$L^p$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients

Abstract:

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 \begin{equation*} \mathcal {L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where the coefficients $a_{ij}(x)$ are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: \begin{equation*} \mu |\xi |^2\leq \sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq \mu ^{-1}|\xi |^2 \end{equation*} 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: \begin{equation*} \left \|X_iX_jf\right \|_{\mathcal {L}^p(\Omega ’)}\leq c\left \{\left \|\mathcal {L}f\right \|_{\mathcal {L}^p(\Omega )}+\left \|f\right \|_{\mathcal {L}^p(\Omega )}\right \} \end{equation*} 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.