Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles
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Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles Eric Schippers1 · Mohammad Shirazi1 · Wolfgang Staubach2 Received: 15 March 2020 / Accepted: 2 September 2020 © The Author(s) 2020
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 , and the union 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 O to the Bergman space of holomorphic forms on 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 by elements of Bergman space and Dirichlet space on fixed regions in R containing . Keywords Approximation · Riemann surfaces · Quasicircles · Schiffer operators Mathematics Subject Classification Primary: 58C99 · 58A10 · 30F10 · 30F15 · 30F30 · 32W05 · Secondary: 58J40
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Wolfgang Staubach [email protected] Eric Schippers [email protected] Mohammad Shirazi [email protected]
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Machray Hall, Department of Mathematics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
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Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden
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E. Schippers et al.
1 Introduction and Statement of the Main Theorems 1.1 Introduction In earlier publications [18–20], two of the authors developed a theory of the transmission of harmonic Dirichlet-bounded functions across curves, and of the related Plemelj–Sokhotski jump formula on such curves. The curves were assumed to be quasicircles, which are not rectifiable in general. This theory involves certain singular integral operators due to M. Schiffer, which are also intimately related to approximations of holomorphic functions and forms, through the Faber and Grunsky operators and their generalizations. A number of results strongly indicate that quasicircles are the natural curves for this circle of ideas. In this paper, we apply these techniques to derive approximation theorems for nested multiply connected domains in Riemann surfaces of arbitrary genus. We show that, given a compact Riemann surface R with nested regions ⊆ ⊂ R obtained by removing disks from R, under certain general conditions, the Dirichlet spaces of functions and Bergman spaces of one-forms on are dense in the Dirichlet spaces and Bergman spaces of , respectively. To do so, we must first extend two of our results from the case of one curve separating the surface to many curves bounding conformal disks. Namely, we must extend the Schiffer isomorphism to this case, as well as the Plemelj–Sokhotski jump isomorphism and decomposition. These results are perhaps interesting o
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