Research Article | Open Access
The Commutators of Fractional Integrals on Generalized Herz Spaces
Let be the fractional integral, and let . We will obtain the weighted estimates for the commutator on the generalized Herz spaces.
1. Introduction and Results
Let , the fractional integral operator is defined by Consider the commutator associated with the fractional integrals and the locally integrable function , The area of the commutators of fractional integrals has been under intensive research. When , and , and Chanillo proved in  that is bounded from to . Segovia and Torrea in  proved that is also bounded from to if . Lu and Yang in  studied Herz spaces; they proved that if ,, and , then maps continuously into .
In , Komori and Matsuoka considered the boundedness of singular integral operators and fractional integral operators on weighted Herz spaces; they introduced generalized Herz spaces, and, as corollaries of their general theory, they obtained the boundedness of these operators on weighted Herz spaces.
For a sequence , we suppose that satisfies doubling condition of order and write if there exists such that for .
Let and let be the characteristic function of the set for any . Suppose that is a weight function on . For , , the generalized Herz space is defined by where
Let . We say (centered reverse doubling) if there is a positive number such that
In 2000, Lu et al. in  proved boundedness results for sublinear operators on weighted Herz spaces with general Muckenhoupt weights. Recently, many authors considered the boundedness of operators on weighted Herz spaces. In , Wang proved the boundedness properties of intrinsic square function on weighted Herz spaces. Tomita in  studied the Strang-Fix theory for approximation order in weighted Herz spaces with general Muckenhoupt weights. In , Hu and Wang consider the boundedness properties of commutator operators generated by function and intrinsic square function on weighted Herz spaces.
In this paper, we investigate the boundedness of commutators generated by fractional integrals and on generalized Herz spaces with general Muckenhoupt weights. Our main results in this paper are formulated as follows.
Theorem 1. Let , , , and let . One assumes that (1), where ;(2), where ;(3).Then, is bounded from to .
Remark 2. When , condition (2) is equivalent to the condition that , but when , (2) is stronger than -condition. Komori and Matsuoka in  showed that the condition (2) is the best possible by a counterexample.
Let in Theorem 1; we have
Corollary 3. Let , , and let . If , , then is bounded from to .
2. Definitions and Preliminaries
We begin with some properties of weights which play a great role in the proofs of our main results.
A weight is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For any ball and , denotes the ball concentric with whose radius is times as long. For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set the weighted measure .
A weight is said to belong to for , if there exists a constant such that for every ball , where is the dual of such that . The class is defined by replacing the above inequality with
A weight is said to belong to if there are positive numbers and so that for all balls and all measurable . It is well known that
The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal function in .
We also need another weight class introduced by Muckenhoupt and Wheeden in  to study weighted boundedness of fractional integral operators.
Given , we say that if there exists a constant such that for every ball , the inequality holds when , and the inequality holds when .
By (13), we have
Lemma 5. Given , (i) if and only if ;(ii) if and only if ;(iii) if and only if ;(iv)If and , then .
A locally integrable function is said to be in if where .
Lemma 6 (John-Nirenberg inequality; see ). Let . Then, for any ball , there exist positive constants and such that, for all ,
Lemma 7 (see ). Let . Then, the norm of is equivalent to the norm of , where
Lemma 8. Suppose and . Then, for any , one has
3. Proof of Theorem 1
Let , . Then, Since , where , then, by (i) of Lemma 5, we know that. Using the fact that is a bounded operator from to (see ), we obtain We now estimate . Obviously, First, we estimate . From , one can see then that Since , then, by Lemma 8, Similar to the proof of Lemma 8, we can deduce that From the definition of , it is easy to see that By Hölder's inequality, we get Then,
We now turn to estimate . Since , then, by (ii) of Lemma 5, we know . Similar to the proof of Lemma 8, we have Using Hölder's inequality and (32), Since , by (iii) of Lemma 4, we have for some . Then, from (15) and (i) of Lemma 4, Hence, Using the condition that , we obtain
Since , we have , and, then, . When , we have When , by Hölder's inequality, we have
Now, we estimate . Since , similar to the estimation of , we have From the fact and (15), we get By , we have Then, Hence, Using the condition that , When , we have When , by the Hölder inequality, we get Combining the above estimates for , , and , the proof of Theorem 1 is completed.
Conflict of Interests
The authors declare that they have no conflict of interests.
The authors wish to thank the referee for his/her useful suggestions for the improvement of the paper.
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