BV and Sobolev homeomorphisms between metric measure spaces and the plane

TL;DR

This paper characterizes BV and Sobolev homeomorphisms between planar domains and metric measure spaces, establishing conditions under which the regularity of a homeomorphism and its inverse are equivalent.

Contribution

It extends the theory of BV and Sobolev homeomorphisms to metric measure spaces, providing new equivalences for regularity of homeomorphisms and their inverses.

Findings

BV regularity of f iff BV regularity of f^{-1} under certain conditions

Sobolev regularity of f iff Sobolev regularity of f^{-1} with Luzin conditions

Applicable to 2-Ahlfors regular metric measure spaces supporting Poincaré inequality

Abstract

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