  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
$X_{1},...,X_{m}$ are real smooth vector fields satisfying H\"{o}rmander 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\left( 1,\infty\right) $, of
the kind: \[ \Vert u\Vert_{W_{X}^{2,p}(\mathbb{R}^{n})}\leq c\left\{ \left\Vert
Lu\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }+\left\Vert u\right\Vert
_{L^{p}\left( \mathbb{R}^{n}\right) }\right\} \] for every $u\in
W_{X}^{2,p}\left( \mathbb{R}^{n}\right) .$ We also prove higher order estimates
and corresponding regularity results: if $a_{ij}\in W_{X}^{k,\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 \[ \Vert
u\Vert_{W_{X}^{k+2,p}(\mathbb{R}^{n})}\leq c\left\{ \Vert
Lu\Vert_{W_{X}^{k,p}(\mathbb{R}^{n})}+\Vert u\Vert_{L^{p}(\mathbb{R}^{n}
)}\right\} . \]
