Emergence of exponentially weighted Lp-norms and Sobolev regularity for the Boltzmann equation

We consider the homogeneous Boltzmann equation for Maxwell and hard potentials, without cutoff, and study the appearance and propagation of L-p-norms, including polynomial and exponential weights. Propagation of Sobolev regularity with such weights is also considered. Classical and novel ideas are combined to elaborate an elementary argument that proves the result in the full range of integrability and singularity . For the case , we use an adaptation of the classical level set method by De Giorgi. All arguments are performed using the weak formulation which simplifies considerably the technicalities.