Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Abstract

Let $\mathcal{L}={\sum }_{j=1}^m{X}_j^2$ be a Hörmander 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 $W_X^{k,p}({\mathbb{R}}^n)$, where $X=\{X_1,\dots ,X_m\}$. In our approach, we combine local results for general Hörmander sums of squares, the homogeneity property of the $X_j$'s, plus a global lifting technique for homogeneous vector fields.