Discrete Fourier restriction associated with Schrodinger equations

TL;DR

This paper provides a new proof for discrete Fourier restriction related to Schrödinger equations, recovering key results and establishing sharp estimates, with applications to boundedness of discrete multilinear maximal functions.

Contribution

It introduces a novel proof technique for discrete Fourier restriction and extends sharp estimates to higher dimensions, with applications to multilinear maximal functions.

Findings

Recovered Bourgain's level set result on Strichartz estimates

Established sharp $L^{rac{2(d+2)}{d}}$ norm estimates for exponential sums

Proved boundedness of certain discrete multilinear maximal functions on $L^2(\

Abstract

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schr\"odinger equations on torus. Some sharp estimates on $L^{\frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(\mathbb Z)$.