Abstract and concrete tangent modules on Lipschitz differentiability spaces

Ikonen, Toni; Pasqualetto, Enrico; Soultanis, Elefterios

doi:10.1090/proc/15656

Abstract and concrete tangent modules on Lipschitz differentiability spaces
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by Toni Ikonen, Enrico Pasqualetto and Elefterios Soultanis PDF

Proc. Amer. Math. Soc. 150 (2022), 327-343 Request permission

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 Bate, Kangasniemi, and Orponen, Cheeger’s differentiation theorem via the multilinear Kakeya inequality, arXiv:1904.00808 (2019), this equivalence is used to show that the $Lip$–$lip$-type condition $lipf\le C|Df|$ self-improves to $lipf =|Df|$.

We also provide a direct proof of a result by Gigli and Pasqualetto, Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces, arXiv:1611.09645 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.

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