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
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
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