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

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