  For the homogeneous Boltzmann equation with (cutoff or non cutoff) hard
potentials, we prove estimates of propagation of Lp norms with a weight $(1+
|x|^2)^q/2$ ($1 < p < +\infty$, $q \in \R\_+$ large enough), as well as
appearance of such weights. The proof is based on some new functional
inequalities for the collision operator, proven by elementary means.
