Résumé
In this article, we establish a "Gehring lemma" for a real function
satisfying a reverse Hölder 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 Poincaré-type inequalities established in previous works.