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Multilinear fractional integral operators on generalized weighted Morrey spaces
Journal of Inequalities and Applications volume 2014, Article number: 323 (2014)
Abstract
Let be multilinear fractional integral operator and let . In this paper, the estimates of , the m-linear commutators and the iterated commutators on the generalized weighted Morrey spaces are established.
MSC:42B35, 42B20.
1 Introduction and results
The classical Morrey spaces were introduced by Morrey [1] in 1938, have been studied intensively by various authors, and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [2–4] for details. Moreover, various Morrey spaces have been defined in the process of this study. Mizuhara [5] introduced the generalized Morrey space ; Komori and Shirai [6] defined the weighted Morrey spaces ; Guliyev [7] gave the concept of generalized weighted Morrey space , which could be viewed as an extension of both and . The boundedness of some operators on these Morrey spaces can be seen in [5–9].
Let be the n-dimensional Euclidean space, be the m-fold product space (), and let be a collection of m functions on . Given and . We consider the multilinear fractional integral operators defined by
The corresponding m-linear commutators and the iterated commutators defined by, respectively,
and
As is well known, multilinear fractional integral operator was first studied by Grafakos [10], subsequently, by Kenig and Stein [11], Grafakos and Kalton [12]. In 2009, Moen [13] introduced weight function and gave weighted inequalities for multilinear fractional integral operators; In 2013, Chen and Wu [14] obtained the weighted norm inequalities for the multilinear commutators and . More results of the weighted inequalities for multilinear fractional integral and its commutators can be found in [15–17].
The aim of the present paper is to investigate the boundedness of multilinear fractional integral operator and its commutator on the generalized weighted Morrey spaces. Our results can be formulated as follows.
Theorem 1.1 Let and let . Suppose , , and , satisfy the condition with , and , , satisfy the condition
where , . If , then there exists a constant C independent of such that
If , and , then there exists a constant C independent of such that
Theorem 1.2 Let and let . Suppose with , and , satisfy the condition with , , and , , satisfy the condition
where , . If , then there exists a constant independent of such that
and
2 Definitions and preliminaries
A weight ω is a nonnegative, locally integrable function on . Let denote the ball with the center and radius . For any ball B and , λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by and set weighted measure .
The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [18]. A weight ω is said to belong to for , if there exists a constant C such that for every ball ,
where is the dual of p such that . The class is defined by replacing the above inequality with
A weight ω is said to belong to if there are positive numbers C and δ so that
for all balls B and all measurable . It is well known that
We need another weight class introduced by Muckenhoupt and Wheeden in [19]. A weight function ω belongs to for if there is a constant such that, for every ball ,
When , ω is in the class with if there is a constant such that, for every ball ,
Let us recall the definition of multiple weights. For m exponents , we write . Let , , and let . Given , set . We say that satisfies the condition if it satisfies
When , is understood as .
Let , and , let , and let . If , then
where .
Lemma 2.2 [20]
Let , and with . Assume that and . Then for any ball B, there exists a constant such that
Let , let φ be a positive measurable function on , and let ω be a nonnegative measurable function on . Following [7], we denote by the generalized weighted Morrey space and the space of all functions with finite norm
where
Furthermore, by we denote the weak generalized weighted Morrey space of all function for which
where
-
(1)
If and with , then is the classical Morrey space.
-
(2)
If , then is the weighted Morrey space.
-
(3)
If , then is the two weighted Morrey space.
-
(4)
If , then is the generalized Morrey space.
-
(5)
If , then .
Let us recall the definition and some properties of . A locally integrable function b is said to be in if
where .
Lemma 2.3 (John-Nirenberg inequality; see [21])
Let . Then for any ball , there exist positive constants and such that for all ,
By Lemma 2.3, it is easy to get the following.
Lemma 2.4 Suppose and . Then for any we have
Lemma 2.5 [22]
Let , , and . Then
where is independent of f, , , and .
By Lemma 2.4 and Lemma 2.5, it is easily to prove the following results.
Lemma 2.6 Suppose and . Then for any and , we have
We also need the following result.
Lemma 2.7 [23]
Let f be a real-valued nonnegative function and measurable on E. Then
At the end of this section, we list some known results about weighted norm inequalities for the multilinear fractional integrals and their commutators.
Lemma 2.8 [13]
Let and let . Suppose , , satisfies the condition. If , then there exists a constant C independent of such that
If , and , then there exists a constant C independent of such that
where .
Lemma 2.9 [14]
Let , let and let . For , , and , if , then there exists a constant such that
and
where .
3 Proof of Theorem 1.1
We first prove the following conclusions.
Theorem 3.1 Let and let . Suppose , , and , satisfy the condition with . If , then there exists a constant C independent of such that
If , and , then there exists a constant C independent of such that
where .
Proof We represent as , where , , and denotes the characteristic function of . Then
where each term of contains at least one . Since is an m-linear operator,
and
Then by (2.17), if , , we get
By (2.18), if , then
Applying Hölder’s inequality, for , , we have
for any ball . Then
Thus, for ,
From (2.7) and Lemma 2.2 we get
Using Hölder’s inequality,
Note that , then
Then for , ,
This gives and are majored by
For the other term, let us first consider the case when . For any , , we have for . Then
Applying Hölder’s inequality, it can be found that is less than
Hence,
Substituting (3.7) and (3.8) into the above, we obtain
Using Hölder’s inequality,
From (3.11) and (3.12) we know and are not greater than (3.10) for , .
Now we consider the case where exactly τ of the are ∞ for some . We only give the arguments for one of the cases. The rest is similar and can easily be obtained from the arguments below by permuting the indices. Then for any ,
Similar to the estimates for , we get
Then and are all less than
Combining the above estimates, the proof of Theorem 3.1 is completed. □
Now, we can give the proof of Theorem 1.1. From the definition of generalized weighted Morrey space, the norm of on equals
By Lemma 2.2 we have
Combining (3.1) and (3.16),
Since , from Lemma 2.7 and the fact are all non-decreasing functions of r, we get
Then
By (1.4) we get
Combining (3.15), (3.17), and (3.20), then
This completes the proof of first part of Theorem 1.1.
Similarly, the norm of on equals
Combining (3.2) and (3.16),
Substituting (3.20) into (3.22),
Then
This completes the proof of second part of Theorem 1.1.
4 Proof of Theorem 1.2
Theorem 4.1 Let and let . Suppose , , and , satisfy the condition with , . If , , then there exists a constant C independent of such that
and
where .
Proof We will give the proof for because the proof for is very similar but easier. Moreover, for simplicity of the expansion, we only present the case .
We represent as , where , , and denotes the characteristic function of . Then
Since bounded from to , we get
Then by (3.9) we get
Owing to the symmetry of II and , we only estimate II. Taking , then
Similar to the estimate of (3.13), for any we can deduce
By Lemma 2.1 we know . Applying Hölder’s inequality and (2.13), we have
Then by (4.6), (4.7), and (3.12), we have
For any , we have
Note that
and
Then
From Lemma 2.1 we know , then by Lemma 2.4 we get
By (3.7) and (3.8) we have
From (4.12), (4.13), and (4.14) we can deduce
Applying (2.13) and (3.12) we have
Then by (4.15) and (4.16),
Similarly, we also have
For any , with the same method of estimate for (4.15) we have
Then
Then combining (4.8), (4.17), (4.18), and (4.20) we get
Finally, we still decompose as follows:
Because each term is completely analogous to , , being slightly different, we get the following estimate without details:
Summing up the above estimates, (4.2) is proved for . □
In the following we give the proof of Theorem 1.2. From (3.16) and (4.2),
Since , , satisfy the condition (1.7), and , by (3.18) we get
Combining (4.24) and (4.25), we have
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The authors would like to thank the referees and the Editors for carefully reading the manuscript and making several useful suggestions.
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Hu, Y., Wang, Y. Multilinear fractional integral operators on generalized weighted Morrey spaces. J Inequal Appl 2014, 323 (2014). https://doi.org/10.1186/1029-242X-2014-323
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DOI: https://doi.org/10.1186/1029-242X-2014-323