On estimates of the Boltzmann collision operator with cutoff

Abstract

We present new estimates of the Boltzmann collision operator in weighted Lebesgue and Bessel potential spaces. The main focus is put on hard potentials under the assumption that the angular part of the collision kernel fulfills some weighted integrability condition. In addition, the proofs for some previously known \mathbb{L}_{p^{-}} estimates have been considerably shortened and carried out by elementary methods. For a class of metric spaces, the collision integral is seen to be a continous operator into the same space. Furthermore, we give a new pointwise lower bound as well as asymptotic estimates for the loss term without requiring that the entropy is finite.