# Stratified groups

A Lie group $\mathbb{G}=(\mathbb{R}^n, \circ)$ is called a stratified group  (or homogeneous Carnot group) if it satisfies the following two conditions:

(1) For some natural numbers $N_1+...+N_r=n$ the decomposition $\mathbb{R}^n = \mathbb{R}^{N_1} \times ... \times \mathbb{R}^{N_r}$ is valid, and for every $\lambda>0$ the dilation $\delta_{\lambda}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ given by

$\delta_{\lambda}(x) \equiv \delta_{\lambda} (x^{(1)},..., x^{(r)}):=(\lambda x^{(1)},...,\lambda^r x^{(r)})$

is an automorphism of the group $\mathbb{G}.$ Here $x^{(k)} \in \mathbb{R}^{N_k}$ for $k=1,...,r.$

(2) Let $N_1$ be as in (a) and let $X_1,...,X_{N_1}$ be the left invariant vector fields on $\mathbb{G}$ such that $X_k(0) = \frac{\partial}{\partial x_k}|_0$ for $k=1,...,N_1.$ Then

$rank (Lie \{X_1,...,X_{N_1}\})=n,$

for every $x \in \mathbb{G},$ i.e. the iterated commutators of the vector fields $X_1,...,X_{N_1}$ from the first stratum of $\mathbb{G}.$ span its Lie algebra.

Main examples of stratified groups is the usual Euclidean space $\mathbb{R}^n$ or the Heisenberg group.

You can read more about stratified groups and related things here (see especially Definition 3.1.5), also with a bunch of examples, including Heisenberg and other groups.

The operator

$\mathcal{L}:=X_1^2+\cdots+ X_{N_1}^2$

is called the sub-Laplacian on $\mathbb{G}.$ It is hypoelliptic by

Hörmander’s sum of the squares theorem: in particular, it means that if $\mathcal{L}f$ is smooth then $f$ is also smooth.

Stratified groups have been extensively studied by Jerry Folland.

You can have a look at the classical potential theory of stratified groups:

M. Ruzhansky, D. Suragan, Layer Potentials, Kac’s problem, and refined Hardy inequality on homogeneous Carnot groups. Adv. Math., 308 (2017), 483-528download