the preimage of a∞(q0) for the local hardy-littlewoodmaximal operator

We describe here all those weight functions u such that mu ∈ a∞ (q) for m the local hardy-littlewood maximal operator restricted to a cube q ⊂ rn. in a recent paper it is shown that for the maximal operator in rn, mu ∈ a∞ implies that mu ∈ a1; here we see that the same is true for the local m but this imposes a stronger condition for weights in q, that is, for m restricted to a finite cube mu ∈ a∞ if and only if u ∈ a∞. this differs from the case in r n where there are weights u not belonging to a∞ such that mu is in a∞. as an application we get a new shorter proof of a result of i. wik. we also give a characterization for those weights in terms the k-functional of peetre.