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 ...

In [7], [1], [2], local decay estimates were proven for the (decoupled) Schrödinger, wave, and Regge-Wheeler equations on the Schwarzschild manifold, using commutator methods. Here, we correct a step in the commutator argument. The corrected argument works either for radial semilinear equations or general linear equations. This recovers the results in [7] and [2], but does not recover the non r...

It is well known, and was first proved by Sylow (1872), that every group of prime-power order p has a presentation of the form {a1, . . . , an | api = vii, 1 ≤ i ≤ n, [ak, aj ] = vjk, 1 ≤ j < k ≤ n}, where vjk is a word in the elements ak+1, . . . , an for 1 ≤ j ≤ k ≤ n . A presentation of this form is now called a power-commutator presentation for the group and written briefly {A | R} . As was...