Dirichlet space of multiply connected domains with Weil-Petersson class boundaries

TL;DR

This paper establishes a jump formula for WP-class quasicircles, characterizes boundary traces of harmonic functions with finite Dirichlet energy, and describes the Dirichlet space of multiply connected domains as a graph of a Grunsky operator.

Contribution

It provides a new boundary value formula and a structural isomorphism for Dirichlet spaces on multiply connected WP-class quasidisks, extending classical analysis results.

Findings

Established Sokhotski-Plemelj jump formula for WP-class quasicircles.

Characterized boundary traces of harmonic functions with finite Dirichlet energy.

Described Dirichlet space of multiply connected domains as a graph of a Grunsky operator.

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-.