New bounds on Cantor maximal operators

TL;DR

This paper establishes new $L^p$ bounds for maximal operators linked to Ahlfors-regular fractal percolation sets, improving previous results and demonstrating boundedness on $L^2$ for certain Salem Cantor sets of dimension greater than 1/2.

Contribution

It introduces sharper bounds for fractal percolation maximal operators and extends boundedness results to a broader class of Salem Cantor sets, combining analytic and probabilistic techniques.

Findings

Improved $L^p$ bounds for fractal percolation maximal operators.

Existence of Salem Cantor sets with dimension > 1/2 where the maximal operator is bounded on $L^2$.

Results are sharp up to the endpoint in some cases.

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.