Improved bounds for Stein's square functions

TL;DR

This paper establishes improved bounds for Stein's square functions, providing new weighted inequalities and $L^p$ bounds for radial Fourier multipliers, with applications to wave and Schrödinger equations.

Contribution

It introduces new weighted norm inequalities for Stein's square functions and derives sharper $L^p$ bounds for radial Fourier multipliers in higher dimensions.

Findings

New weighted norm inequalities for the maximal Bochner--Riesz operator

Enhanced $L^p$ bounds for radial Fourier multipliers for $p \,\geq\, 2+4/d$

Space-time regularity results for wave and Schrödinger equations

Abstract

We prove a weighted norm inequality for the maximal Bochner--Riesz operator and the associated square-function. This yields new $L^p(R^d)$ bounds on classes of radial Fourier multipliers for $p\ge 2+4/d$ with $d\ge 2$, as well as space-time regularity results for the wave and Schr\"odinger equations.