Preprint

Inserted: 1 dec 2020
Last Updated: 1 dec 2020

Pages: 14
Year: 2020

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 ${\rm Lip}-{\rm lip}$ -type condition ${\rm lip} f\leq C
Df
$ implies the existence of a Lipschitz differentiable structure, and moreover self-improves to ${\rm 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.

Keywords: Sobolev space, Lipschitz differentiability space, Rectifiable space, Tangent module