Characterization of AC and Sobolev curves via Lipschitz post-compositions

TL;DR

This paper characterizes p-absolutely continuous and Sobolev curves in metric spaces through their post-compositions with Lipschitz functions, establishing equivalences that connect metric space curves to real-valued Sobolev functions.

Contribution

It provides a novel characterization of AC and Sobolev curves in metric spaces via Lipschitz post-compositions, extending classical analysis concepts.

Findings

p-absolutely continuous curves characterized by Lipschitz post-compositions

Sobolev class equivalence for curves via Lipschitz functions

Results hold in complete, separable metric spaces

Abstract

Let $\operatorname{X}:=(\operatorname{X},\operatorname{d})$ be an arbitrary metric space. For each $p \in [1,\infty]$, we prove that a map $\gamma:[a,b] \to \operatorname{X}$ is $p$-absolutely continuous if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, the post-composition $h \circ \gamma$ is a $p$-absolutely continuous function. Furthermore, if $\operatorname{X}$ 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 $\gamma:[a,b] \to \operatorname{X}$ belongs to the Sobolev $W_{p}^{1}([a,b],\operatorname{X})$-space if and only if, for every Lipschitz function $h:\operatorname{X} \to \mathbb{R}$, 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.