Dirichlet space of domains bounded by quasicircles

Abstract

Consider a multiply-connected domain $\Sigma$ in the sphere bounded by $n$ non-intersecting quasicircles. We characterize the Dirichlet space of $\Sigma$ 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 $\Sigma$ as the graph of the generalized Grunsky operator in direct sums of the space ${\mathcal{H}}^{1/2}({\mathbb{S}}^1)$ on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

References in corpus (1)

• The realization problem for Jørgensen numbers