# A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

- Yoshihiro Sawano
^{1}Email author, - Satoko Sugano
^{2}and - Hitoshi Tanaka
^{3}

**2009**:835865

**DOI: **10.1155/2009/835865

© Yoshihiro Sawano et al. 2009

**Received: **21 July 2009

**Accepted: **13 December 2009

**Published: **17 January 2010

## Abstract

We show some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces.

## 1. Introduction

The present paper is an offspring of [1]. We obtain some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces. They generalize what was shown in [1]. We will go through the same argument as [1].

Here, we use the notation to denote the family of all cubes in with sides parallel to the coordinate axes, , to denote the sidelength of and to denote the volume of . If , , then we have and .

A well-known fact in partial differential equations is that is an inverse of . The operator admits an expression of the form for some . For more details of this operator we refer to [2]. As we will see, these operators will fall under the scope of our main results.

Among other function spaces, it seems that the Morrey spaces reflect the boundedness properties of the fractional integral operators. To describe the Morrey spaces we recall some definitions and notation. All cubes are assumed to have their sides parallel to the coordinate axes. For we use to denote the cube with the same center as , but with sidelength of . denotes the Lebesgue measure of .

(See [1].) Hereafter, we always postulate (1.4) on .

when .

In the present paper, we take up some relations between the generalized fractional integral operator and the generalized fractional maximal operator in the framework of the Morrey spaces (Theorem 1.2). In the last section, we prove a dual version of Olsen's inequality on predual of Morrey spaces (Theorem 3.1). As a corollary (Corollary 3.2), we have the boundedness properties of the operator on predual of Morrey spaces.

Before we formulate our main results, we recall a typical result obtained in [1].

Proposition 1.1 (see [1, Theorem 1.3]).

where the constant is independent of and .

The aim of the present paper is to generalize the function spaces to which and belong. With theorem 1.2, which we will present just below, we can replace with and with . We now formulate our main theorems. In the sequel we always assume that satisfies (1.6) and (1.7), and is used to denote various positive constants.

Theorem 1.2.

where the constant is independent of and .

Remark 1.3.

Hence, Theorem 1.2 generalizes Proposition 1.1.

Letting and in Theorem 1.2, we obtain the result of how controls .

Corollary 1.4.

is bounded.

Corollary 1.5.

We will establish that is bounded on when (Lemma 2.2). Therefore, the second assertion is immediate from the first one.

Theorem 1.6.

where the constant is independent of and .

Theorem 1.6 extends [4, Theorem 2], [1, Theorem 1.1], and [5, Theorem 1]. As the special case and in Theorem 1.6 shows, this theorem covers [1, Remark 2.8].

Corollary 1.7 (see [1, Remark 2.8], see also [6–8]).

Nakai generalized Corollary 1.7 to the Orlicz-Morrey spaces ([9, Theorem 2.2] and [10, Theorem 7.1]).

We dare restate Theorem 1.6 in the special case when is the fractional integral operator . The result holds by letting , and .

Proposition 1.8 (see [1, Proposition 1.7]).

where the constant is independent of and .

Proposition 1.8 extends [4, Theorem 2] (see [1, Remark 1.9]).

Remark 1.9.

The special case and in Proposition 1.8 corresponds to the classical theorem due to Adams (see [11]).

The fractional integral operator , , is bounded from to if and only if the parameters and satisfy and .

is called the trace inequality and is useful in the analysis of the Schrödinger operators. For example, Kerman and Sawyer utilized an inequality of type (1.32) to obtain an eigenvalue estimates of the operators (see [13]). By letting , we obtain a sharp estimate on the constant in (1.32).

In [14], we characterized the range of , which motivates us to consider Proposition 1.8.

Proposition 1.10 (see [14]).

(1) is continuous but not surjective.

holds for , where denotes the Fourier transform.

In view of this proposition is not a good space to describe the boundedness of , although we have (1.29). As we have seen by using Hölder's inequality in Remark 1.9, if we use the space , then we will obtain a result weaker than Proposition 1.8.

Finally it would be interesting to compare Theorem 1.2 with the following Theorem 1.11.

Theorem 1.11.

where the constant is independent of and .

Theorem 1.11 generalizes [1, Theorem 1.7] and the proof remains unchanged except some minor modifications caused by our generalization of the function spaces to which and belong. So, we omit the proof in the present paper.

## 2. Proof of Theorems

### 2.1. Proof of Theorem 1.2

The case and We need the following crucial lemma, the proof of which is straightforward and is omitted (see [15, 16]).

Lemma 2.1.

Summing up all factors, we obtain (2.14), by noticing that is a disjoint family of sets which decomposes .

This is our desired inequality.

This is our desired inequality.

### 2.2. Proof of Theorem 1.6

We need some lemmas.

Lemma 2.2 (see [1, Lemma 2.2]).

Lemma 2.3.

Proof.

The desired inequality then follows.

Proof of Theorem 1.6.

We note that the assumption (1.24) implies . Hence we arrive at the desired inequality by using Lemma 2.3.

## 3. A Dual Version of Olsen's Inequality

In this section, as an application of Theorem 1.6, we consider a dual version of Olsen's inequality on predual of Morrey spaces (Theorem 3.1). As a corollary (Corollary 3.2), we have the boundedness properties of the operator on predual of Morrey spaces. We will define the block spaces following [17].

when . In [17, Theorem 1] and [18, Proposition 5], it was established that the predual space of is . More precisely, if , then is an element of . Conversely, any continuous linear functional in can be realized with some .

Theorem 3.1.

if is a continuous function.

Theorem 3.1 generalizes [1, Theorem 3.1], and its proof is similar to that theorem, hence omitted. As a special case when and , we obtain the following.

Corollary 3.2.

We dare restate Corollary 3.2 in terms of the fractional integral operator . The results hold by letting , , and .

Proposition 3.3 (see [1, Proposition 3.8]).

Remark 3.4 (see [1, Remark 3.9]).

In Proposition 3.3, if is replaced by , then, using the Hardy-Littlewood-Sobolev inequality locally and taking care of the larger scales by the same manner as the proof of Theorem 3.1, one has a naive bound for .

## Declarations

### Acknowledgments

The third author is supported by the Global COE program at Graduate School of Mathematical Sciences, University of Tokyo, and was supported by F*ū* jyukai foundation.

## Authors’ Affiliations

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