Advertisement

Boundary Value Problems

, 2009:835865 | Cite as

A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 Open image in new window the classical fractional integral operator Open image in new window and the classical fractional maximal operator Open image in new window are given by
In the present paper, we generalize the parameter Open image in new window . Let Open image in new window be a suitable function. We define the generalized fractional integral operator Open image in new window and the generalized fractional maximal operator Open image in new window by

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 .

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

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

If Open image in new window , Open image in new window , Open image in new window coincides with the usual Morrey space and we write this for Open image in new window and the norm for Open image in new window . Then we have the inclusion

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.

Let Open image in new window be a function. By the Dini condition we mean that Open image in new window fulfills
while the doubling condition on Open image in new window (with a doubling constant Open image in new window ) is that Open image in new window satisfies
We notice that (1.4) is stronger than the doubling condition. More quantitatively, if we assume (1.4), then Open image in new window satisfies the doubling condition with the doubling constant Open image in new window . A simple consequence that can be deduced from the doubling condition of Open image in new window is that
The key observation made in [1] is that it is frequently convenient to replace Open image in new window satisfying (1.6) and (1.7) by Open image in new window :

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.

Suppose that Open image in new window and Open image in new window are nondecreasing but that Open image in new window and Open image in new window are nonincreasing. Assume also that

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.

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

is bounded.

Corollary 1.5.

Suppose that
In particular, if Open image in new window , then
Here, Open image in new window denotes the Hardy-Littlewood maximal operator defined by

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.

Let Open image in new window . Suppose that Open image in new window and Open image in new window are nondecreasing but that Open image in new window and Open image in new window are nonincreasing. Suppose also that

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]).

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

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 .

Using naively the Adams theorem and Hölder's inequality, one can prove a minor part of Open image in new window in Proposition 1.8. That is, the proof of Proposition 1.8 is fundamental provided Open image in new window Indeed, by virtue of the Adams theorem we have, for any cube Open image in new window ,
These yield
if Open image in new window . In view of inclusion (1.5), the same can be said when Open image in new window . Also observe that Open image in new window Hence we have Open image in new window . Thus, since the condition Open image in new window , Proposition 1.8 is significant only when Open image in new window The case Open image in new window (the case of the Lebesgue spaces) corresponds (so-called) to the Fefferan-Phong inequality (see [12]). An inequality of the form

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.

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

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

For any Open image in new window we will write Open image in new window for the conjugate number defined by Open image in new window . Hereafter, for the sake of simplicity, for any Open image in new window and Open image in new window we will write

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 Open image in new window the family of all dyadic cubes in Open image in new window . We assume that Open image in new window and Open image in new window are nonnegative, which may be done without any loss of generality thanks to the positivity of the integral kernel. We will denote by Open image in new window the ball centered at Open image in new window and of radius Open image in new window . We begin by discretizing the operator Open image in new window following the idea of Pérez (see [16]):
where we have used the doubling condition of Open image in new window for the first inequality. To prove Theorem 1.2, thanks to the doubling condition of Open image in new window , which holds by use of the facts that Open image in new window is nondecreasing and that Open image in new window is nonincreasing, it suffices to show
for all dyadic cubes Open image in new window . Hereafter, we let
and we will estimate

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.

Considering the maximal cubes with respect to inclusion, one can write
where the cubes Open image in new window are nonoverlapping. By virtue of the maximality of Open image in new window one has that
Then Open image in new window is a disjoint family of sets which decomposes Open image in new window and satisfies
Also, one sets
With Lemma 2.1 in mind, let us return to the proof of Theorem 1.2. We need only to verify that
Inserting the definition of Open image in new window , we have
Letting Open image in new window , we will apply Lemma 2.1 to estimate this quantity. Retaining the same notation as Lemma 2.1 and noticing (2.13), we have
We first evaluate
It follows from the definition of Open image in new window that (2.17) is bounded by
By virtue of the support condition and (1.8) we have
If we invoke relations Open image in new window and Open image in new window , then (2.17) is bounded by
Now that we have from the definition of the Morrey norm
we conclude that
Here, we have used the fact that Open image in new window is nondecreasing, that Open image in new window satisfies the doubling condition and that
Similarly, we have

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 .

The case Open image in new window and Open image in new window In this case we establish
by the duality argument. Take a nonnegative function Open image in new window , Open image in new window , satisfying that Open image in new window and that
Letting Open image in new window , we will apply Lemma 2.1 to estimation of this quantity. First, we will insert the definition of Open image in new window ,
First, we evaluate
Going through the same argument as the above, we see that (2.28) is bounded by
Using Hölder's inequality, we have
These yield
Similarly, we have
Summing up all factors we obtain
Another application of Hölder's inequality gives us that
Now that Open image in new window , the maximal operator Open image in new window is Open image in new window -bounded. As a result we have

This is our desired inequality.

The case Open image in new window and Open image in new window By a property of the dyadic cubes, for all Open image in new window we have
As a consequence we obtain
In view of the definition of Open image in new window , for each Open image in new window with Open image in new window there exists a unique cube in Open image in new window whose length is Open image in new window . Hence, inserting these estimates, we obtain
Here, in the last inequality we have used the doubling condition (1.8) and the facts that Open image in new window , Open image in new window , and Open image in new window are nondecreasing and that Open image in new window and Open image in new window satisfy the doubling condition. Thus, we obtain
for all Open image in new window . Inserting this pointwise estimate, we obtain

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 Open image in new window . Suppose that Open image in new window satisfies (1.4), then

Lemma 2.3.

Let Open image in new window . Suppose that Open image in new window satisfies (1.4), then

Proof.

Let Open image in new window be a fixed point. For every cube Open image in new window we see that
This implies
It follows from Lemma 2.2 that for every cube Open image in new window

The desired inequality then follows.

Proof of Theorem 1.6.

We use definition (2.5) again and will estimate
for Open image in new window .The case Open image in new window In the course of the proof of Theorem 1.2, we have established (2.25)
The case Open image in new window It follows that
from the Hölder inequality and the definition of the norm Open image in new window . As a consequence we have
Here, we have used the doubling condition (1.8) and the fact that Open image in new window is nondecreasing in the third inequality. Hence it follows that
Combining (2.48) and (2.51), we obtain

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].

The space Open image in new window is defined by the set of all functions Open image in new window locally in Open image in new window with the norm
where each Open image in new window is a Open image in new window -block and Open image in new window , and the infimum is taken over all possible decompositions of Open image in new window . If Open image in new window , Open image in new window , Open image in new window is the usual block spaces, which we write for Open image in new window and the norm for Open image in new window , because the right-hand side of (3.1) is equal to Open image in new window . It is easy to prove

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.

Let Open image in new window . Suppose that Open image in new window and Open image in new window are nondecreasing but that Open image in new window and Open image in new window are nonincreasing. Suppose also that

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.

Let Open image in new window . Suppose that Open image in new window is nondecreasing but that Open image in new window is nonincreasing. Suppose also that

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 Fū jyukai foundation.

References

  1. 1.
    Sawano Y, Sugano S, Tanaka H: Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces. to appear in Trans. Amer. Math. SocGoogle Scholar
  2. 2.
    Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. Volume 43. Princeton University Press, Princeton, NJ, USA; 1993:xiv+695.Google Scholar
  3. 3.
    Adams DR, Xiao J: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana University Mathematics Journal 2004, 53(6):1629-1663.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Olsen PA: Fractional integration, Morrey spaces and a Schrödinger equation. Communications in Partial Differential Equations 1995, 20(11-12):2005-2055. 10.1080/03605309508821161MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Sugano S, Tanaka H: Boundedness of fractional integral operators on generalized Morrey spaces. Scientiae Mathematicae Japonicae 2003, 58(3):531-540.MathSciNetGoogle Scholar
  6. 6.
    Eridani , Gunawan H: On generalized fractional integrals. Journal of the Indonesian Mathematical Society 2002, 8: 25-28.Google Scholar
  7. 7.
    Eridani , Gunawan H, Nakai E: On generalized fractional integral operators. Scientiae Mathematicae Japonicae 2004, 60(3):539-550.MATHMathSciNetGoogle Scholar
  8. 8.
    Gunawan H: A note on the generalized fractional integral operators. Journal of the Indonesian Mathematical Society 2003, 9(1):39-43.MATHMathSciNetGoogle Scholar
  9. 9.
    Nakai E: Generalized fractional integrals on Orlicz-Morrey spaces. In Banach and Function Spaces. Yokohama Publishers, Yokohama, Japan; 2004:323-333.Google Scholar
  10. 10.
    Nakai E: Orlicz-Morrey spaces and the Hardy-Littlewood maximal function. Studia Mathematica 2008, 188(3):193-221. 10.4064/sm188-3-1MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Adams DR: A note on Riesz potentials. Duke Mathematical Journal 1975, 42(4):765-778. 10.1215/S0012-7094-75-04265-9MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fefferman CL: The uncertainty principle. Bulletin of the American Mathematical Society 1983, 9(2):129-206. 10.1090/S0273-0979-1983-15154-6MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Kerman R, Sawyer E: The trace inequality and eigenvalue estimates for Schrödinger operators. Annales de l'Institut Fourier 1986, 36(4):207-228. 10.5802/aif.1074MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Sawano Y, Sugano S, Tanaka H: Identification of the image of Morrey spaces by the fractional integral operators. Proceedings of A. Razmadze Mathematical Institute 2009, 149: 87-93.MATHMathSciNetGoogle Scholar
  15. 15.
    Tanaka H: Morrey spaces and fractional operators. to appear in Journal of the Australian Mathematical SocietyGoogle Scholar
  16. 16.
    Pérez C:Sharp Open image in new window-weighted Sobolev inequalities. Annales de l'Institut Fourier 1995, 45(3):809-824. 10.5802/aif.1475MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Blasco O, Ruiz A, Vega L: Non-interpolation in Morrey-Campanato and block spaces. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1999, 28(1):31-40.MATHMathSciNetGoogle Scholar
  18. 18.
    Zorko CT: Morrey space. Proceedings of the American Mathematical Society 1986, 98(4):586-592. 10.1090/S0002-9939-1986-0861756-XMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Yoshihiro Sawano et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  • Yoshihiro Sawano
    • 1
  • Satoko Sugano
    • 2
  • Hitoshi Tanaka
    • 3
  1. 1.Department of MathematicsKyoto UniversitySakyoku, KyotoJapan
  2. 2.Kobe City College of TechnologyNishi-ku, KobeJapan
  3. 3.Graduate School of Mathematical SciencesThe University of TokyoMeguro-ku, TokyoJapan

Personalised recommendations