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A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces
Boundary Value Problems volume 2009, Article number: 835865 (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 the classical fractional integral operator
and the classical fractional maximal operator
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ1_HTML.gif)
In the present paper, we generalize the parameter . Let
be a suitable function. We define the generalized fractional integral operator
and the generalized fractional maximal operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ2_HTML.gif)
Here, we use the notation to denote the family of all cubes in
with sides parallel to the coordinate axes,
, to denote the sidelength of
and
to denote the volume of
. If
,
, then we have
and
.
A well-known fact in partial differential equations is that is an inverse of
. The operator
admits an expression of the form
for some
. For more details of this operator we refer to [2]. As we will see, these operators will fall under the scope of our main results.
Among other function spaces, it seems that the Morrey spaces reflect the boundedness properties of the fractional integral operators. To describe the Morrey spaces we recall some definitions and notation. All cubes are assumed to have their sides parallel to the coordinate axes. For we use
to denote the cube with the same center as
, but with sidelength of
.
denotes the Lebesgue measure of
.
Let and
be a suitable function. For a function
locally in
we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ3_HTML.gif)
We will call the Morrey space the subset of all functions
locally in
for which
is finite. Applying Hölder's inequality to (1.3), we see that
provided that
. This tells us that
when
. We remark that without the loss of generality we may assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ4_HTML.gif)
(See [1].) Hereafter, we always postulate (1.4) on .
If ,
,
coincides with the usual Morrey space and we write this for
and the norm for
. Then we have the inclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ5_HTML.gif)
when .
In the present paper, we take up some relations between the generalized fractional integral operator and the generalized fractional maximal operator
in the framework of the Morrey spaces
(Theorem 1.2). In the last section, we prove a dual version of Olsen's inequality on predual of Morrey spaces (Theorem 3.1). As a corollary (Corollary 3.2), we have the boundedness properties of the operator
on predual of Morrey spaces.
Let be a function. By the Dini condition we mean that
fulfills
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ6_HTML.gif)
while the doubling condition on (with a doubling constant
) is that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ7_HTML.gif)
We notice that (1.4) is stronger than the doubling condition. More quantitatively, if we assume (1.4), then satisfies the doubling condition with the doubling constant
. A simple consequence that can be deduced from the doubling condition of
is that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ8_HTML.gif)
The key observation made in [1] is that it is frequently convenient to replace satisfying (1.6) and (1.7) by
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ9_HTML.gif)
Before we formulate our main results, we recall a typical result obtained in [1].
Proposition 1.1 (see [1, Theorem  1.3]).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ10_HTML.gif)
and
. Suppose that
is nonincreasing. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ11_HTML.gif)
where the constant is independent of
and
.
The aim of the present paper is to generalize the function spaces to which and
belong. With theorem 1.2, which we will present just below, we can replace
with
and
with
. We now formulate our main theorems. In the sequel we always assume that
satisfies (1.6) and (1.7), and
is used to denote various positive constants.
Theorem 1.2.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ12_HTML.gif)
Suppose that and
are nondecreasing but that
and
are nonincreasing. Assume also that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ13_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ14_HTML.gif)
where the constant is independent of
and
.
Remark 1.3.
Let and
. Then
and
satisfy the assumption (1.13). Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ15_HTML.gif)
Hence, Theorem 1.2 generalizes Proposition 1.1.
Letting and
in Theorem 1.2, we obtain the result of how
controls
.
Corollary 1.4.
Let . Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ16_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ17_HTML.gif)
Corollary 1.4 generalizes [3, Theorem  4.2]. Letting in Theorem 1.2, we also obtain the condition on
and
under which the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ18_HTML.gif)
is bounded.
Corollary 1.5.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ19_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ20_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ21_HTML.gif)
In particular, if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ22_HTML.gif)
Here, denotes the Hardy-Littlewood maximal operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ23_HTML.gif)
We will establish that is bounded on
when
(Lemma 2.2). Therefore, the second assertion is immediate from the first one.
Theorem 1.6.
Let . Suppose that
and
are nondecreasing but that
and
are nonincreasing. Suppose also that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ24_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ25_HTML.gif)
where the constant is independent of
and
.
Theorem 1.6 extends [4, Theorem  2], [1, Theorem  1.1], and [5, Theorem  1]. As the special case and
in Theorem 1.6 shows, this theorem covers [1, Remark  2.8].
Corollary 1.7 (see [1, Remark  2.8], see also [6–8]).
Let . Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ26_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ27_HTML.gif)
Nakai generalized Corollary 1.7 to the Orlicz-Morrey spaces ([9, Theorem  2.2] and [10, Theorem  7.1]).
We dare restate Theorem 1.6 in the special case when is the fractional integral operator
. The result holds by letting
,
and
.
Proposition 1.8 (see [1, Proposition  1.7]).
Let ,
,
, and
. Suppose that
,
,
,
, and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ28_HTML.gif)
where the constant is independent of
and
.
Proposition 1.8 extends [4, Theorem  2] (see [1, Remark  1.9]).
Remark 1.9.
The special case and
in Proposition 1.8 corresponds to the classical theorem due to Adams (see [11]).
The fractional integral operator ,
, is bounded from
to
if and only if the parameters
and
satisfy
and
.
Using naively the Adams theorem and Hölder's inequality, one can prove a minor part of in Proposition 1.8. That is, the proof of Proposition 1.8 is fundamental provided
Indeed, by virtue of the Adams theorem we have, for any cube
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ29_HTML.gif)
The condition ,
reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ30_HTML.gif)
These yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ31_HTML.gif)
if . In view of inclusion (1.5), the same can be said when
. Also observe that
Hence we have
. Thus, since the condition
, Proposition 1.8 is significant only when
The case
(the case of the Lebesgue spaces) corresponds (so-called) to the Fefferan-Phong inequality (see [12]). An inequality of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ32_HTML.gif)
is called the trace inequality and is useful in the analysis of the Schrödinger operators. For example, Kerman and Sawyer utilized an inequality of type (1.32) to obtain an eigenvalue estimates of the operators (see [13]). By letting , we obtain a sharp estimate on the constant
in (1.32).
In [14], we characterized the range of , which motivates us to consider Proposition 1.8.
Proposition 1.10 (see [14]).
Let ,
, and
. Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ33_HTML.gif)
(1) is continuous but not surjective.
(2)Let be an auxiliary function chosen so that
,
and that
,
,
. Then the norm equivalence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ34_HTML.gif)
holds for , where
denotes the Fourier transform.
In view of this proposition is not a good space to describe the boundedness of
, although we have (1.29). As we have seen by using Hölder's inequality in Remark 1.9, if we use the space
, then we will obtain a result weaker than Proposition 1.8.
Finally it would be interesting to compare Theorem 1.2 with the following Theorem 1.11.
Theorem 1.11.
Let . Suppose that
,
, and
are nondecreasing and that
and
are nonincreasing. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ35_HTML.gif)
where the constant is independent of
and
.
Theorem 1.11 generalizes [1, Theorem  1.7] and the proof remains unchanged except some minor modifications caused by our generalization of the function spaces to which and
belong. So, we omit the proof in the present paper.
2. Proof of Theorems
For any we will write
for the conjugate number defined by
. Hereafter, for the sake of simplicity, for any
and
we will write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ36_HTML.gif)
2.1. Proof of Theorem 1.2
First, we will prove Theorem 1.2. Except for some sufficient modifications, the proof of the theorem follows the argument in [15]. We denote by the family of all dyadic cubes in
. We assume that
and
are nonnegative, which may be done without any loss of generality thanks to the positivity of the integral kernel. We will denote by
the ball centered at
and of radius
. We begin by discretizing the operator
following the idea of Pérez (see [16]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ37_HTML.gif)
where we have used the doubling condition of for the first inequality. To prove Theorem 1.2, thanks to the doubling condition of
, which holds by use of the facts that
is nondecreasing and that
is nonincreasing, it suffices to show
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ38_HTML.gif)
for all dyadic cubes . Hereafter, we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ39_HTML.gif)
Let us define for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ40_HTML.gif)
and we will estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ41_HTML.gif)
The case and
We need the following crucial lemma, the proof of which is straightforward and is omitted (see [15, 16]).
Lemma 2.1.
For a nonnegative function in
one lets
and
. For
let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ42_HTML.gif)
Considering the maximal cubes with respect to inclusion, one can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ43_HTML.gif)
where the cubes are nonoverlapping. By virtue of the maximality of
one has that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ44_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ45_HTML.gif)
Then is a disjoint family of sets which decomposes
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ46_HTML.gif)
Also, one sets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ47_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ48_HTML.gif)
With Lemma 2.1 in mind, let us return to the proof of Theorem 1.2. We need only to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ49_HTML.gif)
Inserting the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ50_HTML.gif)
Letting , we will apply Lemma 2.1 to estimate this quantity. Retaining the same notation as Lemma 2.1 and noticing (2.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ51_HTML.gif)
We first evaluate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ52_HTML.gif)
It follows from the definition of that (2.17) is bounded by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ53_HTML.gif)
By virtue of the support condition and (1.8) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ54_HTML.gif)
If we invoke relations and
, then (2.17) is bounded by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ55_HTML.gif)
Now that we have from the definition of the Morrey norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ56_HTML.gif)
we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ57_HTML.gif)
Here, we have used the fact that is nondecreasing, that
satisfies the doubling condition and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ58_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ59_HTML.gif)
Summing up all factors, we obtain (2.14), by noticing that is a disjoint family of sets which decomposes
.
The case and
In this case we establish
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ60_HTML.gif)
by the duality argument. Take a nonnegative function ,
, satisfying that
and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ61_HTML.gif)
Letting , we will apply Lemma 2.1 to estimation of this quantity. First, we will insert the definition of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ62_HTML.gif)
First, we evaluate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ63_HTML.gif)
Going through the same argument as the above, we see that (2.28) is bounded by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ64_HTML.gif)
Using Hölder's inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ65_HTML.gif)
These yield
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ66_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ67_HTML.gif)
Summing up all factors we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ68_HTML.gif)
Another application of Hölder's inequality gives us that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ69_HTML.gif)
Now that , the maximal operator
is
-bounded. As a result we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ70_HTML.gif)
This is our desired inequality.
The case and
By a property of the dyadic cubes, for all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ71_HTML.gif)
As a consequence we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ72_HTML.gif)
In view of the definition of , for each
with
there exists a unique cube in
whose length is
. Hence, inserting these estimates, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ73_HTML.gif)
Here, in the last inequality we have used the doubling condition (1.8) and the facts that ,
, and
are nondecreasing and that
and
satisfy the doubling condition. Thus, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ74_HTML.gif)
for all . Inserting this pointwise estimate, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ75_HTML.gif)
This is our desired inequality.
2.2. Proof of Theorem 1.6
We need some lemmas.
Lemma 2.2 (see [1, Lemma  2.2]).
Let . Suppose that
satisfies (1.4), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ76_HTML.gif)
Lemma 2.3.
Let . Suppose that
satisfies (1.4), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ77_HTML.gif)
Proof.
Let be a fixed point. For every cube
we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ78_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ79_HTML.gif)
It follows from Lemma 2.2 that for every cube
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ80_HTML.gif)
The desired inequality then follows.
Proof of Theorem 1.6.
We use definition (2.5) again and will estimate
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ81_HTML.gif)
for .The case
In the course of the proof of Theorem 1.2, we have established (2.25)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ82_HTML.gif)
We will use it with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ83_HTML.gif)
The case It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ84_HTML.gif)
from the Hölder inequality and the definition of the norm . As a consequence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ85_HTML.gif)
Here, we have used the doubling condition (1.8) and the fact that is nondecreasing in the third inequality. Hence it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ86_HTML.gif)
Combining (2.48) and (2.51), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ87_HTML.gif)
We note that the assumption (1.24) implies . Hence we arrive at the desired inequality by using Lemma 2.3.
3. A Dual Version of Olsen's Inequality
In this section, as an application of Theorem 1.6, we consider a dual version of Olsen's inequality on predual of Morrey spaces (Theorem 3.1). As a corollary (Corollary 3.2), we have the boundedness properties of the operator on predual of Morrey spaces. We will define the block spaces following [17].
Let and
. Suppose that
satisfies (1.4). We say that a function
on
is a
-block provided that
is supported on a cube
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ88_HTML.gif)
The space is defined by the set of all functions
locally in
with the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ89_HTML.gif)
where each is a
-block and
, and the infimum is taken over all possible decompositions of
. If
,
,
is the usual block spaces, which we write for
and the norm for
, because the right-hand side of (3.1) is equal to
. It is easy to prove
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ90_HTML.gif)
when . In [17, Theorem  1] and [18, Proposition  5], it was established that the predual space of
is
. More precisely, if
, then
is an element of
. Conversely, any continuous linear functional in
can be realized with some
.
Theorem 3.1.
Let . Suppose that
and
are nondecreasing but that
and
are nonincreasing. Suppose also that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ91_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ92_HTML.gif)
if is a continuous function.
Theorem 3.1 generalizes [1, Theorem  3.1], and its proof is similar to that theorem, hence omitted. As a special case when and
, we obtain the following.
Corollary 3.2.
Let . Suppose that
is nondecreasing but that
is nonincreasing. Suppose also that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ93_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ94_HTML.gif)
We dare restate Corollary 3.2 in terms of the fractional integral operator . The results hold by letting
,
,
and
.
Proposition 3.3 (see [1, Proposition  3.8]).
Let ,
, and
. Suppose that
,
, and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F835865/MediaObjects/13661_2009_Article_882_Equ95_HTML.gif)
Remark 3.4 (see [1, Remark  3.9]).
In Proposition 3.3, if is replaced by
, then, using the Hardy-Littlewood-Sobolev inequality locally and taking care of the larger scales by the same manner as the proof of Theorem 3.1, one has a naive bound for
.
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The third author is supported by the Global COE program at Graduate School of Mathematical Sciences, University of Tokyo, and was supported by FÅ« jyukai foundation.
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Sawano, Y., Sugano, S. & Tanaka, H. A Note on Generalized Fractional Integral Operators on Generalized Morrey Spaces. Bound Value Probl 2009, 835865 (2010). https://doi.org/10.1155/2009/835865
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DOI: https://doi.org/10.1155/2009/835865