Hörmander vector fields equipped with dilations: lifting, Lie-group construction, applications

Abstract

Let X = {X1, ... ,Xm} be a set of Hormander vector fields in Rn, where any Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in Rn. If N is the dimension of Lie{X}, we can either lift X to a system of generators of a higher dimensional Carnot group on Rn (if N>n, or we can equip Rn with a Carnot group structure with Lie algebra equal to Lie{X} (if N=n). We shall deduce these facts via a local-to-global procedure (available in the homogeneous setting), starting from more general results on the lifting of finite-dimensional Lie algebras of vector fields. The use of the Baker-Campbell-Hausdorff Theorem is crucial. Due to homogeneity, the lifting procedure is simpler than Rothschild-Stein's lifting technique. We finally provide applications to the study of the fundamental solution Gamma for the Hormander sum of squares Σj=1..m Xj2, including global pointwise estimates of Gamma and of its X-derivatives in terms of the Carnot-Caratheodory distance induced by X on Rn.