On the strongly subdifferentiable points in Lipschitz-free spaces

TL;DR

This paper investigates conditions under which points in Lipschitz-free spaces are strongly subdifferentiable, introducing metric modifications to ensure density of SSD points in these spaces.

Contribution

It provides new sufficient conditions and metric constructions that guarantee the density of SSD points in Lipschitz-free spaces over certain metric spaces.

Findings

Existence of a metric $d_{ extgamma}$ making all finitely supported elements SSD points.

For uniformly discrete spaces, there exists a bi-Lipschitz equivalent metric with dense SSD points.

Main results apply to constructing metrics that enhance the geometric properties of Lipschitz-free spaces.

Abstract

In this paper, we present some sufficient conditions on a metric space $M$ for which every molecule is a strongly subdifferentiable (SSD, for short) point in the Lipschitz-free space $\mathcal{F}(M)$ over $M$. Our main result reads as follows: if $(M,d)$ is a metric space and $\gamma > 0$, then there exists a (not necessarily equivalent) metric $d_{\gamma}$ in $M$ such that every finitely supported element in $\mathcal{F}(M, d_{\gamma})$ is an SSD point. As an application of the main result, it follows that if $M$ is uniformly discrete and $\varepsilon > 0$ is given, there exists a metric space $N$ and a $(1+\varepsilon)$-bi-Lipschitz map $\phi: M \rightarrow N$ such that the set of all SSD points in $\mathcal{F}(N)$ is dense.