  We prove explicit coercivity estimates for the linearized Boltzmann and
Landau operators, for a general class of interactions including any
inverse-power law interactions, and hard spheres. The functional spaces of
these coercivity estimates depend on the collision kernel of these operators.
They cover the spectral gap estimates for the linearized Boltzmann operator
with Maxwell molecules, improve these estimates for hard potentials, and are
the first explicit coercivity estimates for soft potentials (including in
particular the case of Coulombian interactions). We also prove a regularity
property for the linearized Boltzmann operator with non locally integrable
collision kernels, and we deduce from it a new proof of the compactness of its
resolvent for hard potentials without angular cutoff.
