Research Article | Open Access

# Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

**Academic Editor:**John R. Akeroyd

#### Abstract

We will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.

#### 1. Introduction and Main Results

Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. Let be the -dimensional Euclidean space, and let be the -fold product space (). We denote by the space of all Schwartz functions on and by its dual space, the set of all tempered distributions on . Let and be an -linear operator initially defined on the -fold product of Schwartz spaces, and taking values into the space of tempered distributions, Following [2], for given , we say that is an -linear Calderón-Zygmund operator if, for some and with , it extends to a bounded multilinear operator from into and if there exists a kernel function in the class -, defined away from the diagonal in such that whenever and . We say that is a kernel in the class - if it satisfies the size condition for some and all with for some . Moreover, for some , it satisfies the regularity condition that whenever and also that, for each fixed with , whenever . In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calderón-Zygmund singular integral operators were established in [9] by Lerner et al. New more refined multilinear maximal function was defined and used in [9] to characterize the class of multiple weights.

Theorem A (see [9]). *Let and be an -linear Calderón-Zygmund operator. If and with and satisfies the condition, then there exists a constant independent of such that
**
where .*

Theorem B (see [9]). *Let , and let be an -linear Calderón-Zygmund operator. If , and with , and satisfies the condition, then there exists a constant independent of such that
**
where .*

Let , and let . For given , the -linear fractional integral operator is defined by For the boundedness properties of multilinear fractional integrals on various function spaces, we refer the reader to [10–16]. In 2009, Moen [17] considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple weights (see also [18]).

Theorem C (see [17, 18]). *Let , , and let be an -linear fractional integral operator. If , and , and satisfies the condition, then there exists a constant independent of such that
**
where .*

Theorem D (see [17, 18]). *Let , , and let be an -linear fractional integral operator. If , , and , and satisfies the condition, then there exists a constant independent of such that
**
where .*

On the other hand, the classical Morrey spaces were originally introduced by Morrey in [19] to study the local behavior of solutions to second-order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [20–22]. For the properties and applications of classical Morrey spaces, one can see [23–25] and the references therein.

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces which could be viewed as an extension of weighted Lebesgue spaces and studied the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [27–34], we have established the continuity properties of some other operators and their commutators on the weighted Morrey spaces .

The main purpose of this paper is to establish the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. We now formulate our main results as follows.

Theorem 1. *Let , and let be an -linear Calderón-Zygmund operator. If and with and with , then for any , there exists a constant independent of such that
**
where .*

Theorem 2. *Let and be an -linear Calderón-Zygmund operator. If , and with , and with , then for any , there exists a constant independent of such that
**
where .*

Theorem 3. *Let , let , and let be an -linear fractional integral operator. If , , and , and with , then for any , there exists a constant independent of such that
**
where .*

Theorem 4. *Let , let , and let be an -linear fractional integral operator. If , , , and , and with , then, for any , there exists a constant independent of such that
**
where . *

#### 2. Notations and Definitions

The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [35]. A weight is a nonnegative, locally integrable function on ; denotes the ball with the center and radius . For , a weight function is said to belong to if there is a constant such that, for every ball , where denotes the Lebesgue measure of . For the case , , if there is a constant such that for every ball , A weight function if it satisfies the condition for some . We also need another weight class introduced by Muckenhoupt and Wheeden in [36]. A weight function belongs to for if there is a constant such that, for every ball , When , is in the class with if there is a constant such that, for every ball ,

Now let us recall the definitions of multiple weights. For exponents , we will write for the vector . Let , and let with . Given , set . We say that satisfies the condition if it satisfies When , is understood as .

Let , let , and let . Given , set . We say that satisfies the condition if it satisfies When , is understood as .

Given a ball and , denotes the ball with the same center as whose radius is times that of . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and the weighted measure of by , where .

Lemma 5 (see [37]). *Let with . Then, for any ball , there exists an absolute constant such that
*

Lemma 6 (see [38]). *Let . Then for all balls , the following reverse Jensen inequality holds:
*

Lemma 7 (see [37]). *Let . Then, for all balls and all measurable subsets of , there exists such that
*

Lemma 8 (see [9]). *Let , and let . Then if and only if
**
where and the condition in the case is understood as .*

Lemma 9 (see [17, 18]). *Let , and , let , and let . If , then
**
where .*

Given a weight function on , for , the weighted Lebesgue space is defined as the set of all functions such that We also denote by the weighted weak space consisting of all measurable functions such that

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces for . In order to deal with the multilinear case , we will define for all .

*Definition 10. *Let , let , and let be a weight function on . Then the weighted Morrey space is defined by
where
and the supremum is taken over all balls in .

*Definition 11. *Let , let , and let be a weight function on . Then the weighted weak Morrey space is defined by
where

Furthermore, in order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weights.

*Definition 12. *Let and . Then for two weights and , the weighted Morrey space is defined by
where

Throughout this paper, we will use to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. Moreover, we will denote the conjugate exponent of by .

#### 3. Proofs of Theorems 1 and 2

Before proving the main theorems of this section, we need to establish the following lemma.

Lemma 13. *Let , let , and let with . Assume that and ; then, for any ball , there exists a constant such that
*

*Proof. *Since , then, by using Lemma 6, we have
Note that and . Then, by Jensen inequality, we obtain
We are done.

*Proof of Theorem 1. *For any ball and letting , where , , and denotes the characteristic function of , then we write
where each term of contains at least one . Since is an -linear operator, then we have
In view of Lemma 8, we have that . Applying Theorem A and Lemmas 13 and 5, we get
For the other terms, let us first consider the case when . By the size condition, for any , we obtain
where we have used the notation . Furthermore, by using Hölder’s inequality, the multiple condition, and Lemma 13, we deduce that
Since , then it follows directly from Lemma 7 that
Hence,
where the last inequality holds since and . We now consider the case where exactly of the are for some . We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the size condition again, we deduce that, for any ,
and we arrive at the expression considered in the previous case. So for any , we also have
Therefore, by the inequality (42) and the above pointwise inequality, we have
Combining the above estimates and then taking the supremum over all balls , we complete the proof of Theorem 1.

*Proof of Theorem 2. *For any ball and decomposing , where , , then, for any given , we can write
where each term of contains at least one . By Lemma 8 again, we know that with . Applying Theorem B and Lemmas 13 and 5, we have
In the proof of Theorem 1, we have already showed the following pointwise estimate (see (40) and (44)). Consider
Without loss of generality, we may assume that and . Using Hölder’s inequality, the multiple condition, and Lemma 13, we obtain
Observe that with . Thus, it follows from the inequality (42) that, for any ,
If , then the inequality
holds trivially. Now, if instead we suppose that , then, by the pointwise inequality (51), we have
which is equivalent to
Therefore,
Summing up all the above estimates and then taking the supremum over all balls and all , we complete the proof of Theorem 2.

By using Hölder’s inequality, it is easy to check that if each is in , then and this inclusion is strict (see [9]). Thus, as direct consequences of Theorems 1 and 2, we immediately obtain the following.

Corollary 14. *Let , and let be an -linear Calderón-Zygmund operator. If and with and , then, for any , there exists a constant independent of such that
**
where .*

Corollary 15. *Let and let be an -linear Calderón-Zygmund operator. If , and with , and , then, for any , there exists a constant independent of such that
**
where .*

#### 4. Proofs of Theorems 3 and 4

Following along the same lines as those of Lemma 13, we can also show the following result, which plays an important role in our proofs of Theorems 3 and 4.

Lemma 16. *Let , and with . Assume that and ; then, for any ball , there exists a constant such that
*

*Proof of Theorem 3. *Arguing as in the proof of Theorem 1, fix a ball and decompose , where , . Since is an -linear operator, then we have
where each term of contains at least one . In view of Lemma 9, we can see that . Using Theorem C and Lemmas 16 and 5, we get
For the other terms, let us first deal with the case when . By the definition of , for any , we obtain
Moreover, by using Hölder’s inequality, the multiple condition, and Lemma 16, we deduce that
Since , then it follows immediately from Lemma 7 that
Hence,
where in the last inequality we have used the fact that and . We now consider the case where exactly of the are for some