On a kinetic Poincaré inequality and beyond

Abstract

In this article, we give a trajectorial proof of a kinetic Poincaré inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [12] in several directions. We use kinetic trajectories along the vector fields ${\partial }_t+v\cdot {\nabla }_x$ and ${\partial }_{v_i}$, $i=1,\dots ,d$ and do not rely on higher-order commutators such as $[{\partial }_{v_i},{\partial }_t+v\cdot {\nabla }_x]={\partial }_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by investigating a Kolmogorov equation with k steps.