Abstract and concrete tangent modules on Lipschitz differentiability spaces

Abstract

We construct an isometric embedding from Gigli's abstract tangent module into the concrete tangent module of a space admitting a (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when the embedding is an isomorphism. Together with arguments from a recent article by Bate--Kangasniemi--Orponen, this equivalence is used to show that the $\mathrm{Lip}-\mathrm{lip}$ -type condition $\mathrm{lip}f\le C|Df|$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to $\mathrm{lip}f=|Df|$. We also provide a direct proof of a result by Gigli and the second author that, for a space with a strongly rectifiable decomposition, Gigli's tangent module admits an isometric embedding into the so-called Gromov--Hausdorff tangent module, without any a priori reflexivity assumptions.

14 pages, to appear in Proc. Amer. Math. Soc