MAXIMAL OPERATORS FOR CUBE SKELETONS

We study discretized maximal operators associated to averaging over (neighborhoods of) squares in the plane and, more generally, k-skeletons in 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 Keleti, Nagy and Shmerkin, and of Thornton, on sets that contain a scaled k-sekeleton of the unit cube with center in every point of R-n.