Dirichlet space of domains bounded by quasicircles

Consider a multiply-connected domain Σ in the sphere bounded by n non-intersecting 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.