The boundedness of commutators of multilinear Marcinkiewicz integral

Abstract

Let μ be the multilinear Marcinkiewicz integral, b→∈(BMO(Rn))m and w→=(w1,…,wm)∈AP→, where P→=(p1,…,pm), pi∈[1,∞), for i∈{1,…,m}, and AP→ denotes the multiple weight class introduced by Lerner et al. (2009). In this article, for pi∈(1,∞), we prove that the commutators μb→, generated by the multilinear Marcinkiewicz integral μ and b→, are bounded from Lw1p1(Rn)×⋯×Lwmpm(Rn) to Lνw→p(Rn) if b→∈(BMO(Rn))m, where 1∕p=1∕p1+⋯+1∕pm and νw→(x)=∏i=1mwi(x)p∕pi. For the endpoint case, we show two kinds of estimates. One is that if w→∈A(1,…,1) and b→∈(BMO(Rn))m, then we obtain a weighted weak LlogL type estimate for μb→. On the other hand, it is well known that the commutator μb→ may not be bounded from Hw1(Rn)×⋯×Hw1(Rn) to Lw1∕m(Rn) if b→∈(BMO(Rn))m. We obtain that when w∈A1+min{1∕2,γ}∕mn(Rn) satisfying ∫Rnw(x)1+|x|ndx<∞ and b→∈(BMOw(Rn))m⊂(BMO(Rn))m, the commutator μb→ is bounded from Hw1(Rn)×⋯×Hw1(Rn) to Lw1∕m(Rn), where BMOw(Rn) is a subspace of BMO(Rn). This result is new even for the linear case. Let T be the multilinear Calderón–Zygmund operator. We also obtain the boundedness of commutator [b→,T] from Hw1(Rn)×⋯×Hw1(Rn) to Lw1∕m(Rn) if b→∈(BMOw(Rn))m.