Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

Abstract

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 $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ 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 $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ 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 $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

A scattering theory of harmonic one-forms on Riemann surfaces

“…The fact that Θ is an isomorphism was obtained by M. Shirazi [62,60]. We have that the restriction Υ = Υ| ((R 1 A(R)) ⊥ ⊕{0}) takes the form…”

“…This is closely related to the fact that functions can be approximated in the Dirichlet semi-norm by Faber series precisely for domains bounded by quasicircles; see [59] for an overview. The authors showed in [57] that, for a compact Riemann surface divided in two by a quasicircle, T 1,2 is an isomorphism on the orthogonal complement of anti-holomorphic one-forms on R. This was further generalized by M. Shirazi to the case of many curves where all but one of the components is simply connected in [62], [52]. The boundedness of overfare plays a central role in the formulation and proof of this fact.…”

“…The boundedness of overfare plays a central role in the formulation and proof of this fact. This extension of the isomorphism theorem was used by the authors and Shirazi to show that one-forms on a domain in a Riemann surface bounded by quasicircles can be approximated in L 2 on a larger domain [52]. Approximability theorems for general k-differentials with respect to the conformally invariant L 2 norm and less regular boundaries were obtained by N. Askaripour and T. Barron [4,5] using very different methods.…”

Faber series for $L^2$ holomorphic one-forms on Riemann surfaces with boundary

“…Here we are concerned with the development of explicit Faber-type series with a fixed basis. Earlier density results were obtained by the authors with W. Staubach, for nested Riemann surfaces with boundary [14]. It was shown that L 2 holomorphic one-forms on the inner surface can be approximated in L 2 by holomorphic one-forms on the outer surface (and similarly for Dirichlet spaces of functions).…”

“…We call them Faber-Tietz forms, although these are not precisely the same as those defined by Tietz. We use isomorphism theorems for Schiffer and Faber operators of M. Shirazi [20] (see also E. Schippers, M. Shirazi, and W. Staubach [14]) and E. Schippers and W. Staubach [17], generalizing that of Napalkov and Yulmukhametov, in order to produce the series representations and convergence.…”