Remark on dimension-free estimates for discrete maximal functions over $\ell^q$ balls: small dyadic scales

TL;DR

This paper establishes dimension-free bounds for discrete Hardy-Littlewood maximal operators over ^q balls in ^d for small dyadic scales, extending previous results to a broader class of ^q balls.

Contribution

It provides new dimension-free estimates for the maximal operator over ^q balls with small dyadic radii, generalizing prior work to all q .

Findings

Dimension-free bounds for ^p(^d) for p .

Extension of estimates to ^q balls for q .

Combines with prior work to cover all p  and q  cases.

Abstract

We give a dimension-free bound on $\ell^p(\mathbb{Z} ^d)$, $p \in [2, \infty]$ for the discrete Hardy-Littlewood maximal operator over the $\ell^q$ balls in $\mathbb{Z} ^d$ with small dyadic radii. Our result combined with the work of Kosz, Mirek, Plewa, Wr\'obel gives dimension-free estimates on $\ell^p(\mathbb{Z}^d)$, $p \in [2, \infty]$ for the discrete dyadic Hardy-Littlewood maximal operator over $\ell^q$ balls for $q \geq 2$.