The boundedness of commutators of multilinear Marcinkiewicz integral. (June 2020) - He, Sha; Liang, Yiyu
Abstract: Let μ be the multilinear Marcinkiewicz integral, b → ∈ ( BMO ( R n ) ) m and w → = ( w 1, …, w m ) ∈ A P →, where P → = ( p 1, …, p m ), p i ∈ [ 1, ∞ ), for i ∈ { 1, …, m }, and A P → denotes the multiple weight class introduced by Lerner et al. (2009). In this article, for p i ∈ ( 1, ∞ ), we prove that the commutators μ b →, generated by the multilinear Marcinkiewicz integral μ and b →, are bounded from L w 1 p 1 ( R n ) × ⋯ × L w m p m ( R n ) to L ν w → p ( R n ) if b → ∈ ( BMO ( R n ) ) m, where 1 ∕ p = 1 ∕ p 1 + ⋯ + 1 ∕ p m and ν w → ( x ) = ∏ i = 1 m w i ( x ) p ∕ p i . For the endpoint case, we show two kinds of estimates. One is that if w → ∈ A ( 1, …, 1 ) and b → ∈ ( BMO ( R n ) ) m, then we obtain a weighted weak L log L type estimate for μ b → . On the other hand, it is well known that the commutator μ b → may not be bounded from H w 1 ( R n ) × ⋯ × H w 1 ( R n ) to L w 1 ∕ m ( R n ) if b → ∈ ( BMO ( R n ) ) m . We obtain that when w ∈ A 1 + min { 1 ∕ 2, γ } ∕ m n ( R n ) satisfying ∫ R n w ( x ) 1 + | x | n d x < ∞ and b → ∈ ( BMO w ( R n ) ) m ⊂ ( BMO ( R n ) ) m, the commutator μ b → is bounded from H w 1 ( R n ) × ⋯ × H w 1 ( R n ) to L w 1 ∕ m ( R n ), where BMO w ( R n ) is a subspace of BMO ( R n ) . 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 H w 1 ( R n ) × ⋯ × H w 1 ( R n ) to L w 1 ∕ m ( R n ) if b → ∈ ( BMO w ( R n ) ) m .