的紧凑Riemann曲面R，它具有有限数量的不重叠的拟圆，它将R分成两个子集：连通的Riemann曲面\（\ Sigma \）和并集\（\ mathcal {O} \ ）的简单连接区域的有限集合。我们证明，将\（\ mathcal {O} \）上反全纯一形的Bergman空间映射到\（\ Sigma \）上全纯形的Bergman空间的Schiffer积分算子是同构到精确的一形式，当限制为所有R上形式集的正交互补时。然后，我们将其用于证明Plemelj–Sokhotski同构的版本以及针对这种配置的跳跃分解。最后，通过R中包含\（\ Sigma \）的固定区域上的Bergman空间和Dirichlet空间的元素，得出\（\ Sigma \）上一式的Bergman空间和全纯函数Dirichlet空间的一些近似定理。