Planar $W^{1,1}$-extension domains

Abstract

We show that a bounded planar simply connected domain $\Omega$ is a $W^{1,1}$-extension domain if and only if for every pair $x,y$ of points in ${\Omega }^c$ there exists a curve $\gamma \subset {\Omega }^c$ connecting $x$ and $y$ with

Consequently, a planar Jordan domain $\Omega$ 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 $\tilde{\Omega }$ is a $W^{1,\infty }$-extension domain.