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 | References | Similar Articles | Additional Information

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.

- Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré,
*Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces*, Rev. Mat. Iberoam.**29**(2013), no. 3, 969–996. MR**3090143**, DOI 10.4171/RMI/746 - David Bate,
*Structure of measures in Lipschitz differentiability spaces*, J. Amer. Math. Soc.**28**(2015), no. 2, 421–482. MR**3300699**, DOI 10.1090/S0894-0347-2014-00810-9 - D. Bate, I. Kangasniemi, and T. Orponen,
*Cheeger’s differentiation theorem via the multilinear Kakeya inequality*, arXiv:1904.00808, 2019. - David Bate and Sean Li,
*Characterizations of rectifiable metric measure spaces*, Ann. Sci. Éc. Norm. Supér. (4)**50**(2017), no. 1, 1–37 (English, with English and French summaries). MR**3621425**, DOI 10.24033/asens.2314 - David Bate and Sean Li,
*Differentiability and Poincaré-type inequalities in metric measure spaces*, Adv. Math.**333**(2018), 868–930. MR**3818093**, DOI 10.1016/j.aim.2018.06.002 - David Bate and Gareth Speight,
*Differentiability, porosity and doubling in metric measure spaces*, Proc. Amer. Math. Soc.**141**(2013), no. 3, 971–985. MR**3003689**, DOI 10.1090/S0002-9939-2012-11457-1 - Anders Björn and Jana Björn,
*Nonlinear potential theory on metric spaces*, EMS Tracts in Mathematics, vol. 17, European Mathematical Society (EMS), Zürich, 2011. MR**2867756**, DOI 10.4171/099 - J. Cheeger,
*Differentiability of Lipschitz functions on metric measure spaces*, Geom. Funct. Anal.**9**(1999), no. 3, 428–517. MR**1708448**, DOI 10.1007/s000390050094 - Simone Di Marino and Gareth Speight,
*The $p$-weak gradient depends on $p$*, Proc. Amer. Math. Soc.**143**(2015), no. 12, 5239–5252. MR**3411142**, DOI 10.1090/S0002-9939-2015-12641-X - Sylvester Eriksson-Bique,
*Characterizing spaces satisfying Poincaré inequalities and applications to differentiability*, Geom. Funct. Anal.**29**(2019), no. 1, 119–189. MR**3925106**, DOI 10.1007/s00039-019-00479-3 - B. Franchi, P. Hajłasz, and P. Koskela,
*Definitions of Sobolev classes on metric spaces*, Ann. Inst. Fourier (Grenoble)**49**(1999), no. 6, 1903–1924. MR**1738070** - Nicola Gigli,
*Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below*, Mem. Amer. Math. Soc.**251**(2018), no. 1196, v+161. MR**3756920**, DOI 10.1090/memo/1196 - Nicola Gigli,
*Lecture notes on differential calculus on $\mathsf {RCD}$ spaces*, Publ. Res. Inst. Math. Sci.**54**(2018), no. 4, 855–918. MR**3869373**, DOI 10.4171/PRIMS/54-4-4 - N. Gigli and E. Pasqualetto,
*Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces*. Accepted at Communications in Analysis and Geometry, arXiv:1611.09645, 2016. - Nicola Gigli and Enrico Pasqualetto,
*Differential structure associated to axiomatic Sobolev spaces*, Expo. Math.**38**(2020), no. 4, 480–495. MR**4177952**, DOI 10.1016/j.exmath.2019.01.002 - Juha Heinonen and Pekka Koskela,
*Weighted Sobolev and Poincaré inequalities and quasiregular mappings of polynomial type*, Math. Scand.**77**(1995), no. 2, 251–271. MR**1379269**, DOI 10.7146/math.scand.a-12564 - Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson,
*Sobolev spaces on metric measure spaces*, New Mathematical Monographs, vol. 27, Cambridge University Press, Cambridge, 2015. An approach based on upper gradients. MR**3363168**, DOI 10.1017/CBO9781316135914 - Stephen Keith,
*A differentiable structure for metric measure spaces*, Adv. Math.**183**(2004), no. 2, 271–315. MR**2041901**, DOI 10.1016/S0001-8708(03)00089-6 - Stephen Keith,
*Measurable differentiable structures and the Poincaré inequality*, Indiana Univ. Math. J.**53**(2004), no. 4, 1127–1150. MR**2095451**, DOI 10.1512/iumj.2004.53.2417 - Martin Kell,
*On Cheeger and Sobolev differentials in metric measure spaces*, Rev. Mat. Iberoam.**35**(2019), no. 7, 2119–2150. MR**4029797**, DOI 10.4171/rmi/1114 - Bernd Kirchheim,
*Rectifiable metric spaces: local structure and regularity of the Hausdorff measure*, Proc. Amer. Math. Soc.**121**(1994), no. 1, 113–123. MR**1189747**, DOI 10.1090/S0002-9939-1994-1189747-7 - M. E. Mera, M. Morán, D. Preiss, and L. Zajíček,
*Porosity, $\sigma$-porosity and measures*, Nonlinearity**16**(2003), no. 1, 247–255. MR**1950786**, DOI 10.1088/0951-7715/16/1/315 - Danka Lučić and Enrico Pasqualetto,
*The Serre-Swan theorem for normed modules*, Rend. Circ. Mat. Palermo (2)**68**(2019), no. 2, 385–404. MR**4148753**, DOI 10.1007/s12215-018-0366-6 - Kai Rajala,
*Uniformization of two-dimensional metric surfaces*, Invent. Math.**207**(2017), no. 3, 1301–1375. MR**3608292**, DOI 10.1007/s00222-016-0686-0 - Andrea Schioppa,
*On the relationship between derivations and measurable differentiable structures*, Ann. Acad. Sci. Fenn. Math.**39**(2014), no. 1, 275–304. MR**3186817**, DOI 10.5186/aasfm.2014.3910 - Andrea Schioppa,
*Derivations and Alberti representations*, Adv. Math.**293**(2016), 436–528. MR**3474327**, DOI 10.1016/j.aim.2016.02.013 - Andrea Schioppa,
*Metric currents and Alberti representations*, J. Funct. Anal.**271**(2016), no. 11, 3007–3081. MR**3554700**, DOI 10.1016/j.jfa.2016.08.022

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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