  Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a H\"ormander sum of squares of
vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree
$1$ with respect to a family of non-isotropic dilations in space. In this paper
we prove global estimates and regularity properties for $\mathcal{L}$ in the
$X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^n)$, where $X = \{X_1,\ldots,X_m\}$.
In our approach, we combine local results for general H\"ormander sums of
squares, the homogeneity property of the $X_j$'s, plus a global lifting
technique for homogeneous vector fields.
