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

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 po