New bounds for Stein's square functions in higher dimensions

TL;DR

This paper advances the understanding of Stein's square functions in higher dimensions by establishing improved bounds in the context of the Fourier restriction problem, with implications for local smoothing estimates.

Contribution

It provides the best-known bounds for Stein's square functions in dimensions four and higher, enhancing previous results in harmonic analysis.

Findings

Improved $L^p$ bounds for Stein's square functions in $ abla ext{dimensions} ext{ } n ext{ } ext{geq} 4$

Applications to local smoothing estimates in harmonic analysis

Connections to the Fourier restriction problem

Abstract

We improve the $L^p(\mathbb{R}^n)$ bounds on Stein's square function to the best-known range of the Fourier restriction problem when $n\geq4$. Applications including certain local smoothing estimates are also discussed.