Infinite-Dimensional Carnot Groups and Gâteaux Differentiability

Enrico Le Donne; Sean Li; Terhi Moisala

doi:10.1007/s12220-019-00324-x

The Journal of Geometric Analysis

pp 1–30 | Cite as

Infinite-Dimensional Carnot Groups and Gâteaux Differentiability

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Enrico Le Donne
Sean Li
Terhi Moisala

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First Online: 23 November 2019

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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).

Keywords

Carnot groups Differentiability Rademacher Gateaux derivative

Mathematics Subject Classification

28A15 53C17 58C20 46G05

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Notes

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 x, y, z 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 $

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Authors and Affiliations

Enrico Le Donne
- 1
- 3
Sean Li
- 2
Email author
Terhi Moisala
- 3

1.Dipartimento di MatematicaUniversità di PisaPisaItaly
2.University of ConnecticutMansfieldUSA
3.Department of Mathematics and StatisticsUniversity of JyväskyläUniversity of JyväskyläFinland

Infinite-Dimensional Carnot Groups and Gâteaux Differentiability

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Mathematics Subject Classification

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