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We consider a class of nonvariational degenerate elliptic operators of thekind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positivematrix 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 rankcondition in the whole $\mathbb{R}^{n}$ and $1$-homogeneous w.r.t. a family ofnonisotropic dilations. We do not assume that the vector fields are leftinvariant 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) $, ofthe kind: \[ \sum_{i,j=1}^{m}||X_{i}X_{j}u||_{L^{p}\left( \mathbb{R}^{n}\right)} +\sum_{i=1}^{m}||X_{i}u||_{L^{p}\left( \mathbb{R}^{n}\right) }+||u||_{L^{p}\left( \mathbb{R}^{n}\right) }\leq c\{||Lu||_{L^{p}\left(\mathbb{R}^{n}\right) }+||u||_{L^{p}\left( \mathbb{R}^{n}\right) }\} \] forevery $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) .$

Global L^{p} estimates for nonvariational operators with VMO coefficients, built with homogeneous H\"ormander vector fields