"Sobolev-Poincare type" inequalities for kinetic equations

We present a regularity result for kinetic averaging of solutions of kinetic equations, with no a priori assumption on the solution itself, but only assumptions on the data of the problem (source term, initial condition). First, we will recall the history of "kinetic averaging lemmas" in the litterature and show the originality of our result. Then, we will explain our main motivation for such a result : how kinetic interpretation can be useful to understand hyperbolic problems such as conservation laws and micromagnetism. Finally, we will give ideas of proof.