On the strongly subdifferentiable points in Lipschitz-free spaces

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 \(\mathcal {F}(M)\) $\mathcal{F}(M)$ over M. Our main result reads as follows: if (M, d) is a metric space and \(\gamma > 0\) $\gamma >0$ , then there exists a (not necessarily equivalent) metric \(d_{\gamma }\) $d_{\gamma }$ in M such that every finitely supported element in \(\mathcal {F}(M, d_{\gamma })\) $\mathcal{F}(M,d_{\gamma })$ is an SSD point. As an application of the main result, it follows that if M is uniformly discrete and \(\varepsilon > 0\) $\epsilon >0$ is given, there exists a metric space N and a \((1+\varepsilon )\) $(1+\epsilon )$ -bi-Lipschitz map \(\phi : M \rightarrow N\) $\varphi :M\to N$ such that the set of all SSD points in \(\mathcal {F}(N)\) $\mathcal{F}(N)$ is dense.