Commutator method for averaging lemmas

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