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

We show that a bounded planar simply connected domain $\mathrm{\Omega }$ is a $W^{1,1}$-extension domain if and only if for every pair $x,y$ of points in ${\mathrm{\Omega }}^c$ there exists a curve $\gamma \subset {\mathrm{\Omega }}^c$ connecting $x$ and $y$ with $${\int }_{\gamma }\frac{1}{{\chi }_{{\mathbb{R}}^2\setminus \mathrm{\partial }\mathrm{\Omega }}(z)}ds(z)\le C\mathrm{\mid }x-y\mathrm{\mid }\mathrm{.}$$ Consequently, a planar Jordan domain $\mathrm{\Omega }$ 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 $\tilde{\mathrm{\Omega }}$ is a $W^{1,\mathrm{\infty }}$-extension domain.