Global estimates for the fundamental solution of homogeneous Hörmander operators

Abstract

Let $\mathcal{L}={\sum }_{j=1}^m{X}_j^2$ be a Hörmander sum of squares of vector fields in ${\mathbb{R}}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in ${\mathbb{R}}^n$. Then $\mathcal{L}$ 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 ${\mathbb{R}}^n\times {\mathbb{R}}^p$, 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-Carathéodory distance induced by $X=\{X_1,\dots ,X_m\}$ on ${\mathbb{R}}^n$, 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 $\mathcal{L}$ and to singular-integral estimates for the kernel $X_i{X}_j\Gamma$ are also provided. Finally, most of the results about $\Gamma$ are extended to the case of Hörmander operators with drift ${\sum }_{j=1}^m{X}_j^2+X_0$, where $X_0$ is $2$-homogeneous and $X_1,...,X_m$ are $1$-homogeneous.