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Complex Analysis and Operator Theory

, Volume 12, Issue 2, pp 325–354 | Cite as

Riemann Boundary Value Problem on Quasidisks, Faber Isomorphism and Grunsky Operator

  • Eric SchippersEmail author
  • Wolfgang Staubach
Article
  • 141 Downloads

Abstract

Let \(\Gamma \) be a bounded Jordan curve with complementary components \(\Omega ^{\pm }\). We show that the jump decomposition is an isomorphism if and only if \(\Gamma \) is a quasicircle. We also show that the Bergman space of \(L^{2}\) harmonic one-forms on \(\Omega ^{+}\) is isomorphic to the direct sum of the holomorphic Bergman spaces on \(\Omega ^{+}\) and \(\Omega ^{-}\) if and only if \(\Gamma \) is a quasicircle. This allows us to derive various relations between a reflection of harmonic functions in quasicircles and the jump decomposition on the one hand, and the Grunsky operator, Faber series and kernel functions of Schiffer on the other hand. It also leads to new interpretations of the Grunsky and Schiffer operators. We show throughout that the most general setting for these relations is quasidisks.

Keywords

Dirichlet space Bergman space Quasicircles Quasiconformal extension Jump decomposition Faber polynomials Grunsky operator Schiffer operator 

Mathematics Subject Classification

Primary 30F15 31C05 31C25 35Q15 Secondary 30C55 30C62 

Notes

Acknowledgments

Funding was provided by Wenner-Gren Foundation (http://dx.doi.org/10.13039/100001388 and NSERC).

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ManitobaWinnipegCanada
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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