The boundedness of commutators of multilinear Marcinkiewicz integral. (June 2020)

• Record Type: Journal Article
• Authors: He, Sha; Liang, Yiyu
• Is Part Of: Nonlinear analysis. Volume 195(2020)
• Publication Date: 2020-06
• Subjects: primary 47B47 -- secondary 42B20 42B30 42B35; Commutator -- Multilinear Marcinkiewicz integral -- BMO space -- Weighted Hardy space -- Multilinear Calderón–Zygmund operator; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111727
• Languages: English
• ISSNs: 0362-546X

Abstract

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 .