A Harnack inequality and H\"older continuity for weak solutions to parabolic operators involving H\"ormander vector fields

TL;DR

This paper establishes a Harnack inequality and H"older continuity for weak solutions to parabolic equations involving H"ormander vector fields, extending classical results to more general, possibly nonlinear, settings.

Contribution

It adapts Moser's iteration scheme to parabolic operators with H"ormander vector fields and extends these results to nonlinear equations, providing new regularity insights.

Findings

Proved a parabolic Harnack inequality for weak solutions.

Established H"older continuity of solutions under bounded elliptic coefficients.

Extended the iteration scheme to nonlinear parabolic equations.

Abstract

This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with H\"ormander vector fields. Adapting the iteration scheme of J\"urgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show H\"older continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parabolic equations of the form $u_t = -X_i^* A_j(X_j u)$. With this we show both a Harnack inequality and H\"older continuity of weak solutions.