On Weak Compactness and Uniform Regularity in Lipschitz free Spaces

TL;DR

This paper investigates the properties of weakly compact sets in Lipschitz free spaces, introducing uniform regularity and providing characterizations for specific metric spaces like $ ext{R}$-trees.

Contribution

It establishes that weakly precompact sets are uniformly regular and offers intrinsic characterizations of weakly compact sets in Lipschitz free spaces over $ ext{R}$-trees.

Findings

Weakly precompact sets are uniformly regular.

Uniform regularity alone does not characterize weak compactness.

Characterization of norm-compactness via sums of large molecules.

Abstract

We analyze the properties of weakly compact sets in Lipschitz free spaces. Prior research has established that, for a complete metric space $M$, weakly precompact sets in the Lipschitz free space $\mathcal F(M)$ are tight. In this paper, we prove that these sets actually exhibit a stronger property, which we call uniform regularity. However, this condition alone is not sufficient to characterize weakly compact sets, except in the case of scattered metric spaces. On the other hand, if $T$ is an $\mathbb R$-tree, we leverage Godard's isometry between $\mathcal F(T)$ and $L^1(\lambda_T)$ to obtain an intrinsic characterization of weakly compact sets in $\mathcal F(T)$. This approach allows us to identify conditions that may describe weak compactness across a wider range of spaces. In particular, we provide a characterization of norm-compactness in terms of sums of "large molecules'', while we show that sums of "small molecules'' contain an $\ell_1$-basis.