  This paper is concerned with a modified entropy method to establish the
large-time convergence towards the (unique) steady state, for kinetic
Fokker-Planck equations with non-quadratic confinement potentials in whole
space. We extend previous approaches by analyzing Lyapunov functionals with
non-constant weight matrices in the dissipation functional (a generalized
Fisher information). We establish exponential convergence in a weighted
$H^1$-norm with rates that become sharp in the case of quadratic potentials. In
the defective case for quadratic potentials, i.e. when the drift matrix has
non-trivial Jordan blocks, the weighted $L^2$-distance between a
Fokker-Planck-solution and the steady state has always a sharp decay estimate
of the order $\mathcal O\big( (1+t)e^{-t\nu/2}\big)$, with $\nu$ the friction
parameter. The presented method also gives new hypoelliptic regularization
results for kinetic Fokker-Planck equations (from a weighted $L^2$-space to a
weighted $H^1$-space).
