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Some remarks on the sparse dominations for commutators of multi(sub)linear operator
Journal of Inequalities and Applications volume 2021, Article number: 51 (2021)
Abstract
We establish pointwise sparse dominations for the iterated commutators of multi(sub)linear operators satisfying the \(W_{q}\) 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.
1 Introduction
Sparse domination is a relatively new tool to prove weighted estimates for singular integrals. The sparse method has been developed over the past five years by many researchers in harmonic analysis, with significant works by Lerner [1], Lacey [2], Conde-Alonso [3], and the references therein.
A collection \({{\mathcal{S}}}\) is an η-sparse family of cubes in \(\mathbb{R}^{d}\) if for every \(Q\in {{\mathcal{S}}}\) there exists \(E_{Q}\subset Q\) such that \(|E_{Q}|\geq \eta |Q|\) (here \(0<\eta <1\)), and \(E_{Q}\cap E_{Q'}=\varnothing \) when \(Q\neq Q'\). For an η-sparse collection of cubes \({{\mathcal{S}}}\), we use the notation
This averaging form is then easily controlled, facilitating the proof of weighted \(A_{p}\)-type estimates.
Lerner [1] proved a pointwise sparse domination for ω Calderón–Zygmund operators. The key role in his proof is played by the grand maximal operator
Later, Li [4] established sparse domination theorem for multilinear singular integral operators with the kernel satisfying the \({L}^{r}\)-Hörmander condition. Cao and Yabuta [5] developed sparse dominations for the multilinear Littlewood–Paley operators with the same kernel 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 \(\mathcal{M}_{T}(f)(x)\) 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 \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \): there is a nonincreasing function \(\psi _{T,q}\) such that, for any \(f_{j}\in L^{q}(Q)\) with \(j=1,\dots ,m\) and any cube Q,
It is easy to see that \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \) implies that T satisfies the \(W_{q}\) property with \(\psi _{T,q}=\|T\|_{L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty }}\lambda ^{-m/q}\).
Let \(\alpha >0\), we define
Given an operator T, \(\alpha >0\), the iterated commutators of T are defined by
where \(\vec{b}=\vec{b}^{(\ell )}=(b_{1},\dots ,b_{\ell })\) (\(1\leq \ell \leq m\)). Throughout this paper, \(\tau _{m}= \{1,\ldots , m\}\). The symbol \(|\tau |\) denotes the number of the elements in τ. \(\tau '=\tau _{m}\backslash \tau \) is the complementary set.
Our main results of this paper are as follows.
Theorem 1.1
Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), and \(s=\max \{q, r\}\). Then, for any compactly supported functions \(f_{i} \in L^{s}(\mathbb{R}^{d})\), \(i=1,\dots ,m\), there exist \(3^{d}\) sparse families \(\mathcal{S}_{j}\) such that, for a.e. \(x\in \mathbb{R}^{d} \),
where \(C=c_{d,s,\alpha } (\psi _{T,q}(1/12\cdot (2\alpha )^{d}) +\| \mathcal{M}^{\sharp }_{T,\alpha } \|_{L^{r}\times \cdots \times L^{r} \rightarrow L^{r/m,\infty }} )\).
Remark 1.1
Our Theorem 1.1 is the commutators result of [7]. Compared with the hypotheses in [4, 5], the \(\mathcal{M}^{\sharp }_{T,\alpha }\) makes our proof concise and clear. Further, the \(W_{q} \) condition of T is weaker than the assumption \(L^{q}\times \cdots \times L^{q}\rightarrow L^{q/m , \infty } \) of T in [4, 5].
Let \(1\leq p_{1},\dots , p_{m} \leq \infty \), \(\vec{w}=(w_{1},\dots ,w_{m})\), and each \(w_{i}\) is a nonnegative function on \(\mathbb{R}^{d}\). w⃗ is said to satisfy the following \(A_{\vec{p}/r}\) condition if
where \({v_{\vec{w}}}=\prod_{k=1}^{m}w_{k}^{p/p_{k}}\). When \(r=1\), \(A_{{\vec{p}}/{r}}\) is the \(A_{\vec{p}}\) 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 \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), and \(s=\max \{q, r\}\), \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}} \) with \(1\leq s< p_{i}< \infty \). Set \(\sigma _{i}=w_{i}^{1-(p_{i}/s)'}\), \(i=1,\ldots ,m\). Let \(T^{\ell }_{\vec{b}}(\vec{f})\) be defined as (1.2).
\((1)\) For \(\vec{w}\in A_{{\vec{p}}/{s}}\), there exists a constant \(C=C_{m,\vec{p},d,s }\) such that
\((2)\) If \(\vec{w}\in A_{\vec{1}}\), then for any \(\lambda >0\) and \(\Phi _{s,\ell }=t^{s}(1+\log ^{+} t)^{s\ell }\) it holds that
We also obtain the local decay estimate, the Coifman–Fefferman inequality with \(w\in A_{\infty }\) 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 \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), \(s=\max \{q, r\}\), and \(a>s\), then for any functions \(\operatorname{supp} f_{i}\subset Q\), \(i=1,\dots ,m\), there exist constants \(\alpha _{d,m}\) and \(c_{d,m}\) such that
Theorem 1.4
Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), \(s=\max \{q, r\}\), and \(a>s\), then for any \(1\leq p<\infty \) and any weight \(w\in A_{\infty }\),
Theorem 1.5
Assume that the multi(sub)linear integral T satisfies the \(W_{q}\) condition and \(\mathcal{M}^{\sharp }_{T,\alpha }\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \) for some \(\alpha \geq 3\). Let \(1\leq q,r<\infty \), the exponents \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\) with \(p>s\) and \(1< p_{1},\ldots , p_{m}<\infty \), then for all weights \(\vec{w}=(w_{1}, \ldots , w_{m})\), \(\nu _{\vec{w}}=\prod_{i=1}^{m} w_{i}^{p/p_{i}}\),
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.
2 Definitions and main lemmas
We begin by introducing some definitions and notations.
Lemma 2.1
(Generalized Hölder inequality, [9])
Let \({\Phi }_{0},{\Phi }_{1},,\ldots ,{\Phi }_{k} \) be Young functions. If
then the following inequality holds:
for all functions \(f_{1},\ldots ,f_{m}\) and all cubes Q.
In particular, if \(\sum_{i=1}^{k}\frac{1}{s_{i}}=\frac{1}{s}\) with each \(s_{i}\ge 1\), then it holds that
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 \(a\geq 1\),
Lemma 2.2
([5])
Let \(s>1\), \(t>0\), and \(w\in A_{\infty }\), then there holds that
Definition 2.1
The sidelength of Q is denoted by \(\ell (Q)\). Given a cube \(Q_{0} \subset \mathbb{R}^{d}\), let \(\mathcal{D}(Q_{0})\) denote the generation of \(Q_{0}\), that is, the cubes obtained by repeated subdivision of \(Q_{0}\). Given dyadic grids \(\mathcal{D}\), for any \(j\in \mathbb{Z}\), the set \(\mathcal{D}_{j}=\{Q\in \mathcal{D},\ell {(Q)}=2^{j}\}\) forms a partition of \(\mathbb{R}^{d}\).
3 Proof of theorems
Now we devote to proving Theorem 1.1 with the case \(\ell =m\). 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 \(\mathcal{M}^{\sharp }_{T,\alpha }\) makes our proof more convenient than [5].
Proof of Theorem 1.1
Fix a cube \(Q_{0}\in \mathbb{R}^{d}\) and let \(Q_{0}^{*}=\alpha Q_{0}\). Set
We define the set E as follows:
By the weak endpoint estimates of \(M_{s}(\vec{f})\) and (1.1), we can choose \(c=c_{d,s,\alpha }\) and \(A=2\psi _{T,q}(1/12\cdot (2\alpha )^{d}) +\| \mathcal{M}^{\sharp }_{T, \alpha } \|_{L^{r}\rightarrow L^{r,\infty }}\) such that \(|E|\leq \frac{1}{2^{d+2}}|Q_{0}|\). Then, applying the local Calderón–Zygmund decomposition to \(\chi _{E}\) on \(Q_{0}\) at \(\lambda =\frac{1}{2^{d+1}}\), we can get a family of pairwisely disjoint cubes \(\{P_{l}\}\subset \mathcal{D}{(Q_{0})}\) such that
It is easy to have that \(\sum_{l} |P_{l}|\leq \frac{1}{2}|Q_{0}|\) and \(P_{l}\bigcap E^{c} \neq \emptyset \).
From [10, Remark 5.1], there exist \(3^{d}\) dyadic lattices \(\mathcal{D}_{j}\) such that for every cube \(Q \subset \mathbb{R}^{d}\) we can find a cube \(R_{Q}\in \mathcal{D}_{j}\) satisfying \(3Q\subset R_{Q}\) and \(|R_{Q}|\leq 9^{n}|Q|\). Note that \(\prod_{i=1}^{m} (b_{i}(x)-b_{i}(y_{i}) )=\sum_{\tau \subset \tau _{m}}\prod_{i\in \tau } (b_{i}(x)-b_{i, R_{Q_{0}}} ) \prod_{j\in \tau '} (b_{j, R_{Q_{0}}}-b_{j}(y_{j}) ) \). Then we can write
Since \(|E\setminus \bigcup_{l} P_{l}|=0\), it follows that
Now we calculate II in (3.1). For \(x\in P_{l} \) and \(x'\in P_{l}\backslash E\),
Denoting \(\vec{g}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}(x)=\prod_{i \in \tau } f_{i}\chi _{Q_{0}^{*}\backslash P_{l}^{*}}\prod_{j \in \tau '} (b_{j, R_{Q_{0}}}-b_{j} ) f_{j}\chi _{Q_{0}^{*} \backslash P_{l}^{*}}(x)\), we can write
By \(|P_{l}\setminus E|\geq \frac{|P_{l}|}{2}\) and \(| {\{} x\in P_{l}: |T(\vec{f}\chi _{P_{l}^{*}})(x') |>A \prod_{j=1}^{m}\langle f_{j} \rangle _{s,P^{*}_{l}} {\}} |\leq \frac{|P_{l}|}{2^{d+2}}\), we can get that
This allows us to continue (3.3) with
Therefore
\(III\) 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 \(\frac{1}{2}\)-sparse family \(\mathcal{F}\subset \mathcal{D}(Q_{0})\) such that for a.e. \(x\in Q_{0}\)
The remaining procedure which transfers local setting \(Q_{0}\) to global setting \(\mathbb{R}^{d}\) 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 \(\ell =m\). The proof of other cases is similar. Assume \(\operatorname{supp} f_{i}\subset Q_{0}\), \(i=1,\dots ,m\), and denote
It is straightforward to see that we can replace \(b_{R_{Q}}\) with \(b_{Q^{*}}\) in \(\mathcal{A}_{\mathcal{F},\tau }(\vec{b},\vec{f})\). Then, by (3.5), we can get a \(\frac{1}{2}\)-sparse family \(\mathcal{F}\subset \mathcal{D}(Q_{0})\) such that, for almost every \(x\in Q_{0}\),
First we observe that \(|b_{i}(x)-b_{i,Q^{*}}|\lesssim \|b_{i}\|_{BMO}+|b_{i}(x)-b_{i,Q}| \). From [11, Lemma 5.1], we can construct a sparse family \(\tilde{\mathcal{F}}\subset \mathcal{D}(Q_{0})\) such that, for every \(Q\in \mathcal{F}\subset \tilde{\mathcal{F}} \),
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 \(\ell =m\). From Theorem 1.1, it only needs to control \(\mathcal{A}_{\mathcal{S},\tau }(\vec{b},\vec{f})\). By duality, there exists a nonnegative function \(g\in L^{p'}(w)\) 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 \(\mathcal{A}_{\mathcal{S},\tau }(\vec{b},\vec{f})\). Let \(\ell =m\).
We denote \(v_{i}(x):=Mw_{i}(x)\), then it is easy to see that \(\langle w_{i} \rangle _{Q}\leq v_{i}(x) \) 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 family \(\tilde{\mathcal{S}}\) such that, for every \(Q\in \mathcal{S}\subset \tilde{\mathcal{S}} \), it holds that
By Lemma 2.1, it holds that
The sparseness property of collection \({{\mathcal{S}}}\), \({\tilde{\mathcal{S}}}\) allows us to continue with
This finishes the proof of Theorem (1.3). □
4 Variations
Let us introduce some notions first. T is a Calderón–Zygmund operator defined as
Definition 4.1
([4])
Let \(1\leq r <\infty \), \(r'=\frac{r}{r-1}\). The kernel \(K(x, \vec{y})\) is called to satisfy the multilinear \(L^{r}\)-Hörmander condition if
When \(r=1\), it should be understood as
We first show that the multilinear \(L^{r}\)-Hörmander condition implies that \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m,\infty }\) with \(1\leq r<\infty \).
Lemma 4.1
Suppose \(1\leq r<\infty \), \(\alpha \geq 3\). T is an operator whose kernel satisfies the bilinear \(L^{r}\)-Hörmander condition. Then \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m,\infty }\). More specifically, for any \(x\in \mathbb{R}^{d}\), it holds that
where \(\mathcal{M}_{r}(\vec{f})(x)=\mathrm{{M}}(|\vec{f}|^{r})^{\frac{1}{r}}\) and M is the Hardy–Littlewood maximal function.
Proof
Let \(\xi , \xi '\in Q\).
Since \(\mathcal{M}_{r}(\vec{f})(x)\) is bounded from \(L^{r}\times \cdots \times L^{r}\rightarrow L^{r/m , \infty } \), we can get the weak type estimates of \(\mathcal{M}^{\sharp }_{T,\alpha }(\vec{f})\). □
Theorem 4.2
Let the multi(sub)linear integral T satisfy the \(W_{q}\) condition and the kernel satisfy the multilinear \(L^{r}\)-Hörmander condition with \(1\leq q,r<\infty \), then all the estimates in Sect. 1hold for this operator T.
We give another variation of Theorem 1.1. In this case, we do not need to assume that T is a multi(sub)linear operator. Given an operator \(T(\vec{f})\) and \(\alpha >0\), we define
Theorem 4.3
Let \(1\leq q,r<\infty \), \(\alpha \geq 3\). There are nonincreasing functions ψ and ϕ such that, for any cube Q,
Then all the estimates in Sect. 1hold for this operator T.
Remark 4.2
The assumption of \(\mathcal{M}^{\sharp }_{T,\alpha }\vec{f}\) satisfying the \(W_{r}\) condition allows us not to prove Lemma 4.1. In addition, we should use \(T_{\vec{b}}(\vec{f}\chi _{Q_{0}} )(x)-T_{\vec{b}}(\vec{f}\chi _{ P_{l}^{*}} )(x)\) instead of \(T_{\vec{b}}(\vec{f}\chi _{Q_{0}^{*}\backslash P_{l}^{*}} )(x)\) to avoid using the multi(sub)linear property.
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The author wants to express thanks to Qingying Xue for his valuable suggestions on improving this work.
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Wang, Z. Some remarks on the sparse dominations for commutators of multi(sub)linear operator. J Inequal Appl 2021, 51 (2021). https://doi.org/10.1186/s13660-021-02585-z
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DOI: https://doi.org/10.1186/s13660-021-02585-z