On the frequency function of Hardy-Littlewood maximal functions

TL;DR

This paper extends the concept of the frequency function to higher-dimensional and uncentered Hardy-Littlewood maximal functions, analyzing its asymptotic behavior and differences across various function spaces.

Contribution

It introduces a higher-dimensional and uncentered version of the frequency function and investigates its properties, answering open questions and highlighting differences for p>1.

Findings

Asymptotic behavior of the frequency function analyzed

Density of small values characterized for ℓ¹(ℤ) and L¹(ℝ^d)

Significant differences found for p>1 cases

Abstract

We study the frequency function (introduced by Temur) in both the discrete and continuous settings. More precisely, we extend the definition of the frequency function to the higher-dimensional continuous setting and to the uncentered Hardy-Littlewood maximal function. We analyze the asymptotic behavior of the frequency function and the density of its small values for functions in $\ell^1(\mathbb{Z)}$ and $L^1(\mathbb{R}^d)$ answering some questions posed by Temur. Finally, we study the size of the frequency function for functions in $\ell^p(\mathbb{Z})$ with $p>1$, showing that this case differs significantly from the case $p=1$.