On a kinetic Poincar\'e inequality and beyond

In this note, we give an alternative proof of a Poincar\'e inequality due to J. Guerand and C. Mouhot, [4]. We use trajectories along the vector fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$. In contrast to [4] we do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t + v \cdot \nabla_x] = \partial_{x_i}$. Moreover, we improve their result in several directions. The presented method also applies to more general hypoelliptic equations. We illustrate this by studying a Kolmogorov equation with $k$ steps.