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Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz-Hardy spaces
Journal of Inequalities and Applications volume 2012, Article number: 308 (2012)
In this paper, the - and -type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels is obtained.
1 Introduction and definitions
Let T be the Calderón-Zygmund operator and . The commutator generated by T and b is defined by
A classical result of Coifman, Rochberg and Weiss (see [1, 2]) proved that the commutator is bounded on (). However, it was observed that the is not bounded, in general, from to . But if is replaced by a suitable atomic space , then maps continuously into (see ). In addition, we easily know that . In recent years, the theory of Herz-type Hardy spaces have been developed (see [4–7]). The main purpose of this paper is to consider the continuity of multilinear commutators related to Marcinkiewicz operators with variable kernels and functions on certain Hardy and Herz-Hardy spaces. Let us first introduce some definitions (see [3–14]).
Given a positive integer m and , we denote by the family of all finite subsets of of j different elements. For , set . For and , set , and .
Definition 1 Let () be a locally integrable function and . A bounded measurable function a on is said to be a atom if
for any , .
A temperate distribution f is said to belong to if, in the Schwartz distribution sense, it can be written as
where every is atom, and . Moreover, .
Given a set , the characteristic function of E is defined by . Let and and , .
Definition 2 Let , , . For , set and . Denote by the characteristic function of and by the characteristic function of .
The homogeneous Herz space is defined by
The nonhomogeneous Herz space is defined by
Definition 3 Let , , , , . A function a on is called a central -atom (or a central -atom of restrict type) if
(or for some ),
for any , .
A temperate distribution f is said to belong to (or ) if it can be written as (or ), in the Schwartz distribution sense, where is a central -atom (or a central -atom of restrict type) supported on and (or ). Moreover, .
Definition 4 Let Ω be homogeneous of degree zero on such that is defined
If , we say satisfied -Dini condition.
Definition 5 Let , and Ω be homogeneous of degree zero on such that . Assume that , that is, there exists a constant such that for any , . The Marcinkiewicz multilinear commutator is defined by
we also define that
2 Theorems and proofs
We begin with three preliminary lemmas.
Lemma 1 (see )
Let , for and . Then we have
Lemma 2 (see )
Let , and . Then is bounded from to .
Lemma 3 (see )
Let , satisfy () conditions, that is, there exists a constant such that for ,
Theorem 1 Let , , , , , . Then is bounded from to .
Proof It suffices to show that there exists a constant such that for every atom a,
Let a be a atom supported on a ball . We write
For I, taking with and , by Hölder’s inequality and the -boundedness of , we get
For II, denoting with , , where , by Hölder’s inequality and the vanishing moment of a, we get
Note that for , . For , we have
so we have
For , we can obtain
Thus, by Lemma 3, we have
This finishes the proof of Theorem 1. □
Theorem 2 Let , , , , , and , , . Then is bounded from to .
Proof Let and be the atomic decomposition for f as in Definition 3. We write
For JJ, by the -boundedness of , we get
For J, let , , , , we have
For G, noting that , , , we know . Then, similar to the proof of Theorem 1, we obtain
For the sake of simplicity, we denote
This completes the proof of Theorem 2. □
Remark Theorem 2 also holds for nonhomogeneous Herz-type spaces.
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This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar under Grant No. 50925727, National Natural Science Foundation of China under Grant No. 60876022, The National Defense Advanced Research Project Grant No. C1120110004, Hunan Provincial Science and Technology Foundation of China under Grant No. 2010J4 and 2011JK2023, the Key Grant Project of Chinese Ministry of Education under Grant No. 313018.
The authors declare that they have no competing interests.
In this paper, WY carried out the and -type boundedness for the multilinear commutator related to the Marcinkiewicz operator with variable kernels. YH, QL, JZ, XW participated in the analysis. All authors read and revised the final manuscript.
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Yu, W., He, Y., Luo, Q. et al. Boundedness for multilinear commutator of Marcinkiewicz operator with variable kernels on Hardy and Herz-Hardy spaces. J Inequal Appl 2012, 308 (2012). https://doi.org/10.1186/1029-242X-2012-308
- Marcinkiewicz operator
- multilinear commutator
- Hardy space
- Herz-Hardy space