New bounds on Cantor maximal operators

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. Laba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlforsregular Salem Cantor sets of any dimension > 1/2 such that the associated maximal operator is bounded on L 2 (R). We follow the overall scheme of Laba-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.