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Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a H\"ormander sum of squares ofvector 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 paperwe 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 ofsquares, the homogeneity property of the $X_j$'s, plus a global liftingtechnique for homogeneous vector fields.

Global estimates in Sobolev spaces for homogeneous H\"ormander sums of squares