Abstract
Let \(T^{*}\) be a multilinear Calderón–Zygmund maximal operator. In this paper, we study iterated commutators of \(T^{*}\) and pointwise multiplication with functions in Lipschitz spaces. More precisely, we give some new estimates for this kind of commutators under some Dini-type conditions on Lebesgue spaces, homogenous Lipschitz spaces, and homogenous Triebel–Lizorkin spaces.
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1 Introduction and main results
For any \(a>0\), we say that \(\omega \in \operatorname{Dini} (a)\) if
where \(\omega (t):[0,\infty )\mapsto [0,\infty )\) is a nondecreasing function with \(0<\omega (1)<\infty \).
We say that T is a multilinear Calderón–Zygmund operator with kernel of type \(\omega (t)\), denoted by m-linear ω-CZO, if T can be extended to a bounded multilinear operator from \(L^{q_{1}}( \mathbb{R}^{n})\times \cdots \times L^{q_{m}}(\mathbb{R}^{n})\) to \(L^{q,\infty }(\mathbb{R}^{n})\) for some \(1< q,q_{1},\ldots ,q_{m} <\infty \) with \(\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}=\frac{1}{q}\), or from \(L^{q_{1}}(\mathbb{R}^{n})\times \cdots \times L^{q_{m}}( \mathbb{R}^{n})\) to \(L^{1}(\mathbb{R}^{n})\) for some \(1< q_{1}, \ldots ,q _{m}<\infty \) with \(\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}=1\), and if there exists a function K defined off the diagonal \(x=y_{1}=\cdots =y_{m}\) in \((\mathbb{R}^{n})^{m+1}\), satisfying
for all \(x\notin \bigcap_{j=1}^{m} \operatorname{supp} f_{j} \) and \(f_{j}\in C_{c}^{\infty }(\mathbb{R}^{n})\), \(j=1,\ldots ,m\), and if there exists a constant \(A>0\) such that
for all \((x,y_{1},\ldots ,y_{m})\in (\mathbb{R}^{n})^{m+1}\) with \(x\neq y_{j}\) for some \(j\in \{1,2,\ldots ,m\}\), and
whenever \(|x-x'|\leq \frac{1}{m+1}\max_{1\leq j \leq m} |x-y_{j}|\), and
whenever \(|y_{j}-y_{j}'|\leq \frac{1}{m+1}\max_{1\leq j \leq m} |x-y _{j}|\).
When \(\omega (x)=x^{\gamma }\) for some \(\gamma >0\), the m-linear ω-CZO is exactly the multilinear Calderón–Zygmund operator studied by Grafakos and Torres in [11]. The multilinear Calderón–Zygmund operators were introduced and first studied by Coifman and Meyer [5,6,7] and later by Grafakos and Torres [11, 12]. The study of such operators has attracted the interest of many experts; see, for example, [4, 14, 24] and the reference therein. Recently, many mathematicians are concerned to remove or replace the smoothness condition on the kernels; see, for example [1, 8,9,10, 13, 15, 21]. In this paper, we mainly investigate the maximal operator and give some new estimates for its iterated commutators on some function spaces.
The maximal truncated operator \(T^{*}\) is defined by
where \(T_{\delta }\) are the smooth truncations of T, that is,
For the maximal truncated operator \(T^{*}\) and a collection of locally integrable functions \(\vec{b}=(b_{1},\ldots ,b_{m})\), we define the iterated commutator \(T^{*}_{\varPi \vec{b}}\) by
The iterated commutators of multilinear singular integral operators with BMO functions have been studied by a large number of people; see, for example, [2, 18, 19]. On the other hand, commutators of multilinear singular integral operators with Lipschitz functions have been the subject of many recent papers. In 1995, Paluszyński [17] proved that the commutator generated by Calderón–Zygmund operators with classical kernel and Lipschitz functions is bounded from the Lebesgue space to the Lebesgue space and to the homogenous Triebel–Lizorkin space. The multilinear analogues of the results in [17] were given by Wang and Xu [23] and by Mo and Lu [16]. Finally, Sun and Zhang [22] relaxed the smooth condition assumed on the kernel to Dini-type condition. It is natural to ask whether, under the Dini-type condition, the iterated commutators of multilinear Calderón–Zygmund maximal operators and pointwise multiplication with functions in Lipschitz space share similar boundedness properties? In this paper, we give a positive answer. The main result reads as follows.
Theorem 1.1
Suppose \(\omega \in \operatorname{Dini}(1)\) and \(b_{j}\in \operatorname{Lip}_{\beta _{j}}\) with \(0 < \beta _{j} < 1\) for \(j = 1, \ldots ,m\) and \(\beta = \beta _{1} + \cdots + \beta _{m}\). If \(1 < p_{1}, \ldots , p_{m} <\infty \), \(0< q < \infty \), and \(1/p_{j} > \beta _{j}/n\) with \(1/q = 1/p_{1}+\cdots +1/p _{m}-\beta /n\), then
Theorem 1.2
Suppose \(b_{j}\in \operatorname{Lip}_{\beta _{j}}\) with \(0 < \beta _{j} < 1\) for \(j = 1, \ldots ,m\) and \(\beta = \beta _{1} + \cdots + \beta _{m}\). If \(1 < p_{1}, \ldots , p_{m} <\infty \), \(0<1/p_{j}<\beta _{j}/n\), \(0<\beta -n/ p<1\) with \(1/p = 1/p_{1}+\cdots +1/p_{m}\), and ω satisfies
then
Theorem 1.3
Suppose \(b_{j}\in \operatorname{Lip}_{\beta _{j}}\) with \(0 < \beta _{j} < 1\) for \(j = 1, \ldots ,m\) and \(\beta = \beta _{1} + \cdots + \beta _{m}\). If \(1 < p_{1}, \ldots , p_{m} <\infty \) with \(1/p = 1/p_{1}+\cdots +1/p _{m}\) and ω satisfies
then
In the next section, we give some definitions and preliminaries. We focus on the proof of Theorem 1.1 in Sect. 3. The proof of Theorems 1.2 and 1.3 is given in Sect. 4. The notation \(A \lesssim B\) stands for \(A \leq C B\) for some positive constant C independent of A and B.
2 Preliminaries
Definition 2.1
Given a locally integrable function f, define the fractional maximal function by
when \(0\leq \beta < n/ r\). If \(\beta =0\) and \(r=1\), then \(M_{0, 1}f=Mf\) denotes the usual Hardy–Littlewood maximal function. For \(\delta >0\), we denote \(M_{\delta }\) by \(M_{\delta }f=M(|f|^{\delta })^{\frac{1}{ \delta }}\).
The sharp maximal function \(M^{\sharp }\) is given by
where \(f_{Q}\) denotes the average of f over cube Q, and we denote \(M^{\sharp }_{\delta }\) by \(M^{\sharp }_{\delta }f(x)= M^{\sharp }(|f|^{ \delta })^{\frac{1}{\delta }}(x)\).
Definition 2.2
([17])
For \(\beta >0\), the homogenous Lipschitz space \(\operatorname{Lip}_{\beta }( \mathbb{R}^{n})\) is the space of functions f such that
where \(\Delta _{h}^{k}\) denotes the kth difference operator.
To prove Theorems 1.1, 1.2, and 1.3, we need the following lemmas.
Lemma 2.1
([17])
Let \(b\in \operatorname{Lip}_{\beta }(\mathbb{R}^{n})\), \(0<\beta <1\). For any cubes \(Q^{\prime }\), Q in \(\mathbb{R}^{n}\) such that \(Q^{\prime }\subset Q\), we have
Lemma 2.2
([17])
-
(1)
For \(0 <\beta < 1\) and \(1 \leq q <\infty \), we have
$$ \|f\|_{\operatorname{Lip}_{\beta }(\mathbb{R}^{n})}\approx \sup_{Q} \frac{1}{|Q|^{1+n/ \beta }} \int _{Q} |f-f_{Q}|\approx \sup _{Q} \frac{1}{|Q|^{n/ \beta }} \biggl( \int _{Q} |f-f_{Q}|^{q} \biggr)^{\frac{1}{q}}. $$ -
(2)
For \(0 <\beta < 1\) and \(1 \leq p<\infty \), we have
$$ \|f\|_{\dot{F}^{\beta ,\infty }_{p}}\approx \biggl\Vert \sup_{Q} \frac{1}{|Q|^{1+n/ \beta }} \int _{Q} |f-f_{Q}| \biggr\Vert _{L^{p}}. $$
Lemma 2.3
([20])
Let \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots +\frac{1}{p_{m}}\) and \(\vec{\omega }\in A_{\vec{p}}\). Let T be an m-linear ω-CZO with \(\omega \in \operatorname{Dini}(1)\).
-
(1)
If \(1< p_{1}, \ldots , p_{m}<\infty \), then
$$ \bigl\Vert T^{*}\vec{f} \bigr\Vert _{L^{p}(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$ -
(2)
If \(1\leq p_{1}, \ldots , p_{m}<\infty \), then
$$ \bigl\Vert T^{*}\vec{f} \bigr\Vert _{L^{p,\infty }(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$
3 Proof of Theorem 1.1
We borrow some ideas from [19]. Since the proof of Theorem 1.1 follows from similar steps in [22], we omit the proof. We just give three key lemmas.
Let \(u, v\in C^{\infty }([0,\infty ))\) be such that \(|u'(t)|\le Ct ^{-1}\), \(|v'(t)|\le Ct^{-1}\), and
For simplicity, we denote
and
Then we define the maximal operators
It is easy to get \(T^{*}(\vec{f})\le U^{*}(\vec{f})(x)+V^{*}(\vec{f})(x)\). Next, we show that the functions \(K_{u, \eta }\) and \(K_{v,\eta }\) satisfy some smoothness properties.
Lemma 3.1
For any \(j=0,1,2,\ldots ,m\), we have
and
whenever \(|y_{j}-y_{j}'|\leq \frac{1}{m+1}\max_{0\leq j \leq m} |y _{0}-y_{j}|\).
Proof
We just give the estimate for \(K_{u,\eta }\), since \(K_{v,\eta }\) can be estimated in a similar way with slight modifications. Without loss of generality, assuming that \(j=0\), we estimate
Since \(|y_{0}-y_{0}'|\leq \frac{1}{m+1}\max_{0\leq j \leq m} |y_{0}-y _{j}|\), by (1.3) we have
It remains to estimate II. By the mean value theorem there is \(t_{0} \) between \(\frac{|y'_{0}-y_{1}|+\cdots +|y'_{0}-y_{m}|}{\eta }\) and \(\frac{|y_{0}-y_{1}|+\cdots +|y_{0}-y_{m}|}{\eta }\) such that
Again, since \(|y_{0}-y_{0}'|\lesssim \frac{1}{m+1}\max_{0\leq j \leq m} |y_{0}-y_{j}|\), we have
From this,
and therefore
This, together with the size condition (1.2), implies that
This ends the proof of Lemma 3.1. □
Lemma 3.2
Let \(\frac{1}{p}=\frac{1}{p_{1}} +\cdots +\frac{1}{p_{2}}\) and \(\vec{\omega }\in A_{\vec{p}}\). Then we have:
-
(1)
If \(1< p_{1}, \ldots, p_{m}<\infty \), then
$$ \bigl\Vert U^{*}\vec{f} \bigr\Vert _{L^{p}(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$ -
(2)
If \(1\leq p_{1}, \ldots, p_{m}<\infty \), then
$$ \bigl\Vert U^{*}\vec{f} \bigr\Vert _{L^{p,\infty }(\nu _{\vec{\omega }})}\lesssim \prod _{i=1}^{m}\|f_{i} \|_{L^{p_{i}}(\omega _{i})}. $$
Similar estimates hold for \(V^{*}\).
Proof
Lemma 3.2 is a consequence of Lemma 2.3, Lemma 3.1, and Theorem 1.3 in [3]. □
For the maximal truncated operator \(T^{*}\) and a collection of locally integrable functions \(\vec{b}=(b_{1},\ldots ,b_{m})\), we define the commutator \(T^{*}_{\varSigma \vec{b}}\) by
where
Next, we give the key lemma, which plays important role in the proof of Theorem 1.1. We just consider the case \(m=2\) for simplicity.
Lemma 3.3
Let T be an m-linear ω-CZO with \(\omega \in \operatorname{Dini}(1)\). Then we have:
-
(i)
If \(b_{1}\in \operatorname{Lip}_{\beta _{1}} \) and \(b_{2}\in \operatorname{Lip}_{\beta _{2}} \) with \(0<\beta _{1}\), \(\beta _{2}<1\), \(0 <\delta <\epsilon < 1/ 2\), then
$$ \begin{aligned}[b] &M^{\sharp }_{\delta }T^{*}_{\varPi \vec{b}}(f_{1},f_{2}) (x) \\ &\quad \lesssim \Biggl\{ \prod_{i=1}^{2} \|b_{i}\|_{\operatorname{Lip}_{\beta _{i}}} M_{ \epsilon ,\beta } \bigl(T^{*}(f_{1},f_{2}) \bigr) (x)+\|b_{1}\|_{\operatorname{Lip}_{\beta _{1}}} M_{\epsilon ,\beta _{1}} \bigl(T_{\vec{b}}^{*2}(f_{1},f_{2})\bigr) (x) \\ &\qquad {}+\|b_{2}\|_{\operatorname{Lip}_{\beta _{1}}} M_{\epsilon ,\beta _{2}} \bigl(T_{\vec{b}} ^{*1}(f_{1},f_{2})\bigr) (x) \\ &\qquad {}+\prod_{i=1}^{2}\|b_{i} \|_{\operatorname{Lip}_{\beta _{i}}}M _{1,\beta _{1}}(f_{1}) (x) M_{1,\beta _{2}}(f_{2}) (x) \Biggr\} . \end{aligned} $$(3.1) -
(ii)
Suppose that \(b_{j}\in \operatorname{Lip}_{\beta }\), \(j=1,2 \), \(0<\beta <1\), and \(0 <\delta <\epsilon < 1/ 2 <1<n/ \beta \). Then
$$ \begin{aligned}[b] &M^{\sharp }_{\delta }T_{\varSigma \vec{b}}^{*}(f_{1},f_{2}) (x) \\ &\quad \lesssim \|b\|_{\operatorname{Lip}_{\beta }} \bigl\{ M_{\epsilon ,\beta } \bigl(T^{*}(f _{1},f_{2})\bigr) (x)+ M_{1,\beta }(f_{1}) (x)M (f_{2}) (x) \\ &\qquad {}+ M_{1,\beta }(f _{2}) (x)M (f_{1}) (x) \bigr\} . \end{aligned} $$(3.2)
Proof
(i) We need two auxiliary maximal operators. The key role in the proof is played by the maximal operators \(U_{\varPi b}^{*}\) and \(V_{\varPi b}^{*}\) defined by
It is easy to get that \(T^{*}_{\varPi b}(\vec{f})\le U_{\varPi b}^{*}( \vec{f})(x)+V_{\varPi b}^{*}(\vec{f})(x)\). We need to prove (3.1) for \(U^{*}_{\varPi \vec{b}}\) and \(V^{*}_{\varPi \vec{b}}\). We just give the proof for \(U^{*}_{\varPi \vec{b}}\), since the proof for \(V^{*}_{\varPi \vec{b}}\) is almost the same. Fix \(x\in \mathbb{R}^{n} \) and denote by \(Q=Q(x_{Q},l)\) the cube centered at \(x_{Q}\) and containing x with side length l. Denote \(c=\sup_{\eta >0}| c_{\eta }|\) and \(\lambda _{i}={(b_{i})}_{Q^{*}}=\frac{1}{|Q^{*}|}\int _{Q^{*}}b_{i}(y)\,dy\), where \(Q^{*}=8 \sqrt{n}Q\). For any \(z\in \mathbb{R}^{n}\), we have
Thus we have
By Hölder’s inequality,
In a similar way, we can prove that
It remains to estimate the last term \(T_{4}\). Take now \(c_{\eta }= U _{\eta }((b_{1}-\lambda _{1})f_{1}^{\infty },(b_{2}-\lambda _{2})f_{2} ^{\infty })(x)\). Then \(T_{4}\leq T_{41}+T_{42}+T_{43}+T_{44}\), where
By the Kolmogorov inequality and by Lemma 3.2,
Next, by Hölder’s inequality and by the size condition (1.2),
The operator \(T_{43}\) can be estimated in the same way. Finally, we estimate \(T_{44}\). By Lemma 3.1 we have
Combining the obtained estimates proves (3.1).
(ii) It is sufficient to prove (3.2) for the operator with only one symbol. Set
where \(\lambda =b_{Q^{*}}=\frac{1}{|Q^{*}|}\int _{Q^{*}}b(y)\,dy\). Let \(c=\sup_{\eta >0}|c_{\eta }|\). Then
By Hölder’s inequality,
Set \(c_{\eta }= U_{\eta }((b-\lambda )f_{1}^{\infty },f_{2}^{\infty })(x)\). Then \(P_{2}\leq P_{21}+P_{22}+P_{23}+P_{24}\), where
By the Kolmogorov inequality and by Lemma 3.2,
Next, by the size condition (1.2),
Similarly,
By Lemma 3.1 we obtain
Thus we finish the proof of (3.2). Then Lemma 3.3 is proved. □
4 Proofs of Theorems 1.2 and 1.3
The main ideas in this section are from [17] and [19]. We should also mention that the proof of this part is similar to that of Theorem 1.2 and Theorem 1.3 in [22]; we just give the different part of the proof.
We begin with the proof of Theorem 1.2.
Proof
For any cube Q centered at \(x_{Q}\), the theorem will be proved if we can show that
Set \(c=c_{1}+c_{2}+c_{3}\), which will be determined later. We estimate
We estimate these terms separately. For the first term, we can choose \(1< q, q_{j}<\infty \), \(q_{j}< n/\beta _{j} < p_{j}\), \(j=1,2\), with \(1/q=1/q_{1}+1/q_{2}-(\beta _{1}+\beta _{2})/n\). By Hölder’s inequality and by Theorem 1.1 we have
To get \(M_{2}\), we take \(c_{1}=T((b_{1}-\lambda _{1})f_{1}^{0},f_{2} ^{\infty })(x_{Q})\). Then
Using the size condition (1.2) and the estimate in [22, p. 5013], we have
In a similar way, we get \(M_{23}+M_{22}\lesssim \|b_{1}\|_{ \dot{\wedge }_{\beta _{1}}}\|b_{2}\|_{\dot{\wedge }_{\beta _{2}}} \|f _{1}\|_{L^{p_{1}}}\|f_{2}\|_{L^{p_{2}}}\).
By Minkowski’s inequality and by Lemma 3.1,
where we have used assumption (1.5) and the inequality \(1-\beta _{2}/n +1/p_{2}>0\).
Thus \(M_{2}\lesssim \|b_{1}\|_{\dot{\wedge }_{\beta _{1}}}\|b_{2}\|_{ \dot{\wedge }_{\beta _{2}}} \|f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L^{p_{2}}}\). Similarly, \(M_{3}\lesssim \|b_{1}\|_{\dot{\wedge }_{\beta _{1}}}\|b _{2}\|_{\dot{\wedge }_{\beta _{2}}} \|f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L ^{p_{2}}}\).
We deal with \(M_{4}\) as follows:
By Minkowski’s inequality and by the size condition (1.2),
Using Minkowski’s inequality along with Lemma 3.1, we obtain
where we have used assumption (1.5) and the inequality \(0<\beta -n/ p<1\).
Similarly, \(M_{43}\lesssim \|b_{1}\|_{\dot{\wedge }_{\beta _{1}}}\|b _{2}\|_{\dot{\wedge }_{\beta _{2}}} \|f_{1}\|_{L^{p_{1}}}\|f_{2}\|_{L ^{p_{2}}}\).
Now we estimate \(M_{44}\):
Putting the estimates for \(M_{1}\), \(M_{2}\), \(M_{3}\), \(M_{4}\) together, we get (4.1). Thus the proof of Theorem 1.2 is completed. □
Proof of Theorem 1.3.
Proof
We use the same notations as in previous sections. Then we have
For \(1< r< p\), by the Hölder inequality, we have
In what follows, we just give an estimate for \(N_{2}\), since \(N_{3}\) and \(N_{4}\) can be estimated in a similar way. Let
Observe that
From this we have
By the Hölder inequality,
Take \(1< q_{1}< p_{1}\), \(1< q_{2}< p_{2}\), and \(1< q<\infty \) such that \(1/q=1/q_{1}+1/q_{2}\). Then by the Hölder inequality and by Lemma 3.2,
For any \(y_{2}\in (Q^{*})^{c}\), we have \(|y_{2}-x_{Q}|\sim |y_{2}-z|\) and \(|z-x_{Q}|\leq \frac{|y_{2}-z|}{2}\leq \frac{1}{2} \max \{|z-y _{1}|, |z-y_{2}|\}\). Then by Minkowski’s inequality and by Lemma 3.1,
Similarly, \(N_{24}\lesssim \|b_{1}\|_{\dot{\wedge }_{\beta _{1}}}\|b _{2}\|_{\dot{\wedge }_{\beta _{2}}} M(f_{1})(x) M(f_{2})(x)\).
For any \(y_{1},y_{2}\in (Q^{*})^{c}\), we have \(|y_{1}-x_{Q}|\sim |y _{1}-z|\) and \(|y_{2}-x_{Q}|\sim |y_{2}-z|\). Then by Minkowski’s inequality and by Lemma 3.1,
where assumption (1.6) was used.
Combining the estimates for \(N_{21}\), \(N_{22}\), \(N_{23}\), \(N_{24}\), \(N_{25}\), we get
The rest of the proof is the same as in [22], and hence we proved Theorem 1.3. □
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This work was supported partly by the Key Research Project for Higher Education in Henan Province (No. 19A110017) and the National Natural Science Foundation of China (Grant Nos. 11401175, 11571160, and 11471176).
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Si, Z., Zhang, P. Iterated commutators of multilinear Calderón–Zygmund maximal operators on some function spaces. J Inequal Appl 2019, 100 (2019). https://doi.org/10.1186/s13660-019-2051-5
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DOI: https://doi.org/10.1186/s13660-019-2051-5