Global estimates for the fundamental solution of homogeneous Hörmander operators

Let L = Sigma(m)(j=1) X-j(2) be a Hormander sum of squares of vector fields in R-n, where any X-j is homogeneous of degree 1 with respect to a family of non-isotropic dilations in R-n. Then, L is known to admit a global fundamental solution Gamma(x;y) that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space R-n x R-p, equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of Gamma, in terms of the Carnot-Caratheodory distance induced by X = {X-1, ..., X-m} on R-n, as well as global pointwise (upper) estimates for the X-derivatives of any order of Gamma, together with suitable integral representations of these derivatives. The least dimensional case n = 2 presents several peculiarities which are also investigated. Applications to the potential theory for L and to singular-integral estimates for the kernel XiXj Gamma are also provided. Finally, most of the results about Gamma are extended to the case of Hormander operators with drift Sigma(m)(j=1) X-j(2) + X-0, where X-0 is 2-homogeneous and X-1, ..., X-m are 1-homogeneous.