BV and Sobolev homeomorphisms between metric measure spaces and the plane

We show that given a homeomorphism $f:GrightarrowOmega$ 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 $(1,1)$-Poincare inequality, it holds $fin BV_{operatorname{loc}}(G,Omega)$ if and only $f^{-1}in BV_{operatorname{loc}}(Omega,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $fin W^{1,1}_{operatorname{loc}}(G,Omega)$ if and only if $f^{-1}in W^{1,1}_{operatorname{loc}}(Omega,G)$.