Abstract
We give a characterization of weighted Morrey space by using Fefferman and Stein’s sharp maximal function. For this purpose, we consider a local good- inequality.
1. Introduction
The Fefferman-Stein sharp maximal function (see Definition 2) plays an important role in harmonic analysis. For example, we know , where . We also know , where is the weighted space (for further details, see [1, 2]).
On the other hand, many studies have been done for Morrey spaces: Komori and Shirai [3] introduced weighted Morrey spaces (for the precise definition, see Section 2) and proved the boundedness of the Hardy-Littlewood maximal operator and singular integral operators.
Di Fazio and Ragusa [4] proved From this inequality they obtained However, the condition is strong; that is, the left side of inequality (2) is assumed to be finite. In applications (3) is more important than (2). In this paper, we will prove (3) under weaker condition without using (2). Our proof is different from the one in [4]. We use another method, that is, local good- estimate. We think this itself is interesting (see Section 4, Lemma 16), and we also consider weighted estimates.
This paper is organized as follows. In Section 2 we make some definitions for maximal functions, functions spaces, and weights. In Section 3 we state known results and our theorem. In Section 4 we prove our theorems.
2. Preliminaries
The following notation is used: for a set , we denote the Lebesgue measure of by , and for a nonnegative locally integrable function , we write . We denote the characteristic function of by . Throughout this paper, all cubes are assumed to have their sides parallel to the coordinate axes.
2.1. Maximal Functions
First we define some maximal functions.
Definition 1 (the Hardy-Littlewood maximal function). Considerwhere the supremum is taken over all cubes containing .
Definition 2 (the sharp maximal function). Considerwhere .
Next we define the dyadic maximal function. A dyadic cube is a cube of the form
Definition 3 (the dyadic maximal function). Considerwhere the supremum is taken over all dyadic cubes containing .
2.2. Weights
Next we define Muckenhoupt’s weight classes (see [1, 2]).
Definition 4 (). Let . For a nonnegative locally integrable function , one says that if where is a positive constant independent of cubes .
Definition 5 (). One writes
Definition 6 (doubling condition). One says that , if for any cubes, where is the cube that has the same center as with double the side length.
Definition 7 (reverse doubling condition). One says that , if there exists such that and one denotes the supremum of in (11) by .
Definition 8 (-reverse doubling condition). One says that , where , if for any cubes and subsets , and one denotes the infimum of in (12) by .
The notation in Definitions 6, 7, and 8 may not be standard. However, we use these notions frequently; therefore, we use these symbols for short.
The following proposition is well-known (see, e.g., [1, 2]).
Proposition 9. Consider the following:
2.3. Morrey Spaces
Ordinary Morrey spaces are defined as follows (see [4–6]):
Definition 10. Let and . One defines where the supremum is taken over all cubes .
Next we define weighted Morrey spaces (see [3]).
Definition 11. Let and . One defines where the supremum is taken over all cubes .
3. Theorems
The following three theorems are well-known.
Theorem A (see [1, page 161]). Let . If , then
Theorem B (see [1, page 410]). Let . One assumes that . If , then
Theorem C (see [4]). Let and . If , then
As a corollary of this theorem we know the following.
Corollary C. Let and . If , then
Our result is the following.
Theorem 12. Let and . One assumes that . If , then where is a positive constant depending only on , , , and weight .
If , then is bounded on (see [3]). Therefore, we obtain the following corollary.
Corollary 13. Let and . One assumes that . If , then
Furthermore, we obtain the following (compare with Corollary C).
Corollary 14. Let and . If , then
4. Proof of Theorem
First we note that where the supremum is taken over all dyadic cubes. Because every cube can be covered by at most dyadic cubes of comparable size, and satisfies the doubling condition.
We follow the argument in [2, page 148]. Let a dyadic cube be fixed. By (14), we may assume for some .
Here we define another maximal function.
Definition 15. One defines the local dyadic maximal function with respect to as follows: where the supremum is taken over all dyadic cubes containing such that .
Note that for almost all .
We need three lemmas.
Lemma 16. Let and . One has
Remark 17. When we consider a local good- estimate on , the condition is important. See the proof of Lemma 16.
Next we define new symbol.
Definition 18 (dyadic double). Let denote the “parent” of , that is, the dyadic cube containing with double the side length. When is a dyadic cube, one denotes by the parent of . One defines .
Lemma 19. If , then the characteristic function .
Lemma 20. If for some , then one has
Assuming these lemmas, we continue the proof of our theorem. To avoid routine limiting arguments, we assume that (see [2, page 549]).
Let . We write
First we estimate . By Lemma 16, we have
Next we estimate . We have . By (27), Therefore, we have
By (29) and (31), we have Taking sufficiently small, we obtain Since ., this proves the theorem.
5. Proofs of Lemmas
Proof of Lemma 16. Let . As in [2, page 148], for any , there is a maximal dyadic cube that contains such that Since , the maximal dyadic cube satisfies . Therefore, we can write where is a family of disjoint maximal dyadic cubes. We can show By -reverse doubling condition (Definition 8), we obtain the desired result.
Proof of Lemma 19. For any dyadic cubes , and tends to infinity when because (see Proposition 9).
Proof of Lemma 20. Since we haveby reverse doubling condition (Definition 7).
By (39), is a Cauchy sequence; therefore, converges to some constant when . If , then . But this contradicts the fact by Lemma 19. Therefore, we obtain .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.