  We consider the problem of establishing nonlinear smoothing as a general
feature of nonlinear dispersive equations, i.e. the improved regularity of the
integral term in Duhamel's formula, with respect to the initial data and the
corresponding regularity of the linear evolution, and how this property relates
to local well-posedness. In a first step, we show how the problem generally
reduces to the derivation of specific frequency-restricted estimates, which are
multiplier estimates in the spatial frequency alone. Then, using a precise
methodology, we prove these estimates for the specific cases of the modified
Zakharov-Kuznetsov equation, the cubic and quintic nonlinear Schr\"odinger
equation and the quartic Korteweg-de Vries equation.

Analysis of PDEs