Commutator method for averaging lemmas

Abstract

We introduce a commutator method with multiplier to prove averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. Our method requires only elementary techniques in Fourier analysis and highlights a new range of assumptions that are sufficient for the velocity average to be in $L^2([0,T],H_x^{1∕2})$. Our result provides a direct proof (without interpolation) and improves the regularizing result for the measure-valued solutions to scalar conservation laws in space dimension 1.