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arXiv · 1705.01279

Dirichlet space of domains bounded by quasicircles

David Radnell·Eric Schippers·Wolfgang Staubach
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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.

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David Radnell, Eric Schippers, Wolfgang Staubach. 2017-05-03. Dirichlet space of domains bounded by quasicircles. https://doi.org/10.1142/s0219199719500226

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