Embeddings of infinite-dimensional spaces in the sets of norm-attaining Lipschitz functions

Abstract

Motivated by the result of Dantas et. al. (2023) that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of $c_0$, we introduce and study a weaker notion of norm-attainment for Lipschitz functions called the pointwise norm-attainment. As a main result, we show that for every infinite metric space $M$, there exists a metric space $M_0 \subseteq M$ such that the set of pointwise norm-attaining Lipschitz functions on $M_0$ contains an isometric copy of $c_0$. We also observe that there are countable metric spaces $M$ for which the set of pointwise norm-attaining Lipschitz functions contains an isometric copy of $\ell_\infty$, which is a result that does not hold for the set of strongly norm-attaining Lipschitz functions. Several new results on $c_0$-embedding and $\ell_1$-embedding into the set of strongly norm-attaining Lipschitz functions are presented as well. In particular, we show that if $M$ is a subset of an $\mathbb{R}$-tree containing all the branching points, then the set of strongly norm-attaining Lipschitz functions contains $c_0$ isometrically. As a related result, we provide an example of metric space $M$ for which the set of norm-attaining functionals on the Lipschitz-free space over $M$ cannot contain an isometric copy of $c_0$. Finally, we compare the concept of pointwise norm-attainment with the several different kinds of norm-attainment from the literature.