2025

Koskela, Pekka; Rajala, Tapio; Zhang, Yi Ru-Ya

We characterize bounded simply-connected planar W1,p-extension domains for 1<p><2 as those bounded domains Ω⊂R2 for which any two points z1,z2∈R2∖Ω can be connected with a curve γ⊂R2∖Ω satisfying ∫γdist(z,∂Ω)1−pdz≲|z1−z2|2−p. Combined with known results, we obtain the following duality result: a Jordan domain Ω⊂R2 is a W1,p-extension domain, 1</p><p><∞, if and only if the complementary domain R2∖Ω¯ is a W1,p/(p−1)-extension domain.</p>