About $L^p$ estimates for the spatially homogeneous Boltzmann equation
  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. 
Laurent Desvillettes (CMLA), Cl\'ement Mouhot (CEREMADE)