Gain of regularity for the relativistic collision operator

Abstract

We study a regularity property for the gain part of the relativistic Boltzmann collision operator. Our assumptions on the collisional cross-section cover the full range of generic hard and soft potentials with angular cut-off. We first take the Fourier transform of the gain operator and extract the modulus of Fourier symbol of the first order by exploiting the positive-definiteness of the relativistic correction in the post-collisional momentum. We then provide two different proofs based on Young's convolution inequality and the Hardy–Littlewood–Sobolev inequality. © 2018 Elsevier Ltd