Global estimates for the fundamental solution of homogeneous Hörmander operators

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 $Γ(x;y)$, 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 $Γ$, in terms of the Carnot-Carathéodory distance induced by $X=\{X_{1},\ldots ,X_{m}\}$ on $\mathbb{R}^{n}$, as well as global pointwise (upper) estimates for the $X$-derivatives of any order of $Γ$, 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 $X_{i}X_{j}Γ$ are also provided. Finally, most of the results about $Γ$ 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.

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