On the integrability of multi-dimensional rare maximal functions

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 \(B\).