## Abstract

Consider a multiply-connected domain ∑ in the sphere bounded by n non-intersecting quasicircles. We characterize the Dirichlet space of ∑ 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 ∑ as the graph of the generalized Grunsky operator in direct sums of the space 1/2(1) on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

Original language | English |
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Journal | Communications in Contemporary Mathematics |

Volume | 22 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Dirichlet spaces
- Faber operator
- Faber series
- Grunsky operator
- multiply-connected domains
- quasicircles