We focus on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann transport problem in n dimensions assuming in the case of S^n-1-integrability of the angular part of the collision kernel (Grad cut-off assumption) with data near Maxwellian distributions.

We will show convolution estimates of Young's inequality type for the case of hard potentials and Hardy-Littlewood-Sobolev and Brascamp Lieb type inequality for soft potentials. The main technique is radial average symmetrization using classical tools of harmonic analysis. Then, using the Kaniel-Shinbrot iteration we present elementary proofs of existence for initial data near local Maxwellians to obtain globally bounded solutions for soft potentials. We also study the propagation of regularity using the convolution estimates estimate for the gain operator, and an L^p-stability result, with 1 ≤ p ≤ ∞.

This is work in collaboration with Ricardo Alonso, and partly with Emanuel Carneiro as well.