We consider kinetic Fokker-Planck (or Vlasov-Fokker-Planck) equations on the
torus with Maxwellian or fat tail local equilibria. Results based on weak norms
have recently been achieved by S. Armstrong and J.-C. Mourrat in case of
Maxwellian local equilibria. Using adapted Poincar\’e and Lions-type
inequalities, we develop an explicit and constructive method for estimating the
decay rate of time averages of norms of the solutions, which covers various
regimes corresponding to subexponential, exponential and superexponential
(including Maxwellian) local equilibria. As a consequence, we also derive
hypocoercivity estimates, which are compared to similar results obtained by
other techniques.
