# 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…

## 5 Citations

### Weighted Estimates for Maximal Functions Associated to Skeletons

- MathematicsThe Journal of Geometric Analysis
- 2019

We provide quantitative weighted estimates for the $$L^p(w)$$ L p ( w ) norm of a maximal operator associated to cube skeletons in $${\mathbb {R}}^n$$ R n . The method of proof differs from the usual…

### Off-diagonal estimates for cube skeleton maximal operators

- MathematicsColloquium Mathematicum
- 2021

In this work we obtain $L^p \rightarrow L^q$ ($p\leq q$) estimates for the $k$-skeleton maximal operator in $\mathbb{R}^n$. This is achieved by interpolating a weak-type $(1,q^*)$ endpoint estimate,…

### Maximal operators and decoupling for $\Lambda(p)$ Cantor measures

- Mathematics
- 2018

For $2\leq p 2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha 0$, and have no Fourier decay. The proof is based on a decoupling…

### MAXIMAL OPERATORS AND DECOUPLING FOR Λ(p) CANTOR MEASURES

- Mathematics
- 2018

For 2 ≤ p <∞, α′ > 2/p, and δ > 0, we construct Cantor-type measures on R supported on sets of Hausdorff dimension α < α′ for which the associated maximal operator is bounded from Lpδ(R) to L(R).…

### Weighted Estimates for Maximal Functions Associated to Skeletons

- Materials ScienceThe Journal of Geometric Analysis
- 2019

We provide quantitative weighted estimates for theLp(w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

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