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Abstract and concrete tangent modules on Lipschitz differentiability spaces


Authors: Toni Ikonen, Enrico Pasqualetto and Elefterios Soultanis
Journal: Proc. Amer. Math. Soc. 150 (2022), 327-343
MSC (2020): Primary 53C23, 46E35, 49J52
DOI: https://doi.org/10.1090/proc/15656
Published electronically: October 19, 2021
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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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Additional Information

Toni Ikonen
Affiliation: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
ORCID: 0000-0001-5969-7912
Email: toni.m.h.ikonen@jyu.fi

Enrico Pasqualetto
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
MR Author ID: 1358786
Email: enrico.pasqualetto@sns.it

Elefterios Soultanis
Affiliation: Radboud University, Department of Mathematics, PO Box 9010, Postvak 59, 6500 GL Nijmegen, The Netherlands
MR Author ID: 989349
Email: elefterios.soultanis@gmail.com

Keywords: Lipschitz differentiability space, rectifiable space, Sobolev space, tangent module
Received by editor(s): January 11, 2021
Received by editor(s) in revised form: April 25, 2021
Published electronically: October 19, 2021
Additional Notes: The first author was supported by the Academy of Finland, project number 308659, and by the Vilho, Yrjö and Kalle Väisälä Foundation.
The second author was supported by the Academy of Finland, project number 314789, and by the Balzan project led by Prof. Luigi Ambrosio.
The third author was supported by the Swiss National Foundation, grant no. 182423.
Communicated by: Nageswari Shanmugalingam
Article copyright: © Copyright 2021 American Mathematical Society