Global heat kernels for parabolic homogeneous hörmander operators

The aim of this paper is to prove the existence and several selected properties of a global fundamental Heat kernel Gamma for the parabolic operators H = Sigma(m)(j=1) X-j(2)-partial derivative(t), where X-1,..., X-m are smooth vector fields on R-n satisfying Hormander's rank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of G is based on a (algebraic) global lifting technique, together with a representation of G in terms of the integral (performed over the lifting variables) of the Heat kernel for the Heat operator associated with a suitable sub-Laplacian on a homogeneous Carnot group. Among the features of G we prove: homogeneity and symmetry properties; summability properties; its vanishing at infinity; the uniqueness of the bounded solutions of the related Cauchy problem; reproduction and density properties; an integral representation for the higher- order derivatives.