Hormander vector fields equipped with dilations: Lifting, lie-group construction, applications

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 nonisotropic 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 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 G for the Hormander sum of squares including global pointwise estimates of G and of its Xderivatives in terms of the Carnot-Caratheodory distance induced by X.