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.

220px-Bundesarchiv_Bild183-R57262,_Werner_HeisenbergYou 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

HormanderHörmander’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