Infinite-Dimensional Carnot Groups and Gâteaux Differentiability

Abstract

This paper contributes to the generalization of Rademacher’s differentiability result for Lipschitz functions when the domain is infinite dimensional and has nonabelian group structure. We introduce an infinite-dimensional analogue of Carnot groups that are metric groups equipped with dilations (which we call metric scalable groups) admitting a dense increasing sequence of finite-dimensional Carnot subgroups. For such groups, we show that every Lipschitz function has a point of Gâteaux differentiability. As a step in the proof, we show that a certain \(\sigma \)-ideal of sets that are null with respect to this sequence of subgroups cannot contain open sets. We also give a geometric criterion for when such Carnot subgroups exist in metric scalable groups and provide examples of such groups. The proof of the main theorem follows the work of Aronszajn (Stud Math 57(2):147–190, 1976) and Pansu (Ann Math 129(1):1–60, 1989).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Aronszajn, N.: Differentiability of Lipschitzian mappings between Banach spaces. Stud. Math. 57(2), 147–190 (1976)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bate, D., Li, S.: Differentiability and Poincaré-type inequalities in metric measure spaces. Adv. Math. 333, 868–930 (2018)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cheeger, J., Kleiner, B.: On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon–Nikodým property. Geom. Funct. Anal. 19(4), 1017–1028 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Eriksson-Bique, S.: Classifying Poincaré inequalities and the local geometry of RNP-differentiability spaces. Geom. Funct. Anal. (2016) (Accepted)

  6. 6.

    Wofsey, E.: (https://math.stackexchange.com/users/86856/eric wofsey), if \( h \) is a locally compact subgroup of a topological group \( g \), then \( h \) is closed in \( g \), Mathematics Stack Exchange. https://math.stackexchange.com/q/2419383 (version: 2017-09-06)

  7. 7.

    Khukhro, E.I.: \(p\)-Automorphisms of Finite \(p\)-Groups. London Mathematical Society Lecture Note Series, vol. 246. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  8. 8.

    Le Donne, E.: A primer on Carnot groups: homogenous groups, Carnot–Carathéodory spaces, and regularity of their isometries. Anal. Geom. Metr. Spaces 5, 116–137 (2017)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Lindenstrauss, J., Preiss, D.: On Fréchet differentiability of Lipschitz maps between Banach spaces. Ann. Math. 157, 257–288 (2003)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mankiewicz, P.: On the differentiability of Lipschitz mappings in Fréchet spaces. Stud. Math. 45, 15–29 (1973)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mityagin, B.: The Zero Set of a Real Analytic Function, ArXiv e-prints (2015)

  12. 12.

    Magnani, V., Pinamonti, A., Speight, G.: Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups, ArXiv e-prints (2017)

  13. 13.

    Magnani, V., Rajala, T.: Radon–Nikodym property and area formula for Banach homogeneous group targets. Int. Math. Res. Not. 2014(23), 6399–6430 (2014)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Montgomery, D., Zippin, L.: Topological transformation groups, Robert E. Krieger Publishing Co., Huntington, N.Y., Reprint of the 1955 original (1974)

  15. 15.

    Pansu, P.: Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. (2) 129(1), 1–60 (1989)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Preiss, D.: Differentiability of Lipschitz functions on Banach spaces. J. Funct. Anal. 91, 312–345 (1990)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Rudin, W.: Functional Analysis, International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)

    Google Scholar 

  18. 18.

    Schioppa, A.: An example of a differentiability space which is PI-unrectifiable, ArXiv e-prints (2016)

  19. 19.

    Siebert, E.: Contractive automorphisms on locally compact groups. Math. Z. 191(1), 73–90 (1986)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sean Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

E. L. D. and T. M. were partially supported by the Academy of Finland (Grant 288501 ‘Geometry of subRiemannian groups’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). T. M. was also supported by the Finnish Cultural Foundation (Grant 00170709). S. L. was supported by NSF Grant DMS-1812879.

Appendix A: Some Useful Commutator Identities

Appendix A: Some Useful Commutator Identities

Lemma A.1

Let G be a group and \( x,y,z\in G \). Then

$$\begin{aligned}{}[xy,z] = [x,[y,z]][y,z][x,z] \quad \text {and} \quad [z,xy] = [z,x][z,y][[y,z],x] = h[z,x][z,y], \end{aligned}$$

where h is a product of commutators of xyz of weight \( \ge 3 \).

Proof

For the first equation,

$$\begin{aligned}{}[y,z][x,z] = [y,z]xz{x}^{-1}{z}^{-1} = [[y,z],x]xyz{y}^{-1}{z}^{-1}z{x}^{-1}{z}^{-1} = [[y,z],x][xy,z]. \end{aligned}$$

Since \( [a,b]={[b,a]}^{-1} \),

$$\begin{aligned}{}[xy,z] = [x,[y,z]][y,z][x,z]. \end{aligned}$$

Using this and the identity \( [a,b]={[b,a]}^{-1} \),

$$\begin{aligned}{}[z,xy] = [z,x][z,y][[y,z],x]. \end{aligned}$$

The last equation follows by reordering the terms, which produces some higher order commutators into h. \(\square \)

Corollary A.2

If \( [y,z]\in Z(G) \), then

$$\begin{aligned}{}[xy,z] = [x,z][y,z] \quad \text {and} \quad [z,xy] = [z,x][z,y]. \end{aligned}$$

Corollary A.3

Let \( n,m\in {\mathbb {N}}\). Then

$$\begin{aligned}{}[x^n,y^m] = h[x,y]^{nm}, \end{aligned}$$

where h is a product of commutators of x and y of weight \( \ge 3 \).

Proof

The proof is by iterating Lemma A.1 for nm times and reordering the terms, which produces some additional higher order commutators into h. \(\square \)

Lemma A.4

Let G be a group, \( x,y\in G \). Then

$$\begin{aligned} {}[{x}^{-1},y] = [{x}^{-1},[y,x]]{[x,y]}^{-1}. \end{aligned}$$

Proof

The statement follows from

$$\begin{aligned} {}[{x}^{-1},[y,x]] = {x}^{-1}yx{y}^{-1}{x}^{-1}x{[y,x]}^{-1} = [{x}^{-1},y][x,y]. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Le Donne, E., Li, S. & Moisala, T. Infinite-Dimensional Carnot Groups and Gâteaux Differentiability. J Geom Anal 31, 1756–1785 (2021). https://doi.org/10.1007/s12220-019-00324-x

Download citation

Keywords

  • Carnot groups
  • Differentiability
  • Rademacher
  • Gateaux derivative

Mathematics Subject Classification

  • 28A15
  • 53C17
  • 58C20
  • 46G05