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

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 $$\int_\gamma \frac{1}{\chi_{\mathbb R^2\setminus \partial\Omega}(z)}\,ds(z) \le C|x-y|.$$ 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.