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Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces
Journal of Inequalities and Applications volume 2014, Article number: 258 (2014)
Abstract
In this paper, we will obtain the strong type and weak type estimates for vector-valued analogs of intrinsic square functions in the generalized weighted Morrey spaces . We study the boundedness of intrinsic square functions including the Lusin area integral, the Littlewood-Paley g-function and -function, and their multilinear commutators on vector-valued generalized weighted Morrey spaces . In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on without assuming any monotonicity property of on r.
MSC:42B25, 42B35.
1 Introduction
It is well known that the commutator is an important integral operator and it plays a key role in harmonic analysis. In 1965, Calderon [1, 2] studied a kind of commutators, appearing in Cauchy integral problems of Lip-line. Let K be a Calderón-Zygmund singular integral operator and . A well-known result of Coifman et al. [3] states that the commutator operator is bounded on for . The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order (see, for example, [4–8]).
The classical Morrey spaces were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [7–10]. Recently, Komori and Shirai [11] first defined the weighted Morrey spaces and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [12, 13] introduced the generalized weighted Morrey spaces and studied the boundedness of the sublinear operators and their higher order commutators generated by Calderón-Zygmund operators and Riesz potentials in these spaces (see, also [14–16]).
The intrinsic square functions were first introduced by Wilson in [17, 18]. They are defined as follows. For , let be the family of functions such that ϕ’s support is contained in , , and for ,
For and , set
where . Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula
where . Denote .
This function is independent of any particular kernel, such as Poisson kernel. It dominates pointwise the classical square function (Lusin area integral) and its real-variable generalizations. Although the function depends on kernels with uniform compact support, there is pointwise relation between with different β:
We can see details in [17].
The intrinsic Littlewood-Paley g-function and the intrinsic function are defined, respectively, by
When we say that f maps into , we mean that , where each is Lebesgue measurable and, for almost every
Let be a sequence of locally integrable functions on . For any , Wilson [18] also defined the vector-valued intrinsic square functions of by and proved the following result.
Theorem A Let , , and . Then the operators and are bounded from into itself for and from to .
Moreover, in [19], Lerner showed sharp norm inequalities for the intrinsic square functions in terms of the characteristic constant of w for all . Also Huang and Liu [20] studied the boundedness of intrinsic square functions on weighted Hardy spaces. Moreover, they characterized the weighted Hardy spaces by intrinsic square functions. In [21] and [22], Wang and Liu obtained some weak type estimates on weighted Hardy spaces. In [23], Wang considered intrinsic functions and the commutators generated with BMO functions on weighted Morrey spaces. Let and , be locally integrable function on . Setting
the multilinear commutators are defined by
and
In [23], Wang proved the following result.
Theorem B Let , , , and . Then the commutator operators and are bounded from into itself.
Analogously the following result may be proved.
Theorem B′ Let , , . Let also and , . Then the multilinear commutator operators and are bounded from into itself.
In this paper, we will consider the boundedness of the operators , , and their multilinear commutators on vector-valued generalized weighted Morrey spaces. Let be a positive measurable function on and w be non-negative measurable function on . For any , we denote by the vector-valued generalized weighted Morrey spaces, if
When , then coincide the vector-valued generalized Morrey spaces . There are many papers discussed the conditions on to obtain the boundedness of operators on the generalized Morrey spaces. For example, in [24] (see, also [10]), by Guliyev the following condition was imposed on the pair :
where does not depend on x and r. Under the above condition, they obtained the boundedness of Calderón-Zygmund singular integral operators from to . Also, in [25] and [26], Guliyev et al. introduced a weaker condition: If , there exists a constant , such that, for any and ,
If the pair satisfies condition (1.1), then satisfied condition (1.2). But the opposite is not true. We can see Remark 4.7 in [26] for details.
Recently, in [12, 13] (see, also [14–16]), Guliyev introduced a weighted condition: If , there exists a constant , such that, for any and ,
In this paper, we will obtain the boundedness of the vector-valued intrinsic function, the intrinsic Littlewood-Paley g function, the intrinsic function and their multilinear commutators on vector-valued generalized weighted Morrey spaces when and the pair satisfies condition (1.3) or the following inequalities:
where C does not depend on x and r. Our main results in this paper are stated as follows.
Theorem 1.1 Let , , , and satisfy condition (1.3). Then the operator is bounded from to for and from to .
Theorem 1.2 Let , , , , and satisfy condition (1.3). Then the operator is bounded from to for and from to .
Theorem 1.3 Let , , , and satisfy condition (1.4). Let also and , . Then is bounded from to .
Theorem 1.4 Let , , , and satisfy condition (1.4). Let also and , . Then for , is bounded from to .
In [17], the author proved that the functions and are pointwise comparable. Thus, as a consequence of Theorem 1.1 and Theorem 1.3, we have the following results.
Corollary 1.5 Let , , , and satisfy condition (1.3), then is bounded from to for and from to .
Corollary 1.6 Let , , , and satisfy condition (1.4). Let also and , . Then is bounded from to .
Remark 1.7 Note that, in the scalar valued case and for , Theorems 1.1-1.4 and Corollaries 1.5-1.6 was proved in [27]. Also, in the scalar valued case and , , and , Theorems 1.1-1.4 and Corollaries 1.5-1.6 were proved by Wang in [23, 28]. If , then the vector-valued generalized weighed Morrey space coincides with the vector-valued weighed Morrey space and the pair satisfies the two conditions (1.3) and (1.4). Indeed, by Lemma 3.1 there exist and such that for all and :
Then
Throughout this paper, we use the notation to express that there is a positive constant C independent of all essential variables such that . Moreover, C may be different from place to place.
2 Vector-valued generalized weighted Morrey spaces
The classical Morrey spaces were originally introduced by Morrey in [9] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [29, 30].
We denote by the vector-valued Morrey space, the space of all vector-valued functions with finite quasinorm
where and .
Note that and . If or , then , where Θ is the set of all vector-valued functions equivalent to 0 on .
We define the vector-valued generalized weighed Morrey spaces as follows.
Definition 2.1 Let , φ be a positive measurable vector-valued function on and w be non-negative measurable function on . We denote by the vector-valued generalized weighted Morrey space, the space of all vector-valued functions with finite norm
where denotes the vector-valued weighted -space of measurable functions f for which
Furthermore, by we denote the vector-valued weak generalized weighted Morrey space of all functions for which
where denotes the weak -space of measurable functions f for which
Remark 2.2
-
(1)
If , then is the vector-valued generalized Morrey space.
-
(2)
If , then is the vector-valued weighted Morrey space.
-
(3)
If , then is the vector-valued two weighted Morrey space.
-
(4)
If and with , then is the vector-valued Morrey space and is the vector-valued weak Morrey space.
-
(5)
If , then is the vector-valued weighted Lebesgue space.
3 Preliminaries and some lemmas
By a weight function, briefly weight, we mean a locally integrable function on which takes values in almost everywhere. For a weight w and a measurable set E, we define , and denote the Lebesgue measure of E by and the characteristic function of E by . Given a weight w, we say that w satisfies the doubling condition if there exists a constant such that for any ball B, we have . When w satisfies this condition, we write for brevity .
If w is a weight function, we denote by the vector-valued weighted Lebesgue space defined by finiteness of the norm
and by if .
We recall that a weight function w is in the Muckenhoupt class [31], , if
where the sup is taken with respect to all the balls B and . Note that, for all balls B, by Hölder’s inequality
For , the class is defined by the condition with , and for , and .
Lemma 3.1 ([32])
-
(1)
If for some , then . Moreover, for all
-
(2)
If , then . Moreover, for all
-
(3)
If for some , then there exist and such that for any ball B and a measurable set ,
We are going to use the following result on the boundedness of the Hardy operator:
where μ is a non-negative Borel measure on .
Theorem 3.2 ([33])
The inequality
holds for all functions g non-negative and non-increasing on if and only if
and .
We also need the following statement on the boundedness of the Hardy type operator:
where μ is a non-negative Borel measure on .
Theorem 3.3 The inequality
holds for all functions g non-negative and non-increasing on if and only if
and .
Note that Theorem 3.3 can be proved analogously to Theorem 4.3 in [34].
Definition 3.4 is the Banach space modulo constants with the norm defined by
where and
Lemma 3.5 ([35], Theorem 5, p.236)
Let . Then the norm is equivalent to the norm
where
Remark 3.6 (1) The John-Nirenberg inequality: there are constants , such that for all and
(2) For the John-Nirenberg inequality implies that
and for and
Note that by the John-Nirenberg inequality and Lemma 3.1 (part 3) it follows that
for some . Hence
where depends only on , , p, and δ, which implies (3.2).
Also (3.1) is a particular case of (3.2) with .
The following lemma was proved in [13].
Lemma 3.7 (i) Let and . Let also , , , and . Then
where is independent of f, w, x, , and .
(ii) Let and . Let also , , , and . Then
where is independent of f, w, x, , and .
4 Proofs of main theorems
Before proving the main theorems, we need the following lemmas.
Lemma 4.1 [23]
For , denote
Let , , and . Then any , we have
This lemma is easy by the following inequality, which is proved in [17]:
By a similar argument to [2], we can get the following lemma.
Lemma 4.2 Let , , and , then the multilinear commutator is bounded from to itself whenever and , .
Now we are in a position to prove the theorems.
Lemma 4.3 Let , , and .
Then, for , the inequality
holds for any ball and for all .
Moreover, for the inequality
holds for any ball and for all .
Proof The main ideas of these proofs come from [13]. For arbitrary , set , . We decompose , where , . Then
First, let us estimate I. By Theorem A, we obtain
On the other hand,
Therefore from (4.1) and (4.2) we get
Then let us estimate II:
Since , , we have , and
So, we obtain
By Minkowski’s and Hölder’s inequalities and , we have
Thus,
By combining (4.3) and (4.4), we have
□
Proof of Theorem 1.1 By Lemma 4.3 and Theorem 3.2 we have for
and for
□
Lemma 4.4 Let , , , and . Then, for , the inequality
holds for any ball and for all .
Moreover, for the inequality
holds for any ball and for all .
Proof From the definition of , we readily see that
First, let us estimate III:
Now, let us estimate IV:
Thus,
By Lemma 4.3, we have
In the following, we will estimate . We divide into two parts,
where , . For the first part, by Lemma 4.1,
For the second part,
Since , we get
For , so by Fubini’s theorem and Hölder’s inequality, we obtain
So,
Combining (4.7), (4.8), and (4.9), we have
Thus,
Since , by (4.6), (4.10), and (4.11), we have the desired lemma. □
Proof of Theorem 1.2 From inequality (4.5) we have
By Theorem 1.1, we have
In the following, we will estimate . Thus, by substitution of variables and Theorem 3.2, we get
Since , by (4.12), (4.13), and (4.14), we have the desired theorem. □
Lemma 4.5 Let , , , , and , . Then the inequality
holds for any ball and for all .
Proof We decompose , where and . Then
Denote by . By Lemma 4.2, we have
For the term , without loss of generality, we can assume . Thus, the operator can be divided into four parts:
For we have
Then
Let us estimate :
Applying Hölder’s inequality and by Lemma 3.7, we get
Let us estimate :
Applying Hölder’s inequality and by Lemma 3.7, we get
In the same way, we shall get the result of :
In order to estimate note that
By Lemma 3.7, we get
Applying Hölder’s inequality, we get
Thus by (4.16)
Summing up and , for all we get
Finally, from (4.2), (4.15), and (4.17) we get
□
Proof of Theorem 1.3 By substitution of variables, we obtain
By using an argument similar to the above proofs and that of Theorem 1.2, we can also show the boundedness of . □
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Acknowledgements
The research of V Guliyev was partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4003-2.13.007). We thank both referees for some good suggestions, which helped to improve the final version of this paper.
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This work was carried out in collaboration between all authors. VSG raised these interesting problems in the research. VSG and MNO proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, and read and approved the manuscript.
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Guliyev, V.S., Omarova, M. Multilinear commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces. J Inequal Appl 2014, 258 (2014). https://doi.org/10.1186/1029-242X-2014-258
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DOI: https://doi.org/10.1186/1029-242X-2014-258