2013/09/17 by David Radnell, Eric Schippers, Radnell, David +3
Mathematics · Physics and Astronomy · #30C55 #30C62 (primary) #30F60 #31C25 #32G15 #81T40 (secondary) #Analytic and geometric function theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Nonlinear Partial Differential Equations #math-ph #math.CV #math.DG #math.MP #msc:30C55 #msc:30C62 #msc:30F60 #msc:31C25 #msc:32G15 #msc:81T40
paper · pdf · doi:10.48550/arxiv.1309.4337
24 pages. Introductory material revised
openalex publication_date 2013/09/17 · arxiv created 2014/06/16 · arxiv updated 2014/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (f0,...,fn), we also show that (h ∘ f0,...,h ∘ fn) : h ∈ D(Sigma) is the graph of a certain bounded Grunsky operator on D-.