Dirichlet spaces of domains bounded by quasicircles

Abstract

Consider a multiply-connected domain Σ in the sphere bounded by n nonintersecting quasicircles. We characterize the Dirichlet space of Σ 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 Σ as the graph of the generalized Grunsky operator in direct sums of the space H 1/2 (S 1 ) on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

A Model of the Teichmüller space of genus-zero bordered surfaces by period maps

“…These results involve a "reflection" of harmonic Dirichlet functions in quasidisks, obtained by extending to the boundary of the quasidisks, and then extending them to the complementary quasidisk. One may summarize the situation as follows: in the present paper, the use of one-forms creates a clearer geometric picture, whereas in the paper [18], the use of functions created a more clear analytic picture.…”

“…We will not be directly working with Dirichlet spaces in this paper. They will be used only to apply results of the authors [18] for Dirichlet spaces to Bergman spaces, through the use of the isometry (1.4). These results involve a "reflection" of harmonic Dirichlet functions in quasidisks, obtained by extending to the boundary of the quasidisks, and then extending them to the complementary quasidisk.…”

“…Proof. By a result of [18], the operator norm of Gr is strictly bounded by one. The claim thus follows from the fact that d :…”

“…Our construction uses a generalized Grunsky operator, which was shown by the authors to be bounded by one [18], and thus lies in a kind of Siegel disk. We show that this mapping is holomorphic.…”

Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

“…This paper is part of an ongoing investigation of Faber, Grunsky, and Schiffer operators on Riemann surfaces split in pieces by quasicircles. This has applications to Teichmüller theory [11], two-dimensional conformal field theory [13], and approximation theory [12] (as in the present paper).…”