Injectivity of Lipschitz operators

TL;DR

This paper investigates the relationship between the injectivity of Lipschitz maps and their linearised counterparts on Lipschitz-free spaces, identifying conditions under which injectivity is preserved or not.

Contribution

It clarifies when the injectivity of Lipschitz maps implies the injectivity of their linearisations, providing new conditions and counterexamples.

Findings

Injectivity of $ ext{f}$ does not always imply injectivity of $ ext{}\widehat{ ext{f}}$.

For certain metric spaces, injective Lipschitz maps have injective linearisations.

Stronger conditions on $f$ ensure the injectivity of $ ext{}\widehat{ ext{f}}$.

Abstract

Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective.