- Open Access
Strongly singular Calderón-Zygmund operators and commutators on weighted Morrey spaces
© Lin and Sun; licensee Springer. 2014
- Received: 25 September 2014
- Accepted: 25 November 2014
- Published: 12 December 2014
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.
- strongly singular Calderón-Zygmund operator
- weighted Morrey space
- weighted BMO function
The strongly singular non-convolution operator was introduced by Alvarez and Milman in , 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  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.
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.
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  proved the boundedness of the strongly singular Calderón-Zygmund operator on Morrey spaces. Furthermore, Lin and Lu  showed the boundedness of the strongly singular Calderón-Zygmund operator on Herz-type Hardy spaces.
The authors in  obtained the boundedness of the commutators generated by strongly singular Calderón-Zygmund operators and Lipschitz functions on Lebesgue spaces. Lin and Lu  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  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 , 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  established the boundedness of weighted Hardy operators and their commutators on Morrey spaces.
In 2009, Komori and Shirai  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  showed the boundedness of commutators generated by classical Calderón-Zygmund operators and weighted BMO functions on weighted Morrey spaces. In 2013, the authors in  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 ()
It is well known that if with , then for all , and for some .
Definition 1.3 ()
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 .
where and the supremum is taken over all cubes .
Moreover, we denote simply when .
We set , where .
where . We define the t-sharp maximal operator , where .
We also set , where .
Definition 1.6 ()
and the supremum is taken over all cubes Q in .
Definition 1.7 ()
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  and  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.
Before we give the proofs of our main results, we need some lemmas.
Lemma 3.1 ()
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 ()
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 ()
If , , and , then M is bounded on .
Lemma 3.5 ()
Lemma 3.6 ()
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.
for every bounded and compactly supported function f.
Proof For any ball which contains x, there are two cases.
Case 1: .
Case 2: .
If , then the above estimate holds obviously.
This completes the proof of Lemma 3.9. □
Proof For any ball with the center x and radius , there are two cases.
Case 1: .
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 .
Now we are able to prove our main results.
This completes the proof of Theorem 2.1. □
This completes the proof of Theorem 2.2. □
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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