Regularity for rough hypoelliptic equations

TL;DR

This paper develops a general method to establish weak Harnack inequalities for rough hypoelliptic equations, such as kinetic equations, by comparing rough solutions with smooth problems without analyzing commutator structures.

Contribution

Introduces a constructive approach to derive weak Harnack inequalities for rough hypoelliptic equations, bypassing commutator analysis.

Findings

Established weak Harnack inequality for rough hypoelliptic equations

Method applicable to kinetic equations

Avoids detailed commutator structure analysis

Abstract

We present a general approach to obtain a weak Harnack inequality for rough hypoellipitic equations, e.g. kinetic equations. The proof is constructive and does not study the commutator structure but rather compares the rough solution with a smooth problem for which the estimates are assumed.