We characterize the translation invariant monotone collections of multi-dimensional intervals for which the analogue of Stein's criterion for the integrability of the Hardy--Littlewood maximal function is true. Namely, we characterize the collections \(B\) of the mentioned type for which the conditions \(\int_{[0,1]^d}M_B(f)<\infty\) and \(\int_{[0,1]^d}\vert f\vert \log^+\vert f\vert <\infty\) are equivalent for functions \(f\) supported on the unit cube \([0,1]^d\). Here \(M_B\) denotes the maximal operator associated to a collection \(B\).

中文翻译：

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多维稀有极大函数的可积性

我们描述了多维区间的平移不变单调集合，对于该集合，Hardy-Littlewood 极大函数的可积性的 Stein 准则的类似物是正确的。也就是说，我们描述上述类型的集合\(B\)，其条件为\(\int_{[0,1]^d}M_B(f)<\infty\)和\(\int_{[0, 1]^d}\vert f\vert \log^+\vert f\vert <\infty\) 与 单位立方体\([0,1]^d\)支持的函数\(f\ )等效。这里\(M_B\)表示与集合\(B\)关联的最大运算符。