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

Abstract

Let X = {X₁, ... ,X_m} be a set of Hörmander vector fields in Rⁿ, where any X_j 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 Hörmander sum of squares Σ_j=1..m X_j², including global pointwise estimates of Gamma and of its X-derivatives in terms of the Carnot-Carathéodory distance induced by X on Rⁿ.