Schoenflies solutions of conformal boundary values may fail to be Sobolev

We show that there exists a planar Jordan domains $\mathrm{\Omega }$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $\varphi \colon \mathbb D \to \Omega$,anyhomeomorphicextensionof$\varphi$or$\varphi^{-1}$totheentireplaneisnotin$W^{1,,1}_{\rm loc}$ (or even not in $B{V}_{\mathrm{l}\mathrm{o}\mathrm{c}}$).