Abstract
Let \(T_{\vec{b}}\) and \(T_{\Pi b}\) be the commutators in the jth entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well known that the commutators of linear Calderón-Zygmund operators were not of weak type \((1,1)\) and \((H^{1}, L^{1})\), but they did satisfy certain endpoint \(L\log L\) type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that \(T_{\vec{b}}\) and \(T_{\Pi b}\) are bounded from the product Hardy space \(H^{1}\times\cdots\times H^{1}\) to weak \(L^{\frac{1}{m},\infty}\) space, whenever the kernel satisfies a class of Dini type condition. This was done by using a key lemma given by Christ, a very complex decomposition of the integrand domains, and carefully splitting the commutators into several terms.
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1 Introduction
1.1 Commutators of classical C-Z operators
In 1976, Coifman, Rochberg, and Weiss [1] first introduced and studied the commutator of classical linear Calderón-Zygmund singular integrals, which was defined by
The \(L^{p}\) boundedness of \(T_{b}\) was given in [1] for \(1< p<\infty \) when \(b\in BMO(\mathbb{R}^{n})\). It is well known that \(T_{b}\) fails to be of weak type \((1,1)\) and is not bounded from \(H^{1}(\mathbb{R}^{n})\) to \(L^{1}(\mathbb{R}^{n})\). Counterexamples were given by Pérez [2] and Paluszyński [3]. As an alternative result of the weak \((1,1)\) estimate of \(T_{b}\), Pérez [2] obtained the following \(L(\log L)\) type endpoint estimate:
Moreover, alternative results of the \((H^{1}, L^{1})\) boundedness were also considered in the work of Alvarez [4], Pérez [2], and Liang, Ky, and Yang [5], which concerned with the boundedness of \(T_{b}\) on the subspace of atomic Hardy spaces, or concerned with the \((H_{w}^{1}, L_{w}^{1})\) boundedness of \(T_{b}\) if b belongs to a subspace of \(BMO\) which is associated to the weight function w.
On the other hand, another more reasonable and alternative result of weak type \((1,1)\) and \((H^{1}, L^{1})\) estimate was given by Liu and Lu [6] in 2002. The authors [6] showed that \(T_{b}\) is bounded from \(H^{1}(\mathbb{R}^{n})\) to \(L^{1,\infty}(\mathbb{R}^{n})\) if \(b\in BMO(\mathbb{R}^{n})\). We note that \(T_{b}\) also fails to be bounded from \(H^{p}(\mathbb{R}^{n})\) to \(L^{p,\infty}(\mathbb{R}^{n})\) for \(0< p<1\) by the generalized interpolation theorem [7], pp.63. Therefore, the \((H^{1}, L^{1,\infty})\) boundedness of \(T_{b}\) becomes a sharp endpoint estimate. Moreover, always \(L(\log L)(\mathbb{S}^{n-1})\subsetneq H^{1}(\mathbb {S}^{n-1})\) if f vanishes on the unit sphere. However, there is no such inclusion relationship on \(\mathbb{R}^{n}\). Moreover, the inverse including relationship is still not true, since the following example shows that \(H^{1}(\mathbb{R}^{n})\nsubseteq L(\log L)(\mathbb{R}^{n})\).
Example 1.1
Let
Thus, \(f(x)=\sum_{j=1}^{\infty}\lambda_{j}a_{j}(x)\), and it is easy to verify that each \(a_{j}\) is a \((1,\infty,0)\)-atom. Notice that
then we have \(f\in H^{1}(\mathbb{R}^{n})\). Obviously, \(f\notin L(\log L)(\mathbb{R}^{n})\).
Thus, the \((H^{1}, L^{1,\infty})\) boundedness and the \(L\log L\) type estimate of \(T_{b}\) are independent in the sense that one cannot cover the results of the other.
1.2 Commutators of multilinear operators
In recent years, the theory of multilinear Calderón-Zygmund operators with standard kernels have been developed very quickly and a lot of work has been done. Among such achievements is the celebrated work of Coifman and Meyer [8–10], Christ and Journé [11], Kenig and Stein [12], Grafakos and Torres [13, 14], and Lerner et al. [15]. In order to state some well-known results, we need to introduce some definitions.
Definition 1.2
Let \(\omega(t)\) be a non-negative and non-decreasing function on \(\mathbb{R}^{+}\). Let \(K(x, y_{1}, \ldots, y_{m})\) be a locally integrable function defined away from the diagonal \(x= y_{1}=\cdots=y_{m}\) in \((\mathbb{R}^{n})^{m+1}\). Denote \((x, \vec{y})=(x, y_{1}, \ldots, y_{m})\), we say K is an m-linear Calderón-Zygmund kernel of ω type, if there exists a positive constant \(C_{0}\) such that
whenever \(|x-x'|\leq\frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\), and
whenever \(|y_{i}-y_{i}'|\leq\frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\).
Definition 1.3
Multilinear C-Z singular integral operators [16, 17]
Let \(K(x, \vec{y})\) be a C-Z kernel of ω type. For any \(\vec {f}=(f_{1},\ldots,f_{m})\in\mathscr{S}(\mathbb{R}^{n})\times \mathscr{S}(\mathbb{R}^{n}) \times\cdots\times\mathscr{S}(\mathbb{R}^{n})\) and all \(x\notin\bigcap_{j=1}^{m}\) supp \(f_{j}\), we define the multilinear Calderón-Zygmund singular integral operators as follows:
Definition 1.4
Commutators of multilinear C-Z operators
Let \(b_{j}\in BMO(\mathbb{R}^{n})\) and T be the operator defined in Definition 1.3. The commutators in the jth entry and the iterated commutators of T are defined by
and
Remark 1.5
Obviously, in the special case, \(\omega(t)=t^{\varepsilon}\) for some \(\varepsilon>0\), then the operator T defined in Definition 1.3 coincides with the standard multilinear Calderón-Zygmund operator defined and studied by Grafakos and Torres [13]. Moreover, if \(\omega(t)=t^{\varepsilon}\), the weighted strong and \(L(\log L)\) type endpoint estimates for \(T_{\vec{b}} (f_{1}, \ldots, f_{m})(x)=\sum_{j=1}^{m}T_{\vec{b}}^{j}(\vec{f})\) and \(T_{\Pi b}\) have already been studied in [15] and [18], respectively.
Definition 1.6
\(\operatorname{Dini}(a)\) type conditions
Let \(\omega(t)\) be a non-negative and non-decreasing function on \(\mathbb{R}^{+}\). ω is said to satisfy the \(\operatorname{Dini}(a)\) condition if
ω is said to satisfy the \(\log\!\mbox{-}\!\operatorname{Dini}(a)\) condition if the following inequality holds:
Remark 1.7
It is easy to see that the \(\log\!\mbox{-}\!\operatorname{Dini}(a)\) condition is stronger than the \(\operatorname{Dini}(a)\) condition and if \(0< a_{1}< a_{2}\), then \(\operatorname{Dini}(a_{1})\subset \operatorname{Dini}(a_{2})\).
In 2009, Maldonado and Naibo [17] showed that, when ω is concave and \(\omega\in \operatorname{Dini}(1/2)\), the bilinear Calderón-Zygmund operator of ω type is bounded from \(L^{1}\times L^{1}\) to \(L^{\frac{1}{2},\infty}\). In 2014, Lu and Zhang [16] improved the results in [17] by removing the hypothesis that ω is concave and reducing the condition \(\omega\in \operatorname{Dini}(1/2)\) to the weaker condition \(\omega\in \operatorname{Dini}(1)\). Lu and Zhang [16] also extended the weighted strong and \(L(\log L)\) type endpoint estimates to the commutators defined in (1.4) whenever ω satisfies the \(\log\!\mbox{-}\!\operatorname{Dini}(1)\) condition, which is stronger than \(\operatorname{Dini}(1)\) condition but it is much weaker than the standard kernel \(\omega(t)=t^{\varepsilon}\). More previous work on the commutators of multilinear operators with \(\omega(t)=t^{\varepsilon}\) can be found in [18–21] and [22].
1.3 Main results
In this paper, we will consider the sharp endpoint estimates for both the commutator in the jth entry defined in (1.4) and the iterated commutators defined in (1.5) with a C-Z kernel of ω type. We show that they are bounded from a product Hardy space \(H^{1}\times\cdots\times H^{1}\) to a weak \(L^{\frac{1}{m},\infty}\) space, whenever the kernel satisfies a class of Dini type condition. However, the proof is very difficult and complex. In particular, in the case of iterated commutators, we need to control six summations and three integrals at the same time even for \(m=2\). We formulate our main results as follows.
Theorem 1.1
Let T be a multilinear Calderón-Zygmund operators with a C-Z kernel of ω type and \(T_{\vec{b}}\) be the commutators of the jth entries defined in (1.4) with \(\vec{b}\in BMO^{m}\). If \(\omega(t)\) satisfies the \(\log\!\mbox{-}\!\operatorname{Dini}(1)\) condition, then there exists a constant \(C>0\), such that the following inequality holds:
With a stronger condition assumed on the function \(\omega(t)\) than in Theorem 1.1, but a weaker condition than the standard kernel \(\omega(t)=t^{\varepsilon}\), we obtain the following theorem for the iterated commutators.
Theorem 1.2
Let \(\omega(t)\) be a doubling function, satisfying the \(\log\!\mbox{-}\!\operatorname{Dini}(1/2m)\) condition, that is,
Let T be a multilinear Calderón-Zygmund operators with a C-Z kernel of ω type and \(T_{\Pi b}\) be the iterated commutators defined in (1.5) with \(\vec{b}\in BMO^{m}\). Then there exists a constant \(C>0\), such that the following inequality holds:
This article is organized as follows. In Section 2, the proof of Theorem 1.1 will be given. Section 3 will be devoted to the proof of Theorem 1.2.
2 Proofs of Theorem 1.1
To prove Theorem 1.1, we need the following key lemma given by Chirst [23], which provides a foundation for our analysis.
Lemma 2.1
[23]
For any \(\alpha>0\) and any finite collection of dyadic cubes Q and associated positive scalars \(\lambda_{Q}\), there exists a collection of pairwise disjoint dyadic cubes S such that
-
(1)
\(\sum_{Q\subset S}\lambda_{Q}\leq2^{n}\alpha|S|\), for all S;
-
(2)
\(\sum|S|\leq\alpha^{-1}\sum\lambda_{Q}\);
-
(3)
\(\Vert \sum_{Q\nsubseteq\ \mathrm{any}\ S}\lambda _{Q}|Q|^{-1}\chi_{Q} \Vert _{L^{\infty}(\mathbb{R}^{n})}\leq\alpha\).
Proof of Theorem 1.1
For simplicity, we only consider the case for \(m=2\), because there is no essential difference for the general case.
Since \(T_{\vec{b}}\) is bounded from \(L^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\) into \(L^{1}(\mathbb{R}^{n})\) [16], and finite sums of atoms are dense in \(H^{1}(\mathbb{R}^{n})\), we will work with such sums and will obtain desired estimates which is independent of the number of terms in each sum. Thus, for any given \(f_{j}\in H^{1}(\mathbb{R}^{n}) \) (\(j=1, 2\)), we may assume that \(f_{j}=\sum_{k_{j}}\lambda_{k_{j}}a_{k_{j}}\) is a finite sum of \(H^{1}\)-atoms, where each \(a_{k_{j}}\) is a \((1,\infty,0)\) atom, with \(\sum_{k_{j}}|\lambda_{k_{j}}|\leq C\Vert f_{j} \Vert _{H^{1}(\mathbb{R}^{n})}\). Set \(C_{1}=\Vert T_{\vec{b}} \Vert _{L^{2}\times L^{2}\rightarrow L^{1,\infty}}\) and \(C_{2}=\Vert T \Vert _{L^{1}\times L^{1}\rightarrow L^{\frac {1}{2},\infty}}\). By linearity, it is sufficient to consider the commutator of T with only one symbol, that is, for \(\vec{b}=b\in BMO(\mathbb{R}^{n})\), we will consider the operator
To prove inequality (1.7), without loss of generality, we may assume that \(\Vert f_{j} \Vert _{H^{1}(\mathbb{R}^{n})}=1\) for \(j=1, 2\). For fix \(\lambda>0\), we only need to show that there is a constant \(C>0\), independent on the variables and \(f_{j} \) (\(j=1,2\)), such that
Let γ be a positive number to be determined later. Take the finite collection of dyadic cubes \(Q_{j,k_{j}}\), which is associated with the positive scalars \(\lambda_{Q_{j,k_{j}}}\) in the given atomic decomposition of \(f_{j}\). Now, we take \(\alpha=(\gamma\lambda)^{1/2}\) in Lemma 2.1. Then there exists a collection of pairwise disjoint dyadic cubes \(S_{j,l_{j}}\), such that
Denote \(S_{j,l_{j}}^{*} = 8\sqrt{n}S_{j,l_{j}}\), \(S_{j}^{*}=\bigcup _{l_{j}}S_{j,l_{j}}^{*} \) for \(j=1, 2\), and \(S^{*}=\bigcup_{j=1}^{2}S_{j}^{*} \). Set
By the definition of \(g_{j}\) and \(h_{j}\), (III), and the properties of the \((1,\infty,0)\) atoms, we have
Now, we introduce some more notations as follows:
By (II), it follows that
From the \(L^{2}\times L^{2}\rightarrow L^{1,\infty}\) boundedness of \(T_{\vec{b}}\), the Chebyshev inequality, and \(\Vert g_{j} \Vert _{L^{\infty }(\mathbb{R}^{n})}\leq(\gamma\lambda)^{1/2}\), one may obtain
Therefore, we get
Hence, to finish the proof of Theorem 1.1, we only need to consider the contributions of each \(|E_{s}|\) for \(2\le s\le4\), separately.
• Estimate for \(|E_{2}|\) . By the definition of \(g_{j}\) and \(h_{j}\), the moment condition of \(H^{1}\)-atoms, and employing the linearity of \(T_{b}\), it now follows that
Therefore, we have
Thus, to show the contributions of \(E_{2}\), we only need to consider the contributions of \(E_{2,1}\) and \(E_{2,2}\), respectively.
To estimate \(|E_{2,1}|\), we fix \(k_{1}\) and denote \(\mathscr{R}_{1, k_{1}}^{i}=(2^{i+2}\sqrt{n}Q_{1, k_{1}})\backslash(2^{i+1}\sqrt{n}Q_{1, k_{1}})\), \(i=1,2,\ldots\) . Then it is obvious that \(\mathbb{R}^{n}\backslash S^{*}\subset \mathbb{R}^{n}\backslash Q_{1, k_{1}}^{*}\subset\bigcup _{i=1}^{\infty}\mathscr{R}_{1, k_{1}}^{i}\). Let \(c_{1,k_{1}}\) be the center of cube \(Q_{1,k_{1}}\), \(l_{Q_{1,k_{1}}}\) be the side length of cube \(Q_{1,k_{1}}\) Then, for any \(y_{1}\in Q_{1, k_{1}}\) and \(x\in\mathscr {R}_{1, k_{1}}^{i}\), we have
By the Chebychev inequality and (1.3), it follows that
Since \(\mathbb{R}^{n}\backslash S^{*}\subset\bigcup_{i=1}^{\infty}\mathscr {R}_{1, k_{1}}^{i}\) and ω is non-decreasing, together with (2.6) and noticing that \(a_{1,k_{1}}\in L^{1}(\mathbb{R}^{n})\), one obtains
Putting the above estimate into (2.7) and noticing the fact that \(\Vert g_{j} \Vert _{L^{\infty}(\mathbb{R}^{n})}\leq(\gamma \lambda )^{1/2}\), we have
Now, we are in the position to estimate \(|E_{2,2}|\). The \(L^{1}\times L^{1}\rightarrow L^{\frac{1}{2}, \infty}\) boundedness of T implies that
Therefore in all, combining (2.8) and the above estimate, we conclude that
• Estimate for \(|E_{3}|\) . The estimate of \(|E_{3}|\) is similar to \(|E_{2}|\). In fact,
Repeating the same steps as we have done for \(|E_{2}|\), we may obtain
• Estimate for \(|E_{4}|\) . First, we split \(T_{b}(h_{1},h_{2})\) in the form as follows:
Hence, we have
For fixed \(k_{2}\), denote \(\mathscr{R}_{2, k_{2}}^{h}=(2^{h+2}\sqrt{n}Q_{2, k_{2}})\backslash(2^{h+1}\sqrt{n}Q_{2, k_{2}})\), \(h=1,2,\ldots\) . Recalling the definition of \(\mathscr{R}_{1, k_{1}}^{i}\), it is easy to check
Therefore, one may obtain
By the Chebychev inequality, (1.3), and (2.11), it follows that
Moreover, by (2.11), the integrals in the above summations can be controlled by
For fixed \(x\in(S^{*})^{c}\), and any \(y_{1}, y_{2}\in S\), we have
This implies that
Note that \(\{S_{j,l_{j}}\}_{l_{j}}\) are pairwise disjoint dyadic cubes, by (I) and (2.14), it now follows that
Combining (2.12), (2.13), and (2.15), we obtain
The estimate of \(|\{x\in \mathbb{R}^{n}\backslash S^{*}: |I_{4,2}(x)| >\lambda /8\}|\) is similar to (2.9). In fact, we only need to replace \(g_{2}\) by \(h_{2}\) in (2.9), and noting that \(\Vert h_{2} \Vert _{L^{1}}\leq C\Vert f_{2} \Vert _{H^{1}}\), we have
Putting (2.16) and (2.17) into (2.10), it yields
Thus, we have proved that
Set \(\gamma=(C_{0}+C_{1}+C_{2})^{-1}\), by (2.4) and (2.18), we have
The proof of (2.1) is finished. Since we have reduced the proof of Theorem 1.1 to (2.1), the proof of Theorem 1.1 is completed. □
3 Proof of Theorem 1.2
Proof of Theorem 1.2
Since there is no essential difference for the general case, we will also only consider Theorem 1.2 for the case \(m = 2\). Thus, it is sufficient to consider the following operator:
where \(f_{j}\in H^{1}(\mathbb{R}^{n}) \) (\(j=1, 2\)) with \(\Vert f_{j} \Vert _{H^{1}(\mathbb{R}^{n})}=1\) for \(j=1, 2\). Since \(T_{\pi b}(f_{1},f_{2})(x)\) is bounded from \(L^{2}(\mathbb{R}^{n})\times L^{2}(\mathbb{R}^{n})\) into \(L^{1}(\mathbb{R}^{n})\) [18], we may set \(C_{1}'=\Vert T_{\pi b} \Vert _{L^{2}\times L^{2}\rightarrow L^{1,\infty}}\). Recall \(C_{2}=\Vert T \Vert _{L^{1}\times L^{1}\rightarrow L^{\frac {1}{2},\infty }}\), following the same argument as in the proof of Theorem 1.1, it is also sufficient to show that
The same decomposition for \(f_{j}\in H^{1}(\mathbb{R}^{n}) \) (\(j=1, 2\)) as in Theorem 1.1 yields
where \(g_{j}\) and \(h_{j}\) enjoy the same properties as in Theorem 1.1.
With abuse of notations, we may still set
Then (2.2) still gives
Note that \(C_{1}'=\Vert T_{\pi b} \Vert _{L^{2}\times L^{2}\rightarrow L^{1,\infty }}\), repeating the arguments as in the estimates of (2.3), we may obtain
Therefore,
Thus, to show Theorem 1.2 is true, we only have to show that
In fact, let \(\gamma=(C_{0}+C_{1}'+C_{2})^{-\frac{1}{2}}\), it is easy to check that the inequality (3.1) is true.
• Estimate for \(|E_{2}|\) . Employing the linearity of \(T_{\pi b}\) and the atomic decomposition of \(h_{1}\), we may get
Thus
By the definition of \(I_{2,1}\) and the moment condition of \(H^{1}\)-atoms, we have
Putting the above identity into the definition of \(|E_{2,1}|\) and noting that \(\Vert g_{2} \Vert _{L^{\infty}(\mathbb{R}^{n})}\leq (\gamma\lambda )^{1/2}\), \(\mathbb{R}^{n}\backslash S^{*}\subset\bigcup_{i=1}^{\infty}\mathscr {R}_{1, k_{1}}^{i}\), together with the Chebyshev inequality and condition (1.3), we have
By (2.6) and the non-decreasing property of ω, we have
By the Hölder inequality, one obtains
Combining (3.5) and (3.6), we get
Now we begin to estimate \(|E_{2,2}|\).
Similarly to our dealing with \(|E_{2,1}|\), and together with the size condition of \(H^{1}\)-atoms, it follows that
The estimate for \(|E_{2,3}|\) is more complicated, and we need to split the domain of the variable \(y_{2}\). First, similar to our dealing with \(|E_{2,1}|\) in (3.4) and (3.5), we may get
Denote \(\mathscr{R}_{1, k_{1}}^{h}=(2^{h+2}\sqrt{n}Q_{1, k_{1}})\backslash (2^{h+1}\sqrt{n}Q_{1, k_{1}})\) and recall that \(Q_{1, k_{1}}^{*}=4\sqrt {n}Q_{1, k_{1}}\), then
Thus \(|E_{2,3}|\) can be controlled by
For any \(h\in\mathbb{N}\), if \(y_{2}\in\mathscr{R}_{1, k_{1}}^{h}\), note that \(y_{1}\in Q_{1,k_{1}}\), then
On the other hand, for any \(i\in\mathbb{N}\), if \(x\in\mathscr {R}_{1, k_{1}}^{i}\) and \(y_{1}\in Q_{1,k_{1}}\), then
By the geometric properties of \(y_{1}\), \(y_{2}\), x above, we may obtain
It is easy to see that
Since \(a(y_{1})\in L^{1}(\mathbb{R}^{n})\), putting the above estimate into (3.8), we have
If \(y_{2}\in Q_{1, k_{1}}^{*}\), note that \(x\in(8\sqrt{n}Q_{1, k_{1}})^{c}\), then
By the definition of \(|E_{2,3}^{2}|\) and (3.7), we have
Hence, we obtain
Now we begin to consider \(|E_{2,4}|\). Similarly,
Repeating the same steps as in the estimate of \(|E_{2,3}|\), we have
By the definition of \(|E_{2,4}^{1}|\), one may obtain
By (3.9), and taking the integral for x first, we have
The estimate for \(|E_{2,4}^{2}|\) is quite similar to \(|E_{2,3}^{2}|\), we may get \(|E_{2,4}^{2}|\leq CC_{0}\gamma^{1/2}\lambda^{-1/2}\).
• Estimate for \(|E_{3}|\) . Since \(|E_{3}|\) is a symmetrical case of \(|E_{2}|\), we can obtain
• Estimate for \(|E_{4}|\) .
Thus, we obtain
Now we begin considering \(|E_{4,1}|\). By the definition of \(I_{4,1}(x)\), we can write
Fix for a moment \(k_{1}\), \(k_{2}\) and assume, without loss of generality, that \(l(Q_{1,k_{1}})\leq l(Q_{2,k_{2}})\). By the moment condition of \(H^{1}\)-atoms and the regularity condition (1.3) of the kernel K, we have
Recalling the definition of \(\mathscr{R}_{1, k_{1}}^{i}\), \(\mathscr {R}_{2, k_{2}}^{h}\), and note that \(y_{1}\in Q_{1,k_{1}}\), \(y_{2}\in Q_{2,k_{2}}\), it is obvious that, for any fixed \(i, h, k_{1}, k_{2}\), if \(x\in (S^{*})^{c}\cap\mathscr{R}_{1, k_{1}}^{i}\cap\mathscr{R}_{2, k_{2}}^{h}\), then we have
This and the non-decreasing property of ω give
By (2.11), the Chebychev inequality and the estimate above, we control \(|E_{4,1}|\) by
Let us first consider the inside integrals, by the Hölder inequality, we may have
Note that \(a_{2,k_{2}}(y_{2})\in L^{1}(\mathbb{R}^{n})\), a similar argument to (2.15) yields
Note that the integrals in the above inequality are independent of \(S_{2,l_{2}}\) and \(Q_{2,k_{2}}\) and ω is doubling, similar to what we have done with (2.14), for fixed \(x\in(S^{*})^{c}\) and any \(y_{1}, y_{2}\in S\), we have
Recalling (I) in Theorem 1.1 and putting the inequality above into (3.10), we may get
Now we begin with the estimate for \(|E_{4,2}|\).
Recalling the definition of \(I_{4,2}(x)\), the moment condition of \(H^{1}\)-atoms and smoothness condition (1.3). Similar to the estimates in (3.10), we may obtain
First, we consider the following summation.
Property (I) in Theorem 1.1, inequality (3.12), and the size condition of \(H^{1}\)-atoms, that is, \(\Vert a_{Q_{2,k_{2}}} \Vert _{L^{\infty}}\leq|Q_{2,k_{2}}|^{-1}\), together with the Hölder inequality, enable us to obtain
Therefore, by (3.13) and noting that \(a_{Q_{1,k_{1}}}(y_{2})\in L^{1}(\mathbb{R}^{n})\), we have
Since \(|E_{4,3}|\) is a symmetrical case of \(|E_{4,2}|\) we may also obtain
A similar argument still works as in (2.9), we may have
This completes the estimate for \(|E_{4}|\). Thus, we have proved inequality (3.3) and the proof of Theorem 1.2 is finished. □
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Acknowledgements
The authors were supported partly by NSFC (No. 11471041 and No. 11671039), the Fundamental Research Funds for the Central Universities (No. 2014kJJCA10) and NCET-13-0065. The authors want to express their sincere thanks to the referees for their significant comments and suggestions.
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Li, Z., Xue, Q. Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels. J Inequal Appl 2016, 252 (2016). https://doi.org/10.1186/s13660-016-1201-2
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DOI: https://doi.org/10.1186/s13660-016-1201-2