In this talk we present several global results concerning the class of the homogeneous Hörmander sums of squares. As the name suggests, the operators falling in this class are sums of squares of smooth vector fields which are homogeneous of degree 1 with respect to a family of non-isotropic diagonal maps (usually called dilations); moreover, these operators intervene in several contexts of interest (Lie group Theory, sub-Riemannian manifolds, Mathematical Finance, etc.).

After a brief introduction on general sub-elliptic operators (of which any homogeneous sum of squares is a particular case), we properly introduce the class of the homogeneous Hörmander sums of squares and we discuss some global qualitative aspects regarding these operators: global lifting on Carnot groups; existence/global estimates for the associated fundamental solution and heat kernel; maximum principles on unbounded domains.

The results presented in this talk are contained in several papers in collaboration with A. Bonfiglioli, M. Bramanti and E. Lanconelli.