  This note is devoted to a simple method for proving hypocoercivity of the
solutions of a kinetic equation involving a linear time relaxation operator,
i.e. the construction of an adapted Lyapunov functional satisfying a
Gronwall-type inequality. The method clearly distinguishes the coercivity at
microscopic level, which directly arises from the properties of the relaxation
operator, and a spectral gap inequality at the macroscopic level for the
spatial density, which is connected to the diffusion limit. It improves on
previously known results. Our approach is illustrated by the linear BGK model
and a relaxation operator which corresponds at macroscopic level to the
linearized fast diffusion.
