BV and Sobolev homeomorphisms between metric measure spaces and the plane

We show that, given a homeomorphism f: G → ω {f:G\rightarrow\Omega} where G is an open subset of 2 {\mathbb{R}{2}} and ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak (1, 1) {(1,1)} -Poincaré inequality, it holds f BV loc (G, ω) {f\in{\operatorname{BV{\mathrm{loc}}}}(G,\Omega)} if and only if f - 1 BV loc (ω, G) {f{-1}\in{\operatorname{BV{\mathrm{loc}}}}(\Omega,G)}. Further, if f satisfies the Luzin N and N - 1 {{}{-1}} conditions, then f W loc 1, 1 (G, ω) {f\in\operatorname{W{\mathrm{loc}}{1,1}}(G,\Omega)} if and only if f - 1 W loc 1, 1 (ω, G) {f{-1}\in\operatorname{W{\mathrm{loc}}{1,1}}(\Omega,G)}.