Shmerkin, P., & Suomala, V. (2022). New bounds on Cantor maximal operators. Revista de La Unión Matemática Argentina, 69–86. https://doi.org/10.33044/revuma.3170

### New bounds on Cantor maximal operators

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Author: Shmerkin, Pablo1; Suomala, Ville2
Organizations: 1Department of Mathematics, the University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada
2Research Unit of Mathematical Sciences, P.O. Box 8000, FI-90014, University of Oulu, Finland
Format: article
Version: published version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Language: English
Published: Unión Matemática Argentina, 2022
Publish Date: 2023-01-03
Description:

# Abstract

We prove $$L^p$$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors-regular Salem Cantor sets of any dimension $$> 1/2$$ such that the associated maximal operator is bounded on $$L^2(\mathbb{R})$$. We follow the overall scheme of Łaba–Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures.

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Series: Revista de la Unión Matemática Argentina
ISSN: 0041-6932
ISSN-E: 1669-9637
ISSN-L: 0041-6932
Volume: 64
Issue: 1
Pages: 69 - 86
DOI: 10.33044/revuma.3170