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Open Access Research Article

A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Yoshihiro Sawano1*, Satoko Sugano2 and Hitoshi Tanaka3

Author Affiliations

1 Department of Mathematics, Kyoto University, Kitasir-akawa, Sakyoku, Kyoto 606-8502, Japan

2 Kobe City College of Technology, 8-3 Gakuen-higashimachi, Nishi-ku, Kobe 651-2194, Japan

3 Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

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Boundary Value Problems 2009, 2009:835865  doi:10.1155/2009/835865

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2009/1/835865


Received:21 July 2009
Revisions received:31 August 2009
Accepted:13 December 2009
Published:17 January 2010

© 2009 The Author(s)

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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].

For the classical fractional integral operator and the classical fractional maximal operator are given by

(11)

In the present paper, we generalize the parameter . Let be a suitable function. We define the generalized fractional integral operator and the generalized fractional maximal operator by

(12)

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 .

Let and be a suitable function. For a function locally in we set

(13)

We will call the Morrey space the subset of all functions locally in for which is finite. Applying Hölder's inequality to (1.3), we see that provided that . This tells us that when . We remark that without the loss of generality we may assume

(14)

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

If , , coincides with the usual Morrey space and we write this for and the norm for . Then we have the inclusion

(15)

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.

Let be a function. By the Dini condition we mean that fulfills

(16)

while the doubling condition on (with a doubling constant ) is that satisfies

(17)

We notice that (1.4) is stronger than the doubling condition. More quantitatively, if we assume (1.4), then satisfies the doubling condition with the doubling constant . A simple consequence that can be deduced from the doubling condition of is that

(18)

The key observation made in [1] is that it is frequently convenient to replace satisfying (1.6) and (1.7) by :

(19)

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

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

Let

(110)

and . Suppose that is nonincreasing. Then

(111)

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.

Let

(112)

Suppose that and are nondecreasing but that and are nonincreasing. Assume also that

(113)

then

(114)

where the constant is independent of and .

Remark 1.3.

Let and . Then and satisfy the assumption (1.13). Indeed,

(115)

Hence, Theorem 1.2 generalizes Proposition 1.1.

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

Corollary 1.4.

Let . Suppose that

(116)

then

(117)

Corollary 1.4 generalizes [3, Theorem  4.2]. Letting in Theorem 1.2, we also obtain the condition on and under which the mapping

(118)

is bounded.

Corollary 1.5.

Let

(119)

Suppose that

(120)

then

(121)

In particular, if , then

(122)

Here, denotes the Hardy-Littlewood maximal operator defined by

(123)

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

Theorem 1.6.

Let . Suppose that and are nondecreasing but that and are nonincreasing. Suppose also that

(124)

then

(125)

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 [68]).

Let . Suppose that

(126)

then

(127)

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]).

Let , , , and . Suppose that , , , , and then

(128)

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 .

Using naively the Adams theorem and Hölder's inequality, one can prove a minor part of in Proposition 1.8. That is, the proof of Proposition 1.8 is fundamental provided Indeed, by virtue of the Adams theorem we have, for any cube ,

(129)

The condition , reads

(130)

These yield

(131)

if . In view of inclusion (1.5), the same can be said when . Also observe that Hence we have . Thus, since the condition , Proposition 1.8 is significant only when The case (the case of the Lebesgue spaces) corresponds (so-called) to the Fefferan-Phong inequality (see [12]). An inequality of the form

(132)

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]).

Let , , and . Assume that

(133)

(1) is continuous but not surjective.

(2)Let be an auxiliary function chosen so that , and that , , . Then the norm equivalence

(134)

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.

Let . Suppose that , , and are nondecreasing and that and are nonincreasing. Then

(135)

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

For any we will write for the conjugate number defined by . Hereafter, for the sake of simplicity, for any and we will write

(21)

2.1. Proof of Theorem 1.2

First, we will prove Theorem 1.2. Except for some sufficient modifications, the proof of the theorem follows the argument in [15]. We denote by the family of all dyadic cubes in . We assume that and are nonnegative, which may be done without any loss of generality thanks to the positivity of the integral kernel. We will denote by the ball centered at and of radius . We begin by discretizing the operator following the idea of Pérez (see [16]):

(22)

where we have used the doubling condition of for the first inequality. To prove Theorem 1.2, thanks to the doubling condition of , which holds by use of the facts that is nondecreasing and that is nonincreasing, it suffices to show

(23)

for all dyadic cubes . Hereafter, we let

(24)

Let us define for

(25)

and we will estimate

(26)

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

Lemma 2.1.

For a nonnegative function in one lets and . For let

(27)

Considering the maximal cubes with respect to inclusion, one can write

(28)

where the cubes are nonoverlapping. By virtue of the maximality of one has that

(29)

Let

(210)

Then is a disjoint family of sets which decomposes and satisfies

(211)

Also, one sets

(212)

Then

(213)

With Lemma 2.1 in mind, let us return to the proof of Theorem 1.2. We need only to verify that

(214)

Inserting the definition of , we have

(215)

Letting , we will apply Lemma 2.1 to estimate this quantity. Retaining the same notation as Lemma 2.1 and noticing (2.13), we have

(216)

We first evaluate

(217)

It follows from the definition of that (2.17) is bounded by

(218)

By virtue of the support condition and (1.8) we have

(219)

If we invoke relations and , then (2.17) is bounded by

(220)

Now that we have from the definition of the Morrey norm

(221)

we conclude that

(222)

Here, we have used the fact that is nondecreasing, that satisfies the doubling condition and that

(223)

Similarly, we have

(224)

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

The case and In this case we establish

(225)

by the duality argument. Take a nonnegative function , , satisfying that and that

(226)

Letting , we will apply Lemma 2.1 to estimation of this quantity. First, we will insert the definition of ,

(227)

First, we evaluate

(228)

Going through the same argument as the above, we see that (2.28) is bounded by

(229)

Using Hölder's inequality, we have

(230)

These yield

(231)

Similarly, we have

(232)

Summing up all factors we obtain

(233)

Another application of Hölder's inequality gives us that

(234)

Now that , the maximal operator is -bounded. As a result we have

(235)

This is our desired inequality.

The case and By a property of the dyadic cubes, for all we have

(236)

As a consequence we obtain

(237)

In view of the definition of , for each with there exists a unique cube in whose length is . Hence, inserting these estimates, we obtain

(238)

Here, in the last inequality we have used the doubling condition (1.8) and the facts that , , and are nondecreasing and that and satisfy the doubling condition. Thus, we obtain

(239)

for all . Inserting this pointwise estimate, we obtain

(240)

This is our desired inequality.

2.2. Proof of Theorem 1.6

We need some lemmas.

Lemma 2.2 (see [1, Lemma  2.2]).

Let . Suppose that satisfies (1.4), then

(241)

Lemma 2.3.

Let . Suppose that satisfies (1.4), then

(242)

Proof.

Let be a fixed point. For every cube we see that

(243)

This implies

(244)

It follows from Lemma 2.2 that for every cube

(245)

The desired inequality then follows.

Proof of Theorem 1.6.

We use definition (2.5) again and will estimate

(246)

for .The case In the course of the proof of Theorem 1.2, we have established (2.25)

(247)

We will use it with

(248)

The case It follows that

(249)

from the Hölder inequality and the definition of the norm . As a consequence we have

(250)

Here, we have used the doubling condition (1.8) and the fact that is nondecreasing in the third inequality. Hence it follows that

(251)

Combining (2.48) and (2.51), we obtain

(252)

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].

Let and . Suppose that satisfies (1.4). We say that a function on is a -block provided that is supported on a cube and satisfies

(31)

The space is defined by the set of all functions locally in with the norm

(32)

where each is a -block and , and the infimum is taken over all possible decompositions of . If , , is the usual block spaces, which we write for and the norm for , because the right-hand side of (3.1) is equal to . It is easy to prove

(33)

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.

Let . Suppose that and are nondecreasing but that and are nonincreasing. Suppose also that

(34)

then

(35)

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.

Let . Suppose that is nondecreasing but that is nonincreasing. Suppose also that

(36)

then

(37)

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]).

Let , , and . Suppose that , , and , then

(38)

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 .

Acknowledgments

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

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