  The purpose of this paper is to extend the hypocoercivity results for the
kinetic Fokker-Planck equation in $H^1$ space in Villani's memoir
\cite{Villani} to higher order Sobolev spaces. As in the $L^2$ and $H^1$
setting, there is lack of coercivity in $H^k$ for the associated operator. To
remedy this issue, we shall modify the usual $H^k$ norm with certain
well-chosen mixed terms and with suitable coefficients which are constructed by
induction on $k$. In parallel, a similar strategy but with coefficients
depending on time (c.f. \cite{Herau}), usually referred as H\'erau's method,
can be employed to prove global hypoellipticity in $H^k$. The exponents in our
regularity estimates are optimal in short time. Moreover, as in our recent work
\cite{GLWZ}, the general results here can be applied in the mean-field setting
to get estimates independent of the dimension; in particular, an application to
the Curie-Weiss model is presented.
