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

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## 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.

## Keywords

Morrey Space Boundedness Property Doubling Condition Fractional Integral Operator Dyadic Cube## 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 Open image in new window to denote the family of all cubes in Open image in new window with sides parallel to the coordinate axes, Open image in new window , to denote the sidelength of Open image in new window and Open image in new window to denote the volume of Open image in new window . If Open image in new window , Open image in new window , then we have Open image in new window and Open image in new window .

A well-known fact in partial differential equations is that Open image in new window is an inverse of Open image in new window . The operator Open image in new window admits an expression of the form Open image in new window for some Open image in new window . 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 Open image in new window we use Open image in new window to denote the cube with the same center as Open image in new window , but with sidelength of Open image in new window . Open image in new window denotes the Lebesgue measure of Open image in new window .

(See [1].) Hereafter, we always postulate (1.4) on Open image in new window .

when Open image in new window .

In the present paper, we take up some relations between the generalized fractional integral operator Open image in new window and the generalized fractional maximal operator Open image in new window in the framework of the Morrey spaces Open image in new window (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 Open image in new window 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 Open image in new window is independent of Open image in new window and Open image in new window .

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

Theorem 1.2.

where the constant Open image in new window is independent of Open image in new window and Open image in new window .

Remark 1.3.

Hence, Theorem 1.2 generalizes Proposition 1.1.

Letting Open image in new window and Open image in new window in Theorem 1.2, we obtain the result of how Open image in new window controls Open image in new window .

Corollary 1.4.

is bounded.

Corollary 1.5.

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

Theorem 1.6.

where the constant Open image in new window is independent of Open image in new window and Open image in new window .

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

Corollary 1.7 (see [1, Remark 2.8], see also [6, 7, 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 Open image in new window is the fractional integral operator Open image in new window . The result holds by letting Open image in new window , Open image in new window and Open image in new window .

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

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

Remark 1.9.

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

The fractional integral operator Open image in new window , Open image in new window , is bounded from Open image in new window to Open image in new window if and only if the parameters Open image in new window and Open image in new window satisfy Open image in new window and Open image in new window .

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 Open image in new window , we obtain a sharp estimate on the constant Open image in new window in (1.32).

In [14], we characterized the range of Open image in new window , which motivates us to consider Proposition 1.8.

Proposition 1.10 (see [14]).

(1) Open image in new window is continuous but not surjective.

holds for Open image in new window , where Open image in new window denotes the Fourier transform.

In view of this proposition Open image in new window is not a good space to describe the boundedness of Open image in new window , although we have (1.29). As we have seen by using Hölder's inequality in Remark 1.9, if we use the space Open image in new window , 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.

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 Open image in new window and Open image in new window belong. So, we omit the proof in the present paper.

## 2. Proof of Theorems

### 2.1. Proof of Theorem 1.2

The case Open image in new window and Open image in new window 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 Open image in new window is a disjoint family of sets which decomposes Open image in new window .

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 Open image in new window . 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 Open image in new window on predual of Morrey spaces. We will define the block spaces following [17].

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

Theorem 3.1.

if Open image in new window 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 Open image in new window and Open image in new window , we obtain the following.

Corollary 3.2.

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

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

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

In Proposition 3.3, if Open image in new window is replaced by Open image in new window , 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 Open image in new window .

## Notes

### 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.

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