Probabilistic Fourier extension and the Schrödinger equation

We give applications of the probabilistic Fourier extension theorem for the paraboloid in d dimensions (see [Saw7] for the sphere, and the proof there holds for a smooth compact piece of the paraboloid) to the global initial value problem for the Schrödinger equation with initial data f, in which we obtain sharp averaged Sobolev regularity results over certain `martingale transforms’ of the intial data f. The deterministic analogue in three dimensions has been obtained recently by OpenAI [Ope].