In this paper, the authors establish the boundedness of commutators generated by weighted Lipschitz functions and strongly singular Calderón-Zygmund operators or generalized Calderón-Zygmund operators on weighted Morrey spaces.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
1 Introduction
The study of generalized Calderón-Zygmund operators and strongly singular non-convolution operators originated in the classical Calderón-Zygmund operator. The theory of the Calderón-Zygmund operator was one of the important achievements of classical analysis in the last century, and it has many important applications in Fourier analysis, complex analysis, operator theory and so on.
The introduction of the strongly singular Calderón-Zygmund operator is motivated by a class of multiplier operators whose symbol is given by \(e^{i|\xi|^{a}}/|\xi|^{\beta}\) away from the origin, \(0< a<1\), \(\beta>0\). Fefferman and Stein [1] enlarged the multiplier operators onto a class of convolution operators. Coifman [2] also considered a related class of operators for \(n=1\). The strongly singular non-convolution operator, whose properties are similar to those of the classical Calderón-Zygmund operator, but the kernel is more singular near the diagonal than that of the standard one, was introduced by Alvarez and Milman in [3]. Furthermore, following a suggestion of Stein, the authors in [3] showed that the pseudo-differential operator with symbol in the Hörmander class \(S^{-\beta}_{\alpha, \delta}\), where \(0<\delta\leq\alpha<1\) and \(n(1-\alpha)/2\leq\beta< n/2\), is included in the strongly singular Calderón-Zygmund operator. Thus, the strongly singular Calderón-Zygmund operator correlates closely with both the theory of Calderón-Zygmund singular integrals in harmonic analysis and the theory of pseudo-differential operators in partial differential equations.
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Suppose that T is a strongly singular Calderón-Zygmund operator or a generalized Calderón-Zygmund operator, whose strict definitions will be given later, and b is a locally integrable function on \(\mathbf{R}^{n}\). The commutator \([b, T]\) generated by b and T is defined as follows:
$$[b, T](f) (x)=b(x)Tf(x)-T(bf) (x). $$
In 1986, Alvarez and Milman [3, 4] discussed the boundedness of the strongly singular Calderón-Zygmund operator on Lebesgue spaces. In 2006, in [5], the behavior of Toeplitz operators related to strongly singular Calderón-Zygmund operators and Lipschitz functions was discussed on Lebesgue spaces. Furthermore, in 2007, Lin and Lu [6] proved the boundedness of the commutators of strongly singular Calderón-Zygmund operators on Hardy type spaces. In 2008, the authors in [7] obtained two kinds of endpoint estimates for strongly singular Calderón-Zygmund operators. And the pointwise estimate for the sharp maximal function of commutators generated by strongly singular Calderón-Zygmund operators and BMO functions was also established.
In 2007, the authors in [8] obtained the boundedness of generalized Calderón-Zygmund operators on weighted Lebesgue spaces and weighted Hardy spaces. In 2011, the pointwise estimates for the sharp maximal functions of commutators generated by generalized Calderón-Zygmund operators and BMO functions or Lipschitz functions were established in [9].
The classical Morrey spaces which were introduced by Morrey in [10] came from [10, 11] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, one can refer to [10, 12]. In 1987, Chiarenza and Frasca [13] proved the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator and the Calderón-Zygmund singular integral operator on Morrey spaces. In 2010, Fu and Lu [14] established the boundedness of weighted Hardy operators and their commutators on Morrey spaces. Lin [9, 15] discussed the commutators of strongly singular Calderón-Zygmund operators and generalized Calderón-Zygmund operators on Morrey spaces, respectively.
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In 2009, Komori and Shirai [16] defined the weighted Morrey spaces and studied the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator and the classical Calderón-Zygmund singular integral operator on these weighted spaces. In 2012, Wang [17] researched the behavior of commutators generated by classical Calderón-Zygmund operators and weighted Lipschitz functions or weighted BMO functions on weighted Morrey spaces. In 2013, the authors in [18] proved the boundedness of some sublinear operators and their commutators on weighted Morrey spaces. In 2014, Lin and Sun [19] established the boundedness of commutators generated by strongly singular Calderón-Zygmund operators and weighted BMO functions on weighted Morrey spaces. In 2015, the authors in [20] studied the properties of commutators generated by generalized Calderón-Zygmund operators and weighted BMO functions on weighted Morrey spaces.
Inspired by the above results, in this paper we are interested in the boundedness of the commutators generated by weighted Lipschitz functions and strongly singular Calderón-Zygmund operators or generalized Calderón-Zygmund operators on weighted Morrey spaces.
Before stating our main results, let us first recall some necessary definitions and notations.
Definition 1.1
Let \(\mathcal{S}\) be the space of all Schwartz functions on \(\mathbf{R}^{n}\) and \(\mathcal{S}'\) be its dual space, the class of all tempered distributions on \(\mathbf{R}^{n}\). Let \(T:\mathcal{S}\rightarrow\mathcal{S}'\) be a bounded linear operator. T is called a strongly singular Calderón-Zygmund operator if the following three conditions are satisfied.
(1)
T can be extended into a continuous operator from \(L^{2}(\mathbf{R}^{n})\) into itself.
(2)
There exists a function \(K(x, y)\) continuous away from the diagonal \(\{(x, y): x=y\}\) such that
for \(f, g\in\mathcal{S}\) with disjoint supports.
(3)
For some \(n(1-\alpha)/2\leq\beta< n/2\), both T and its conjugate operator \(T^{*}\) can be extended into continuous operators from \(L^{q}\) to \(L^{2}\), where \(1/q=1/2+\beta/n\).
Definition 1.2
Suppose that \(T:\mathcal{S}\rightarrow\mathcal{S}'\) is a linear operator with kernel \(K(\cdot,\cdot)\) defined initially by
where \(|x-y|>2|z-y|\) for some \(\delta>0\), we can find out that the classical Calderón-Zygmund operator is a generalized Calderón-Zygmund operator defined as in Definition 1.2 with \(C_{j}=2^{-j\delta}\), \(j\in\mathbf{N}\), and any \(1< q<\infty\).
A non-negative measurable function ω is said to be in the Muckenhoupt class \(A_{p}\) with \(1< p<\infty\) if for every cube Q in \(\mathbf{R}^{n}\), there exists a positive constant C independent of Q such that
where Q denotes a cube in \(\mathbf{R}^{n}\) with the side parallel to the coordinate axes and \(1/p+1/p'=1\). When \(p=1\), a non-negative measurable function ω is said to belong to \(A_{1}\) if there exists a constant \(C>0\) such that for any cube Q,
It is well known that if \(\omega\in A_{p}\) with \(1< p<\infty\), then \(\omega\in A_{r}\) for all \(r>p\) and \(\omega\in A_{q}\) for some \(1< q< p\).
and the supremum is taken over all cubes Q in \(\mathbf{R}^{n}\).
Definition 1.6
Let \(1\leq p<\infty\), \(0<\beta _{0}<1\) and ω be a weighted function. A locally integrable function b is said to be in the weighted Lipschitz space \(\operatorname{Lip}_{\beta_{0}}^{p}(\omega)\) if
where \(b_{Q}=\frac{1}{|Q|}\int_{Q}b(y)\,dy\) and the supremum is taken over all cubes \(Q\subset\mathbf{R}^{n}\). Moreover, we denote simply by \(\operatorname{Lip}_{\beta_{0}}(\omega)\) when \(p=1\).
Definition 1.7
The Hardy-Littlewood maximal operator M is defined by
A weighted function ω belongs to the reverse Hölder class \(\mathit{RH}_{r}\) if there exist two constants \(r>1\) and \(C>0\) such that the following reverse Hölder inequality
$$\biggl(\frac{1}{|Q|}\int_{Q} \omega(x)^{r} \,dx \biggr)^{\frac{1}{r}}\leq C \biggl(\frac{1}{|Q|}\int _{Q} \omega(x)\,dx \biggr) $$
holds for every cube Q in \(\mathbf{R}^{n}\). Denote by \(r_{\omega}\) the critical index of ω for the reverse Hölder condition. That is, \(r_{\omega}=\sup\{r>1: \omega\in \mathit{RH}_{r}\}\).
Definition 1.9
For \(0<\beta_{0}<n\), \(1\leq r<\infty\), the fractional maximal operator \(M_{\beta_{0},r}\) is defined by
where the above supremum is taken over all cubes Q containing x.
2 Main results
In what follows, we will give the main results in this paper.
Theorem 2.1
LetTbe a strongly singular Calderón-Zygmund operator, α, β, δbe given as in Definition 1.1, and\(\frac{n(1-\alpha)}{2}<\beta<\frac {n}{2}\) (\(n\geq2\)). Suppose\(0<\beta_{0}<1\), \(\frac{n(1-\alpha)+2\beta}{2\beta}< p<\frac{n}{\beta_{0}}\), \(\frac {1}{q}=\frac{1}{p}-\frac{\beta_{0}}{n}\), \(0< k<\frac{p}{q}\), \(\omega\in A_{1}\)and\(r_{\omega}>\frac{(n(1-\alpha )+2\beta)(p-1)}{2\beta p-n(1-\alpha)-2\beta}\). If\(b\in \operatorname{Lip}_{\beta_{0}}(\omega)\), then\([b, T]\)is bounded from\(L^{p, k}(\omega)\)to\(L^{q, kq/p}(\omega ^{1-q}, \omega)\).
Theorem 2.2
LetTbe a generalized Calderón-Zygmund operator and\(q'\)be given as in Definition 1.2. Suppose\(0<\beta_{0}<1\), \(q'< p< n/\beta _{0}\), \(1/s=1/p-\beta_{0}/n\), \(0< k< p/s\), \(\{jC_{j}\}\in l^{1}\), \(\omega^{s/p}\in A_{1}\)and\(r_{\omega}>\max\{\frac{(p-1)q'}{p-q'}, \frac{1-k}{p/s-k}\}\). If\(b\in \operatorname{Lip}_{\beta_{0}}(\omega)\), then\([b, T]\)is bounded from\(L^{p, k}(\omega)\)to\(L^{s, ks/p}(\omega ^{1-s}, \omega)\).
Noticing that the classical Calderón-Zygmund operator is a generalized Calderón-Zygmund operator defined as in Definition 1.2 with \(C_{j}=2^{-j\delta}\) (\(j\in\mathbf{N}\)) and any \(1< q<\infty\), we can obtain the following result as a corollary.
As matter of fact, the result of Corollary 2.1 was obtained in [17] with the special case \(\delta =1\). Thus Theorem 2.2 can be regarded as a generalization of the corresponding result in [17]. And from this point of view, the range of the index in Theorem 2.2 is reasonable.
3 Preliminaries
In order to obtain our main results, first we introduce some requisite lemmas.
IfTis a strongly singular Calderón-Zygmund operator, thenTis the type of weak\((1,1)\)and can be defined to be a continuous operator from\(L^{\infty}\)to BMO.
It follows from Definition 1.1, Lemma 3.1 and the interpolation theory that the strongly singular Calderón-Zygmund operator T is bounded on \(L^{p}\) for \(1< p<\infty\), and T is bounded from \(L^{u}\) to \(L^{v}\), \(\frac{n(1-\alpha)+2\beta}{2\beta}\leq u<\infty\) and \(0<\frac{u}{v}\leq \alpha\). In particular, if we restrict \(\frac{n(1-\alpha)}{2}<\beta<\frac{n}{2}\) in (3) of Definition 1.1, then T is bounded from \(L^{u}\) to \(L^{v}\), where \(\frac{n(1-\alpha)+2\beta }{2\beta}< u<\infty\) and \(0<\frac{u}{v}<\alpha\).
LetTbe a generalized Calderón-Zygmund operator and the sequence\(\{C_{j}\}\in l^{1}\), thenTis bounded on\(L^{p}(\mathbf{R}^{n})\)and the type of weak\((1, 1)\), where\(1< p<\infty\).
Let\(\mu\in A_{1}\), then there are constants\(C_{1}\), \(C_{2}\)and\(0<\delta<1\)depending only on\(A_{1}\)-constant ofμsuch that for any measurable subsetEof a ballB,
LetTbe a generalized Calderón-Zygmund operator, \(q'\)be the same as in Definition 1.2, and the sequence\(\{C_{j}\}\in l^{1}\). If\(q'< p<\infty\), \(0< k<1\)and\(\omega\in A_{p/q'}\), thenTis bounded on\(L^{p, k}(\omega)\).
Let\(0<\beta_{0}<n\), \(1< p< n/\beta_{0}\), \(1/s=1/p-\beta_{0}/n\)and\(\omega^{s/p}\in A_{1}\). Then if\(0< k< p/s\)and\(r_{\omega}>\frac{1-k}{p/s-k}\), we have
Let\(0<\beta_{0}<1\)and\(\omega\in A_{1}\). Then, for any\(1\leq p<\infty\), there exists an absolute constant\(C>0\)such that\(\|b\|_{ \operatorname{Lip}_{\beta_{0}}^{p}(\omega)}\leq C\|b\|_{ \operatorname{Lip}_{\beta_{0}}(\omega)}\).
Lemma 3.11
Let\(0<\beta_{0}<1\), \(\omega\in A_{1}\)andfbe a function in\(\operatorname{Lip}_{\beta_{0}}(\omega)\). Suppose\(1\leq p<\infty\), \(x\in\mathbf{R}^{n}\), and\(r_{1}, r_{2}>0\). Then
Without loss of generality, we may assume that \(0< r_{2}\leq r_{1}\) and omit the case \(0< r_{1}< r_{2}\) due to their similarity. For \(0< r_{2}\leq r_{1}\), there are \(k_{1}, k_{2}\in\mathbf{Z}\) such that \(2^{k_{1}-1}< r_{1}\leq2^{k_{1}}\) and \(2^{k_{2}-1}< r_{2}\leq2^{k_{2}}\). Then \(k_{2}\leq k_{1}\) and
LetTbe a strongly singular Calderón-Zygmund operator, α, β, δbe given as in Definition 1.1and\(\frac{n(1-\alpha)}{2}<\beta<\frac {n}{2}\). Let\(0<\beta_{0}<1\), \(0< t<1\), \(\frac{n(1-\alpha)+2\beta}{2\beta }< s<\infty\), \(\omega\in A_{1}\cap \mathit{RH}_{r}\)with\(r>\frac{(n(1-\alpha)+2\beta )(s-1)}{2\beta s-n(1-\alpha)-2\beta}\), and\(b\in \operatorname{Lip}_{\beta_{0}}(\omega)\), then we have, for a.e. \(x\in\mathbf{R}^{n}\),
Since \(r>\frac{(n(1-\alpha)+2\beta)(s-1)}{2\beta s-n(1-\alpha)-2\beta }\), then \(\frac{n(1-\alpha)+2\beta}{2\beta}<\frac{rs}{s+r-1}\). There exists \(s_{0}\) such that \(\frac{n(1-\alpha)+2\beta}{2\beta}< s_{0}<\frac{rs}{s+r-1}\). For this index \(s_{0}\), there exists \(l_{0}\) such that T is bounded from \(L^{s_{0}}\) to \(L^{l_{0}}\) and \(0<\frac{s_{0}}{l_{0}}<\alpha\). Then we can take θ satisfying \(0<\frac{s_{0}}{l_{0}}<\theta<\alpha\). Let \(\tilde{B}=B(x, r_{B}^{\theta})\). Write \(f=f_{3}+f_{4}\), where \(f_{3}=f\chi_{2\tilde{B}}\), then
Since \(1< s_{0}< s<\infty\), there exists l (\(1< l<\infty\)) such that \(\frac{1}{s_{0}}=\frac{1}{s}+\frac{1}{l}\). It follows from Hölder’s inequality and the \((L^{s_{0}}, L^{l_{0}})\)-boundedness of T that
Denote \(p_{0}=\frac{(r-1)(s-s_{0})}{s(s_{0}-1)}\). The fact \(s_{0}<\frac {rs}{s+r-1}\) implies \(1< p_{0}<\infty\). Then we get \(p'_{0}=\frac{(r-1)(s-s_{0})}{rs-(s+r-1)s_{0}}\) and \(1< p'_{0}<\infty\). By Hölder’s inequality, Lemma 3.11, and noticing that \(r=\frac{lp_{0}}{s'}-\frac{p_{0}}{p'_{0}}\), we have
Denote \(\varepsilon_{1}=n(\frac{\theta}{s_{0}}-\frac{1}{l_{0}})\). The inequality \(0<\frac{s_{0}}{l_{0}}<\theta\) implies that \(\varepsilon_{1}>0\). By Lemma 3.7, we have
Let \(\varepsilon_{2}=\frac{\delta}{\alpha}(\alpha-\theta)\). It follows from \(\theta<\alpha\) that \(\varepsilon_{2}>0\). For any \(y\in B\) and \(z\in(2\tilde{B})^{c}\), there is \(2|y-x|^{\alpha}\leq2r_{B}^{\alpha}\leq2r_{B}^{\theta}\leq|z-x|\) since \(0< r_{B}\leq1\). Applying (2) of Definition 1.1, Hölder’s inequality, Lemma 3.7 and Lemma 3.11, we get
LetTbe a generalized Calderón-Zygmund operator, \(q'\)be the same as in Definition 1.2and the sequence\(\{jC_{j}\}\in l^{1}\). Let\(0<\delta<1\), \(0<\beta_{0}<1\), \(\omega\in A_{1}\), \(r_{\omega}>q'\), and\(b\in \operatorname{Lip}_{\beta_{0}}(\omega)\), then, for all\(r>\frac{r_{\omega}-1}{r_{\omega}-q'}q'\), we have
Since \(r>\frac{r_{\omega}-1}{r_{\omega}-q'}q'>q'\), there exists \(1< l<\infty\) such that \(\frac{1}{q}+\frac{1}{l}+\frac{1}{r}=1\). By Hölder’s inequality for the three numbers q, l and r, we can get
Denote \(p_{0}=\frac{s-1}{\frac{q'(r-1)}{r-q'}-1}\). It follows from \(r>\frac{r_{\omega}-1}{r_{\omega}-q'}q'\) that \(r_{\omega}>\frac{r-1}{r-q'}q'\). Since \(r_{\omega}=\sup\{s>1:\omega\in \mathit{RH}_{s}\}\), there exists s such that \(\omega\in \mathit{RH}_{s}\) and \(s>\frac{r-1}{r-q'}q'\). We can get \(1< p_{0}<\infty\) and \(s=\frac{lp_{0}}{r'}-\frac{p_{0}}{p_{0}'}\). Applying Hölder’s inequality for \(p_{0}\) and \(p'_{0}\), we have
Let \(\tilde{\beta}=\beta_{0}r\), \(\tilde{s}=s/r\), and \(\tilde{p}=p/r\). We have \(0<\tilde{\beta}<n\) and \(1<\tilde{p}<n/\tilde{\beta}\). Then \(1/\tilde{s}=1/\tilde{p}-\tilde{\beta}/n\), \(\omega^{\tilde{s}/\tilde{p}}\in A_{1}\), \(0< k<\tilde{p}/\tilde{s}\), and \(r_{\omega}>\frac {1-k}{\tilde{p}/\tilde{s}-k}\). Using Lemma 3.8, we obtain
Since \(r_{\omega}>\frac {(n(1-\alpha)+2\beta)(p-1)}{2\beta p-n(1-\alpha)-2\beta}\), there is r such that \(r>\frac{(n(1-\alpha)+2\beta)(p-1)}{2\beta p-n(1-\alpha)-2\beta}\) and \(\omega\in \mathit{RH}_{r}\). Then \(p>\frac{(n(1-\alpha)+2\beta)(r-1)}{2\beta r-n(1-\alpha)-2\beta}\). There exists s such that \(p>s>\frac{(n(1-\alpha)+2\beta)(r-1)}{2\beta r-n(1-\alpha)-2\beta}>\frac{n(1-\alpha)+2\beta}{2\beta}\). The fact \(s>\frac{(n(1-\alpha)+2\beta)(r-1)}{2\beta r-n(1-\alpha)-2\beta }\) implies that \(r>\frac{(n(1-\alpha)+2\beta)(s-1)}{2\beta s-n(1-\alpha)-2\beta}\). By Lemma 3.3 and Lemma 3.12, we have
It follows from \(r_{\omega }>\frac{p-1}{p-q'}q'\) that \(p>\frac{r_{\omega}-1}{r_{\omega}-q'}q'\). There exists r such that \(p>r>\frac{r_{\omega}-1}{r_{\omega}-q'}q'\). By Lemma 3.3 and Lemma 3.13, we have
Choose q such that \(\max\{\frac{pr_{\omega}}{(p-1)(r_{\omega}-1)}, 2, p'\}< q<\infty\), then \(1< q'<2\), \(q'< p\) and \(q'<\frac{pr_{\omega}}{p+r_{\omega}-1}\). It follows from \(q'<\frac{pr_{\omega}}{p+r_{\omega}-1}\) that \(r_{\omega }>\frac{(p-1)q'}{p-q'}\).
Since T is a generalized Calderón-Zygmund operator defined as in Definition 1.2 with \(C_{j}=2^{-j\delta}\) (\(j\in\mathbf{N}\)) and this pair of numbers \((q,q')\), then by Theorem 2.2 we have that \([b, T]\) is bounded from \(L^{p, k}(\omega)\) to \(L^{s, ks/p}(\omega^{1-s}, \omega)\). □
Acknowledgements
This work was supported by the National Natural Science Foundation of China (11171345), Beijing Higher Education Young Elite Teacher Project (YETP0946) and the Fundamental Research Funds for the Central Universities (2009QS16).
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.