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).

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

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

$\square$