Commutator method for averaging lemmas

We introduce a commutator method with multipliers to prove averaging lemmas,\nthe regularizing effect for the velocity average of solutions for kinetic\nequations. This method requires only elementary techniques in Fourier analysis\nand shows a new range of assumptions that are sufficient for the velocity\naverage to be in $L^2([0,T],H^{1/2}_x).$ This result not only shows an\ninteresting connection between averaging lemmas and local smoothing property of\ndispersive equations, but also provide a direct proof for the regularizing\neffect for the measure-valued solutions of scalar conservation laws in space\ndimension one.\n