On some maximal multipliers in $L^p$

TL;DR

This paper generalizes Bourgain's $L^2$ maximal multiplier result to all $L^p$ spaces for $1<p< \infty$, broadening the applicability of maximal multiplier theorems in harmonic analysis.

Contribution

The paper extends Bourgain's maximal multiplier theorem from $L^2$ to all $L^p$ spaces, filling a significant gap in the theory.

Findings

Maximal multiplier bounds established for all $L^p$ spaces

Extension of Bourgain's $L^2$ result to $1<p<\infty$

Enhanced understanding of multipliers in harmonic analysis

Abstract

We extend an $L^2$ maximal multiplier result of Bourgain to all $L^p$ spaces, $1<p<\infty$.