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.