Maximal estimates for averages over space curves

TL;DR

This paper establishes the boundedness of a maximal operator over smooth space curves with nonvanishing curvature and torsion on L^p spaces, precisely for p>3, advancing understanding of geometric maximal functions.

Contribution

It proves the L^p boundedness of the maximal operator associated with smooth space curves with nonvanishing curvature and torsion, for p>3, filling a gap in harmonic analysis.

Findings

Maximal operator is bounded on L^p for p>3

Boundedness fails for p≤3

Results extend understanding of geometric maximal functions

Abstract

Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$.