#### Abstract

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.

#### 1. Introduction

The aim of this paper is to investigate the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on weighted Morrey type spaces. The Hardy-Littlewood maximal function is given by where , is locally integrable function, and the description is an integral mean on : In this paper, for every cube , we assume that the sides of are parallel to the coordinate axes and describes the volume of . A weight is a nonnegative locally integrable function on that takes values in almost everywhere. Given a weight and a measurable set , we use the notation to denote the -measure of the set . We recall weighted Lebesgue spaces, Morrey spaces, weighted Morrey spaces, and multi-Morrey spaces.

*Definition 1 (weighted Lebesgue spaces). *Let and let be a weight. The weighted Lebesgue space is defined by the norm (or quasinorm)

*Definition 2 (Morrey spaces). *For , the Morrey space is defined by the norm (or quasi-norm)

*Definition 3 (weighted Morrey spaces). *Let and be weights. For , the weighted Morrey space is defined by the norm (or quasinorm)

*Definition 4 (multi-Morrey space). *Let and . The multi-Morrey space is defined by the quantity for vector valued function :

Hence it is natural to consider multiple weighted Morrey spaces.

*Definition 5 (multiple weighted Morrey spaces). *Let and . Let be a multiple weight. Let be a weight. The multiple weighted Morrey space is defined by the quantity for vector valued function :

In this framework, we investigate the well-known results. The following theorem which is elementary result was discovered by Muckenhoupt [1].

Theorem A. *Let be a weight and :
**
if and only if
**
where
*

Komori and Shirai [2] introduced the weighted Morrey spaces and proved the following theorem. The following theorem gives us the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.

Theorem B. *Let be a weight and . If , then one has
*

On the other hand, in [3], the following theorem was proved. The following result gives also the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.

Theorem C. *Let be a weight and . If
**
then one has
*

Moreover we can extend Theorem C to the multilinear version (see [3, 4]).

Theorem D. *Let and . Let be a multiple weight. If
**
Then one has
**
where and
*

Because the ways of the evaluation by the weighted Morrey norm are different, Theorems B and C are independent. Therefore, it is natural to consider unifying Theorems B and C. The question is not settled yet. In this paper, we unify Theorems B–D.

#### 2. Main Results

Theorem 6. *Let and let and be weights. Additionally assume that satisfies the doubling condition: there exists such that
**
Assume that the following condition holds:
**
Then, one has
*

*Remark 7. *If condition (19) holds, then . In fact, for every cube ,
Therefore, we obtain .

Theorem 8. *Let and be weights. Let . Additionally assume that satisfies the doubling condition. If , then one has
*

*Remark 9. *Theorem 8 includes Theorems B and C. In Theorem 8, if , then Theorem 8 corresponds with Theorem C. In Theorem 8, if , then Theorem 8 corresponds with Theorem B. In fact, condition (19) corresponds with . We state the detail:

Theorem 10. *Let , and be weights. Additionally assume that satisfies the doubling condition. Let , , and
**
Then one has
*

We can extend Theorem 10 as follows.

Theorem 11. *Let , and be weights. Additionally assume that satisfies the doubling condition. Let , , and
**
Then one has
*

By considering the multilinear version, we obtain the following theorems.

Theorem 12. *Let , , and . Let and be weights. Let be a multiple weight. Additionally assume that satisfies the doubling condition and
**
Then one has
**
where .*

We can extend Theorem 12 as follows.

Theorem 13. *Let , , , and . Let and be weights. Let be a multiple weight. Additionally assume that satisfies the doubling condition and
**
Then one has
**
where .*

#### 3. Some Lemmas

We use the following lemma (see [3, 5–9]).

Lemma 14. *Let . Let be a weight. For every cube , one has
**
where
*

Moreover we can extend Lemma 14 to the multilinear maximal function.

Lemma 15. *Let . Let be a weight. For every cube , one has
**
where
*

*Remark 16. *In [3], there is not the restriction in the maximal function . However, reexamining the argument of the proof in [3], we can remove the restriction. Hence the norm inequalities in Lemmas 14 and 15 are sharper than the norm inequalities in [3].

#### 4. Proof of Theorems

##### 4.1. Onelinear Version

We prove Theorem 11. We omit the proof of Theorems 6, 8, and 10.

*Proof of Theorem 11. *For every cube , by sublinearity, . Let and . By Lemma 14,
By Hölder’s inequality, we have
By condition (26), we have
Since , we have
Since satisfies the doubling condition, we obtain the following inequality; for every cube ,
Since , for pair of cubes , we have
Hence we have
This implies that
By the boundedness of , we obtain
Since , by the doubling condition of , we have
Taking the weighted Morrey norm, we obtain the following inequality:

Next, we estimate . By routine geometric observation, for , we have
For pair of cubes , by Hölder’s inequality, we obtain
Therefore we have
By condition (26), we have
This implies that
Therefore we obtain the desired result.

##### 4.2. Multilinear Version

We prove Theorem 13.

*Proof of Theorem 13. *For every cube , we have . Let , , and :
Firstly, we estimate . By Lemma 15, we have
where
By Hölder’s inequality, we obtain
By condition (30), we obtain
Since , we have
Since satisfies the doubling condition, we have
Since , for pair of cubes , we have
Therefore we obtain
This implies that
Since , by Hölder’s inequality, we have
By the boundedness of on , we have
Since , by the doubling condition of , we have
Taking the multiple weighted Morrey quantity, we have

Next, we estimate . By routine geometric observation, for , we have
By Hölder’s inequality, we have
Taking the multiple weighted Morrey quantity, we obtain
This implies that
By condition (28), we obtain
Therefore we obtain the desired result.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.