Sharp weak type estimates for a family of Córdoba bases

Let B be a collection of rectangular parallelepipeds in R³ whose sides are parallel to the coordinate axes and such that B consists of parallelepipeds with sidelengths of the form s, t, 2 ^Nst, where s, t> 0 and N lies in a nonempty subset S of the natural numbers. In this paper, we prove the following: If S is a finite 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)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))=0.Alternatively, 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)2but 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.