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This paper is dealing with two $L^2$ hypocoercivity methods based on Fourierdecomposition and mode-by-mode estimates, with applications to rates ofconvergence or decay in kinetic equations on the torus and on the wholeEuclidean space. The main idea is to perturb the standard $L^2$ norm by a twistobtained either by a nonlocal perturbation build upon diffusive macroscopicdynamics, or by a change of the scalar product based on Lyapunov matrixinequalities. We explore various estimates for equations involving aFokker-Planck and a linear relaxation operator. We review existing results insimple cases and focus on the accuracy of the estimates of the rates. The twomethods are compared in the case of the Goldstein-Taylor model inone-dimension.

Sharpening of decay rates in Fourier based hypocoercivity methods