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Strongly singular Calderón-Zygmund operators and commutators on weighted Morrey spaces
Journal of Inequalities and Applications volume 2014, Article number: 519 (2014)
Abstract
In this paper, the authors establish the boundedness of the strongly singular Calderón-Zygmund operator on weighted Morrey spaces. Moreover, the boundedness of the commutator generated by the strongly singular Calderón-Zygmund operator and the weighted BMO function on weighted Morrey spaces is also obtained.
MSC:42B20, 42B25, 42B35.
1 Introduction
The strongly singular non-convolution operator was introduced by Alvarez and Milman in [1], whose properties are similar to those of the Calderón-Zygmund operator, but the kernel is more singular near the diagonal than that of the standard case. Furthermore, following a suggestion of Stein, the authors in [1] showed that the pseudo-differential operators with symbols in the class , where and , are 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 PDE.
Definition 1.1 Let be a bounded linear operator. T is called a strongly singular Calderón-Zygmund operator if the following conditions are satisfied.
-
(1)
T can be extended into a continuous operator from into itself.
-
(2)
There exists a function continuous away from the diagonal such that
if for some and . And , for with disjoint supports.
-
(3)
For some , both T and its conjugate operator can be extended to continuous operators from to , where .
Alvarez and Milman [1, 2] discussed the boundedness of the strongly singular Calderón-Zygmund operator on Lebesgue spaces. Lin [3] proved the boundedness of the strongly singular Calderón-Zygmund operator on Morrey spaces. Furthermore, Lin and Lu [4] showed the boundedness of the strongly singular Calderón-Zygmund operator on Herz-type Hardy spaces.
Suppose that T is a strongly singular Calderón-Zygmund operator and b is a locally integrable function on . The commutator generated by b and T is defined as follows:
The authors in [5] obtained the boundedness of the commutators generated by strongly singular Calderón-Zygmund operators and Lipschitz functions on Lebesgue spaces. Lin and Lu [4] proved the boundedness of the commutators of strongly singular Calderón-Zygmund operators on Hardy-type spaces. Moreover, Lin and Lu [3, 6] discussed the boundedness of the commutator on Morrey spaces when b is a BMO function or a Lipschitz function, respectively.
The classical Morrey space was originally introduced by Morrey in [7] to study the local behavior of solutions of second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, one can refer to [7, 8]. In [9], Chiarenza and Frasca showed 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 [10] established the boundedness of weighted Hardy operators and their commutators on Morrey spaces.
In 2009, Komori and Shirai [11] 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 [12] showed the boundedness of commutators generated by classical Calderón-Zygmund operators and weighted BMO functions on weighted Morrey spaces. In 2013, the authors in [13] proved the boundedness of some sublinear operators and their commutators on weighted Morrey spaces.
Inspired by the above results, the main purpose of this paper is to overcome the stronger singularity near the diagonal and establish the boundedness properties of the strongly singular Calderón-Zygmund operators and their commutators on weighted Morrey spaces.
Let us first recall some necessary definitions and notations.
Definition 1.2 ([14])
A non-negative measurable function ω is said to be in the Muckenhoupt class with if for every cube Q in , there exists a positive constant C independent of Q such that
where Q denotes a cube in with the side parallel to the coordinate axes and . When , a non-negative measurable function ω is said to belong to , if there exists a constant such that for any cube Q,
It is well known that if with , then for all , and for some .
Definition 1.3 ([15])
A weighted function ω belongs to the reverse Hölder class if there exist two constants and such that the reverse Hölder inequality
holds for every cube Q in .
It is well known that if with , then there exists a such that . It follows directly from Hölder’s inequality that implies for all .
Definition 1.4 Let and ω be a weighted function. A locally integrable function b is said to be in the weighted BMO space if
where and the supremum is taken over all cubes .
Moreover, we denote simply when .
Definition 1.5 The Hardy-Littlewood maximal operator M is defined by
We set , where .
The sharp maximal operator is defined by
where . We define the t-sharp maximal operator , where .
Let ω be a weight. The weighted maximal operator is defined by
We also set , where .
Definition 1.6 ([11])
Let , , and ω be a weighted function. Then the weighted Morrey space is defined by
where
and the supremum is taken over all cubes Q in .
Definition 1.7 ([11])
Let and . Then for two weighted functions u and v, the weighted Morrey space is defined by
where
2 Main results
Now we state our main results as follows.
Theorem 2.1 Let T be a strongly singular Calderón-Zygmund operator, and α, β, δ be given in Definition 1.1. If , , and , then T is bounded on .
Theorem 2.2 Let T be a strongly singular Calderón-Zygmund operator, α, β, δ be given in Definition 1.1 and . Suppose , , and with . If , then is bounded from to .
If we consider the extreme cases and in Definition 1.1, then the strongly singular Calderón-Zygmund operator comes back to the classical Calderón-Zygmund operator. Thus, we get the boundedness of the classical Calderón-Zygmund operator and its commutator on weighted Morrey spaces as corollaries of Theorem 2.1 and Theorem 2.2.
Corollary 2.1 Let T be a classical Calderón-Zygmund operator. If , , and , then T is bounded on .
Corollary 2.2 Let T be a classical Calderón-Zygmund operator, , and . If , then is bounded from to .
Remark 2.1 Actually, Corollary 2.1 and Corollary 2.2 have been exactly obtained in [11] and [12] in the special case . Thus, from this perspective, Theorem 2.1 and Theorem 2.2 generalized the corresponding results in [11, 12], and the range of the index in Theorem 2.1 and Theorem 2.2 is reasonable.
3 Preliminaries
Before we give the proofs of our main results, we need some lemmas.
Lemma 3.1 ([1])
If T is a strongly singular Calderón-Zygmund operator, then T can be defined to be a continuous operator from to BMO.
Lemma 3.2 ([2])
If T is a strongly singular Calderón-Zygmund operator, then T is of weak type.
By Lemma 3.1, Lemma 3.2, Definition 1.1, and interpolation theory, we find that T is bounded on , . Besides the -boundedness, the strongly singular Calderón-Zygmund operator T still has other kinds of boundedness properties on Lebesgue spaces. By interpolating between and , where q is given in Definition 1.1 and , T is bounded from to with and . It is easy to see that in this situation. Then we interpolate between and weak to obtain the boundedness of T from to , where and . In this situation, if and only if . In a word, the boundedness properties of the strongly singular Calderón-Zygmund operator on Lebesgue spaces can be summarized as follows.
Remark 3.1 The strongly singular Calderón-Zygmund operator T is bounded on for . And T is bounded from to , and . In particular, if we restrict in (3) of Definition 1.1, then T is bounded from to , , and .
Let . Then for any , there exists an absolute constant such that .
Lemma 3.4 ([11])
If , , and , then M is bounded on .
Lemma 3.5 ([12])
Let , , and , then for any , we have
Lemma 3.6 ([12])
Let , , and . If , then we have
for all functions f such that the left-hand side is finite. In particular, when and , we have
for all functions f such that the left-hand side is finite.
Lemma 3.7 Given , we have , for all .
Let , . The above result comes from the monotone property of the function φ.
Lemma 3.8 If T is a strongly singular Calderón-Zygmund operator, α, β, δ are given in Definition 1.1, and , then for all , there exists a positive constant C such that
for every bounded and compactly supported function f.
Proof For any ball which contains x, there are two cases.
Case 1: .
We have
For , by Hölder’s inequality and the -boundedness of T, we get
Since and for any , , by Hölder’s inequality and (2) of Definition 1.1, we have
Case 2: .
Denote . There is
Since , by Remark 3.1, there exists an l such that T is bounded from into and . It follows from Hölder’s inequality that
Since and for any , , similarly to , we have
Therefore, combining the estimates in both cases, there is
□
Lemma 3.9 Let and f be a function in . Suppose , , and . Then
Proof Without loss of generality, we may assume that and omit the case since their similarity. For , there are such that and . Then and
Thus, we have
Write
and
If , then by the fact , we have
If , then the above estimate holds obviously.
Thus,
This completes the proof of Lemma 3.9. □
Lemma 3.10 Let T be a strongly singular Calderón-Zygmund operator, α, β, δ be given in Definition 1.1 and . Let , , with , and , then we have
Proof For any ball with the center x and radius , there are two cases.
Case 1: .
We decompose , where and denotes the characteristic function of 2B. Observe that
Since , we have
We are now going to estimate each term, respectively. Since , it follows from Hölder’s inequality and Lemma 3.3 that
Applying Kolmogorov’s inequality [15], Lemma 3.2, Hölder’s inequality, and Lemma 3.3, we get
Since and for any , , by (2) of Definition 1.1, we have
Applying Hölder’s inequality and Lemma 3.9, we get
Case 2: .
Since , that is, , there exists an such that . For the index which we chose, by Remark 3.1, there exists an such that T is bounded from to and . Then we can take a θ satisfying .
Let . We decompose , where and denotes the characteristic function of . Write
Since , we have
Similarly to estimate , we have
Since , there exists an l () such that . By Hölder’s inequality and the -boundedness of T, we have
Let . Since , we get . So and . Applying Hölder’s inequality for and , Lemma 3.9, and noticing that , we get
The inequality implies that . By Lemma 3.7, we have
The fact implies that . For any and , we have since . It follows from (2) of Definition 1.1, Hölder’s inequality, Lemma 3.9, and Lemma 3.7 that
Combining the estimates in both cases, we have
□
4 Proof of the main results
Now we are able to prove our main results.
Proof of Theorem 2.1 Since and , there exists an l such that and . Since , there exists an s such that . It follows from that . Applying Lemma 3.6, Lemma 3.8, and Lemma 3.4, we have
This completes the proof of Theorem 2.1. □
Proof of Theorem 2.2 Since , that is , there exists an s such that . Since , we have . Applying Lemma 3.6 and Lemma 3.10, we thus have
Therefore, by using Lemma 3.5 and Theorem 2.1, we obtain
This completes the proof of Theorem 2.2. □
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 11001266 and 11171345), Beijing Higher Education Young Elite Teacher Project (YETP0946), and the Fundamental Research Funds for the Central Universities (2009QS16).
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YL put forward the ideas of the paper, and the authors completed the paper together. They also read and approved the final manuscript.
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Lin, Y., Sun, G. Strongly singular Calderón-Zygmund operators and commutators on weighted Morrey spaces. J Inequal Appl 2014, 519 (2014). https://doi.org/10.1186/1029-242X-2014-519
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DOI: https://doi.org/10.1186/1029-242X-2014-519