On the frequency function of Hardy-Littlewood maximal functions

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 ${\mathrm{\ell }}^1(\mathbb{Z})$ and $L^1({\mathbb{R}}^d)$ answering some questions posed by Temur. Finally, we study the size of the frequency function for functions in ${\mathrm{\ell }}^p(\mathbb{Z})$ with $p>1$, showing that this case differs significantly from the case $p=1$.