A new approach to velocity averaging lemmas in {B}esov spaces

TL;DR

This paper introduces a novel dispersive approach to velocity averaging lemmas in Besov spaces, achieving sharp results and handling complex integrability cases relevant to kinetic equations.

Contribution

It presents a new dispersive method for velocity averaging lemmas, improving sharpness and applicability in Besov spaces, including degenerate cases.

Findings

Achieved sharp velocity averaging results with a gain of one full derivative.

Extended the analysis to $L_x^rL^p_v$ integrability with $r \,\leq\, p$.

Controlled concentrations in the degenerate $L_{x,v}^1$ case for Boltzmann equation analysis.

Abstract

We develop a new approach to velocity averaging lemmas based on the dispersive properties of the kinetic transport operator. This method yields unprecedented sharp results, which display, in some cases, a gain of one full derivative. Moreover, the study of dispersion allows to treat the case of $L_x^rL^p_v$ integrability with $r\leq p$. We also establish results on the control of concentrations in the degenerate $L_{x,v}^1$ case, which is fundamental in the study of the hydrodynamic limit of the Boltzmann equation.