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 ℝ2{\\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∈BVloc\u2061(G,Ω){f\\in{\\operatorname{BV_{\\mathrm{loc}}}}(G,\\Omega)} if and only if f-1∈BVloc\u2061(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∈Wloc1,1\u2061(G,Ω){f\\in\\operatorname{W_{\\mathrm{loc}}^{1,1}}(G,\\Omega)} if and only if f-1∈Wloc1,1\u2061(Ω,G){f^{-1}\\in\\operatorname{W_{\\mathrm{loc}}^{1,1}}(\\Omega,G)}.