BV and Sobolev homeomorphisms between metric measure spaces and the plane

Abstract

We show that given a homeomorphism $f:G\to \Omega$ where $G$ is a open subset of ${\mathbb{R}}^2$ and $\Omega$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincaré inequality, it holds $f\in B{V}_{\mathrm{loc}}(G,\Omega )$ if and only $f^{-1}\in B{V}_{\mathrm{loc}}(\Omega ,G)$. Further if $f$ satisfies the Luzin N and N${}^{-1}$ conditions then $f\in W_{\mathrm{loc}}^{1,1}(G,\Omega )$ if and only if $f^{-1}\in W_{\mathrm{loc}}^{1,1}(\Omega ,G)$.