Hypoelliptic Regularity in Kinetic Equations

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 HH ormander 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 HH ormander's com-mutators, 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 HH ormander's commutator.