We establish new regularity estimates, in terms of Sobolev spaces, of the solution $f$ to a kinetic equation. The right-hand side can contain partial derivatives in time, space and velocity, as in classical averaging, and $f$ is assumed to have a certain amount of regularity in velocity. The result is that $f$ is also regular in time and space, and this is related to a commutator identity introduced by Hörmander for hypoelliptic operators. In contrast with averaging, the number of derivatives does not depend on the $L^p$ space considered. Three type of proofs are provided: one relies on the Fourier transform, another one uses Hörmander's commutators, and the last uses a characteristics commutator. Regularity of averages in velocity are deduced. We apply our method to the linear Fokker-Planck operator and recover the known optimal regularity, by direct estimates using Hörmander's commutator.

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