Dirichlet spaces of domains bounded by quasicircles

Consider a multiply-connected domain [Formula: see text] in the sphere bounded by [Formula: see text] non-intersecting quasicircles. We characterize the Dirichlet space of [Formula: see text] 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 [Formula: see text] as the graph of the generalized Grunsky operator in direct sums of the space [Formula: see text] on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.