Planar W^{1, 1}-extension domains

2025/12/09 by Pekka Koskela, Koskela, Pekka, Tapio Rajala +3
Mathematics · #Holomorphic and Operator Theory #Rings, Modules, and Algebras #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2512.09167

Abstract

We show that a bounded planar simply connected domain Ω is a W^{1, 1}-extension domain if and only if for every pair x,y of points in Ω^c there exists a curve γ⊂ Ω^c connecting x and y with ∫_γ\frac1χ_{\mathbb R²∖ ∂Ω}(z) ds(z) ≤ C|x-y|. Consequently, a planar Jordan domain Ω is a W^{1, 1}-extension domain if and only if it is a BV-extension domain, and if and only if its complementary domain Ω is a W^{1, ∞}-extension domain.

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