H\"older regularity for weak solutions of H\"ormander type operators

TL;DR

This paper establishes H"older regularity for weak solutions of H"ormander type operators by adapting the Moser iteration technique to a hypoelliptic setting, using a parabolic Poincaré inequality and geometric insights.

Contribution

It introduces a parabolic Poincaré inequality and demonstrates how elliptic Moser iteration can be applied to hypoelliptic operators, unifying various existing results.

Findings

Proves H"older regularity for weak solutions of H"ormander type operators.

Develops a parabolic Poincaré inequality tailored for hypoelliptic equations.

Shows the applicability of elliptic Moser iteration in a parabolic hypoelliptic context.

Abstract

Motivated by recent results on the (possibly conditional) regularity for time-dependent hypoelliptic equations, we prove a parabolic version of the Poincar\'e inequality, and as a consequence, we deduce a version of the classical Moser iteration technique using in a crucial way the geometry of the equation. The point of this contribution is to emphasize that one can use the {\sl elliptic} version of the Moser argument at the price of the lack of uniformity, even in the {\sl parabolic } setting. This is nevertheless enough to deduce H\"older regularity of weak solutions. The proof is elementary and unifies in a natural way several results in the literature on Kolmogorov equations, subelliptic ones and some of their variations.