Multiple Weighted Estimates for Vector-Valued Multilinear Singular Integrals with Non-Smooth Kernels and Its Commutators

Abstract

This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.

1. Introduction

It is well known that multiple weighted norm inequalities for multilinear operators and their related commutators on various spaces of function is a center topic of harmonic analysis, which recently attracts a lot of attention, see [1–3] et al.

In this paper, we will focus on the multiple weighted estimates for vector valued multilinear singular integral with nonsmooth kernel and its commutators. Now we give some information on multilinear Calderón-Zygmund operators.

The multilinear operator 𝑇 we study is initially defined on the 𝑚-fold product of Schwartz space 𝒮(ℝ^𝑛)×⋯×𝒮(ℝ^𝑛) and taking values into the space of tempered distributions 𝒮^′(ℝ^𝑛); that is, 𝑇∶𝒮ℝ^𝑛×⋯×𝒮ℝ^𝑛⟶𝒮^′ℝ^𝑛. (1)

A locally integrable function 𝐾(𝑥,𝑦₁,…,𝑦_𝑚) defined away from the diagonal 𝑥=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1 is called an associated kernel of 𝑇 if 𝑇𝑓₁,…,𝑓_𝑚(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚, (2) where 𝑓₁,…,𝑓_𝑚 are 𝐶^∞ functions with compact support and for all 𝑥∉⋂^𝑚_𝑗=1 supp𝑓_𝑗.

Moreover, we assume the associated kernel 𝐾 satisfies the following standard estimates: 𝐾𝑦₀,𝑦₁,…,𝑦_𝑚≤𝐴∑^𝑚_𝑘,𝑙=0𝑦_𝑘−𝑦_𝑙^𝑚𝑛, (3) for some 𝐴>0 and all (𝑦₀,…,𝑦_𝑚) with 𝑦₀≠𝑦_𝑗 for some 𝑗∈{1,…,𝑚}, and 𝐾𝑦₀,…,𝑦_𝑗,…,𝑦_𝑚−𝐾𝑦₀,…,𝑦^′_𝑗,…,𝑦_𝑚≤𝐴𝑦_𝑗−𝑦^′_𝑗^𝜀∑^𝑚_𝑘,𝑙=0𝑦_𝑘−𝑦_𝑙^{𝑚𝑛+𝜀}, (4) for some 𝜀>0 and all 0≤𝑗≤𝑚, where |𝑦_𝑗−𝑦^′_𝑗|≤(1/2)max_{0≤𝑘≤𝑚}|𝑦_𝑗−𝑦_𝑘|. These facts can be founded in [4].

We now turn to present the definitions of 𝑚-linear commutator and iterated commutator of multilinear singular integral.

For the multilinear operator 𝑇 and 𝑏=(𝑏₁,…,𝑏_𝑚) in 𝐵𝑀𝑂^𝑚, we define the 𝑚-linear commutator 𝑇_Σ𝑏 as the following form: 𝑇_Σ𝑏𝑓(𝑥)=^𝑚_𝑗=1𝑇^𝑗_𝑏𝑗𝑓(𝑥)∶=^𝑚_𝑗=1𝑏_𝑗,𝑇_𝑗𝑓₁,…,𝑓_𝑚(𝑥), (5) where 𝑇^𝑗_𝑏𝑗𝑓(𝑥)=𝑏_𝑗(𝑥)𝑇𝑓₁,…,𝑓_𝑚(𝑥)−𝑇𝑓₁,…,𝑏_𝑗𝑓_𝑗,…,𝑓_𝑚(𝑥).𝑇_Π𝑏𝑓(𝑥)∶=𝑏₁,𝑏₂,…𝑏_𝑚−1,𝑏_𝑚,𝑇_𝑚𝑚−1,…₂₁×𝑓₁,…,𝑓_𝑚(𝑥). (6)

If 𝑇 is associated with a distribution kernel, which coincides with the function 𝐾 defined away from the diagonal 𝑥=𝑦₁=⋯=𝑦_𝑚 in (ℝ^𝑛)^𝑚+1, then 𝑇^𝑗_𝑏𝑗𝑓(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗⋯𝑓_𝑚𝑦_𝑚𝑑𝑦₁⋯𝑑𝑦_𝑚. (7) And we also present the iterated commutator 𝑇_Π𝑏, 𝑇_Π𝑏𝑓(𝑥)=_{(ℝ𝑛)𝑚}𝐾𝑥,𝑦₁,…,𝑦_𝑚×^𝑚_𝑗=1𝑏_𝑗(𝑥)−𝑏_𝑗𝑦_𝑗𝑓_𝑗𝑦_𝑗𝑑𝑦₁⋯𝑑𝑦_𝑚. (8)

Here the notations of commutators are taken from [5, 6].

The following class of weights were introduced in [1]. Let 1≤𝑝₁,…,𝑝_𝑚<∞, and 1/𝑚≤𝑝<∞, with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚, and 𝑃=(𝑝₁,…,𝑝_𝑚); given 𝜔=(𝜔₁,…,𝜔_𝑚), we say that 𝜔 satisfies the 𝐴_𝑝 condition if sup_𝑄1|𝑄|_𝑄𝜈_𝜔^𝑚1/𝑝_𝑗=11|𝑄|_𝑄𝜔^{1−𝑝′𝑗}_𝑗^{1/𝑝′𝑗}<∞, (9) where 𝜈_𝜔=∏^𝑚_𝑗=1𝜔^{𝑝/𝑝𝑗}_𝑗, when 𝑝_𝑗=1,    ((1/|𝑄|)∫_𝑄𝜔^{1−𝑝′𝑗}_𝑗)^{1/𝑝′𝑗} is understood as (inf_𝑄𝜔_𝑗)⁻¹.

Observe that 𝐴_(1,…,1) is contained in 𝐴_𝑃 for each 𝑃. However, the class 𝐴_𝑃 is not increasing with the natural partial order, see [1] for detail. Lerner et al. [1] established multiple weighted estimates for multilinear C-Z operators and that for 𝑚-linear commutator of multilinear C-Z commutator. In 2012, Chen and Wu [2] extend their results to 𝑚-linear commutator and iterated commutator of multilinear C-Z operator with nonsmooth kernel satisfying Assumptions (H1) and (H2).

Next we define the vector-valued multilinear operator 𝑇_𝑞(𝑞>0) associated with the operator 𝑇 by 𝑇_𝑞𝑓₁,…,𝑓_𝑚(𝑥)=𝑇𝑓₁,…,𝑓_𝑚𝑞(𝑥)=^∞_𝑘=1𝑇𝑓_1𝑘,…,𝑓_𝑚𝑘(𝑥)^𝑞1/𝑞, (10) where 𝑓_𝑖={𝑓_𝑖𝑘}^∞_𝑘=1, for 𝑖=1,…,𝑚.

This operator was first studied by Grafakos and Martell in [7]. Later Cruz-Uribe et al. gained the weak boundedness of this one in [8]. We list them as follows.

Theorem A (see [7]). Let 𝑇 be a multilinear Calderón-Zygmund operator as before, and let 1/𝑚<𝑝<∞,  1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚 with 1<𝑝₁,…,𝑝_𝑚<∞,1/𝑚<𝑞<∞, and 1/𝑞=1/𝑞₁+⋯+1/𝑞_𝑚 with 1<𝑞₁,…,𝑞_𝑚<∞. There exists a constant 𝐶>0 such that 𝑇_𝑞𝑓_{𝐿𝑝(ℝ𝑛)}≤𝐶^𝑚_𝑗=1𝑓_{𝑗𝑞𝑗𝐿𝑝𝑗(ℝ𝑛)}. (11)

For the sequence {𝑓_𝑘}^∞_𝑘=1=(𝑓_1𝑘,…,𝑓_𝑚𝑘)^∞_𝑘=1, the vector-valued version of the commutators 𝑇_Σ𝑏,𝑞 and 𝑇_Π𝑏,𝑞 are given by 𝑇_Σ𝑏,𝑞𝑓(𝑥)=𝑇_Σ𝑏𝑓_𝑞(𝑥)=^∞_𝑘=1𝑇_Σ𝑏𝑓_1𝑘,…,𝑓_𝑚𝑘(𝑥)^𝑞1/𝑞,𝑇_Π𝑏,𝑞𝑓(𝑥)=𝑇_Π𝑏𝑓_𝑞(𝑥)=^∞_𝑘=1𝑇_Π𝑏𝑓_1𝑘,…,𝑓_𝑚𝑘(𝑥)^𝑞1/𝑞. (12)(13)

In 2008, Tang established weighed norm inequalities for the commutators of vector-valued multilinear operator in [6], but their results are not the multiple weighted estimates that are obtained by Lerner et al. in [1].

Now we restore to give some information on the kernel 𝐾 which satisfy Assumptions (H1) and (H2). Let {𝐴_𝑡}_𝑡>0 be a class of integral operators which play the role of the approximation to the identity. We always assume that the operators 𝐴_𝑡 are associated with kernels 𝑎_𝑡(𝑥,𝑦) in the sense that for all 𝑓∈⋃_{𝑝∈[1,∞]}𝐿^𝑝 and 𝑥∈ℝ^𝑛𝐴_𝑡𝑓(𝑥)=_ℝ𝑛𝑎_𝑡𝑥,𝑦𝑓𝑦𝑑𝑦, (14) and that the kernels 𝑎_𝑡(𝑥,𝑦) satisfy the following conditions 𝑎_𝑡𝑥,𝑦≤ℎ_𝑡𝑥,𝑦∶=𝑡^−𝑛/𝑠ℎ𝑥−𝑦𝑡^1/𝑠, (15) where 𝑠 is a positive fixed constant and ℎ is a positive, bounded, decreasing function satisfying that for some 𝜂>0, lim_𝑟→∞𝑟^𝑛+𝜂ℎ𝑟^𝑠=0. (16)

Recall that the 𝑗th transpose 𝑇^∗,𝑗 of the 𝑚-linear operator 𝑇 is defined via 𝑇^∗,𝑗𝑓₁,…,𝑓_𝑚=𝑇𝑓,𝑔₁,…,𝑓_𝑗−1,𝑔,𝑓_𝑗+1,…,𝑓_𝑚,𝑓_𝑗, (17) for all 𝑓₁,…,𝑓_𝑚,𝑔 in 𝒮(ℝ^𝑛). Notice that the kernel 𝐾^∗,𝑗 of 𝑇^∗,𝑗 is related to the kernel 𝐾 of 𝑇 via the identity 𝐾^∗,𝑗𝑥,𝑗₁,…,𝑦_𝑗,…,𝑦_𝑚=𝐾𝑦_𝑗,𝑦₁,…,𝑥,…,𝑦_𝑚. (18)

If an 𝑚-linear operator 𝑇 maps a product of Banach Spaces 𝑋_𝑚×⋯×𝑋_𝑚 to another Banach Space 𝑋; then transpose 𝑇^∗,𝑗 maps the product of Banach Spaces 𝑋₁×⋯×𝑋_𝑗−1×𝑋^∗_𝑗×𝑋_𝑗+1×⋯×𝑋_𝑚 into 𝑋^∗_𝑗. Moreover, the norms of 𝑇 and 𝑇^∗,𝑗 are equal. To maintain uniform notation, we may occasionally denote 𝑇 by 𝑇^∗,0 and 𝐾 by 𝐾^∗,0.

Assumption (H1). Assume that for each 𝑖=1,…,𝑚, there exist operators {𝐴^𝑖_𝑡}_𝑡>0 with kernel 𝑎^𝑖_𝑡(𝑥,𝑦) satisfying conditions 𝑠 and 𝜂 and that for every 𝑗=0,1,…,𝑚, there exist kernels 𝐾_𝑡^{∗,𝑗(𝑖)}(𝑥,𝑦₁,…,𝑦_𝑚) such that 𝑇^∗,𝑗𝑓₁,…,𝐴_𝑡^(𝑖)𝑓_𝑗,…,𝑓_𝑚=,𝑔_{ℝ𝑛(ℝ𝑛)𝑚}𝐾_𝑡^{∗,𝑗(𝑖)}𝑥,𝑦₁,…,𝑦_𝑚𝑓₁𝑦₁⋯𝑓_𝑚𝑦_𝑚𝑔(𝑥)𝑑𝑦𝑑𝑥, (19) for all 𝑓₁,…,𝑓_𝑚 in 𝒮(ℝ^𝑛) with ⋂^𝑚_𝑘=1supp𝑓_𝑘∩supp𝑔=∅. Also assume that for every 𝑗=0,1,…,𝑚 and every 𝑖=1,…,𝑚, we have 𝐾^∗,𝑗𝑥,𝑦−𝐾_𝑡^{∗,𝑗(𝑖)}𝑦≤𝐴𝑥,∑^𝑚_𝑘=1𝑥−𝑦_𝑘^𝑚𝑛×^𝑚_{𝑘=1,𝑘≠𝑖}Φ𝑦_𝑖−𝑦_𝑘𝑡^1/𝑠+𝐴𝑡^𝜖/𝑠∑^𝑚_𝑘=1𝑥−𝑦_𝑘^{𝑚𝑛+𝜖}, (20) where 𝑡^1/𝑠≤|𝑥−𝑦_𝑖|/2,𝑦=(𝑦₁,…,𝑦_𝑚).

If 𝑇 satisfies Assumption (H1), we will say that 𝑇 is an 𝑚-linear operator with generalized Caderón-Zygmund kernel 𝐾. The collection of function 𝐾 satisfying (19) and (20) with parameters 𝑚,𝐴,𝑠,𝜂 and 𝜖 will be denoted by 𝑚-GCZK(𝐴,𝑠,𝜂,𝜖), we say that 𝑇 is of class 𝑚-GCZK(𝐴,𝑠,𝜂,𝜖) if 𝑇 has an associated kernel 𝐾 in 𝑚-GCZK(𝐴,𝑠,𝜂,𝜖).

Assumption (H2). Assume that there exist operators {𝐵_𝑡}_𝑡>0 with kernel 𝑏_𝑡(𝑥,𝑦) satisfying conditions (15) and (16) with constants 𝑠 and 𝜂. Let 𝐾_𝑡⁽⁰⁾𝑥,𝑦₁,…,𝑦_𝑚=_ℝ𝑛𝐾𝑧,𝑦₁,…,𝑦_𝑚𝑏_𝑡(𝑥,𝑧)𝑑𝑧, (21) whenever 2𝑡^1/𝑠≤min_{1≤𝑗≤𝑚}|𝑥−𝑦_𝑗| and 𝐾𝑥,𝑦−𝐾_𝑡⁽⁰⁾𝑥^′,𝑦≤𝐴∑^𝑚_𝑘=1𝑥−𝑦_𝑘^𝑚𝑛×^𝑚_{𝑘=1,𝑘≠𝑖}Φ𝑦_𝑖−𝑦_𝑘𝑡^1/𝑠+𝐴𝑡^𝜖/𝑠∑^𝑚_𝑘=1𝑥−𝑦_𝑘^{𝑚𝑛+𝜖}, (22) whenever 2|𝑥−𝑥^′|≤𝑡^1/𝑠 and 2𝑡^1/𝑠≤max_{1≤𝑗≤𝑚}|𝑥−𝑦_𝑗|.

When 𝑇 is of 𝑚−GCZK(𝐴,𝑠,𝜂,𝜖) and its kernel also satisfies Assumption (H2), Duong et al. in [5] proved that multilinear singular integral 𝑇 is bounded from 𝐿^𝑝1(ℝ^𝑛)×⋯×𝐿^𝑝𝑚(ℝ^𝑛) to 𝐿^𝑝(ℝ^𝑛)(𝐿^𝑝,∞(ℝ^𝑛)) for 0<𝑝<∞, 1≤𝑝₁,…,𝑝_𝑚<∞ with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚. And they also remarked that the above kernel which they studied has weaker regularity. It is natural to ask whether the vector-valued multilinear operator 𝑇_𝑞 with kernels satisfying the same conditions as in [5] and its commutators 𝑇_Σ𝑏,𝑞 and 𝑇_Π𝑏,𝑞 have multiple weighted estimates or not. These problems will be addressed by our next theorems.

Now we can formulate our results as follows.

Theorem 1. Assume that 𝑇_𝑞 is a vector-valued multilinear operator defined as (10) associated with 𝑇 being an 𝑚-𝐺𝐶𝑍𝐾(𝐴,𝑠,𝜂,𝜖) whose kernel 𝐾 satisfies Assumption (H2). If there exist 1≤𝑝₁,…,𝑝_𝑚<∞, with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚 and  𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑃 with 𝑃=(𝑝₁,…,𝑝_𝑚) then(i)𝑇_𝑞 can be extended to a bounded operator from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝(𝜈_𝜔) if all exponents 𝑝_𝑗 are strictly greater than 1;(ii)𝑇_𝑞 can be extended to a bounded operator from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝,∞(𝜈_𝜔) if 1≤𝑝_𝑗<∞, 𝑗=1,…,𝑚, and at least one of the 𝑝_𝑗=1.

Theorem 2. Assume that 𝑇_𝑞 is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the multilinear commutator 𝑇_Σ𝑏,𝑞 is defined as (12). Let 𝑏∈𝐵𝑀𝑂^𝑚, 1<𝑝₁,…,𝑝_𝑚<∞,1<𝑝<∞ with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚, and 𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑃 with 𝑃=(𝑝₁,…,𝑝_𝑚). Then there exists a constant 𝐶>0 such that 𝑇_Σ𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖^𝑚_{𝐵𝑀𝑂𝑗=1}𝑓_{𝑗𝑞𝑗𝐿𝑝𝑗(𝜔𝑗)}. (23)

Theorem 3. Assume that 𝑇_𝑞 is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the iterated commutator 𝑇_Π𝑏,𝑞 is defined as (13). Let 𝑏∈𝐵𝑀𝑂^𝑚, 1<𝑝₁,…,𝑝_𝑚<∞,  1<𝑝<∞ with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚  𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑃 with 𝑃=(𝑝₁,…,𝑝_𝑚). Then there exists a constant 𝐶>0 such that 𝑇_Π𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶^𝑚_𝑖=1𝑏_𝑖^𝑚_{𝐵𝑀𝑂𝑗=1}𝑓_{𝑗𝑞𝑗𝐿𝑝𝑗(𝜔𝑗)}, (24) where 𝜈_𝜔=∏^𝑚_𝑗=1𝜔^{𝑝/𝑝𝑗}_𝑗, 1<𝑞₁,…,𝑞_𝑚<∞, 1/𝑚<𝑞<∞,  and 1/𝑞=1/𝑞₁+⋯+1/𝑞_𝑚.

The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas. In Section 3, we introduce a class of new maximal functions and prove some useful estimates which will play key roles in the proofs of our theorems. In Section 4, it is devoted to the proof our theorems. Throughout this paper, we use the letter 𝐶 to denote a positive constant that varies from line to line, but it is independent of the essential variable.

2. Some Preliminaries

Lemma 4 (see [1, Lemma 6.1]). Assume that 𝜔=(𝜔₁,…,𝜔_𝑚) satisfies 𝐴_𝑃 condition, then there exists a finite constant 𝑟>1 such that 𝜔∈𝐴_𝑃/𝑟.

We recall the definition of the Hardy-Littlewood maximal function and the sharp maximal function: 𝑀𝑓(𝑥)=sup_𝑄∋𝑥1|𝑄|_𝑄𝑓𝑦𝑀𝑑𝑦,_♯𝑓(𝑥)=sup_𝑄∋𝑥inf_𝑥1|𝑄|_𝑄𝑓𝑦−𝑐𝑑𝑦≈sup_𝑄∋𝑥1|𝑄|_𝑄𝑓𝑦−𝑓_𝑄𝑑𝑦, (25) and their variants 𝑀_𝛿(𝑓)(𝑥)=𝑀(|𝑓|^𝛿)^1/𝛿(𝑥) and 𝑀^♯_𝛿(𝑓)(𝑥)=𝑀^♯(|𝑓|^𝛿)^1/𝛿(𝑥).

We will use the following inequality (see [9]): _ℝ𝑛𝑀_𝛿𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥≤𝐶_ℝ𝑛𝑀^♯_𝛿𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥, (26) all functions 𝑓 for which the left-hand side is finite, and where 0<𝑝,  𝛿<∞,  𝜔∈𝐴_∞.

Now we introduce some facts on Orlicz spaces. Let Φ: [0,∞)→[0,∞) be a Young function, that is, a continuous, convex, increasing function with Φ(0)=0 and such that Φ(𝑡)→∞ as 𝑡→∞. In this paper, any Young function Φ will be doubling, namely Φ(2𝑡)≤𝐶Φ(𝑡) for 𝑡>0. We define the Φ-average of function 𝑓 over a cube 𝑄 by 𝑓_Φ,𝑄1=inf𝜆>0∶|𝑄|_𝑄Φ𝑓(𝑥)𝜆𝑑𝑥≤1. (27) It is a simple but important observation that 𝑓_Φ,𝑄1>1iff|𝑄|_𝑄Φ𝑓(𝑥)𝜆𝑑𝑥>1. (28)

A particular case of interest, and especially in this paper, is the Young function Φ(𝑡)=𝑡(1+log⁺𝑡), the average ‖𝑓‖_Φ,𝑄 of a function 𝑓 given by the Luxemburg norm ‖𝑓‖_{𝐿(log𝐿),𝑄}.

Associated with this average, we have a maximal function 𝑀_{𝐿(log𝐿)𝑓(𝑥)=sup𝑄∋𝑥}𝑓_{𝐿(log𝐿),𝑄}, (29) where the supremum is taken over all the cubes containing 𝑥.

By the generalized Hölder’s inequality, we also get 1|𝑄|_𝑄𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤𝐶𝑓_𝐵𝑀𝑂𝑔_{𝐿(log𝐿),𝑄}. (30)

3. New Maximal Functions

In this section, we will introduce certain variant multilinear maximal functions and establish the multiple weighted estimates for such functions, which are one of the main parts in this paper.

Recall the definitions of these maximal functions, which are introduced by Lerner et al. in [1] ℳ𝑓(𝑥)=sup^𝑚_{𝑄∋𝑥𝑗=1}1|𝑄|_𝑄𝑓_𝑗𝑦ℳ𝑑𝑦,_𝑟𝑓(𝑥)=sup^𝑚_{𝑄∋𝑥𝑗=1}1|𝑄|_𝑄𝑓_𝑗𝑦^𝑟𝑑𝑦^1/𝑟,ℳ_{𝐿(log𝐿)}𝑓(𝑥)=sup^𝑚_{𝑄∋𝑥𝑗=1}𝑓_{𝑗𝐿(log𝐿),𝑄}. (31)

The fact that 𝑟>1, there exists a constant 𝐶>0 such that 𝑓_{𝐿(log𝐿),𝑄}1≤𝐶|𝑄|_𝑄𝑓_𝑗𝑦^𝑟𝑑𝑦^1/𝑟, (32) so it is easy to check that ℳ𝑓(𝑥)≤ℳ_{𝐿(log𝐿)}𝑓(𝑥)≤ℳ_𝑟𝑓(𝑥). (33)

Characterizations of the multiple weights in terms of ℳ were proven in Theorems 3.3 and 3.7 in [1].

Lemma 5. Let 1≤𝑝₁,…,𝑝_𝑚<∞, 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚, and 𝑃=(𝑝₁,…,𝑝_𝑚),(i)If 1<𝑝₁,…,𝑝_𝑚<∞, then ℳ is bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝(𝜈_𝜔) if and only if 𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑃;(ii)If 1≤𝑝₁,…,𝑝_𝑚<∞, then ℳ is bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝,∞(𝜈_𝜔) if and only if 𝜔=(𝜔₁,…,𝜔_𝑚)∈𝐴_𝑝.

In the following, we introduce the modified multilinear maximal functions.

Let 𝑟>1, 1≤𝑙<𝑚, 𝜚={𝑗₁,…,𝑗_𝑙}⊆{1,…,𝑚} and 𝜚^′={1,…,𝑚}⧵𝜚. We defined the following multilinear maximal functions: ℳ_𝜚𝑓_𝑞(𝑥)∶=sup^∞_{𝑄∋𝑥𝜈=0}2^{−𝜈𝑛𝑙}_𝑗∈𝜚1|𝑄|_𝑄𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×_{𝑗∈𝜚′}1|2^𝜈𝑄|₂^𝜈_𝑄𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗,ℳ_{𝜚,𝐿(log𝐿)}𝑓_𝑞(𝑥)∶=sup^∞_{𝑄∋𝑥𝜈=0}2^{−𝜈𝑛𝑙}_𝑗∈𝜚𝑓_{𝑗𝑞𝑗𝐿(log𝐿),𝑄}×_{𝑗∈𝜚′}𝑓_{𝑗𝑞𝑗𝐿(log𝐿),2𝜈𝑄},ℳ_𝜚,𝑟𝑓_𝑞(𝑥)∶=sup^∞_{𝑄∋𝑥𝜈=0}2^{−𝜈𝑛𝑙}_𝑗∈𝜚1|𝑄|_𝑄𝑓_𝑗^𝑟_𝑞𝑗𝑑𝑦_𝑗^1/𝑟×_{𝑗∈𝜚′}1|2^𝜈𝑄|₂^𝜈_𝑄𝑓_𝑗^𝑟_𝑞𝑗𝑑𝑦_𝑗^1/𝑟, (34)(35)(36)

where |𝑓_𝑗|_𝑞𝑗=(∑^∞_𝑘=1|𝑓_𝑗𝑘(𝑦_𝑗)|^𝑞𝑗)^1/𝑞𝑗.

We remark that when 𝜚=1,…,𝑙 and 𝑓=(𝑓₁,…,𝑓_𝑚), ℳ_𝜚 was first introduced by Grafakos el al. in [10] and denoted ℳ_𝜚 by ℳ_𝑙. Chen and Wu [2] proved the multiple weighted norm inequality for ℳ_𝜚,𝑟. Similarly to (33), for any 𝑟>0, we have ℳ_𝜚𝑓_𝑞≤ℳ_{𝜚,𝐿(log𝐿)}𝑓_𝑞≤ℳ_𝜚,𝑟𝑓_𝑞. (37)

Lemma 6. Let 1<𝑝₁,…,𝑝_𝑚<∞, 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚, 𝑃=(𝑝₁,…,𝑝_𝑚), 𝜔∈𝐴_𝑝, and 𝜚={𝑗₁,…,𝑗_𝑙}⊆{1,…,𝑚}, 1≤𝑙<𝑚. Then for some 𝑟>1 (𝑟 depending only on 𝜔), ℳ_𝑟 and ℳ_𝜚,𝑟 defined by (36) are bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝(𝜈_𝜔). ℳ, ℳ_{𝐿(log𝐿)}, ℳ_𝜚, and ℳ_𝜚,𝑟 are bounded from 𝐿^𝑝1(𝜔₁)×⋯×𝐿^𝑝𝑚(𝜔_𝑚) to 𝐿^𝑝(𝜈_𝜔).

4. Weighted Inequalities for Vector-Valued Singular Integral and Its Commutators

To prove our theorems, we first give two Lemmas about vector-valued operator 𝑇_𝑞 associated with 𝑇 in 𝑚-𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) which were obtained by Duong et al. in [5].

Lemma 7. Let 𝑇 be a multilinear operator in 𝑚-𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) with 𝐾 satisfying Assumption (H2). And let 1/𝑚<𝑝<∞, 1<𝑝₁,…,𝑝_𝑚<∞ with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚,  1/𝑚<𝑞<∞, and 1<𝑞₁,…,𝑞_𝑚<∞ with 1/𝑞=1/𝑞₁+⋯+1/𝑞_𝑚. Then there exists a constant 𝐶>0 such that 𝑇𝑓_{𝐿𝑝(ℝ𝑛)}≤𝐶^𝑚_𝑗=1𝑓_𝐿^𝑝_{𝑗(ℝ𝑛)}. (38)

Lemma 8. Let 𝑇 be a multilinear operator in 𝑚-𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) with 𝐾 satisfying Assumption (H2). And let 1/𝑚<𝑝<∞, 1≤𝑝₁,…,𝑝_𝑚<∞ with 1/𝑝=1/𝑝₁+⋯+1/𝑝_𝑚, and 1/𝑚<𝑞<∞, 1<𝑞₁,…,𝑞_𝑚<∞ with 1/𝑞=1/𝑞₁+⋯+1/𝑞_𝑚. Then there exists a constant 𝐶>0 such that 𝑇𝑓_{𝐿𝑝,∞(ℝ𝑛)}≤𝐶^𝑚_𝑗=1𝑓_𝐿^𝑝_{𝑗(ℝ𝑛)}. (39)

Before proving Theorem 1, we first present the estimates on the pointwise estimates for sharp Fefferman-stein maximal function action on 𝑇_𝑞.

Proposition 9. Let 𝑇 be an 𝑚-linear operator in 𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) and 𝑇_𝑞 satisfy the assumption in Theorem 1. Assume that 𝜚₀={1,…,𝑚} and 0<𝛿<1/𝑚. For any {𝑓_1𝑚,…,𝑓_𝑚𝑘}^∞_𝑘=1 in the product spaces 𝐿^𝑝1(𝑙^𝑞1)×⋯×𝐿^𝑝𝑚(𝑙^𝑞𝑚) with 1<𝑝_𝑗,  𝑞_𝑗<∞ for 𝑗=1,…,𝑚, then these exists a constant 𝐶>0 such that 𝑀^♯_𝛿𝑇_𝑞𝑓(𝑥)≤𝐶ℳ𝑓_𝑞(𝑥)+ℳ_𝜚𝑓_𝑞(𝑥). (40)
The ideas and arguments used in the proof are similar to those in [1] with some modifications. For completeness, we give the proof as follows.

Proof of Proposition 9. For a point 𝑥 and a cube 𝑄∋𝑥, since ||𝛼|^𝑟−|𝛽|^𝑟|≤|𝛼−𝛽|^𝛽, for 0<𝑟<1, to obtain (40), it suffices to prove for 0<𝛿<1/𝑚1|𝑄|_𝑄𝑇_𝑞𝑓(𝑧)−𝐶_𝑄^𝛿𝑑𝑧^1/𝛿≤𝐶ℳ𝑓_𝑞(𝑥)+ℳ_𝜚𝑓_𝑞(𝑥), (41) for some constant 𝐶_𝑄 to be determined later.
Let {𝑓_1𝑘,…,𝑓_𝑚𝑘}^∞_𝑘=1 be any smooth vector-valued functions. Set each 𝑓_𝑘=𝑓⁰_𝑘+𝑓^∞_𝑘, where {𝑓⁰_𝑘}^∞_𝑘=1={𝑓_𝑘𝜒_𝑄∗}^∞_𝑘=1={𝑓_1𝑘𝜒_𝑄∗,…,𝑓_𝑚𝑘𝜒_𝑄∗}^∞_𝑘=1 and 𝑄^∗=(8√𝑛+4)𝑄, then we can write ^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗=^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗𝑓⁰_𝑗𝑘𝑦_𝑗+𝑓^∞_𝑗𝑘𝑦_𝑗=_{𝛼1,…,𝛼𝑚}𝑓^𝛼1_1𝑘𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑘𝑦_𝑚=^𝑚_𝑗=1𝑓⁰_𝑗𝑘𝑦_𝑗+^′𝑓^𝛼1_1𝑘𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑘𝑦_𝑚, (42) where each term of ∑^′ contains at least one 𝛼_𝑗≠0.
Hence, we can write 𝑇_𝑞𝑓(𝑧)=𝑇_𝑞𝑓⁰(𝑧)+^′𝑇_𝑞𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑧). (43) Applying Kolmogorov’s inequality to the term 𝑇_𝑞𝑓⁰(𝑧)=𝑇_𝑞𝑓⁰_1𝑘,…,𝑓⁰_𝑚𝑘(𝑧), (44) with 𝑝=𝛿 and 𝑞=1/𝑚, we derive 1|𝑄|_𝑄𝑇_𝑞𝑓⁰(𝑧)^𝛿𝑑𝑧^1/𝛿≤𝐶𝑇_𝑞𝑓⁰_{𝐿1/𝑚,∞(𝑄,𝑑𝑥/|𝑄|)}≤𝐶^𝑚_𝑗=11𝑄^∗_𝑄^∗𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶ℳ𝑓_𝑞(𝑥), (45) since 𝑇∶𝐿¹×⋯×𝐿¹→𝐿^1/𝑚,∞.
To estimate the remaining terms, we now set 𝐶_𝑄=∑^′𝑇_𝑞(𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘)(𝑥) and will show that, for any 𝑧∈𝑄, ^′𝑇_𝑞𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑧)−𝑇_𝑞𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘𝑥𝑓()≤𝐶ℳ_𝑞(𝑥). (46)
Consider the case when 𝛼₁=⋯=𝛼_𝑚=∞ and define 𝑇_𝑞𝑓^∞=𝑇_𝑞𝑓^∞_1𝑘,…,𝑓^∞_𝑚𝑘. (47) So 𝑇_𝑞𝑓^∞_1𝑘,…,𝑓^∞_𝑚𝑘(𝑧)−𝑇_𝑞𝑓^∞_1𝑘,…,𝑓^∞_𝑚𝑘(𝑥≤_{∞𝑘=1(ℝ𝑛⧵𝑄∗)𝑚})𝐾𝑧,𝑦𝑓_1𝑘𝑦₁⋯𝑓_𝑚𝑘𝑦_𝑚𝑑𝑦^𝑞1/𝑞−^∞_{𝑘=1(ℝ𝑛⧵𝑄∗)𝑚}𝐾𝑥,𝑦𝑓_1𝑘𝑦₁⋯𝑓_𝑚𝑘𝑦_𝑚𝑑𝑦^𝑞1/𝑞≤^∞_{𝑘=1(ℝ𝑛⧵𝑄∗)𝑚}𝐾𝑦𝑦×𝑓𝑧,−𝐾𝑥,_1𝑘𝑦₁⋯𝑓_𝑚𝑘𝑦_𝑚𝑑𝑦^𝑞1/𝑞≤^∞_{𝑘=1(ℝ𝑛⧵𝑄∗)𝑚}𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑦+𝐾𝑧,_𝑡(0)𝑦𝑦×𝑓𝑧,−𝐾𝑥,_1𝑘𝑦₁⋯𝑓_𝑚𝑘𝑦_𝑚𝑑𝑦^𝑞1/𝑞≤^∞_𝑘=1^∞_{𝜈=0(2𝜈+1𝑄∗⧵2𝜈𝑄∗)𝑚}𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑦+𝐾𝑧,_𝑡(0)𝑦𝑦×𝑓𝑧,−𝐾𝑥,_1𝑘𝑦₁⋯𝑓_𝑚𝑘𝑦_𝑚𝑑𝑦^𝑞1/𝑞∶=𝐼. (48) Since 𝑥,𝑧∈𝑄,  𝑦_𝑗∈ℝ^𝑛⧵(8√𝑛+4)𝑄, and 𝑡=(2√𝑛𝑙(𝑄))^𝑠, then |𝑥−𝑦_𝑗|∼|𝑧−𝑦_𝑗|, |𝑧−𝑦_𝑗|>(4√𝑛+1)𝑙(𝑄)>2𝑡^1/𝑠, |𝑥−𝑦_𝑗|>2𝑡^1/𝑠, and |𝑥−𝑧|<2𝑡^1/𝑠. Hence,𝜙(|𝑦_𝑗−𝑧|/𝑡^1/𝑠)=0 for 𝑗=1,2. By Assumption (H2), we can infer 𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑧,𝑦≤𝐴𝑡^𝜖/𝑠∑^𝑚_𝑖=1𝑧−𝑦_𝑗^{𝑚𝑛+𝜖},𝐾_𝑡⁽⁰⁾𝑧,𝑦−𝐾𝑥,𝑦≤𝐴𝑡^𝜖/𝑠∑^𝑚_𝑖=1𝑥−𝑦_𝑗^{𝑚𝑛+𝜖}. (49) Thus, 𝐼≤𝐶^∞_𝑘=1^∞_{𝜈=0(2𝜈+1𝑄∗⧵2𝜈𝑄∗)𝑚}𝐴𝑡^𝜖/𝑠∑^𝑚_𝑖=1𝑧−𝑦_𝑗^{𝑚𝑛+𝜖}×^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤𝐶^∞_𝑘=1^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝑛𝑄^{∗1/𝑛𝑚𝑛+𝜖}×_{(2𝜈+1𝑄∗)𝑚}^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤𝐶^∞_𝑘=1^∞_𝜈=012^𝜈𝜖12^𝜈+1𝑄^{∗1/𝑛𝑚𝑛}×_{(2𝜈+1𝑄∗)𝑚}^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤𝐶^∞_𝜈=112^𝑚𝜈𝜖_𝑗=1^∞_𝑘=112^𝜈+1𝑄^{∗1/𝑛𝑛}×_{2𝜈+1𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}≤𝐶^∞_𝜈=112^𝑚𝜈𝜖_𝑗=112^𝜈+1𝑄^{∗1/𝑛𝑛2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶ℳ𝑓_𝑞(𝑥). (50)It remains to estimate the terms in (46) with 𝛼_𝑗1=⋯=𝛼_𝑗𝑙=0, for some 𝜚={𝑗₁,…,𝑗_𝑙}⊂{1,…,𝑚}=𝜚₀ for 1≤𝑙<𝑚 and 𝜚^′=𝜚₀⧵𝜚, by Assumption (H2), we have 𝑇_𝑞𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑧)−𝑇_𝑞𝑓^𝛼1₁,…,𝑓^𝛼𝑚_𝑚(𝑥≤_{∞𝑘=1(ℝ𝑛)𝑚})𝐾𝑦𝑦×𝑓𝑧,−𝐾𝑥,_𝛼11𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑦_𝑚𝑑𝑦^𝑞1/𝑞≤^∞_{𝑘=1(ℝ𝑛)𝑚}𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑦+𝐾𝑧,_𝑡(0)𝑦𝑦×𝑓𝑧,−𝐾𝑥,_𝛼11𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑦_𝑚𝑑𝑦^𝑞1/𝑞≤𝐶^∞_{𝑘=1𝑗∈𝜚𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗×_{(ℝ𝑛⧵𝑄∗)𝑚}𝑡^𝜖/𝑡∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^{𝑚𝑛+𝜖}+1∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^𝑚𝑛×_{𝑗∈𝜚′}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^𝑞1/𝑞≤𝐶^∞_{𝑘=1𝑗∈𝜚𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗×^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝜈𝑄^{∗1/𝑛𝑚𝑛+𝜖}+12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{(2𝜈+1𝑄∗)𝑚𝑗∈𝜚′}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^𝑞1/𝑞≤𝐶^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝜈𝑄^{∗1/𝑛𝑚𝑛+𝜖}+12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_𝑗∈𝜚^∞_{𝑘=1𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}×_{𝑗∈𝜚′}^∞_{𝑘=12𝜈+1𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}≤𝐶^∞_𝜈=012^𝜈𝜖12^𝜈𝑄^{∗1/𝑛𝑚𝑛𝑗∈𝜚𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×_{𝑗∈𝜚′2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗+𝐶^∞_𝜈=012^𝜈𝑄^{∗1/𝑛𝑚𝑛𝑗∈𝜚𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×_{𝑗∈𝜚′2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶^∞_𝜈=012^𝜈𝜖12^𝜈+1𝑄^{∗1/𝑛𝑚𝑚𝑛𝑗=12𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗+𝐶^∞_𝜈=012^𝜈𝑛𝑙_𝑗∈𝜚1𝑄^∗_𝑄^∗𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×_{𝑗∈𝜚′}12^𝜈+1𝑄^{∗1/𝑛𝑛2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶ℳ𝑓_𝑞(𝑥)+ℳ_𝜚𝑓_𝑞(𝑥). (51)
This finishes the proof of Proposition 9.

Now we restore to prove Theorem 1.

Proof of Theorem 1. By the definition of 𝜔∈𝐴_𝑃 implies that 𝜈_𝜔∈𝐴_∞. Using Proposition 9 and the Fefferman-Stein inequality (26) and observing that 0<𝛿<1/𝑚, we have 𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝑀_𝛿𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝑀^♯_𝛿𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤ℳ𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_𝜚𝑓_{𝑞𝐿𝑝(𝜈𝜔)}. (52)
Then following from (52) and Lemma 6, Theorem 1 can be proved.
We are left to check that ‖𝑀_𝛿(𝑇_𝑞(𝑓))‖_{𝐿𝑝(𝜈𝜔)} is finite. The method of proof is similar to that of Theorem 3.19 in [2]. So we omit it here.

Here is a crucial proposition on commutator 𝑇_Σ𝑏,𝑞 to prove Theorem 2.

Proposition 10. Let 𝑇_𝑞 be a vector-valued multilinear operator associated with an 𝑚-linear operator 𝑇 in 𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) whose kernels satisfie the Assumption (H2). Suppose that 𝑇_Σ𝑏,𝑞 is the corresponding commutator of 𝑇_𝑞 with 𝑏∈𝐵𝑀𝑂^𝑚. Let 0<𝛿<min{𝜖,1/𝑚}, 𝑟>1. Then there exists a constant 𝐶>0 depending on 𝛿 and 𝜖, such that 𝑀^♯_𝛿𝑇_Σ𝑏,𝑞𝑓(𝑥)≤𝐶^𝑚_𝑖=1𝑀_𝜖𝑇_𝑞𝑓(𝑥)+ℳ_{𝐿(log𝐿)}𝑓_𝑞(𝑥)+_{∅≠𝜚⊂𝜚0}ℳ_{𝜚,𝐿(log𝐿)}𝑓_𝑞(𝑥) (53) holds for all bounded measurable vector functions {𝑓_𝑘}^∞_𝑘=1={𝑓_1𝑘,…,𝑓_𝑚𝑘}^∞_𝑘=1.

Proof of Proposition 10. By linearity, it suffices to consider the operator with only symbols. Without loss of generality, we only consider the case: 𝑗=1 and denote 𝑏₁ by 𝑏 for convenience.
Note that for any constant 𝜆 we have 𝑇_Σ𝑏,𝑞𝑓(𝑧)=(𝑏(𝑧)−𝜆)𝑇_𝑞𝑓₁,…,𝑓_𝑚(𝑧)−𝑇_𝑞𝑏𝑓₁,…,𝑓_𝑚(𝑧). (54) Fix 𝑥, for any cube 𝑄 centered at 𝑥 and a constant 𝑐 determined later, we have 1|𝑄|_𝑄𝑇_𝑏,𝑞𝑓(𝑧)^𝛿−|𝑐|^𝛿𝑑𝑧^1/𝛿≤1|𝑄|_𝑄𝑇_𝑏,𝑞𝑓(𝑧)−𝑐^𝛿𝑑𝑧^1/𝛿≤𝐶|𝑄|_𝑄(𝑏(𝑧)−𝜆)𝑇_𝑞𝑓(𝑧)^𝛿𝑑𝑧^1/𝛿+𝐶|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓₁,…,𝑓_𝑚(𝑧)−𝑐^𝛿𝑑𝑧^1/𝛿∶=𝐼+𝐼𝐼. (55) We analyze each term separately. Recall that 𝑄^∗=(8√𝑛+4)𝑄 and 𝜆=𝑏_𝑄^∗  and thanks to Hölder’s inequality and note that 1<𝑞<𝜖/𝛿, it follows that 1𝐼≤𝐶|𝑄|_𝑄|𝑏(𝑧)−𝜆|^𝛿𝑞′𝑑𝑧^{1/𝛿𝑞′}×1|𝑄|_𝑄𝑇_𝑞𝑓(𝑧)^𝛿𝑞𝑑𝑧^1/𝛿𝑞≤𝐶‖𝑏‖_𝐵𝑀𝑂𝑀_𝜖𝑇_𝑞𝑓(𝑥). (56)
To estimate 𝐼𝐼, we split again each 𝑓_𝑘 into 𝑓⁰_𝑘+𝑓^∞_𝑘, where 𝑓⁰_𝑘=𝑓_𝑘𝜒_𝑄∗={𝑓_1𝑘𝜒_𝑄∗,…,𝑓_𝑚𝑘𝜒_𝑄∗}. This yields ^𝑚_𝑗=1𝑓_𝑗𝑘𝑦_𝑗=_{𝛼1,…,𝛼𝑚∈{}0,∞}𝑓_𝛼1𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑘𝑦_𝑚=^𝑚_𝑗=1𝑓⁰_𝑗𝑘𝑦_𝑗+^′𝑓^𝛼1_1𝑘𝑦₁⋯𝑓^𝛼𝑚_𝑚𝑘𝑦_𝑚, (57) where each term in ∑^′ contains at least one 𝛼_𝑗≠0.
Choose 𝑐=∑^′𝐶_{𝛼1,…,𝛼𝑚} with 𝐶_{𝛼1,…,𝛼𝑚}=𝑇_𝑞((𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘)(𝑥), then 1𝐼𝐼≤𝐶|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓⁰_1𝑘,…,𝑓⁰_𝑚𝑘(𝑧)^𝛿𝑑𝑧^1/𝛿+^′1|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑧)−𝐶_{𝛼1,…,𝛼𝑚}^𝛿𝑑𝑧^1/𝛿∶=𝐼𝐼₀+^′𝐼𝐼_{𝛼1,…,𝛼𝑚}. (58) Noting that 𝛿<1/𝑚 and 𝑇_𝑞∶𝐿¹×⋯×𝐿¹→𝐿^1/𝑚,∞ and using Hölder’ inequality, we obtain 𝐼𝐼₀1=𝐶|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓⁰_1𝑘,…,𝑓⁰_𝑚𝑘(𝑧)^𝛿𝑑𝑧^1/𝛿≤𝐶𝑇_𝑞(𝑏−𝜆)𝑓⁰_1𝑘,…,𝑓⁰_{𝑚𝑘𝐿1/𝑚,∞(𝑄,𝑑𝑥/|𝑄|)}1≤𝐶|𝑄|_𝑄𝑏𝑦₁−𝜆𝑓⁰_1𝑞𝑑𝑦₁^𝑚_𝑗=21|𝑄|_𝑄𝑓⁰_𝑗𝑞𝑗𝑑𝑦_𝑗1≤𝐶|𝑄|_𝑄^∗𝑏𝑦₁−𝜆𝑓_1𝑞𝑑𝑦₁^𝑚_𝑗=21|𝑄|_𝑄^∗𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶‖𝑏‖_𝐵𝑀𝑂𝑓_{1𝑞𝐿(log𝐿),𝑄∗}^𝑚_𝑗=2𝑓_{𝑗𝑞𝑗𝑄∗}≤𝐶‖𝑏‖_𝐵𝑀𝑂ℳ_{𝐿(log𝐿)}𝑓_𝑞(𝑥). (59)
Now consider the term 𝐼𝐼_∞,…,∞. By Hölder’s and Minkoswki’s inequalities as well as Assumptions (H1) and (H2), we obtain 𝐼𝐼_∞,…,∞=1|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓^∞_1𝑘,…,𝑓^∞_𝑚𝑘(𝑧)−𝑇_𝑞(𝑏−𝜆)𝑓^∞_1𝑘,…,𝑓^∞_𝑚𝑘(𝑥)^𝛿𝑑𝑧^1/𝛿1≤𝐶|𝑄|_𝑄^∞_𝑘=1(ℝ^𝑛_⧵𝑄^∗₎^𝑚𝐾𝑦𝑦×𝑏𝑦𝑧,−𝐾𝑥,₁−𝜆𝑓_1𝑘𝑦₁×^𝑚_𝑗=2𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞1𝑑𝑧≤𝐶|𝑄|_𝑄^∞_𝑘=1(ℝ^𝑛_⧵𝑄^∗₎^𝑚𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑦+𝐾𝑧,_𝑡(0)𝑦𝑦×𝑏𝑦𝑧,−𝐾𝑥,₁−𝜆𝑓_1𝑘𝑦₁×^𝑚_𝑗=2𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞𝑑𝑧∶=𝐼𝐼^′_∞,…,∞. (60)
Since 𝑥,𝑧∈𝑄,𝑦_𝑗∈ℝ^𝑛⧵(8√𝑛+4)𝑄, and 𝑡=(2√𝑛𝑙(𝑄))^𝑠, then |𝑥−𝑦_𝑗|∼|𝑧−𝑦_𝑗|, |𝑧−𝑦_𝑗|>(4√𝑛+1)𝑙(𝑄)>2𝑡^1/𝑠, |𝑥−𝑦_𝑗|>2𝑡^1/𝑠 and |𝑥−𝑧|<2𝑡^1/𝑠. Hence,𝜙(|𝑦_𝑗−𝑧|/𝑡^1/𝑠)=0 for 𝑗=1,2.
Thus from the Assumption (H2), we can follow that 𝐼𝐼^′_∞,…,∞≤𝐶|𝑄|_𝑄^∞_𝑘=1(ℝ^𝑛_⧵𝑄^∗₎^𝑚𝐴𝑡^𝜖/𝑠∑^𝑚_𝑖=1𝑧−𝑦_𝑗^{𝑚𝑛+𝜖}×𝑏𝑦₁−𝜆𝑓_1𝑘𝑦₁×^𝑚_𝑗=2𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞𝑑𝑧≤𝐶^∞_𝑘=1^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝜈𝑄^{∗1/𝑛𝑚𝑛+𝜖}×_{(2𝜈+1𝑄∗)𝑚}𝑏₁𝑦₁−𝜆𝑓₁𝑦₁×^𝑚_𝑗=1𝑓_𝑗𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤𝐶^∞_𝜈=012^𝜈𝜖12^𝜈+1𝑄^{∗1/𝑛𝑚𝑛}×^∞_{𝑘=12𝜈+1𝑄∗}𝑏₁𝑦₁×𝑓−𝜆₁𝑦₁𝑑𝑦₁^𝑞11/𝑞1×^𝑚_𝑗=2^∞_{𝑘=12𝜈+1𝑄∗}𝑓_𝑗𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}≤𝐶^∞_𝜈=012^𝜈𝜖12^𝜈+1𝑄^{∗1/𝑛𝑚𝑛}×_{2𝜈+1𝑄∗}𝑏₁𝑦₁−𝜆𝑓_1𝑞1𝑑𝑦₁×^𝑚_{𝑗=22𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶^∞_𝜈=0𝜈2^𝜈𝜖‖𝑏‖_𝐵𝑀𝑂𝑓_{1𝑞1𝐿(log𝐿),2𝜈+1𝑄∗}×^𝑚_𝑗=2𝑓_{𝑗𝑞𝑗𝐿(log𝐿),2𝜈+1𝑄∗}≤𝐶‖𝑏‖_𝐵𝑀𝑂ℳ_{𝐿(log𝐿)}𝑓_𝑞(𝑥), (61)
where we use the fact that ‖𝑏−𝜆‖_exp𝐿,2^𝜈+1_𝑄≤𝐶𝜈‖𝑏‖_exp𝐿,𝑄.
What remains to be considered is the term 𝐼𝐼_{𝛼1,…,𝛼𝑚} such that 𝛼_𝑗1=⋯=𝛼_𝑗𝑙=0, for some 𝜚={𝑗₁,…,𝑗_𝑙}⊂{1,…,𝑚} for 1≤𝑙<𝑚 and denote 𝜚^′=𝜚₀⧵𝜚. We consider only the case 1∈𝜚, by Assumption (H2), we have 𝐼𝐼_{𝛼1,…,𝛼𝑚}=1|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑧)−𝑇_𝑞(𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑥)^𝛿𝑑𝑧^1/𝛿≤1|𝑄|_𝑄𝑇_𝑞(𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑧)−𝑇_𝑞(𝑏−𝜆)𝑓^𝛼1_1𝑘,…,𝑓^𝛼𝑚_𝑚𝑘(𝑥≤1)𝑑𝑧|𝑄|_𝑄^∞_𝑘=1(ℝ^𝑛₎^𝑚𝐾𝑦𝑦×𝑏𝑦𝑧,−𝐾𝑥,₁−𝜆𝑓⁰_1𝑘𝑦₁×^𝑚_𝑗=2𝑓^𝛼𝑗_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤1𝑑𝑧|𝑄|_𝑄^∞_𝑘=1(ℝ^𝑛₎^𝑚𝐾𝑧,𝑦−𝐾_𝑡⁽⁰⁾𝑦+𝐾𝑥,_𝑡(0)𝑦𝑦×𝑏𝑦𝑧,−𝐾𝑥,₁−𝜆𝑓⁰_1𝑘𝑦₁×^𝑚_𝑗=2𝑓^𝛼𝑗_𝑗𝑘𝑦_𝑗𝑑𝑦^𝑞1/𝑞≤𝐶𝑑𝑧|𝑄|_𝑄^∞_𝑘=1𝑄^∗𝑏𝑦₁−𝜆𝑓_1𝑘𝑦₁𝑑𝑦₁×_{{1}𝑗∈𝜚⧵𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗×_{(ℝ𝑛⧵𝑄∗)𝑚−𝑙}×𝑡^𝜖/𝑡∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^{𝑚𝑛+𝜖}+1∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^𝑚𝑛×_{𝑗∈𝜚′}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^𝑞1/𝑞≤𝐶𝑑𝑧|𝑄|_𝑄^∞_𝑘=1𝑄^∗𝑏𝑦₁−𝜆𝑓_1𝑘𝑦₁𝑑𝑦₁×_{{1}𝑗∈𝜚⧵𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗×^∞_{𝜈=0(2𝜈+1𝑄∗⧵2𝜈𝑄∗)𝑚−𝑙}×𝑡^𝜖/𝑡∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^{𝑚𝑛+𝜖}+1∑_{𝑗∈𝜚′}𝑧−𝑦_𝑗^𝑚𝑛×_{𝑗∈𝜚′}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^𝑞1/𝑞𝑑𝑧≤𝐶^∞_{𝑘=1𝑄∗}𝑏𝑦₁−𝜆𝑓_1𝑘𝑦₁𝑑𝑦₁×_{{1}𝑗∈𝜚⧵Q∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗×^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝜈𝑄^{∗1/𝑛𝑚𝑛+𝜖}+12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{(2𝜈+1𝑄∗)𝑚−𝑙𝑗∈𝜚′}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^𝑞1/𝑞≤𝐶^∞_{𝑘=1𝑄∗}𝑏𝑦₁−𝜆𝑓_1𝑘𝑦₁𝑑𝑦₁^𝑞11/𝑞1×_{{1}𝑗∈𝜚⧵}^∞_{𝑘=1𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}×^∞_𝜈=0𝑄^∗𝜖/𝑛2^𝜈𝑄^{∗1/𝑛𝑚𝑛+𝜖}+12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{𝑗∈𝜚′}^∞_{𝑘=12𝜈+1𝑄∗}𝑓_𝑗𝑘𝑦_𝑗𝑑𝑦_𝑗^{𝑞𝑗1/𝑞𝑗}≤_𝑄∗𝑏𝑦₁−𝜆𝑓_1𝑞1𝑑𝑦₁×_{{1}𝑗∈𝜚⧵𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×^∞_𝜈=012^𝜈𝜖12^𝜈𝑄^{∗1/𝑛𝑚𝑛}+12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{𝑗∈𝜚′2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶‖𝑏‖_𝐵𝑀𝑂𝑓_{1𝑞1𝐿(log𝐿),𝑄∗}𝑄^∗×_{{1}𝑗∈𝜚⧵𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗×^∞_𝜈=012^𝜈𝜖12^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{𝑗∈𝜚′2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗+^∞_𝜈=012^𝜈𝑄^{∗1/𝑛𝑚𝑛}×_{𝑗∈𝜚′2𝜈+1𝑄∗}𝑓_𝑗𝑞𝑗𝑑𝑦_𝑗≤𝐶‖𝑏‖_𝐵𝑀𝑂^∞_𝜈=012^𝑚𝜈𝜖_𝑗=1𝑓_{𝑗𝑞𝑗𝐿(log𝐿),2𝜈+1𝑄∗}+^∞_𝜈=012^𝜈𝑛𝑙_𝑗∈𝜚𝑓_{𝑗𝑞𝑗𝐿(log𝐿),𝑄∗}×_{𝑗∈𝜚′}𝑓_{𝑗𝑞𝑗𝐿(log𝐿),2𝜈+1𝑄∗}≤𝐶‖𝑏‖_𝐵𝑀𝑂ℳ_{𝐿(log𝐿)}𝑓_𝑞(𝑥)+ℳ_{𝜚,𝐿(log𝐿)}𝑓_𝑞(𝑥). (62) This finishes the proof of Proposition 10.

The next paragraph we will present the proof of Theorem 2.

Proof of Theorem 2. By Lemma 4, 𝜔∈𝐴_𝑃 implies that 𝜈_𝜔∈𝐴_∞. For simplicity, we may assume that ∑^𝑚_𝑖=1‖𝑏_𝑖‖_𝐵𝑀𝑂=1. Using Proposition 10 and the Fefferman-Stein inequality (26), with 0<𝛿<𝜖<1/𝑚, we have 𝑇_Σ𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀_𝛿𝑇_Σ𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀^♯_𝛿𝑇_Σ𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀_𝜖𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}≤𝐶𝑀^♯_𝜖𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}≤𝐶ℳ𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+ℳ_𝜚𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}. (63)
Then following from (63) and Lemma 5, Theorem 2 is proved.

The reminder that we are left to do is similar to the proof of Theorem 1. Here we omit it.

The proof of Theorem 3 heavily relied on Proposition 11.

Proposition 11. Let 𝑇_𝑞 be a vector-valued multilinear operator associated with an 𝑚-linear operator 𝑇 in 𝐺𝐶𝑍𝑂(𝐴,𝑠,𝜂,𝜖) whose kernel satisfies the Assumption (H2). Suppose that 𝑇_Π𝑏,𝑞 is the corresponding iterated commutator of 𝑇_𝑞 with 𝑏∈𝐵𝑀𝑂². Let 0<𝛿<min{𝜖,1/6}, 𝑟>1, and 𝜚⊂𝜚₀={1,…,𝑚}. Then there exists a constant 𝐶>0 depending on 𝛿 and 𝜖, such that 𝑀^♯_𝛿𝑇_Π𝑏𝑓(𝑥)≤𝐶𝑏_{1𝐵𝑀𝑂}𝑏_{2𝐵𝑀𝑂}×𝑀_𝜖𝑇_𝑞𝑓(𝑥)+ℳ_{𝐿(log𝐿)}𝑓_𝑞(𝑥)+ℳ_{{1},𝐿(log𝐿)}𝑓_𝑞(𝑥)+ℳ_{{2},𝐿(log𝐿)}𝑓_𝑞(𝑥). (64)

The proof of Proposition 11 may be omitted since it is easy to get by the similar arguments of the proof of Proposition 10 and [3].

Finally we will give the proof of Theorem 3.

Proof of Theorem 3. By Lemma 4, 𝜔∈𝐴_𝑃 implies that 𝜈_𝜔∈𝐴_∞. For simplicity, we may assume that ∑^𝑚_𝑖=1‖𝑏_𝑖‖_𝐵𝑀𝑂=1. Using Proposition 11 and the Fefferman-Stein inequality (26) with 0<𝛿<𝜖<1/𝑚, we have 𝑇_Π𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀_𝛿𝑇_Π𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀^♯_𝛿𝑇_Π𝑏,𝑞𝑓_{𝐿𝑝(𝜈𝜔)}≤𝐶𝑀_𝜖𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}≤𝐶𝑀^♯_𝜖𝑇_𝑞𝑓_{𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}≤𝐶ℳ𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+ℳ_𝜚𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+ℳ_{𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}+_𝜚⊂𝜚0ℳ_{𝜚,𝐿(log𝐿)}𝑓_{𝑞𝐿𝑝(𝜈𝜔)}. (65) Then following from (65) and Lemma 5, Theorem 3 is proved.

How to check that ‖𝑀_𝛿(𝑇_Σ𝑏,𝑞(𝑓))‖_{𝐿𝑝(𝜈𝜔)} and ‖𝑀_𝜖(𝑇_𝑞(𝑓))‖_{𝐿𝑝(𝜈𝜔)} are finite is similar to the proof of Theorem  3.19 in [2]. We omit it here, too.