BV and Sobolev homeomorphisms between metric measure spaces and the plane

Abstract We show that, given a homeomorphism f : G → Ω {f:G\rightarrow\Omega} where G is an open subset of ℝ 2 {\mathbb{R}^{2}} and Ω 2-Ahlfors regular metric measure space supporting weak ( 1 , stretchy="false">) {(1,1)} -Poincaré inequality, it holds ∈ BV loc ⁡ {f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)} if only - {f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)} . Further, f satisfies the Luzin N /> {{}^{-1}} conditions, then mathvariant="normal">W {f\in\operatorname{W_{\mathrm{loc}}^{1,1}}(G,\Omega)} {f^{-1}\in\operatorname{W_{\mathrm{loc}}^{1,1}}(\Omega,G)}