Schoenflies solutions of conformal boundary values may fail to be Sobolev

Abstract

We show that there exists a planar Jordan domains $\Omega$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $\phi :\mathbb{D}\to \Omega$, any homeomorphic extension of $\phi$ or ${\phi }^{-1}$ to the entire plane is not in $W_{\mathrm{loc}}^{1,1}$ (or even not in $B{V}_{\mathrm{loc}}$).