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.