$W^{1,p}_{X}$ interior estimates for variational hypoelliptic operator with $VMO_X$ coefficients

TL;DR

This paper establishes interior $W^{1,p}_X$ estimates for solutions to hypoelliptic divergence form operators with coefficients in the VMO space relative to the Carnot--Caratheodory metric, under H"ormander's condition.

Contribution

It extends interior regularity results for hypoelliptic operators to coefficients in VMO_X, a significant generalization beyond continuous or smooth coefficients.

Findings

Interior $L^{p}$ estimates for weak solutions are obtained.

Results apply to operators with VMO_X coefficients under H"ormander condition.

The estimates hold for $2 \,\leq\, p < \infty$.

Abstract

We consider a divergence form hypoelliptic operator consisting of a system of real smooth vector fields $X_{1},..., X_{q}$ satisfying H\"ormander condition in some domain $\Omega\subseteq\erren$. Interior $L^{p}$ estimates, $2\leq p<\infty$, can be obtained for weak solutions of the equation $X_j^T(a^{ij}X_iu)=X_j^T F^j,$ by assuming that the coefficients $a^{ij}$ belong locally to the space $VMO_X$ with respect to the Carnot--Caratheodory metric induced by the vector fields.