  This paper is dealing with two $L^2$ hypocoercivity methods based on Fourier
decomposition and mode-by-mode estimates, with applications to rates of
convergence or decay in kinetic equations on the torus and on the whole
Euclidean space. The main idea is to perturb the standard $L^2$ norm by a twist
obtained either by a nonlocal perturbation build upon diffusive macroscopic
dynamics, or by a change of the scalar product based on Lyapunov matrix
inequalities. We explore various estimates for equations involving a
Fokker-Planck and a linear relaxation operator. We review existing results in
simple cases and focus on the accuracy of the estimates of the rates. The two
methods are compared in the case of the Goldstein-Taylor model in
one-dimension.
