BV and Sobolev homeomorphisms between metric measure spaces and the plane

Abstract

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 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)} .