Abstract
In this paper, we discuss the properties for commutators and iterated commutators generated by the multilinear ω-CZO and weighted Lipschitz functions on Lebesgue space.
Similar content being viewed by others
1 Introduction and results
The singular integral operator theory plays an important role in many aspects of harmonic analysis. The Calderón–Zygmund operator theory is one of the most important achievements of classical analysis in the last century, which has many important applications in Fourier analysis, complex analysis, operator theory, and so on. The multilinear Calderón–Zygmund theory was introduced by Coifman and Meyer in [1, 2]. This theory was then further studied by Grafakos and Torres [7, 8], who considered the multilinear Calderón–Zygmund operator with classical standard kernels. And this topic keeps attracting many researchers.
In 1985, Yabuta [23] firstly considered Calderón–Zygmund operators with kernels of type ω as the generalizations of Calderón–Zygmund operators when studied pseudodifferential operator. In 2009, Maldonado and Naibo [15] studied the bilinear Calderón–Zygmund operators of type ω. In 2014, Lu and Zhang [14] considered the multilinear case. Assume that \(\omega (t):[0,\infty )\rightarrow [0,\infty )\) is a nondecreasing function with \(0<\omega (1)<\infty \). For \(a>0\), we say \(\omega \in \mathrm{{Dini}}\mathit{{(a)}}\) if
Definition 1.1
([14])
A locally integrable function \(K(x,y_{1},\ldots ,y_{m})\), defined away from the diagonal \(x=y_{1}=\cdots =y_{m}\) in \((\mathbb{R}^{n})^{m+1}\), is called an m-linear Calderón–Zygmund kernel of type \(\omega (t)\) if there exists a constant \(A>0\) such that
or 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 \(|x-x'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\), and
whenever \(|y_{j}-y_{j}'|\leq \frac{1}{2}\max_{1\leq j\leq m}|x-y_{j}|\).
We say \(T:\mathscr{S}(\mathbb{R}^{n})\times \cdots \times \mathscr{S}( \mathbb{R}^{n})\rightarrow \mathscr{S}'(\mathbb{R}^{n})\) is an m-linear operator with an m-linear Calderón–Zygmund kernel \(K(x,y_{1},\ldots ,y_{m})\) of type \(\omega (t)\) if
whenever \(f_{1},\ldots ,f_{m}\in C_{c}^{\infty}(\mathbb{R}^{n})\) and \(x\notin \cap _{j=1}^{m} \operatorname{supp}f_{j}\).
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 \({1}/{q_{1}}+\cdots +{1}/{q_{m}}={1}/{q}\) 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 \({1}/{q_{1}}+\cdots +{1}/{q_{m}}=1\), then T is called an m-linear Calderón–Zygmund operator of type ω, abbreviated to m-linear ω-CZO.
Obviously, when \(\omega (t)=t^{\varepsilon}\) for some \(\varepsilon >0\), the m-linear ω-CZO is exactly the multilinear Calderón–Zygmund operator studied by Grafakos and Torres [7] and Lerner et al. [12].
To shorten the notation, we denote \(\vec{f}=(f_{1},\ldots ,f_{m})\) and \(\mathrm{d}\vec{y}=\mathrm{d}{y_{1}}\cdots \mathrm{d}{y_{m}}\) in the following.
In 2014, Lu and Zhang [14] gave the endpoint estimate for the m-linear ω-CZO under some weaker assumptions of \(\omega (t)\) and also got the following multiple weighted estimates.
Theorem A
([14])
Let T be an m-linear ω-CZO with \(\omega \in \mathrm{Dini}{(1)}\). Let \({1}/{p}={1}/{p_{1}}+\cdots +{1}/{p_{m}}\) and \({\mu}\in A_{\min \{p_{1},p_{2},\ldots ,p_{m}\}}(\mathbb{R}^{n})\). If \(1< p_{j}<\infty \) for all \(j=1,\ldots ,m\), then
Theorem B
([14])
Let T be an m-linear ω-CZO with \(\omega \in \mathrm{{Dini}{(1)}}\). Then T can be extended to a bounded operator from \(L^{1}(\mathbb{R}^{n})\times \cdots \times L^{1}(\mathbb{R}^{n})\) to \(L^{1/m,\infty}(\mathbb{R}^{n})\).
Let \(\vec{b}=(b_{1},\ldots ,b_{m})\) be a collection of locally integrable functions, the commutator generated by m-linear ω-CZO, and b⃗ is defined by
where
Let \(\vec{b}=(b_{1},\ldots ,b_{m})\) be a collection of locally integrable functions, the commutator generated by m-linear ω-CZO, and b⃗ is defined by
where
The iterated commutator \(T_{\Pi \vec{b}}(\vec{f})\) is defined as follows:
which can also be given formally by
When \(m=1\), \(T_{\Sigma \vec{b}}(\vec{f})=T_{\Pi \vec{b}}(\vec{f})=[b,T]f=bT(f)-T(bf)\), which is the well-known classical commutator studied in [3]. In 1995, Paluszynski [17] proved that the commutator \([b,T]\) generated by Calderón–Zygmund operators T with classical kernel and \(b\in \mathrm{Lip}_{\beta}(\mathbb{R}^{n})\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\) whenever \(0<\beta <1\), \({1}/{q}={1}/{p}-{\beta}/{n}\), and \(1< p< q<\infty \), and from \(L^{p}\) to homogenous Triebel–Lizorkin spaces \(\dot{F}_{p}^{\beta ,\infty}(\mathbb{R}^{n})\) which is defined in [20].
For the weighted case, Hu and Gu [9] proved when \(b\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})\), the commutators \([b,T]\) is bounded from \(L^{p}(\mu )\) to \(L^{q}(\mu ^{1-q})\). In 2011, Lian, Ma, and Wu [13] studied the m-linear commutators generated by the multilinear Calderón–Zygmund operators with nonsmooth kernels and weighted Lipschitz functions bounded from the product of weighted Lebesgue spaces to the weighted Lebesgue space. For more articles about multilinear operators, see [1, 2, 4, 6, 7, 10–12, 14, 16, 24, 25], and [26].
In this paper, we will discuss the mapping properties of multilinear commutators generated by m-linear Dini’s type Calderón–Zygmund operators and weighted Lipschitz functions on some function spaces. We obtain the following results.
Theorem 1.1
Let T be an m-linear ω-CZO satisfying
Suppose \(0<\beta <1\) and \(1/r=1/p-\beta /n\), \(1< p< r<\infty \) for \(1/{p_{1}}+\cdots +1/{p_{m}}=1/p\) with \(1< p_{i}<\infty \), \(i=1,\ldots ,m\). If \(\mu \in A_{1}(\mathbb{R}^{n})\) and \(b_{j}\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})(1\leq j\leq m)\), then \(T_{b_{j}}^{j}(\vec{f})\) is bounded from \(L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )\) to \(L^{r}(\mu ^{1-r})\).
Furthermore, \(T_{\sum \vec{b}}(\vec{f})\) is bounded from \(L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )\) to \(L^{r}(\mu ^{1-r})\).
Theorem 1.2
Let T be an m-linear ω-CZO satisfying
Suppose \(0<\beta _{i}<1\), \(i=1,\ldots ,m\), \(1/{r_{i}}=1/{p_{i}}-{\beta _{i}}/n\), \(1< p_{i}< r_{i}<\infty \) with \(1/{p_{1}}+\cdots +1/{p_{m}}=1/p\), \(1/{r_{1}}+\cdots +1/{r_{m}}=1/r\), \({\beta _{1}}+\cdots +{\beta _{m}}=\beta \), and \(0<\beta <1\). If \(\mu \in A_{1}(\mathbb{R}^{n})\), \(b_{i}\in \mathrm{Lip}_{\beta _{i},\mu}(\mathbb{R}^{n})(1\leq i\leq m)\), then \(T_{\Pi \vec{b}}(\vec{f})\) is bounded from \(L^{p_{1}}(\mu )\times \cdots \times L^{p_{m}}(\mu )\) to \(L^{r}(\mu ^{1-mr})\).
Remark 1.1
Theorem 1.1 and 1.2 are also valid for commutators of multilinear Calderón–Zygmund operator with standard kernels.
Remark 1.2
Theorem 1.2 extends the corresponding result in [19] and [22].
The rest of this paper is organized as follows. After recalling some notations and lemmas in Sect. 2, we prove our results in Sect. 3.
Throughout this paper, we denote by \(p'\) the conjugate index of p, that is, \(1/p+1/p'=1\). The letter C, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but independent of the main parameters.
2 Preliminaries and lemmas
A nonnegative locally integrable function is called a weight function.
Definition 2.1
([5])
Let μ be a weight function, \(1< p<\infty \). If there is a constant \(C>0\) such that, for every ball \(B\subseteq \mathbb{R}^{n}\),
then we say \(\mu \in A_{p}\). We say \(\mu \in A_{1}\) if there is a constant \(C>0\) such that, for every ball \(B\subseteq \mathbb{R}^{n}\),
A weight function \(\mu \in A_{\infty}\) if it satisfies the \(A_{p}\) condition for some \(1 < p < \infty \). The smallest constant satisfying the formulas above is called \(A_{p}\) constant of w, we denote it by \([\mu ]_{A_{p}}\).
For \(1\leq p< q<\infty \), we have \(A_{1}\subset A_{p}\subset A_{q}\). And \(A_{\infty}=\cup _{1\leq p<\infty}A_{p}\).
If \(\mu \in A_{r}\), \(1< r<\infty \), then \(\mu ^{1-r'}\in A_{r'}\).
For a function \(f\in L_{loc}(\mathbb{R}^{n})\), the Hardy–Littlewood maximal and the sharp maximal functions are defined by
and
where \(f_{Q}\) denotes the average of f over cube Q, that is, \(f_{Q}=\frac{1}{|Q|}\int _{Q}f(x)\,\mathrm{d}x\).
For \(\delta >0\), we denote \(M_{\delta}(f)\) and \(M_{\delta}^{\sharp}(f)\) by \(M_{\delta}(f)=M(|f|^{\delta})^{{1}/{\delta}}\) and \(M_{\delta}^{\sharp}(f)=[M^{\sharp}(|f|^{\delta})]^{1/{\delta}}\). We denote the following fractional maximal operator:
Recall that \(M_{\alpha}:=M_{\alpha ,1,1}\) is the fractional maximal operator
Definition 2.2
([5])
Let \(1\leq p\leq \infty \), \(\ 0<\beta <1\), \(\mu \in A_{\infty}\), the weighted Lipschitz space \(\mathrm{Lip}_{ \beta ,\mu}^{p}\) contains all locally integrable functions f satisfying
where \(f_{B}=\frac{1}{|B|}\int _{B}f(y)\,\mathrm{d}y\), the supremum is taken over all balls B in \(\mathbb{R}^{n}\).
The smallest number C satisfying the above inequality was denoted by \(\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{p}}\), and we also denote by \(\|f\|_{\mathrm{Lip}_{\beta ,\mu}}=\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{1}}\). Obviously, when \(\mu =1\), \(\mathrm{Lip}_{\beta ,\mu}=\mathrm{Lip}_{\beta}\).
\(A\sim B\) means there exist \(C_{1}>0\), \(C_{2}>0\) such that \(C_{1}A \leq B\leq C_{2}A\). When \(\mu \in A_{1}\), García–Cuerva in [5] proved that for \(1\leq p,q\leq \infty \), \(\|f\|_{\mathrm{Lip}_{\beta ,\mu}^{p}}\sim \|f\|_{\mathrm{Lip}_{\beta ,\mu}^{q}}\).
As usual, we denote \(\|f\|_{L^{p}(\mu )}=(\int _{\mathbb{R}^{n}}|f(x)|^{p}\mu (x) \,\mathrm{d}x)^{{1}/{p}} \) for \(1< p<\infty \) and \(p=\infty \), \(\|f\|_{L^{\infty}(\mu )}=\|f\|_{L^{\infty}}\).
We will use the following Kolmogorov inequality:
where \(0< p< q<\infty \). See [12, 21].
Lemma 2.1
([18])
Let \(0< p,\delta <\infty \), \(\mu \in A_{\infty}\), then there exists a constant C such that
for any function f for which the left-hand side is finite.
Lemma 2.2
([13])
Suppose \(\mu \in A_{1}\), \(0<\beta <1\), \(b\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})\).
(i) For \(k\geq 1\),
(ii) For any \(1\leq s<\infty \) and any ball \(B\ni x\), we have
(iii) For any \(1< s<\infty \) and any ball \(B\ni x\), we have
Lemma 2.3
([13])
Suppose that \(0<\alpha <n\), \(0< s< p< n/\alpha \), \(1/q=1/p-\alpha /n\). If \(\mu \in A_{\infty}(\mathbb{R}^{n})\), then
Lemma 2.4
([13])
Suppose that \(0<\alpha <n\), \(1< p< n/\alpha \), and \(1/q=1/p-\alpha /n\). If \(\mu \in A_{1+q/p'}(\mathbb{R}^{n})\), then
3 Proofs of theorems
For simplicity, we only prove for the case \(m=2\). The argument for the case \(m>2\) is similar. We first establish the following lemmas.
Lemma 3.1
Let T be a 2-linear ω-CZO satisfying (1.4). Suppose \(\mu \in A_{1}(\mathbb{R}^{n})\) and \(b_{j}\in \mathrm{Lip}_{\beta ,\mu}(\mathbb{R}^{n})\) with \(0<\beta <1\), \(j=1,2\). Let \(0<\delta <1/2<1<s<n/\beta \). Then we have
for \(j=1,2\).
Proof
We only estimate \(M_{\delta}^{\sharp} (T_{b_{1}}^{1}(f_{1},f_{2}) )\) and write \(b_{1}=b\) for simplicity. A similar discussion also works for \(M_{\delta}^{\sharp} (T_{b_{2}}^{2}(f_{1},f_{2}) )\).
Fix \(x\in \mathbb{R}^{n}\) for any cube \(Q(x_{Q},l_{Q})\) containing x with side-length \(l_{Q}\), set \(Q^{*}=8\sqrt{n}Q=Q(x_{Q},8\sqrt{n}l_{Q})\). We decompose \(f_{j}=f_{j}^{0}+f_{j}^{\infty}\), where \(f_{j}^{0}=f_{j}\chi _{Q^{*}}\) and \(f_{j}^{\infty}=f\chi _{{\mathbb{R}^{n}}\setminus {Q^{*}}}\), \(j=1,2\).
Since \(0<\delta <1/2\), then for any constant c, we have
Since \(0<\delta <1\), \(\mu \in A_{1}\), and \(b\in \mathrm{Lip}_{\beta ,\mu}\), by Lemma 2.2(iii) and Hölder’s inequality, we get
For the second term \(I_{2}\), since \(0 <\delta < 1/2\), by Kolmogorov’s inequality, Theorem B, and Lemma 2.2(iii), we obtain
For the term \(I_{3}\), noting the fact that \(|z-y_{1}|\sim |y_{1}-x_{Q}|\) for any \(y_{1}\in {(Q^{*})}^{c}\) and \(z\in Q\), then by (1.1) and Lemma 2.2(i)(ii), we obtain
Similarly, we have
For the last term \(I_{5}\), since \((\mathbb{R}^{n}\setminus Q^{*})^{2}\subseteq \mathbb{R}^{2n} \setminus {(Q^{*})^{2}}\subseteq \cup _{k=1}^{\infty}(2^{k+3}\sqrt{n}Q)^{2} \setminus (2^{k+2}\sqrt{n}Q)^{2}\), making use of assumption (1.2), we have
□
Proof of Theorem 1.1
Since \(\mu \in A_{1}\subset A_{r'}\) and \(\mu ^{1-r}\in A_{r}\subset A_{\infty}\), by Lemma 2.1 and Lemma 3.1, for any \(j=1,2\), we get
For \(U_{1}\), since \(1/r=1/p-\beta /n\) and select s satisfying \(1< s< p< n/\beta \), by Theorem A and Lemma 2.3, we have
For \(U_{2}\), let \(1/r=1/p_{2}+1/l\), then we have \(1/l=1/p_{1}-\beta /n\). Then, by Hölder’s inequality and Lemma 2.3, we obtain
Similarly as the estimate of \(U_{2}\), we may get
Thus Theorem 1.1 is proved. □
Lemma 3.2
Let T be a 2-linear ω-CZO satisfying \(\int _{0}^{1}\frac{\omega (t)}{t}(1+\log \frac{1}{t})^{2}dt<\infty \). Suppose \(\mu \in A_{1}(\mathbb{R}^{n})\) and \(b_{j}\in \mathrm{Lip}_{\beta _{j},\mu}(\mathbb{R}^{n})\), \(j=1,2\). Let \(\beta _{1}+\beta _{2}=\beta \), \(0<\beta <1\), and \(0<\delta <1/3<1<s<n/\beta \). Then we have
Proof
We fix \(x\in \mathbb{R}^{n}\) for any cube \(Q(x_{Q},l_{Q})\) containing x with side-length \(l_{Q}\), \(i=1,2\), \(Q^{*}=8\sqrt {n} Q\). Set \(\lambda _{i}=(b_{i})_{Q^{*}}\), let c be a constant to be fixed along the proof. Since \(0<\delta <1/3\), we have
Firstly we consider \(K_{1}\). For \(0<\delta <1/3\), it follows from Hölder’s inequality and Lemma 2.2(ii) that
For the terms \(K_{2}\), \(K_{3}\), notice that \(0<\delta <1/3\), we use the facts \(1/\delta =1+(1-\delta )/\delta \) and \(0<\frac{\delta}{1-\delta}<1/2\). By Hölder’s inequality, we get
Similarly, we have
Now, we consider the last term \(K_{4}\). For each \(i = 1, 2\), we decompose \(f_{i}=f_{i}^{0}+f_{i}^{\infty}\), where \(f_{i}^{0}=f_{i}\chi _{Q^{*}}\), then
We first estimate \(K_{41}\). Applying Kolmgorov’s inequality, Theorem B, and Lemma 2.2(iii), we have
Next, we consider the term \(K_{42}\). Note that for any \(z\in Q\), \(y_{2}\in (Q^{*})^{c}\), \(|z-y_{2}|\sim |y_{2}-x_{Q}|\), by (1.1) and Lemma 2.2(iii), we have
Similarly,
Finally, we consider the term \(K_{44}\). For any \(z\in Q\) and \((y_{1},y_{2})\in (2^{k+3}\sqrt{n}Q)^{2}\setminus (2^{k+2}\sqrt{n}Q)^{2}\),
Set \(c=T((b_{1}-\lambda _{1})f_{1}^{\infty},(b_{2}-\lambda _{2})f_{2}^{ \infty})(x)\), then by (3.2) and Lemma 2.2(iii), we have
This, together with the estimates for \(K_{1}\), \(K_{2}\), \(K_{3}\), \(K_{4}\) gives
Thus we finish the proof of Lemma 3.2. □
Proof of Theorem 1.2
Similarly as the proof of Theorem 1.1, since \(\mu \in A_{1}\), by Lemma 2.1 and Lemma 3.2, for \(0<\delta <1/2\),
For \(V_{1}\), by Lemma 2.3 and Theorem A, we have
For \(V_{2}\), since \(1/{r_{1}}+1/{r_{2}}=1/r\), by Hölder’s inequality and Lemma 2.3, we get
For the term \(V_{3}\), by Theorem 1.1 and Lemma 2.4, let \(1/r=1/{l_{1}}-{\beta _{1}}/n\), and then \(1+r/{{l_{1}}'}=r-{r\beta _{1}}/n>1\), \(1/{l_{1}}=1/p-{\beta _{2}}/n\). Since \(\mu \in A_{1}\), then \(\mu ^{1-r+{r\beta _{1}}/n}\in A_{1+r/{{l_{1}}'}}\), then we have
Similarly as the discussion of \(V_{3}\), we have
By combining the estimates of \(V_{1}\), \(V_{2}\), \(V_{3}\), \(V_{4}\), we finish the proof of Theorem 1.2. □
Availability of data and materials
Not applicable.
References
Coifman, R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)
Coifman, R., Meyer, Y.: Au delá des opérateurs pseudo-différentiels. Astérisque 1978(57)
Coifman, R., Rochberg, R., Weiss, G.: Fractorization theorems for Hardy spaces in several variables. Ann. Math. 103(3), 611–635 (1976)
Duong, X.T., Gong, R.M., Grafakos, L., Yan, L.X.: Maximal operators for multilinear singular integrals with non-smooth kernels. Indiana Univ. Math. J. 58(6), 2517–2542 (2009)
García-Cuerva, J.: Weighted \(H^{p}\) spaces. Diss. Math. (1979)
Grafakos, L., Liu, L.G., Maldonado, D., Yang, D.C.: Multilinear Analysis on Metric Spaces. Diss. Math., vol. 497, p. 121 (2014)
Grafakos, L., Torres, R.H.: Multilinear Calderón-Zygmund theory. Adv. Math. 165(1), 124–164 (2002)
Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51(5), 1261–1276 (2002)
Hu, B., Gu, J.: Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions. J. Math. Anal. Appl. 340(1), 598–605 (2008)
Kenig, C.E., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6(1), 1–15 (1999)
Lacey, M., Thiele, C.: \(L^{p}\) estimates on the bilinear Hilbert transform for \(2 < p <\infty \). Ann. Math. 146(3), 693–724 (1997)
Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weighted for the multilinear Calderón-Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)
Lian, J.L., Ma, B.L., Wu, H.X.: Commutators of weighted Lipschitz functions and multilinear singular integrals with non-smooth kernels. Appl. Math. J. Chin. Univ. 26(3), 353–367 (2011)
Lu, G.Z., Zhang, P.: Multilinear Calderón-Zygmund operators with kernels of Dini’s type and applications. Nonlinear Anal. 107(9), 92–117 (2014)
Maldonado, D., Naibo, V.: Weighted norm inequalities for paraproducts and bilinear pseudodifferential operators with mild regularity. J. Fourier Anal. Appl. 15(2), 218–261 (2009)
Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60, 213–274 (2009)
Paluszynski, M.: Characterization of the Besov spaces via the commutator operator of Coifman Rochberg and Weiss. Indiana Univ. Math. J. 44, 1–18 (1995)
Pérez, C., Trujillo-González, R.: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc. 65, 672–692 (2002)
Sun, J., Zhang, P.: Commutators of multilinear Calderón-Zygmund operators with Dini type kernels on some function spaces. J. Nonlinear Sci. Appl. 10, 5002–5019 (2017)
Triebel, H.: Theory of Function Spaces. Monograph in Math., vol. 78. Birkhäuser, Basel (1983)
Wilson, M.: Littlewood-Paley Theory and Exponential-Square Integrability. Lecture Notes in Math., pp. 40–92. Springer, Berlin (2007)
Xie, R., Shu, L.S.: On multilinear commutators of θ-type Calderón-Zygmund operators. Anal. Theory Appl. 24(3), 260–270 (2008)
Yabuta, K.: Generalization of Calderón-Zygmund operators. Stud. Math. 82(1), 17–31 (1985)
Zhang, P.: Multiple weighted estimates for commutators of multilinear maximal function. Acta Math. Sin. Engl. Ser. 31(16), 973–994 (2015)
Zhang, P., Sun, J.: Commutators of multilinear Calderón-Zygmund operators with kernels of Dini’s type and applications. J. Math. Inequal. 13(4), 1071–1093 (2019)
Zhao, Y., Zhou, J.: New weighted norm inequalities for multilinear Calderón-Zygmund operators with kernels of Dini’s type and their commutators. J. Inequal. Appl. 29, 1–24 (2021)
Acknowledgements
The author would like to thank the referee for his/her invaluable comments and suggestions.
Funding
This work was supported by a Research Project of the Basic Scientific Research Expenses for Heilongjiang Provincial Higher Institutions (No.1355ZD010), the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Excellent Young Talents Project of Heilongjiang Province)(2020YQ07), and the Project of Mudanjiang Normal University (No. GP2019006).
Author information
Authors and Affiliations
Contributions
Jie Sun finished the manscuript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Sun, J. Weighted Lipschitz estimates for commutators of multilinear Calderón–Zygmund operators with Dini type kernels. J Inequal Appl 2022, 149 (2022). https://doi.org/10.1186/s13660-022-02888-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-022-02888-9