Refined Strichartz estimates and their orthornomal counterparts for Schr\"odinger equations on torus

Abstract

The aim of the paper is twofold. We establish refined Strichartz estimates for the Schr\"odinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schr\"odinger equation has better regularity in mixed Lebesgue spaces. This complements the well-established theory over the past few decades, where initial data comes from the Sobolev space with respect to all spatial variables. As an application, we obtain local well-posedness for non-gauge-invariant nonlinearities with partially regular initial data. On the other hand, we extend refined Strichartz estimates for infinite systems of orthonormal functions, which generalizes the classical orthonormal Strichartz estimates on the torus by Nakamura [41] . As an application, we establish well-posedness for the Hartree equation for infinitely many fermions in some Schatten spaces. In the process, we develop several harmonic analysis tools for mixed Lebesgue spaces, e.g. Fourier multiplier transference principle, vector-valued Bernstein inequality, and vector-valued Littlewood--Paley theory for densities of operators, which may be of independent interest and complement the results of [45,55].