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

Abstract

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 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 $p\in \left(1,\infty \right)$, 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 \[ \Vert u\Vert_{W_{X}^{k+2,p}(\mathbb{R}^{n})}\leq c\left\{ \Vert Lu\Vert_{W_{X}^{k,p}(\mathbb{R}^{n})}+\Vert u\Vert_{L^{p}(\mathbb{R}^{n} )}\right\} . \]