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\begin{document}$$\mathcal {B}$$\end{document} be a collection of rectangular parallelepipeds in R3\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^3$$\end{document} whose sides are parallel to the coordinate axes and such that B\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {B}$$\end{document} consists of parallelepipeds with sidelengths of the form s,t,2Nst\documentclass[12pt]{minimal}
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\begin{document}$$s, t, 2^N st$$\end{document}, where s,t>0\documentclass[12pt]{minimal}
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\begin{document}$$s, t > 0$$\end{document} 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 MB\documentclass[12pt]{minimal}
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\begin{document}$$M_\mathcal {B}$$\end{document} satisfies the weak type estimate x∈R3:MBf(x)>α≤C∫R3|f|α1+log+|f|α\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \frac{|f|}{\alpha }\left( 1 + \log ^+ \frac{|f|}{\alpha }\right) \; \end{aligned}$$\end{document}but does not satisfy an estimate of the form x∈R3:MBf(x)>α≤C∫R3ϕ|f|α\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \phi \left( \frac{|f|}{\alpha }\right) \end{aligned}$$\end{document}for any convex increasing function ϕ:[0,∞)→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\phi : [0, \infty ) \rightarrow [0, \infty )$$\end{document} satisfying the condition limx→∞ϕ(x)x(log(1+x))=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{\phi (x)}{x (\log (1 + x))} = 0\;. \end{aligned}$$\end{document}Alternatively, if S is an infinite set, then the associated geometric maximal operator MB\documentclass[12pt]{minimal}
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\begin{document}$$M_\mathcal {B}$$\end{document} satisfies the weak type estimate x∈R3:MBf(x)>α≤C∫R3|f|α1+log+|f|α2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \frac{|f|}{\alpha } \left( 1 + \log ^+ \frac{|f|}{\alpha }\right) ^{2} \end{aligned}$$\end{document}but does not satisfy an estimate of the form x∈R3:MBf(x)>α≤C∫R3ϕ|f|α\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left| \left\{ x \in \mathbb {R}^3 : M_{\mathcal {B}}f(x) > \alpha \right\} \right| \le C \int _{\mathbb {R}^3} \phi \left( \frac{|f|}{\alpha }\right) \end{aligned}$$\end{document}for any convex increasing function ϕ:[0,∞)→[0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\phi : [0, \infty ) \rightarrow [0, \infty )$$\end{document} satisfying the condition limx→∞ϕ(x)x(log(1+x))2=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \lim _{x \rightarrow \infty }\frac{\phi (x)}{x (\log (1 + x))^2} = 0. \end{aligned}$$\end{document}
