Characterization of AC and Sobolev curves via Lipschitz post-compositions

Let X: = (X, d) be an arbitrary metric space. For each p 1, , we prove that a map: a, b X is p-absolutely continuous if and only if, for every Lipschitz function h: X R, the post-composition h is a p-absolutely continuous function. Furthermore, if X is complete and separable, then, for each p (1, ), we show that the equivalence class (up to L^1-a. e. equality) of a Borel map: a, b X belongs to the Sobolev W₏^1 (a, b, X) -space if and only if, for every Lipschitz function h: X R, the equivalence class (up to L^1-a. e. equality) of the post-composition h belongs to the Sobolev W₏^1 (a, b, R) -space.