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$liesinanonemptysubset$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 $$\mid \{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \}\mid \le C{\int }_{{\mathbb{R}}^3}\frac{\mathrm{\mid }f\mathrm{\mid }}{\alpha }{(1+{\log }^{+}\frac{\mathrm{\mid }f\mathrm{\mid }}{\alpha })}^2$$ but does not satisfy an estimate of the form $$\mid \{x\in {\mathbb{R}}^3:M_{\mathcal{B}}f(x)>\alpha \}\mid \le C{\int }_{{\mathbb{R}}^3}\varphi \left(\frac{\mathrm{\mid }f\mathrm{\mid }}{\alpha }\right)$$ for any convex increasing function $\phi: \mathbb[0, \infty) \rightarrow [0, \infty)$satisfyingthecondition$$\lim_{x \rightarrow \infty}\frac{\phi(x)}{x (\log(1 + x))^2} = 0;.$$