Gehring's Lemma for kinetic Fokker-Planck equations

TL;DR

This paper extends Gehring's lemma to kinetic Fokker-Planck equations, showing that weak solutions' velocity gradients have higher integrability than previously guaranteed, using a novel hypoelliptic approach.

Contribution

It introduces a kinetic version of Gehring's lemma, localizes it in hypoelliptic settings, and applies it to prove higher integrability of solutions' velocity gradients.

Findings

Velocity gradients are more integrable than energy estimates suggest.

The kinetic Gehring lemma is established with a streamlined proof.

Application to linear kinetic equations shows improved regularity.

Abstract

In this article, we establish a "Gehring lemma" for a real function satisfying a reverse H\"older inequality on all "kinetic cylinders" contained in a large one: it asserts that the integrability degree of the function improves under such an assumption. The kinetic cylinders are derived from the non-commutative group of invariances of the Kolmogorov equation. Our contributions here are (1) the extension of Gehring's Lemma to this kinetic (hypoelliptic) scaling used to generate the cylinders, (2) the localisation of the lemma in this hypoelliptic context (using ideas from the elliptic theory), (3) the streamlining of a short and quantitative proof. We then use this lemma to establish that the velocity gradient of weak solutions to linear kinetic equations of Fokker-Planck type with rough coefficients have Lebesgue integrability strictly greater than two, while the natural energy estimate merely ensures that it is square integrable. Our argument here is new but relies on Poincar\'e-type inequalities established in previous works.