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.

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