Maximal Operators for cube skeletons

TL;DR

This paper investigates discretized maximal operators related to averaging over cube skeletons in Euclidean space, providing nearly optimal bounds and extending recent geometric measure theory results.

Contribution

It introduces nearly sharp $L^p$ bounds for discretized maximal operators associated with cube skeletons, extending prior work on geometric configurations in Euclidean spaces.

Findings

Established nearly sharp $L^p$ bounds for small discretization scales.

Extended recent results on sets containing scaled $k$-skeletons of the cube.

Connected maximal operator bounds to geometric measure theory problems.

Abstract

We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, $k$-skeletons in $\mathbb{R}^n$. Although these operators are known not to be bounded on any $L^p$, we obtain nearly sharp $L^p$ bounds for every small discretization scale. These results are motivated by, and partially extend, recent results of T. Keleti, D. Nagy and P. Shmerkin, and of R. Thornton, on sets that contain a scaled $k$-sekeleton of the unit cube with center in every point of $\mathbb{R}^n$.