A trilinear approach to square function and local smoothing estimates for the wave operator

TL;DR

This paper advances the understanding of square function and local smoothing estimates for the wave operator in low-dimensional spaces by employing a novel trilinear approach based on recent decoupling and restriction theorems.

Contribution

It introduces a trilinear method to improve estimates for the wave operator, combining decoupling and restriction theorems in a new way.

Findings

Enhanced estimates for Mockenhaupt's square function in R^3

Improved local smoothing estimates in R^{2+1} spacetime

Application of trilinear techniques to wave operator analysis

Abstract

The purpose of this paper is to improve the known estimates for Mockenhaupt's square function in $\mathbb R^3$ and for Sogge's local smoothing in $\mathbb R^{2+1}$ spacetime. For this we use the trilinear approach of S. Lee and A. Vargas for the cone multiplier with some trilinear estimates obtained from the $\ell^2$ decoupling theorem and multilinear restriction theorem.