Some remarks on the sparse dominations for commutators of multi(sub)linear operator

We establish pointwise sparse dominations for the iterated commutators of multi(sub)linear operators satisfying the Wq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W_{q}$\end{document} condition. As consequences, we present some quantitative weighted estimates for the commutators. In addition, we also obtain the Fefferman–Stein inequality, the Coifman–Fefferman inequality, and the local decay estimates regarding the iterated commutators.

condition. Wen, Wu, and Xue [6] gave a sparse domination for the iterated commutators of multilinear pseudo-differential operators. Recently Lerner and Ombrosi [7] improved the results in [1] by weakening the assumption on T and by replacing M T (f )(x) with a more flexible operator. Motivated by the above works, the purpose of this paper is to establish a sparse domination for the iterated commutators of multi(sub)linear operator with weaker hypotheses than [4,5].
Motivated by Lerner and Ombrosi [7], we assume that T is an operator satisfying the following W q property instead of assuming T is bounded from L q × · · · × L q → L q/m,∞ : there is a nonincreasing function ψ T,q such that, for any f j ∈ L q (Q) with j = 1, . . . , m and any cube Q, (1.1) It is easy to see that L q × · · · × L q → L q/m,∞ implies that T satisfies the W q property with ψ T,q = T L q ×···×L q →L q/m,∞ λ -m/q . Let α > 0, we define Given an operator T, α > 0, the iterated commutators of T are defined by . Throughout this paper, τ m = {1, . . . , m}. The symbol |τ | denotes the number of the elements in τ . τ = τ m \τ is the complementary set.
Our main results of this paper are as follows.
Remark 1.1 Our Theorem 1.1 is the commutators result of [7]. Compared with the hypotheses in [4,5], the M T,α makes our proof concise and clear. Further, the W q condition of T is weaker than the assumption L q × · · · × L q → L q/m,∞ of T in [4,5].
Let 1 ≤ p 1 , . . . , p m ≤ ∞, w = (w 1 , . . . , w m ), and each w i is a nonnegative function on R d . w is said to satisfy the following A p/r condition if where v w = m k=1 w p/p k k . When r = 1, A p/r is the A p weight class defined by Lerner et al. [8]. With the pointwise sparse domination result, we can obtain the following quantitative weighted estimates and endpoint estimates from Sect. 3 in [5] immediately.

Corollary 1.2 Assume that the multi(sub)linear integral T satisfies the W q condition and
M T,α is bounded from L r × · · · × L r → L r/m,∞ for some α ≥ 3. Let 1 ≤ q, r < ∞, and s = max{q, r}, 1 (1) For w ∈ A p/s , there exists a constant C = C m, p,d,s such that (2) If w ∈ A 1 , then for any λ > 0 and s, = t s (1 + log + t) s it holds that v w x : We also obtain the local decay estimate, the Coifman-Fefferman inequality with w ∈ A ∞ weight, and the Fefferman-Stein inequality with arbitrary weights regarding the iterated commutators. To the best knowledge of the author, these results are new for the iterated commutators of multi(sub)linear operator.

Theorem 1.3 Assume that the multi(sub)linear integral T satisfies the W q condition and
and a > s, then for any functions supp f i ⊂ Q, i = 1, . . . , m, there exist constants α d,m and c d,m such that

Theorem 1.4 Assume that the multi(sub)linear integral T satisfies the W q condition and
M T,α is bounded from L r ×· · ·×L r → L r/m,∞ for some α ≥ 3. Let 1 ≤ q, r < ∞, s = max{q, r}, and a > s, then for any 1 ≤ p < ∞ and any weight w ∈ A ∞ ,

Theorem 1.5 Assume that the multi(sub)linear integral T satisfies the W q condition
and M T,α is bounded from L r × · · · × L r → L r/m,∞ for some α ≥ 3. Let 1 ≤ q, r < ∞, the exponents 1 p = 1 p 1 + · · · + 1 p m with p > s and 1 < p 1 , . . . , p m < ∞, then for all weights The article is organized as follows. Section 2 contains some definitions and main lemmas. The proof of theorems is given in Sect. 3. We present some variations of Theorem 1.1 in Sect. 4.

Definitions and main lemmas
We begin by introducing some definitions and notations.
then the following inequality holds: for all functions f 1 , . . . , f m and all cubes Q.
We introduce the weighted maximal operator and the multi(sub)linear maximal operator which will be used in the proof of our theorem. Let w be a weight and a ≥ 1,

Definition 2.1
The sidelength of Q is denoted by (Q). Given a cube Q 0 ⊂ R d , let D(Q 0 ) denote the generation of Q 0 , that is, the cubes obtained by repeated subdivision of Q 0 . Given dyadic grids D, for any j ∈ Z, the set D j = {Q ∈ D, (Q) = 2 j } forms a partition of R d .

Proof of theorems
Now we devote to proving Theorem 1.1 with the case = m. The other cases are similar so we omit their proof here. The basic idea of our proof is borrowed from Sect. 3 in [5], but the definition of W q and M T,α makes our proof more convenient than [5].
Proof of Theorem 1.1 Fix a cube Q 0 ∈ R d and let Q * 0 = αQ 0 . Set We define the set E as follows: By the weak endpoint estimates of M s ( f ) and (1.1), we can choose c = c d,s,α and A = 2ψ T,q (1/12 · (2α) d ) + M T,α L r →L r,∞ such that |E| ≤ 1 2 d+2 |Q 0 |. Then, applying the local Calderón-Zygmund decomposition to χ E on Q 0 at λ = 1 2 d+1 , we can get a family of pairwisely disjoint cubes {P l } ⊂ D(Q 0 ) such that It is easy to have that l |P l | ≤ 1 2 |Q 0 | and P l E c = ∅.
From [10, Remark 5.1], there exist 3 d dyadic lattices D j such that for every cube Q ⊂ R d we can find a cube R Q ∈ D j satisfying 3Q ⊂ R Q and |R Q | ≤ 9 n |Q|. Note that ). Then we can write Since |E \ l P l | = 0, it follows that Now we calculate II in (3.1). For x ∈ P l and x ∈ P l \E, By |P l \ E| ≥ |P l | 2 and |{x ∈ P l : This allows us to continue (3.3) with III in (3.1) is the term we need. Combining (3.2) with (3.4), it follows that Integrating the above estimates, we can get a 1 2 -sparse family F ⊂ D(Q 0 ) such that for a.e. x ∈ Q 0 The remaining procedure which transfers local setting Q 0 to global setting R d can be referred to Sect. 4.2 in [5]. We omit the details to avoid redundancy. The proof of Theorem 1.1 is finished now.
Proof of Theorem 1.3 Let = m. The proof of other cases is similar. Assume supp f i ⊂ Q 0 , i = 1, . . . , m, and denote It is straightforward to see that we can replace b R Q with b Q * in A F ,τ ( b, f ). Then, by (3.5), we can get a 1 2 -sparse family F ⊂ D(Q 0 ) such that, for almost every x ∈ Q 0 , [11,Lemma 5.1], we can construct a sparse familyF ⊂ D(Q 0 ) such that, for every Q ∈ F ⊂F , For any a > s, Lemma 2.1 gives that

Now we can write
Hence, it is straightforward to have that [12, Lemma 2.1] gives that This finishes the proof of Theorem 1.3.
Proof of Theorem 1.4 Let = m. From Theorem 1.1, it only needs to control A S,τ ( b, f ). By duality, there exists a nonnegative function g ∈ L p (w) satisfying g L p (w) = 1. Then we can write where s < a. The second inequality follows from Lemma 2.1, and we have used Lemma 2.2 in the last inequality. By [13], we know M w (|g| q )(x) 1/q is L p (w j ) bounded when 1 < q < p . We can continue writing above display as where 1 < q < p . This finishes the proof of Theorem 1.4.
Proof of Theorem 1.5. The basic idea of our proof is borrowed from Sect. 4.3 in [14] or [15]. From Theorem 1.1, it is enough to control A S,τ ( b, f ). Let = m.
We denote v i (x) := Mw i (x), then it is easy to see that w i Q ≤ v i (x) for any cube Q containing x. We can choose constants a, b with s < a < p < b. [11,Lemma 5.1] gives us a sparse familyS such that, for every Q ∈ S ⊂S, it holds that