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

Let X 1 , X 2 , … , X q X_1,X_2,\ldots ,X_q be a system of real smooth vector fields, satisfying Hörmander’s condition in some bounded domain Ω ⊂ R n \Omega \subset \mathbb {R}^n ( n > q n>q ). We consider the differential operator L = ∑ i = 1 q a i j ( x ) X i X j , \begin{equation*} \mathcal {L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where the coefficients a i j ( x ) a_{ij}(x) are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: μ | ξ | 2 ≤ ∑ i , j = 1 q a i j ( x ) ξ i ξ j ≤ μ − 1 | ξ | 2 \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 ∈ Ω x\in \Omega , every ξ ∈ R q \xi \in \mathbb {R}^q , some constant μ \mu . Moreover, we assume that the coefficients a i j 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 , … , X q X_1,X_2,\ldots ,X_q . We prove the following local L p \mathcal {L}^p -estimate: ‖ X i X j f ‖ L p ( Ω ′ ) ≤ c { ‖ L f ‖ L p ( Ω ) + ‖ f ‖ L p ( Ω ) } \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 Ω ′ ⊂ ⊂ Ω \Omega ’\subset \subset \Omega , 1 > p > ∞ 1>p>\infty . We also prove the local Hölder continuity for solutions to L f = g \mathcal {L}f=g for any g ∈ L p g\in \mathcal {L}^p with p p large enough. Finally, we prove L p \mathcal {L}^p -estimates for higher order derivatives of f f , whenever g g and the coefficients a i j a_{ij} are more regular.