Dirichlet space of multiply connected domains with Weil-Petersson class boundaries
Abstract
The restricted class of quasicircles sometimes called the "Weil-Petersson-class" has been a subject of interest in the last decade. In this paper we establish a Sokhotski-Plemelj jump formula for WP-class quasicircles, for boundary data in a certain conformally invariant Besov space. We show that this Besov space is precisely the set of traces on the boundary of harmonic functions of finite Dirichlet energy on the WP-class quasidisk. We apply this result to multiply connected domains, Sigma, which are the complement of n+1 WP-class quasidisks. Namely, we give a bounded isomorphism between the Dirichlet space D(Sigma) of Sigma and a direct sum of Dirichlet spaces, D-, of the unit disk. Writing the quasidisks as images of the disk under conformal maps (f_0,...,f_n), we also show that {(h \circ f_0,...,h \circ f_n) : h \in D(Sigma)} is the graph of a certain bounded Grunsky operator on D-.
Keywords
Cite
@article{arxiv.1309.4337,
title = {Dirichlet space of multiply connected domains with Weil-Petersson class boundaries},
author = {David Radnell and Eric Schippers and Wolfgang Staubach},
journal= {arXiv preprint arXiv:1309.4337},
year = {2014}
}
Comments
24 pages. Introductory material revised