A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

  • Yoshihiro Sawano1Email author,

    Affiliated with

    • Satoko Sugano2 and

      Affiliated with

      • Hitoshi Tanaka3

        Affiliated with

        Boundary Value Problems20102009:835865

        DOI: 10.1155/2009/835865

        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].

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq1_HTML.gif the classical fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq2_HTML.gif and the classical fractional maximal operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq3_HTML.gif are given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ1_HTML.gif
        (1.1)
        In the present paper, we generalize the parameter http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq4_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq5_HTML.gif be a suitable function. We define the generalized fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq6_HTML.gif and the generalized fractional maximal operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq7_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ2_HTML.gif
        (1.2)

        Here, we use the notation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq8_HTML.gif to denote the family of all cubes in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq9_HTML.gif with sides parallel to the coordinate axes, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq10_HTML.gif , to denote the sidelength of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq12_HTML.gif to denote the volume of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq13_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq15_HTML.gif , then we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq16_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq17_HTML.gif .

        A well-known fact in partial differential equations is that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq18_HTML.gif is an inverse of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq19_HTML.gif . The operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq20_HTML.gif admits an expression of the form http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq21_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq22_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq23_HTML.gif we use http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq24_HTML.gif to denote the cube with the same center as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq25_HTML.gif , but with sidelength of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq26_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq27_HTML.gif denotes the Lebesgue measure of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq28_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq29_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq30_HTML.gif be a suitable function. For a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq31_HTML.gif locally in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq32_HTML.gif we set
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ3_HTML.gif
        (1.3)
        We will call the Morrey space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq33_HTML.gif the subset of all functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq34_HTML.gif locally in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq35_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq36_HTML.gif is finite. Applying Hölder's inequality to (1.3), we see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq37_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq38_HTML.gif . This tells us that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq39_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq40_HTML.gif . We remark that without the loss of generality we may assume
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ4_HTML.gif
        (1.4)

        (See [1].) Hereafter, we always postulate (1.4) on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq41_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq43_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq44_HTML.gif coincides with the usual Morrey space and we write this for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq45_HTML.gif and the norm for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq46_HTML.gif . Then we have the inclusion
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ5_HTML.gif
        (1.5)

        when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq47_HTML.gif .

        In the present paper, we take up some relations between the generalized fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq48_HTML.gif and the generalized fractional maximal operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq49_HTML.gif in the framework of the Morrey spaces http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq50_HTML.gif (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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq51_HTML.gif on predual of Morrey spaces.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq52_HTML.gif be a function. By the Dini condition we mean that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq53_HTML.gif fulfills
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ6_HTML.gif
        (1.6)
        while the doubling condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq54_HTML.gif (with a doubling constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq55_HTML.gif ) is that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq56_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ7_HTML.gif
        (1.7)
        We notice that (1.4) is stronger than the doubling condition. More quantitatively, if we assume (1.4), then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq57_HTML.gif satisfies the doubling condition with the doubling constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq58_HTML.gif . A simple consequence that can be deduced from the doubling condition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq59_HTML.gif is that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ8_HTML.gif
        (1.8)
        The key observation made in [1] is that it is frequently convenient to replace http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq60_HTML.gif satisfying (1.6) and (1.7) by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq61_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ9_HTML.gif
        (1.9)

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

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

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ10_HTML.gif
        (1.10)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq62_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq63_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq64_HTML.gif is nonincreasing. Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ11_HTML.gif
        (1.11)

        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq65_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq66_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq67_HTML.gif .

        The aim of the present paper is to generalize the function spaces to which http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq69_HTML.gif belong. With theorem 1.2, which we will present just below, we can replace http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq70_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq71_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq72_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq73_HTML.gif . We now formulate our main theorems. In the sequel we always assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq74_HTML.gif satisfies (1.6) and (1.7), and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq75_HTML.gif is used to denote various positive constants.

        Theorem 1.2.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ12_HTML.gif
        (1.12)
        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq76_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq77_HTML.gif are nondecreasing but that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq79_HTML.gif are nonincreasing. Assume also that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ13_HTML.gif
        (1.13)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ14_HTML.gif
        (1.14)

        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq80_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq82_HTML.gif .

        Remark 1.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq84_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq86_HTML.gif satisfy the assumption (1.13). Indeed,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ15_HTML.gif
        (1.15)

        Hence, Theorem 1.2 generalizes Proposition 1.1.

        Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq88_HTML.gif in Theorem 1.2, we obtain the result of how http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq89_HTML.gif controls http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq90_HTML.gif .

        Corollary 1.4.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq91_HTML.gif . Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ16_HTML.gif
        (1.16)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ17_HTML.gif
        (1.17)
        Corollary 1.4 generalizes [3, Theorem  4.2]. Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq92_HTML.gif in Theorem 1.2, we also obtain the condition on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq94_HTML.gif under which the mapping
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ18_HTML.gif
        (1.18)

        is bounded.

        Corollary 1.5.

        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ19_HTML.gif
        (1.19)
        Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ20_HTML.gif
        (1.20)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ21_HTML.gif
        (1.21)
        In particular, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq95_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ22_HTML.gif
        (1.22)
        Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq96_HTML.gif denotes the Hardy-Littlewood maximal operator defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ23_HTML.gif
        (1.23)

        We will establish that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq97_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq98_HTML.gif when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq99_HTML.gif (Lemma 2.2). Therefore, the second assertion is immediate from the first one.

        Theorem 1.6.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq100_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq101_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq102_HTML.gif are nondecreasing but that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq104_HTML.gif are nonincreasing. Suppose also that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ24_HTML.gif
        (1.24)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ25_HTML.gif
        (1.25)

        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq105_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq107_HTML.gif .

        Theorem 1.6 extends [4, Theorem  2], [1, Theorem  1.1], and [5, Theorem  1]. As the special case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq109_HTML.gif in Theorem 1.6 shows, this theorem covers [1, Remark  2.8].

        Corollary 1.7 (see [1, Remark  2.8], see also [68]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq110_HTML.gif . Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ26_HTML.gif
        (1.26)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ27_HTML.gif
        (1.27)

        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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq111_HTML.gif is the fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq112_HTML.gif . The result holds by letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq115_HTML.gif .

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq118_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq119_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq120_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq122_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq123_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq124_HTML.gif then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ28_HTML.gif
        (1.28)

        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq125_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq127_HTML.gif .

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

        Remark 1.9.

        The special case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq129_HTML.gif in Proposition 1.8 corresponds to the classical theorem due to Adams (see [11]).

        The fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq130_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq131_HTML.gif , is bounded from http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq132_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq133_HTML.gif if and only if the parameters http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq135_HTML.gif satisfy http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq137_HTML.gif .

        Using naively the Adams theorem and Hölder's inequality, one can prove a minor part of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq138_HTML.gif in Proposition 1.8. That is, the proof of Proposition 1.8 is fundamental provided http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq139_HTML.gif Indeed, by virtue of the Adams theorem we have, for any cube http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq140_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ29_HTML.gif
        (1.29)
        The condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq142_HTML.gif reads
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ30_HTML.gif
        (1.30)
        These yield
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ31_HTML.gif
        (1.31)
        if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq143_HTML.gif . In view of inclusion (1.5), the same can be said when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq144_HTML.gif . Also observe that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq145_HTML.gif Hence we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq146_HTML.gif . Thus, since the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq147_HTML.gif , Proposition 1.8 is significant only when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq148_HTML.gif The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq149_HTML.gif (the case of the Lebesgue spaces) corresponds (so-called) to the Fefferan-Phong inequality (see [12]). An inequality of the form
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ32_HTML.gif
        (1.32)

        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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq150_HTML.gif , we obtain a sharp estimate on the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq151_HTML.gif in (1.32).

        In [14], we characterized the range of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq152_HTML.gif , which motivates us to consider Proposition 1.8.

        Proposition 1.10 (see [14]).

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq154_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq155_HTML.gif . Assume that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ33_HTML.gif
        (1.33)

        (1) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq156_HTML.gif is continuous but not surjective.

        (2)Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq157_HTML.gif be an auxiliary function chosen so that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq159_HTML.gif and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq162_HTML.gif . Then the norm equivalence
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ34_HTML.gif
        (1.34)

        holds for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq163_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq164_HTML.gif denotes the Fourier transform.

        In view of this proposition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq165_HTML.gif is not a good space to describe the boundedness of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq166_HTML.gif , although we have (1.29). As we have seen by using Hölder's inequality in Remark 1.9, if we use the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq167_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq168_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq169_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq170_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq171_HTML.gif are nondecreasing and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq173_HTML.gif are nonincreasing. Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ35_HTML.gif
        (1.35)

        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq174_HTML.gif is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq175_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq176_HTML.gif .

        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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq178_HTML.gif belong. So, we omit the proof in the present paper.

        2. Proof of Theorems

        For any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq179_HTML.gif we will write http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq180_HTML.gif for the conjugate number defined by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq181_HTML.gif . Hereafter, for the sake of simplicity, for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq183_HTML.gif we will write
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ36_HTML.gif
        (2.1)

        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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq184_HTML.gif the family of all dyadic cubes in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq185_HTML.gif . We assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq186_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq187_HTML.gif are nonnegative, which may be done without any loss of generality thanks to the positivity of the integral kernel. We will denote by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq188_HTML.gif the ball centered at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq189_HTML.gif and of radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq190_HTML.gif . We begin by discretizing the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq191_HTML.gif following the idea of Pérez (see [16]):
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ37_HTML.gif
        (2.2)
        where we have used the doubling condition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq192_HTML.gif for the first inequality. To prove Theorem 1.2, thanks to the doubling condition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq193_HTML.gif , which holds by use of the facts that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq194_HTML.gif is nondecreasing and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq195_HTML.gif is nonincreasing, it suffices to show
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ38_HTML.gif
        (2.3)
        for all dyadic cubes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq196_HTML.gif . Hereafter, we let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ39_HTML.gif
        (2.4)
        Let us define for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq197_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ40_HTML.gif
        (2.5)
        and we will estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ41_HTML.gif
        (2.6)

        The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq199_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq200_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq201_HTML.gif one lets http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq203_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq204_HTML.gif let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ42_HTML.gif
        (2.7)
        Considering the maximal cubes with respect to inclusion, one can write
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ43_HTML.gif
        (2.8)
        where the cubes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq205_HTML.gif are nonoverlapping. By virtue of the maximality of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq206_HTML.gif one has that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ44_HTML.gif
        (2.9)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ45_HTML.gif
        (2.10)
        Then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq207_HTML.gif is a disjoint family of sets which decomposes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq208_HTML.gif and satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ46_HTML.gif
        (2.11)
        Also, one sets
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ47_HTML.gif
        (2.12)
        Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ48_HTML.gif
        (2.13)
        With Lemma 2.1 in mind, let us return to the proof of Theorem 1.2. We need only to verify that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ49_HTML.gif
        (2.14)
        Inserting the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq209_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ50_HTML.gif
        (2.15)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq210_HTML.gif , we will apply Lemma 2.1 to estimate this quantity. Retaining the same notation as Lemma 2.1 and noticing (2.13), we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ51_HTML.gif
        (2.16)
        We first evaluate
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ52_HTML.gif
        (2.17)
        It follows from the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq211_HTML.gif that (2.17) is bounded by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ53_HTML.gif
        (2.18)
        By virtue of the support condition and (1.8) we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ54_HTML.gif
        (2.19)
        If we invoke relations http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq213_HTML.gif , then (2.17) is bounded by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ55_HTML.gif
        (2.20)
        Now that we have from the definition of the Morrey norm
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ56_HTML.gif
        (2.21)
        we conclude that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ57_HTML.gif
        (2.22)
        Here, we have used the fact that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq214_HTML.gif is nondecreasing, that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq215_HTML.gif satisfies the doubling condition and that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ58_HTML.gif
        (2.23)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ59_HTML.gif
        (2.24)

        Summing up all factors, we obtain (2.14), by noticing that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq216_HTML.gif is a disjoint family of sets which decomposes http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq217_HTML.gif .

        The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq218_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq219_HTML.gif In this case we establish
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ60_HTML.gif
        (2.25)
        by the duality argument. Take a nonnegative function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq221_HTML.gif , satisfying that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq222_HTML.gif and that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ61_HTML.gif
        (2.26)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq223_HTML.gif , we will apply Lemma 2.1 to estimation of this quantity. First, we will insert the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq224_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ62_HTML.gif
        (2.27)
        First, we evaluate
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ63_HTML.gif
        (2.28)
        Going through the same argument as the above, we see that (2.28) is bounded by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ64_HTML.gif
        (2.29)
        Using Hölder's inequality, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ65_HTML.gif
        (2.30)
        These yield
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ66_HTML.gif
        (2.31)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ67_HTML.gif
        (2.32)
        Summing up all factors we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ68_HTML.gif
        (2.33)
        Another application of Hölder's inequality gives us that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ69_HTML.gif
        (2.34)
        Now that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq225_HTML.gif , the maximal operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq226_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq227_HTML.gif -bounded. As a result we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ70_HTML.gif
        (2.35)

        This is our desired inequality.

        The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq229_HTML.gif By a property of the dyadic cubes, for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq230_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ71_HTML.gif
        (2.36)
        As a consequence we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ72_HTML.gif
        (2.37)
        In view of the definition of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq231_HTML.gif , for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq232_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq233_HTML.gif there exists a unique cube in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq234_HTML.gif whose length is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq235_HTML.gif . Hence, inserting these estimates, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ73_HTML.gif
        (2.38)
        Here, in the last inequality we have used the doubling condition (1.8) and the facts that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq237_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq238_HTML.gif are nondecreasing and that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq240_HTML.gif satisfy the doubling condition. Thus, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ74_HTML.gif
        (2.39)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq241_HTML.gif . Inserting this pointwise estimate, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ75_HTML.gif
        (2.40)

        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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq242_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq243_HTML.gif satisfies (1.4), then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ76_HTML.gif
        (2.41)

        Lemma 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq244_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq245_HTML.gif satisfies (1.4), then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ77_HTML.gif
        (2.42)

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq246_HTML.gif be a fixed point. For every cube http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq247_HTML.gif we see that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ78_HTML.gif
        (2.43)
        This implies
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ79_HTML.gif
        (2.44)
        It follows from Lemma 2.2 that for every cube http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq248_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ80_HTML.gif
        (2.45)

        The desired inequality then follows.

        Proof of Theorem 1.6.

        We use definition (2.5) again and will estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ81_HTML.gif
        (2.46)
        for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq249_HTML.gif .The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq250_HTML.gif In the course of the proof of Theorem 1.2, we have established (2.25)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ82_HTML.gif
        (2.47)
        We will use it with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq251_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ83_HTML.gif
        (2.48)
        The case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq252_HTML.gif It follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ84_HTML.gif
        (2.49)
        from the Hölder inequality and the definition of the norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq253_HTML.gif . As a consequence we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ85_HTML.gif
        (2.50)
        Here, we have used the doubling condition (1.8) and the fact that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq254_HTML.gif is nondecreasing in the third inequality. Hence it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ86_HTML.gif
        (2.51)
        Combining (2.48) and (2.51), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ87_HTML.gif
        (2.52)

        We note that the assumption (1.24) implies http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq255_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq256_HTML.gif on predual of Morrey spaces. We will define the block spaces following [17].

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq257_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq258_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq259_HTML.gif satisfies (1.4). We say that a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq260_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq261_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq262_HTML.gif -block provided that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq263_HTML.gif is supported on a cube http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq264_HTML.gif and satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ88_HTML.gif
        (3.1)
        The space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq265_HTML.gif is defined by the set of all functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq266_HTML.gif locally in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq267_HTML.gif with the norm
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ89_HTML.gif
        (3.2)
        where each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq268_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq269_HTML.gif -block and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq270_HTML.gif , and the infimum is taken over all possible decompositions of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq271_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq272_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq273_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq274_HTML.gif is the usual block spaces, which we write for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq275_HTML.gif and the norm for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq276_HTML.gif , because the right-hand side of (3.1) is equal to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq277_HTML.gif . It is easy to prove
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ90_HTML.gif
        (3.3)

        when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq278_HTML.gif . In [17, Theorem  1] and [18, Proposition  5], it was established that the predual space of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq279_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq280_HTML.gif . More precisely, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq281_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq282_HTML.gif is an element of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq283_HTML.gif . Conversely, any continuous linear functional in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq284_HTML.gif can be realized with some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq285_HTML.gif .

        Theorem 3.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq286_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq287_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq288_HTML.gif are nondecreasing but that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq289_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq290_HTML.gif are nonincreasing. Suppose also that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ91_HTML.gif
        (3.4)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ92_HTML.gif
        (3.5)

        if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq291_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq292_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq293_HTML.gif , we obtain the following.

        Corollary 3.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq294_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq295_HTML.gif is nondecreasing but that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq296_HTML.gif is nonincreasing. Suppose also that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ93_HTML.gif
        (3.6)
        then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ94_HTML.gif
        (3.7)

        We dare restate Corollary 3.2 in terms of the fractional integral operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq297_HTML.gif . The results hold by letting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq298_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq299_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq301_HTML.gif .

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq302_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq303_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq304_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq305_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq306_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq307_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ95_HTML.gif
        (3.8)

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

        In Proposition 3.3, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq308_HTML.gif is replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq309_HTML.gif , 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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq310_HTML.gif .

        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 Fū jyukai foundation.

        Authors’ Affiliations

        (1)
        Department of Mathematics, Kyoto University
        (2)
        Kobe City College of Technology
        (3)
        Graduate School of Mathematical Sciences, The University of Tokyo

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        © 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.