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

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 $γ\subset Ω^c$ connecting $x$ and $y$ with $$ \int_γ\frac{1}{χ_{\mathbb R^2\setminus \partialΩ}(z)}\,ds(z) \le C|x-y|.$$ 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 $\tilde Ω$ is a $W^{1,\,\infty}$-extension domain.