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 integral([0,1]d) M-B(f) < infinity and integral([0,1]d) vertical bar f vertical bar log(+) vertical bar f vertical bar < infinity 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.