[HG_AP] Lipschitz-free spaces over purely 1-unrectifiable metric spaces
ABSTRACT
The Lipschitz-free space $\mathcal{F}(M)$ is a canonical linearization of a complete metric space $M$ whose dual is the space of Lipschitz functions on $M$. In this talk, we will review the properties of $\mathcal{F}(M)$ when the underlying space $M$ is purely 1-unrectifiable (that is, it contains no bi-Lipschitz copy of a subset of $\mathbb{R}$ with positive measure) and relate them to the properties of locally flat functions on $M$. On the isomorphic side, pure 1-unrectifiability of $M$ is equivalent to $\mathcal{F}(M)$ having the Radon-Nikodým and Schur properties and, in the compact case, also to admitting a predual. On the isometric side, every element of such a Lipschitz-free space can be expressed as a convex integral of elementary molecules. Time permitting, we will discuss some open problems and perspectives about these spaces.