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
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
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 wellknown 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
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
(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
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
while the doubling condition on (with a doubling constant ) is that satisfies
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
The key observation made in [1] is that it is frequently convenient to replace satisfying (1.6) and (1.7) by :
Before we formulate our main results, we recall a typical result obtained in [1].
Proposition 1.1 (see [1, Theorem 1.3]).
Let
and . Suppose that is nonincreasing. Then
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
Suppose that and are nondecreasing but that and are nonincreasing. Assume also that
then
where the constant is independent of and .
Remark 1.3.
Let and . Then and satisfy the assumption (1.13). Indeed,
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
then
Corollary 1.4 generalizes [3, Theorem 4.2]. Letting in Theorem 1.2, we also obtain the condition on and under which the mapping
is bounded.
Corollary 1.5.
Let
Suppose that
then
In particular, if , then
Here, denotes the HardyLittlewood maximal operator defined by
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
then
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
then
Nakai generalized Corollary 1.7 to the OrliczMorrey 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
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 ,
The condition , reads
These yield
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 (socalled) to the FefferanPhong inequality (see [12]). An inequality of the form
is called the trace inequality and is useful in the analysis of the Schrödinger operators. For example, Kerman and Sawyer utilized an inequality of type (1.32) to obtain an eigenvalue estimates of the operators (see [13]). By letting , 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
(1) is continuous but not surjective.
(2)Let be an auxiliary function chosen so that , and that , , . Then the norm equivalence
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
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
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]):
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
for all dyadic cubes . Hereafter, we let
Let us define for
and we will estimate
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
Considering the maximal cubes with respect to inclusion, one can write
where the cubes are nonoverlapping. By virtue of the maximality of one has that
Let
Then is a disjoint family of sets which decomposes and satisfies
Also, one sets
Then
With Lemma 2.1 in mind, let us return to the proof of Theorem 1.2. We need only to verify that
Inserting the definition of , we have
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
We first evaluate
It follows from the definition of that (2.17) is bounded by
By virtue of the support condition and (1.8) we have
If we invoke relations and , then (2.17) is bounded by
Now that we have from the definition of the Morrey norm
we conclude that
Here, we have used the fact that is nondecreasing, that satisfies the doubling condition and that
Similarly, we have
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
by the duality argument. Take a nonnegative function , , satisfying that and that
Letting , we will apply Lemma 2.1 to estimation of this quantity. First, we will insert the definition of ,
First, we evaluate
Going through the same argument as the above, we see that (2.28) is bounded by
Using Hölder's inequality, we have
These yield
Similarly, we have
Summing up all factors we obtain
Another application of Hölder's inequality gives us that
Now that , the maximal operator is bounded. As a result we have
This is our desired inequality.
The case and By a property of the dyadic cubes, for all we have
As a consequence we obtain
In view of the definition of , for each with there exists a unique cube in whose length is . Hence, inserting these estimates, we obtain
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
for all . Inserting this pointwise estimate, we obtain
This is our desired inequality.
2.2. Proof of Theorem 1.6
We need some lemmas.
Lemma 2.2 (see [1, Lemma 2.2]).
Let . Suppose that satisfies (1.4), then
Lemma 2.3.
Let . Suppose that satisfies (1.4), then
Proof.
Let be a fixed point. For every cube we see that
This implies
It follows from Lemma 2.2 that for every cube
The desired inequality then follows.
Proof of Theorem 1.6.
We use definition (2.5) again and will estimate
for .The case In the course of the proof of Theorem 1.2, we have established (2.25)
We will use it with
The case It follows that
from the Hölder inequality and the definition of the norm . As a consequence we have
Here, we have used the doubling condition (1.8) and the fact that is nondecreasing in the third inequality. Hence it follows that
Combining (2.48) and (2.51), we obtain
We note that the assumption (1.24) implies . 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
The space is defined by the set of all functions locally in with the norm
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 righthand side of (3.1) is equal to . It is easy to prove
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
then
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
then
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
Remark 3.4 (see [1, Remark 3.9]).
In Proposition 3.3, if is replaced by , then, using the HardyLittlewoodSobolev inequality locally and taking care of the larger scales by the same manner as the proof of Theorem 3.1, one has a naive bound for .
Acknowledgments
The third author is supported by the Global COE program at Graduate School of Mathematical Sciences, University of Tokyo, and was supported by Fūjyukai foundation.
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