# Global estimates for the fundamental solution of homogeneous H\"ormander operators

## TL;DR

This paper establishes precise global estimates for the fundamental solution of homogeneous Hörmander operators, including derivatives, and explores their applications in potential theory and singular integrals, with special attention to the two-dimensional case.

## Contribution

It provides the first comprehensive global pointwise bounds for the fundamental solution and its derivatives of homogeneous Hörmander operators, extending to operators with drift and addressing the two-dimensional case.

## Key findings

- Derived global upper and lower bounds for the fundamental solution
- Obtained estimates for derivatives of the fundamental solution
- Extended results to operators with drift and special cases in two dimensions

## Abstract

Let $\mathcal{L}=\sum_{j=1}^{m}X_{j}^{2}$ be a H\"{o}rmander 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-Carath\'{e}odory 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 $\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 H\"{o}rmander 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.