Global Sobolev regularity for nonvariational operators built with homogeneous Hörmander vector fields

We consider a class of nonvariational degenerate elliptic operators of the kind $$Lu=\sum _{i,j=1}^m{a}_{ij}\left(x\right)X_i{X}_{j}u$$ where ${\left\{a_{ij}\left(x\right)\right\}}_{i,j=1}^m$ is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole ${\mathbb{R}}^n$ ($n>m$), possibly discontinuos but satisfying a $VMO$ assumption, and $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_m$ are real smooth vector fields satisfying Hörmander rank condition in the whole ${\mathbb{R}}^n$ and $1$-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global $W_X^{2,p}$ a-priori estimates, for every $p\in (1,\mathrm{\infty })$, of the kind: $$\mathrm{\parallel }u{\mathrm{\parallel }}_{W_X^{2,p}({\mathbb{R}}^n)}\le c\{{\parallel Lu\parallel }_{L^p\left({\mathbb{R}}^n\right)}+{\parallel u\parallel }_{L^p\left({\mathbb{R}}^n\right)}\}$$ for every $u\in W_X^{2,p}\left({\mathbb{R}}^n\right)\mathrm{.}$ We also prove higher order estimates and corresponding regularity results: if $a_{ij}\in W_X^{k,\mathrm{\infty }}\left({\mathbb{R}}^n\right)$, $u\in W_X^{2,p}\left({\mathbb{R}}^n\right)$, $Lu\in W_X^{k,p}\left({\mathbb{R}}^n\right)$, then $u\in W_X^{k+2,p}\left({\mathbb{R}}^n\right)$ and $$\mathrm{\parallel }u{\mathrm{\parallel }}_{W_X^{k+2,p}({\mathbb{R}}^n)}\le c\{\mathrm{\parallel }Lu{\mathrm{\parallel }}_{W_X^{k,p}({\mathbb{R}}^n)}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\}\mathrm{.}$$