Sharp weak‐type estimate for maximal operators associated to Cartesian families under an arithmetic condition

Abstract

Given a set of integers , we consider the smallest family invariant by translations that contains the rectangles for any and where for integer. We prove that if contains arbitrary large arithmetic progression, then the maximal operator associated to the family satisfies a weak‐type estimate of the form but does not satisfy a weak‐type estimate of the form for any nonnegative convex increasing function such that as tends to infinity.