On a kinetic Poincaré inequality and beyond

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 [10] in several directions. We use kinetic trajectories along the vector fields ${\mathrm{\partial }}_t+v\cdot {\mathrm{\nabla }}_x$ and ${\mathrm{\partial }}_{v_i}$, $i=1,\dots ,d$ and do not rely on higher-order commutators such as $[{\mathrm{\partial }}_{v_i},{\mathrm{\partial }}_t+v\cdot {\mathrm{\nabla }}_x]={\mathrm{\partial }}_{x_i}$ or on the fundamental solution. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps.