# Dirichlet space of domains bounded by quasicircles

## TL;DR

This paper characterizes the Dirichlet space of multiply-connected domains bounded by quasicircles using generalized Faber and Grunsky operators, linking boundary value spaces and Fourier decompositions.

## Contribution

It introduces a novel isomorphic representation of the Dirichlet space via a generalized Faber operator and provides a second characterization through a Grunsky operator, extending classical theories.

## Key findings

- Dirichlet space characterized as an isomorphic image of disk Dirichlet spaces
- Construction of a generalized Faber operator using jump formulas
- Representation of Dirichlet space as the graph of a Grunsky operator

## 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.