  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.
