Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces

TL;DR

This paper develops new moment inequalities for trigonometric polynomials with frequencies on curved hypersurfaces, using multilinear oscillatory integral theory, leading to improved estimates for discrete exponential sums and toral eigenfunctions.

Contribution

It introduces a novel approach applying multilinear oscillatory integral theory to establish moment inequalities for polynomials with spectrum on curved hypersurfaces.

Findings

New square function inequality for oscillatory integrals

Distributional estimates for toral eigenfunctions

Enhanced bounds for discrete exponential sums

Abstract

Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a new distributional estimate for toral eigenfunctions is established.