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arXiv · 2106.14818

New bounds on Cantor maximal operators

Pablo Shmerkin·Ville Suomala
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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. {\L}aba 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 {\L}aba-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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Pablo Shmerkin, Ville Suomala. 2021-06-28. New bounds on Cantor maximal operators. https://arxiv.org/abs/2106.14818

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