  Let $\mathcal{B}$ be a collection of rectangular parallelepipeds in
$\mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that
$\mathcal{B}$ consists of parallelepipeds with side lengths of the form $s, 2^j
s, t $, where $s, t > 0$ and $j$ lies in a nonempty subset $S$ of the integers.
In this paper, we prove the following:
  If $S$ is a finite set, then the associated geometric maximal operator
$M_\mathcal{B}$ satisfies the weak type estimate of the form
  $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) >
\alpha\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}{\alpha}\left(1 +
\log^+ \frac{|f|}{\alpha}\right)\;$$ but does not satisfy an estimate of the
form
  $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) >
\alpha\right\}\right| \leq C \int_{\mathbb{R}^3}
\phi\left(\frac{|f|}{\alpha}\right)$$ for any convex increasing function $\phi:
\mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x
\rightarrow \infty}\frac{\phi(x)}{x (\log(1 + x))} = 0\;.$$
  On the other hand, if $S$ is an infinite set, then the associated geometric
maximal operator $M_\mathcal{B}$ satisfies the weak type estimate
  $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) >
\alpha\right\}\right| \leq C \int_{\mathbb{R}^3} \frac{|f|}{\alpha} \left(1 +
\log^+ \frac{|f|}{\alpha}\right)^{2}$$
  but does not satisfy an estimate of the form
  $$\left|\left\{x \in \mathbb{R}^3 : M_{\mathcal{B}}f(x) >
\alpha\right\}\right| \leq C \int_{\mathbb{R}^3}
\phi\left(\frac{|f|}{\alpha}\right)$$ for any convex increasing function $\phi:
\mathbb[0, \infty) \rightarrow [0, \infty)$ satisfying the condition $$\lim_{x
\rightarrow \infty}\frac{\phi(x)}{x (\log(1 + x))^2} = 0\;.$$
