Global estimates for the fundamental solution of homogeneous Hormander operators

Let $L=sum_{j=1}^m X_j^2$ be a Hörmander 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 Γ(x;y) that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $R^n×R^p$, 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 $X={X_1,…,X_m}$ on $R^n$, 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 L and to singular-integral estimates for the kernel $X_iX_jΓ$ are also provided. Finally, most of the results about Γ are extended to the case of Hörmander operators with drift $L+X_0$, where $X_0$ is 2-homogeneous and $X_1,…,X_m$ are 1-homogeneous.