$L^{\lowercase{p}}$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients

Let $X_1,X_2,\ldots,X_q$ be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain $\Omega\subset\mathbb{R}^n$ ($n>q$). We consider the differential operator

where the coefficients $a_{ij}(x)$ are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. $x\in\Omega$, every $\xi\in\mathbb{R}^q$, some constant $\mu$. Moreover, we assume that the coefficients $a_{ij}$ belong to the space VMO (``Vanishing Mean Oscillation''), defined with respect to the subelliptic metric induced by the vector fields $X_1,X_2,\ldots,X_q$. We prove the following local $\mathcal{L}^p$-estimate:

for every $\Omega'\subset\subset\Omega$, $1<p<\infty$. We also prove the local Hölder continuity for solutions to $\mathcal{L}f=g$ for any $g\in\mathcal{L}^p$ with $p$ large enough. Finally, we prove $\mathcal{L}^p$-estimates for higher order derivatives of $f$, whenever $g$ and the coefficients $a_{ij}$ are more regular.