Sharp $L^p$ bounds for the helical maximal function

TL;DR

This paper proves sharp $L^p$ bounds for the helical maximal function in three dimensions for all $p>3$, improving previous bounds and introducing a new microlocal smoothing estimate through square function analysis.

Contribution

It establishes the $L^p$ boundedness of the helical maximal function for $p>3$, refining earlier results and employing novel microlocal smoothing techniques.

Findings

Boundedness of the helical maximal function for $p>3$

Improved bounds over previous results for $p>4$

Introduction of a new microlocal smoothing estimate

Abstract

We establish the $L^p(\mathbb{R}^3)$ boundedness of the helical maximal function for the sharp range $p>3$. Our results improve the previous known bounds for $p>4$. The key ingredient is a new microlocal smoothing estimate for averages along dilates of the helix, which is established via a square function analysis.