Abstract: 
In this talk, we will present an extension of Villani's hypocoercivity theorem for the kinetic FokkerPlanck equation in the $H^1$ space to the $H^k$ spaces. As in the $L^2$ and $H^1$ setting, there is lack of coercivity in $H^k$ for the associated operator. To overcome that, we will modify the usual $H^k$ norm with certain mixed terms and suitable coefficients which are constructed by induction on $k$. In parallel, a similar strategy but with coefficients depending on time, sometimes referred as H\'erau's method, can be employed to prove global hypoellipticity in $H^k$ for the solutions to the kinetic FokkerPlanck equation. Such regularity results provide a second proof of hypocoercivity in $H^k$ spaces, in light of the hypocoercivity theorems due to Villani or Dolbeault, Mouhot and Schmeiser. 
