On the frequency function of Hardy-Littlewood maximal functions
Carlos Garzón, José Madrid
TL;DR
This work extends the notion of Temur's frequency function to higher-dimensional continuous settings and the uncentered Hardy-Littlewood maximal operator, examining its asymptotic behavior and the density of small values for $f$ in $L^1$-type spaces. It establishes that, in the discrete and higher-dimensional continuous settings, the ratio $r_x/|x|$ tends to $0$ or $1$ outside a finite region, and it proves density-zero results for the occurrence of small radii, while also constructing counterexamples in $\\ell^p(\\Z)$ for $p>1$ showing significant deviations from this behavior. The paper further demonstrates that density results fail in the $p>1$ regime, providing explicit constructions with nontrivial limsup densities and addressing the uncentered case with analogous phenomena. Overall, the results delineate the limits of the dichotomy observed for $p=1$, illuminate the regularity and extremal-parameter structure of maximal-operator radii, and answer several open questions posed by Temur.
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 $p=1$.