Smoothed scattering: Randomized initial data and the deterministic case

In this joint word with Nicolas Burq, Nicola Visciglia and Nikolay Tzvetkov we consider two different cases for scattering and prove that the scattering operator minus the identity is more regular than the initial data. For randomized initial data we use the lens transform to transform the problem to a finite interval. In the deterministic case for the nonlinear Schrödinger equation and the Korteweg-de Vries equation with nonlinearity |u|^p u, p >4 we use a local smoothing estimate and a maximal estimate to gain up to one derivative for the difference above.