University of Oulu

Shmerkin, P., & Suomala, V. (2022). New bounds on Cantor maximal operators. Revista de La Unión Matemática Argentina, 69–86.

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)
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Language: English
Published: Unión Matemática Argentina, 2022
Publish Date: 2023-01-03


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
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: PS was supported by project PICT 2015-3675 (ANPCyT) and by an NSERC discovery grant. VS was in part supported by the Academy of Finland. We also acknowledge support from the Institut Mittag-Leffler via the “Fractal Geometry and Dynamics” research program, where this project started.
Copyright information: © The Author(s). This work is licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.