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

Abstract

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