L^p and Schauder estimates for nonvariational operators structured on H\"ormander vector fields with drift

TL;DR

This paper establishes new regularity estimates for a class of second-order nonvariational PDEs structured on Hörmander vector fields, including operators with drift terms like Kolmogorov-Fokker-Planck, under various coefficient regularity conditions.

Contribution

It proves Schauder and L^p estimates for these operators with coefficients in Hölder or VMO spaces, extending regularity theory to operators with drift terms.

Findings

Established local Schauder estimates for Hölder continuous coefficients.

Proved local L^p estimates for VMO coefficients.

Included operators with drift, such as Kolmogorov-Fokker-Planck.

Abstract

We consider linear second order nonvariational partial differential operators of the kind a_{ij}X_{i}X_{j}+X_{0}, on a bounded domain of R^{n}, where the X_{i}'s (i=0,1,2,...,q, n>q+1) are real smooth vector fields satisfying H\"ormander's condition and a_{ij} (i,j=1,2,...,q) are real valued, bounded measurable functions, such that the matrix {a_{ij}} is symmetric and uniformly positive. We prove that if the coefficients a_{ij} are H\"older continuous with respect to the distance induced by the vector fields, then local Schauder estimates on X_{i}X_{j}u, X_{0}u hold; if the coefficients belong to the space VMO with respect to the distance induced by the vector fields, then local L^{p} estimates on X_{i}_{j}u, X_{0}u hold. The main novelty of the result is the presence of the drift term X_{0}, so that our class of operators covers, for instance, Kolmogorov-Fokker-Planck operators.