A New insight into the Ambrosio--Reshetnyak approach to Sobolev maps

Abstract

Given $p \in (1,\infty)$, let $\operatorname{X}=(\operatorname{X},ρ,μ)$ be a metric measure space such that the measure $μ$ is uniformly locally doubling and $\operatorname{X}$ supports a weak local $(1,p)$-Poincaré inequality. Let $(\operatorname{Y},\operatorname{d},\underline{y})$ be a complete pointed metric space. We prove that the equivalence class of a Borel map $ u:\operatorname{X} \to \operatorname{Y}$ (modulo coincidence $μ$-a.e.) belongs to the Ambrosio--Reshetnyak--Sobolev class $W_{p}^{1}(\operatorname{X},\operatorname{Y})$ if and only if $h \circ u$ belongs to the Sobolev space $W_{p}^{1}(\operatorname{X})$ for every $1$-Lipschitz function $h:\operatorname{Y} \to \mathbb{R}$.