Sharp weak type estimates for a family of Soria bases

Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in $\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $\mathcal{B}$ contains parallelepipeds with side lengths of the form $s, \frac{2^N}{s} , t$, 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 $M_\mathcal{B}$ satisfies the weak type estimate $\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}α \left(1 + \log^+ \frac{|f|}α\right)^{2}$ but does not satisfy an estimate of the form $\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) > α\right\}\right| \leq C \int_{\mathbb{R}^3} ϕ\left(\frac{|f|}α\right)$ for any convex increasing function $ϕ: \mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $\lim_{x \rightarrow \infty}\frac{ϕ(x)}{x (\log(1 + x))^2} = 0\;.$

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