Dirichlet spaces of domains bounded by quasicircles

David Radnell*, Eric Schippers, Wolfgang Staubach

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

3 Citations (Scopus)

Abstract

Consider a multiply-connected domain ∑ in the sphere bounded by n non-intersecting quasicircles. We characterize the Dirichlet space of ∑ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary values. Thereafter, we define a Grunsky operator on direct sums of Dirichlet spaces of the disk, and give a second characterization of the Dirichlet space of ∑ as the graph of the generalized Grunsky operator in direct sums of the space 1/2(1) on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

Original languageEnglish
JournalCommunications in Contemporary Mathematics
DOIs
Publication statusE-pub ahead of print - 1 Jan 2019
MoE publication typeA1 Journal article-refereed

Keywords

  • Dirichlet spaces
  • Faber operator
  • Faber series
  • Grunsky operator
  • multiply-connected domains
  • quasicircles

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