A Lie group is called a stratified group (or homogeneous Carnot group) if it satisfies the following two conditions:
(1) For some natural numbers the decomposition is valid, and for every the dilation given by
is an automorphism of the group Here for
(2) Let be as in (a) and let be the left invariant vector fields on such that for Then
for every i.e. the iterated commutators of the vector fields from the first stratum of span its Lie algebra.
Main examples of stratified groups is the usual Euclidean space 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.
is called the sub-Laplacian on It is hypoelliptic by
Hörmander’s sum of the squares theorem: in particular, it means that if is smooth then 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), 483-528. download