Open Access

A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq1_HTML.gif the classical fractional integral operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq2_HTML.gif and the classical fractional maximal operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq3_HTML.gif are given by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq4_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq6_HTML.gif and the generalized fractional maximal operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq7_HTML.gif by
https://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 https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq10_HTML.gif , to denote the sidelength of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq12_HTML.gif to denote the volume of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq13_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq15_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq16_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq18_HTML.gif is an inverse of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq19_HTML.gif . The operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq20_HTML.gif admits an expression of the form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq21_HTML.gif for some https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq23_HTML.gif we use https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq25_HTML.gif , but with sidelength of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq26_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq27_HTML.gif denotes the Lebesgue measure of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq28_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq29_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq31_HTML.gif locally in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq32_HTML.gif we set
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq33_HTML.gif the subset of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq34_HTML.gif locally in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq35_HTML.gif for which https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq37_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq38_HTML.gif . This tells us that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq39_HTML.gif when https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq41_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq43_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq45_HTML.gif and the norm for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq46_HTML.gif . Then we have the inclusion
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ5_HTML.gif
(1.5)

when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq48_HTML.gif and the generalized fractional maximal operator https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq51_HTML.gif on predual of Morrey spaces.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq53_HTML.gif fulfills
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq54_HTML.gif (with a doubling constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq55_HTML.gif ) is that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq56_HTML.gif satisfies
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq59_HTML.gif is that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq61_HTML.gif :
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ10_HTML.gif
(1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq63_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq64_HTML.gif is nonincreasing. Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ11_HTML.gif
(1.11)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq65_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq66_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq68_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq70_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq72_HTML.gif with https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ12_HTML.gif
(1.12)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq77_HTML.gif are nondecreasing but that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq79_HTML.gif are nonincreasing. Assume also that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ13_HTML.gif
(1.13)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ14_HTML.gif
(1.14)

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

Remark 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq84_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq86_HTML.gif satisfy the assumption (1.13). Indeed,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq87_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq89_HTML.gif controls https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq90_HTML.gif .

Corollary 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq91_HTML.gif . Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ16_HTML.gif
(1.16)
then
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq94_HTML.gif under which the mapping
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ19_HTML.gif
(1.19)
Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ20_HTML.gif
(1.20)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ21_HTML.gif
(1.21)
In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq95_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ22_HTML.gif
(1.22)
Here, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ23_HTML.gif
(1.23)

We will establish that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq97_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq98_HTML.gif when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq100_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq102_HTML.gif are nondecreasing but that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq104_HTML.gif are nonincreasing. Suppose also that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ24_HTML.gif
(1.24)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ25_HTML.gif
(1.25)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq105_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq106_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq108_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq110_HTML.gif . Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ26_HTML.gif
(1.26)
then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq111_HTML.gif is the fractional integral operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq112_HTML.gif . The result holds by letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq113_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq114_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq118_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq119_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq123_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq124_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ28_HTML.gif
(1.28)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq125_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq126_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq128_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq131_HTML.gif , is bounded from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq132_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq133_HTML.gif if and only if the parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq135_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq136_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq140_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ29_HTML.gif
(1.29)
The condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq142_HTML.gif reads
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ30_HTML.gif
(1.30)
These yield
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ31_HTML.gif
(1.31)
if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq144_HTML.gif . Also observe that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq145_HTML.gif Hence we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq146_HTML.gif . Thus, since the condition https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq148_HTML.gif The case https://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
https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq154_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq155_HTML.gif . Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ33_HTML.gif
(1.33)

(1) https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq158_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq159_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq161_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq162_HTML.gif . Then the norm equivalence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ34_HTML.gif
(1.34)

holds for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq163_HTML.gif , where https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq168_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq169_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq170_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq171_HTML.gif are nondecreasing and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq173_HTML.gif are nonincreasing. Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ35_HTML.gif
(1.35)

where the constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq174_HTML.gif is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq175_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq177_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq179_HTML.gif we will write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq180_HTML.gif for the conjugate number defined by https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq183_HTML.gif we will write
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq185_HTML.gif . We assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq186_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq188_HTML.gif the ball centered at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq189_HTML.gif and of radius https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq190_HTML.gif . We begin by discretizing the operator https://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]):
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq194_HTML.gif is nondecreasing and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq195_HTML.gif is nonincreasing, it suffices to show
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ38_HTML.gif
(2.3)
for all dyadic cubes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq196_HTML.gif . Hereafter, we let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ39_HTML.gif
(2.4)
Let us define for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq197_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ40_HTML.gif
(2.5)
and we will estimate
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ41_HTML.gif
(2.6)

The case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq198_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq200_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq201_HTML.gif one lets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq203_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq204_HTML.gif let
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ43_HTML.gif
(2.8)
where the cubes https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq206_HTML.gif one has that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ44_HTML.gif
(2.9)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ45_HTML.gif
(2.10)
Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq208_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ46_HTML.gif
(2.11)
Also, one sets
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ47_HTML.gif
(2.12)
Then
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ49_HTML.gif
(2.14)
Inserting the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq209_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ50_HTML.gif
(2.15)
Letting https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ51_HTML.gif
(2.16)
We first evaluate
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq211_HTML.gif that (2.17) is bounded by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ54_HTML.gif
(2.19)
If we invoke relations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq213_HTML.gif , then (2.17) is bounded by
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ56_HTML.gif
(2.21)
we conclude that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq214_HTML.gif is nondecreasing, that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq215_HTML.gif satisfies the doubling condition and that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ58_HTML.gif
(2.23)
Similarly, we have
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq217_HTML.gif .

The case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq218_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq219_HTML.gif In this case we establish
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq221_HTML.gif , satisfying that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq222_HTML.gif and that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ61_HTML.gif
(2.26)
Letting https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq224_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ62_HTML.gif
(2.27)
First, we evaluate
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ65_HTML.gif
(2.30)
These yield
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ66_HTML.gif
(2.31)
Similarly, we have
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ69_HTML.gif
(2.34)
Now that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq225_HTML.gif , the maximal operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq226_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq227_HTML.gif -bounded. As a result we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq228_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq230_HTML.gif we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq231_HTML.gif , for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq232_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq233_HTML.gif there exists a unique cube in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq234_HTML.gif whose length is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq235_HTML.gif . Hence, inserting these estimates, we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq237_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq238_HTML.gif are nondecreasing and that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq239_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ74_HTML.gif
(2.39)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq241_HTML.gif . Inserting this pointwise estimate, we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq242_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq243_HTML.gif satisfies (1.4), then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ76_HTML.gif
(2.41)

Lemma 2.3.

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

Proof.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq247_HTML.gif we see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ78_HTML.gif
(2.43)
This implies
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq248_HTML.gif
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ81_HTML.gif
(2.46)
for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq249_HTML.gif .The case https://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)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq251_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ83_HTML.gif
(2.48)
The case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq252_HTML.gif It follows that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq253_HTML.gif . As a consequence we have
https://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 https://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
https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq257_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq258_HTML.gif . Suppose that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq260_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq261_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq262_HTML.gif -block provided that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq263_HTML.gif is supported on a cube https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq264_HTML.gif and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ88_HTML.gif
(3.1)
The space https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq266_HTML.gif locally in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq267_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ89_HTML.gif
(3.2)
where each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq268_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq269_HTML.gif -block and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq271_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq272_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq273_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq275_HTML.gif and the norm for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq277_HTML.gif . It is easy to prove
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ90_HTML.gif
(3.3)

when https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq279_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq280_HTML.gif . More precisely, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq281_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq282_HTML.gif is an element of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq283_HTML.gif . Conversely, any continuous linear functional in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq284_HTML.gif can be realized with some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq285_HTML.gif .

Theorem 3.1.

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

if https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq292_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq294_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq295_HTML.gif is nondecreasing but that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq296_HTML.gif is nonincreasing. Suppose also that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ93_HTML.gif
(3.6)
then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq297_HTML.gif . The results hold by letting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq298_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq299_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq300_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq302_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq303_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq304_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq305_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq306_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq307_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_IEq308_HTML.gif is replaced by https://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 https://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

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