Continuity for maximal commutator of Bochner-Riesz operators on some weighted Hardy spaces

which is the Bochner-Riesz operator (see [8]). Let E be the space E = {h : ‖h‖ = supr>0 |h(r)| <∞}, then, for each fixed x ∈ Rn, Bδ r ( f )(x) may be viewed as a mapping from [0,+∞) to E, and it is clear that Bδ ∗( f )(x) = ‖Bδ r ( f )(x)‖ and B ∗,b( f )(x) = ‖b(x)Bδ r ( f )(x)−Bδ r (b f )(x)‖. As well known, a classical result of Coifman et al. [4] proved that the commutator [b,T] generated by BMO(Rn) functions and the Calderón-Zygmund operator is bounded on Lp(Rn) (1 < p <∞). However, it was observed that [b,T] is not bounded, in general, from Hp(Rn) to Lp(Rn) and from L1(Rn) to L1,∞ (Rn) for p ≤ 1. But, if Hp(Rn) is replaced by some suitable atomic space H p b (R n) and H B(R n) (see [1, 6, 7, 9]), then [b,T] maps continuously H p b (R n) into Lp(Rn) andH B(R n) into weak L1(Rn) for p ∈ (n/(n+ 1),1]. The main purpose of this paper is to establish the weighted boundedness of the commutators