Characterization of AC and Sobolev curves via Lipschitz post-compositions

Abstract

Let $X:=(X,d)$ be an arbitrary metric space. For each $p\in [1,\infty ]$, we prove that a map $\gamma :[a,b]\to X$ is $p$-absolutely continuous if and only if, for every Lipschitz function $h:X\to \mathbb{R}$, the post-composition $h\circ \gamma$ is a $p$-absolutely continuous function. Furthermore, if $X$ is complete and separable, then, for each $p\in (1,\infty )$, we show that the equivalence class (up to ${\mathcal{L}}^1$-a.e. equality) of a Borel map $\gamma :[a,b]\to X$ belongs to the Sobolev $W_p^1([a,b],X)$-space if and only if, for every Lipschitz function $h:X\to \mathbb{R}$, the equivalence class (up to ${\mathcal{L}}^1$-a.e. equality) of the post-composition $h\circ \gamma$ belongs to the Sobolev $W_p^1([a,b],\mathbb{R})$-space.