Maximal operators for cube skeletons

@article{Olivo2018MaximalOF,
  title={Maximal operators for cube skeletons},
  author={Andrea Olivo and Pablo Shmerkin},
  journal={Annales Academiae Scientiarum Fennicae Mathematica},
  year={2018}
}
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… 

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