  We show trilinear Strichartz estimates in one and two dimensions on
frequency-dependent time intervals. These improve on the corresponding linear
estimates of periodic solutions to the Schr\"odinger equation. The proof
combines decoupling iterations with bilinear short-time Strichartz estimates.
Secondly, we use decoupling to show new linear Strichartz estimates on
frequency dependent time intervals. We apply these in case of the Airy
propagator to obtain the sharp Sobolev regularity for the existence of
solutions to the modified Korteweg-de Vries equation.

Analysis of PDEs;Classical Analysis and ODEs