- Research Article
- Open Access
Commutators of Littlewood-Paley Operators on the Generalized Morrey Space
© Yanping Chen et al. 2010
- Received: 6 May 2010
- Accepted: 11 July 2010
- Published: 28 July 2010
Let , , and denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized Littlewood-Paley function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators of , , and on the generalized Morrey space .
- Positive Constant
- Linear Operator
- Harmonic Analysis
- High Dimension
- Lebesgue Measure
Let be the unit sphere in equipped with the Lebesgue measure . Suppose that satisfies the following conditions.
Many important operators gave a characterization of BMO space. In 1976, Coifman et al.  gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo  studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space.
The purpose of this paper is to give a characterization of BMO space by the boundedness of the commutators of , , and on the generalized Morrey space .
The main result in this paper is as follows.
If is bounded on for some , then
Let and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.9), satisfies (1.1), (1.2), and (1.15). If is a bounded operator on for some , then
Let , , and . Assume that is nonincreasing and is nondecreasing. Suppose that is defined as (1.10), satisfies (1.1), (1.2), and (1.15). If is on for some , then
It is easy to check that (see, e.g., the proof of ( ) in [15, page 89]), we therefore give only the proofs of Theorem 1.2 for and Theorem 1.3 for .
It is easy to see that the condition (1.15) is weaker than for . In the proof of Theorems 1.2 and 1.3, we will use some ideas in . However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof.
Let us begin with recalling some known conclusion.
Similar to the proof of , we can easily get the following.
Now let us return to the proof of Theorem 1.2. Suppose that is a bounded operator on , we are going to prove that
Now, we consider the norm of in the following two cases.
Case 1 ( ).
Case 2 ( ).
Now, the estimate of is divided into two cases, namely, 1: ; 2: .
Case 1 ( ).
Case 2 ( ).
Then, Theorem 1.2 is proved.
Similar to the proof of Theorem 1.2, we only give the outline.
Suppose that is a bounded operator on , we are going to prove that
Then, Theorem 1.3 is proved.
The authors wish to express their gratitude to the referee for his/her valuable comments and suggestions. The research was supported by NSF of China (Grant nos.: 10901017, 10931001), SRFDP of China (Grant no.: 20090003110018), and NSF of Zhenjiang (Grant no.: Y7080325).
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