Sharp Weak Type Estimates for a Family of Soria Bases

Let B be a collection of rectangular parallelepipeds in R³ whose sides are parallel to the coordinate axes and such that B contains parallelepipeds with side lengths of the form s,2Ns,t, where s, t> 0 and N lies in a nonempty subset S of the natural numbers. We show that if S is an infinite set, then the associated geometric maximal operator M_B satisfies the weak type estimate |{x∈R3:MBf(x)>α}|≤C∫R3|f|α(1+log+|f|α)2,but does not satisfy an estimate of the form |{x∈R3:MBf(x)>α}|≤C∫R3ϕ(|f|α)for any convex increasing function ϕ: [0 , ∞) → [0 , ∞) satisfying the condition limx→∞ϕ(x)x(log(1+x))2=0.