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

Abstract

We show that a bounded planar simply connected domain $\hat{I}\text{c◯}$ is a $W^{1,1}$-extension domain if and only if for every pair $x,y$ of points in $\hat{I}{\text{c◯}}^c$ there exists a curve ${\hat{I}}^3\subset \hat{I}{\text{c◯}}^c$ connecting $x$ and $y$ with $${\int }_{\hat{I}}^3\frac{1}{\ddot{I}{‡}_{{\mathbb{R}}^2\setminus \partial \hat{I}\text{c◯}}(z)}ds(z)\le C|x-y|.$$ Consequently, a planar Jordan domain $\hat{I}\text{c◯}$ 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 $\tilde{\hat{I}}\text{c◯}$ is a $W^{1,\infty }$-extension domain.

56 pages, 5 figures