- Research Article
- Open Access
A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces
© Yoshihiro Sawano et al. 2009
- Received: 21 July 2009
- Accepted: 13 December 2009
- Published: 17 January 2010
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.
- Morrey Space
- Boundedness Property
- Doubling Condition
- Fractional Integral Operator
- Dyadic Cube
The present paper is an offspring of . 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 . We will go through the same argument as .
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 . 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 .) Hereafter, we always postulate (1.4) on .
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 .
Proposition 1.1 (see [1, Theorem 1.3]).
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.
Hence, Theorem 1.2 generalizes Proposition 1.1.
Proposition 1.8 (see [1, Proposition 1.7]).
The special case and in Proposition 1.8 corresponds to the classical theorem due to Adams (see ).
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 ). By letting , we obtain a sharp estimate on the constant in (1.32).
In , we characterized the range of , which motivates us to consider Proposition 1.8.
Proposition 1.10 (see ).
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 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.
2.1. Proof of Theorem 1.2
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]).
The desired inequality then follows.
Proof of Theorem 1.6.
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 .
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 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.
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 .
The third author is supported by the Global COE program at Graduate School of Mathematical Sciences, University of Tokyo, and was supported by Fū jyukai foundation.
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