Sharp local well-posedness and nonlinear smoothing for dispersive equations through frequency-restricted estimates

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.