Global Heat Kernels for Parabolic Homogeneous Hörmander Operators

Abstract

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 $\mathcal{H}={\sum }_{j=1}^m{X}_j^2-{\partial }_t$, where $X_1,\dots ,X_m$ are smooth vector fields on ${\mathbb{R}}^n$ satisfying Hörmander'snrank condition, and enjoying a suitable homogeneity assumption with respect to a family of non-isotropic dilations. The proof of the existence of $\Gamma$ is based on a (algebraic) global lifting technique, together with a representation of $\Gamma$ 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 $\Gamma$ 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.