A Note on Inclusions of Discrete Morrey Spaces
Hendra Gunawan, Denny Ivanal Hakim, and Mochammad Idris

TL;DR
This paper investigates inclusion relations among discrete Morrey spaces, providing necessary conditions, generalizations, and properness of these relations, along with connections to classical Morrey spaces and their weak type variants.
Contribution
It introduces new necessary conditions for inclusion, generalizes known properties, and explores the properness and connections of these relations in discrete Morrey spaces.
Findings
Established necessary inclusion conditions.
Proved a generalization of p-summable sequence spaces.
Showed all inclusion relations are proper.
Abstract
We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in \cite{GKS,HS2}. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of -summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Mathematical Analysis and Transform Methods · Advanced Banach Space Theory
A Note on Inclusions of Discrete Morrey Spaces
Hendra Gunawan1,∗
1,2Analysis and Geometry Group,
Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology,
Bandung 40132, INDONESIA
*∗*Email: [email protected]
*†*Email: [email protected]
3Department of Mathematics,
Faculty of Mathematics and Natural Sciences,
Universitas Lambung Mangkurat,
Banjarbaru 70714, INDONESIA
Email: [email protected]
Denny I. Hakim2,†
1,2Analysis and Geometry Group,
Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology,
Bandung 40132, INDONESIA
*∗*Email: [email protected]
*†*Email: [email protected]
3Department of Mathematics,
Faculty of Mathematics and Natural Sciences,
Universitas Lambung Mangkurat,
Banjarbaru 70714, INDONESIA
Email: [email protected]
and Mochammad Idris3
1,2Analysis and Geometry Group,
Faculty of Mathematics and Natural Sciences,
Bandung Institute of Technology,
Bandung 40132, INDONESIA
*∗*Email: [email protected]
*†*Email: [email protected]
3Department of Mathematics,
Faculty of Mathematics and Natural Sciences,
Universitas Lambung Mangkurat,
Banjarbaru 70714, INDONESIA
Email: [email protected]
Abstract
We give a necessary condition for inclusion relations between discrete Morrey spaces which can be seen as a complement of the results in [3, 7]. We also prove another inclusion property of discrete Morrey spaces which can be viewed as a generalization of the inclusion property of the spaces of -summable sequences. Analogous results for weak type discrete Morrey spaces is also presented. In addition, we show that each of these inclusion relations is proper. Some connections between inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.
**MSC (2010): 42B35, 46A45, 46B45.
Key words: Discrete Morrey spaces, inclusion properties, Morrey spaces.**
1 Introduction
Let . The discrete Morrey space , introduced in [3], is defined to be the set of all sequences such that
[TABLE]
where , , and . This space is a Banach space with the norm
[TABLE]
We remark that, for , we have . Moreover, it is shown in [3] that is a proper subset of whenever . In the same paper, the authors also prove that if , then
[TABLE]
In addition, it is shown that
[TABLE]
whenever , , and . Recently, these inclusion results are extended to discrete Morrey spaces on in [7]. Moreover, the authors in [7] give a necessary condition for inclusion relations of discrete Morrey spaces on . They also prove that the embedding (1.2) is never compact.
One of the aims of this present paper is to give an alternative proof of a necessary condition of inclusion of discrete Morrey spaces. In particular, we show that is necessary for the inclusion relation (1.2). Besides (1.2), we also discuss another inclusion property of discrete Morrey spaces which generalizes
[TABLE]
for every , namely whenever and . We prove this result by using the fact that contains for all .
We also see later that coincides with the space where . Here, is defined to be the set of all sequences for which
[TABLE]
In addition to the results for discrete Morrey spaces, we give the corresponding results for weak type discrete Morrey spaces in Section 2.2. The proper inclusion relation between a discrete Morrey space and its weak type is given in Section 2.3.
In Section 3, we discuss the relation between discrete Morrey spaces and ‘continuous’ Morrey spaces on . Recall that, for , the Morrey space is defined to be the set of all functions such that the norm
[TABLE]
is finite. We apply the result in [8] to recover the inclusion properties of discrete Morrey spaces from those of Morrey spaces. We also reprove some necessary condition for the inclusion relations between Morrey spaces in [2] by combining the results in Section 2 and [8]. Our main result in this section is a necessary condition for the proper inclusion relation between weak type discrete Morrey spaces and discrete Morrey spaces.
Let us mention some previous works related to this paper. The inclusion properties of discrete Morrey spaces, their weak type spaces and their generalization are initially studied in [3]. The analogous results on Morrey spaces can be found in [1, 2, 6, 9, 10, 11]. The boundedness of the Hardy-Littlewood maximal operator on discrete Morrey spaces is investigated in [4].
Throughout this paper, we denote by a positive constant which is independent of the sequence and its value may be different from line to line. We write if there exists a positive constant such that . Meanwhile, means . In addition, we denote by if and .
2 Main results
In this section, we consider three types of inclusion properties of discrete Morrey spaces.
2.1 Inclusion property of discrete Morrey spaces
Our first result is a necessary condition for the inclusion property of the first kind.
Theorem 2.1**.**
Let and . If , then .
Remark 2.2*.*
As mentioned before, a similar result for discrete Morrey spaces on can be found in [7]. Here, we give a different proof of the necessary condition for this inclusion property.
Proof.
Assume to the contrary that . Choose such that
[TABLE]
Let be the smallest positive integer such that . Define by
[TABLE]
Let . Observe that, for every we have
[TABLE]
Therefore, for , we have
[TABLE]
According to (2.1), we see that , so that as . Combining this with (2.1), we have
[TABLE]
which tells us that .
We shall show that . For every , define and let . We also define
[TABLE]
Observe that
[TABLE]
Since , we have
[TABLE]
If , then
[TABLE]
On the other hand, if , then there exists such that
[TABLE]
Hence we obtain
[TABLE]
Note that (2.1) implies . As a consequence of this inequality and (2.1), we have
[TABLE]
Combining (2.6) and (2.9), we get
[TABLE]
which means that . Hence, but . This contradicts . ∎
Remark 2.3*.*
Note that the condition and is not a necessary condition for the inclusion because when .
We now move on to the inclusion property of the second kind. As a preparation, we show that contains where Recall that is the set of all sequences such that
[TABLE]
Theorem 2.4**.**
If , then (i) and (ii) Moreover, the inclusion is proper.
Remark 2.5*.*
The inclusion is obtained in [8]. For the reader’s convenience, we provide our proof and also show that the inclusion is proper.
Proof.
(i) Suppose that . Take an arbitrary . Note that for every , we have
[TABLE]
Because , we obtain .
Next we shall show that . Define by for every . We obtain
[TABLE]
On the other hand,
[TABLE]
Thus, .
(ii) Now, take an arbitrary . For every , holds. Since , we have
[TABLE]
Meanwhile, take an arbitrary . Observe that
[TABLE]
Therefore, . Hence, ∎
Using the relation between and in Theorem 2.4, we shall prove the inclusion property of discrete Morrey spaces of the second type.
Theorem 2.6**.**
Let and . If and , then with
[TABLE]
Moreover, whenever ( and ) or ( and ).
Proof.
Take an arbitrary . Then for every and , we have
[TABLE]
where we use Theorem 2.4 and the definition of . Since , we get
[TABLE]
and hence with
[TABLE]
We now prove the second part of the theorem. Note that our assumption implies . Define where dan for every . Then, for every and for , we have
[TABLE]
Therefore
[TABLE]
Meanwhile, we obtain
[TABLE]
Thus, . This completes the proof. ∎
As a corollary of Theorem 2.6, we obtain the following result.
Theorem 2.7**.**
If and then if and only if with . Furthermore, is a proper subset of whenever .
Proof.
In view of Theorem 2.6, we only need to prove that the inclusion with imply . Let and take an example where for and for . Observe that
[TABLE]
Since , we conclude that . ∎
Remark 2.8*.*
The argument in the proof of Theorem 2.7 cannot be applied to ‘continuous’ Morrey spaces , , because the case occurs in the formula of the -norm.
2.2 Incusion property of weak type discrete Morrey spaces
Let us recall the definition of weak type discrete Morrey spaces.
Definition 2.9**.**
Let . The weak type discrete Morrey spaces is defined to be the set of all sequences for which the quasi-norm
[TABLE]
is finite.
The quasi-norm in Definition 2.9 can be rewritten as
[TABLE]
where and
[TABLE]
Note that, for , the space is the weak type space. It is shown in [3, Example 3.1] that the space is a proper subset of . More general, the discrete Morrey space is a subset of (see [3, Theorem 3.2]).
For the inclusion between weak type discrete Morrey spaces, it is proven in [3] that whenever . Now we show that is a necessary condition for these inclusions.
Theorem 2.10**.**
Let and . If , then .
Proof.
Assume to the contrary that . Let be defined by (2.2). Since and , we have . If we can prove that , then we obtain a contradiction, so we must have . Now we show that . Observe that
[TABLE]
Since either or , we have
[TABLE]
Consequently,
[TABLE]
as desired. ∎
Similar with the (strong type) discrete Morrey spaces, we also show the inclusion property of weak type discrete Morrey spaces of the second kind.
Theorem 2.11**.**
Let and . If and , then with
[TABLE]
Moreover, whenever ( and ) or ( and ).
Proof.
Let . According to Theorem 2.6, we have
[TABLE]
which tells us that . Hence, with .
Now we prove the second part of this theorem. By our assumption, we have . Let be defined as in the proof of Theorem 2.6. Note that . Therefore,
[TABLE]
Hence, . On the other hand, for every , we have
[TABLE]
Consequently,
[TABLE]
Hence, . Thus, . ∎
Necessary and sufficient conditions for the inclusion property of weak type discrete Morrey spaces of the second kind are presented in the following theorem.
Theorem 2.12**.**
Let and . Then if and only if with . Furthermore, is a proper subset of whenever .
Proof.
If , then by taking in Theorem 2.11, we get with . Conversely, assume that with . Let be defined by
[TABLE]
Then
[TABLE]
Therefore, . Since , we have , so that . The second part of this theorem follows from Theorem 2.11 by taking . ∎
2.3 Inclusion relation between discrete Morrey spaces and weak type discrete Morrey spaces
Theorem 2.13**.**
Let . Then the inclusion is proper.
Proof.
Let . We shall show that there exists such that
[TABLE]
for every and . Observe that
[TABLE]
where
[TABLE]
and is chosen later. The estimate for is
[TABLE]
Meanwhile, by using Definition 2.9 and , we have
[TABLE]
Combining (2.13) and (2.3) and taking , we get
[TABLE]
Hence, (2.12) follows by taking -th root of (2.15) and multiplying by . Since (2.12) holds for arbitrary and , we have with
[TABLE]
as desired.
We shall now prove that the inclusion is proper. Choose such that
[TABLE]
Let be the smallest positive integer such that . Define by the formula
[TABLE]
By the same calculation as in the proof of Theorem 2.1, we obtain but . Moreover, repeating the calculation in the proof of Theorem 2.10, we obtain , so that . This shows that . Thus, we conclude that the inclusion is proper. ∎
3 Relation between discrete Morrey spaces and their continuous counterpart
In this section, we discuss the relation between the inclusion property of discrete Morrey spaces and that of Morrey spaces. In particular, we reprove the inclusion property of discrete Morrey spaces by using the inclusion property of Morrey spaces, and we recover some necessary conditions for the inclusion property of Morrey spaces. We also give a necessary condition for the proper inclusion relation between discrete Morrey spaces and weak type discrete Morrey spaces. Our results are based on some recent results in [8].
We now recall some definitions and notation. Let . The Morrey space is defined to be the set of all functions for which
[TABLE]
is finite. The weak Morrey space is defined to be the set of all measurable functions on such that the quasi-norm
[TABLE]
is finite. From these definitions, it is clear that . The relation between and is given as follows. For every sequence , define a function by
[TABLE]
Then, the discrete Morrey space can be realized as a closed subspace of in the following sense.
Theorem 3.1**.**
[8, Theorem 2.1]* Let . Then, there exist positive constants and such that*
[TABLE]
for every and
[TABLE]
for every sequence .
The analogous result for weak type discrete Morrey spaces is presented in the following.
Theorem 3.2**.**
[8, Theorem 2.3]* Let . Then, there exist positive constants and such that for every and for every sequence the inequalities*
[TABLE]
and
[TABLE]
hold.
We apply Theorems 3.1 and 3.2 with inclusion of Morrey spaces to recover the inclusion of discrete Morrey spaces obtained in [3].
Corollary 3.3**.**
[3]* Let . Then and .*
Proof.
Our proof is an alternative to that in [3]. Let . Then, it follows from (3.1) that . Since , we have , so that . According to (3.1), we have . Thus, . To prove that , let . Then, as a consequence of (3.3), we have . The inclusion implies . By virtue of (3.4), we have . Thus, . ∎
As a consequence of Theorems 2.1, 2.10, 3.1, and 3.2, we may recover a necessary condition for the inclusion property of Morrey spaces and that of weak Morrey spaces obtained in [2].
Corollary 3.4**.**
[2, Theorem 1.6]* Let and .*
If , then . 2. 2.
If , then .
Proof.
Let . Then, according to Theorem 3.1, we have . Since , we obtain . This fact and (3.2) imply . Therefore, . Consequently, by virtue of Theorem 2.1, we conclude that . 2. 2.
Let . By virtue of Theorem 3.2, we have , so that . Combining this with (3.4), we obtain . Consequently, . Thus, the conclusion follows from Theorem 2.10.
∎
Finally, we present our main result in this section, namely a necessary condition for the proper inclusion relation between weak type discrete Morrey spaces and discrete Morrey spaces.
Theorem 3.5**.**
Let and . If the inclusion is proper, then .
Remark 3.6*.*
Note that Theorem 3.5 can be viewed as a complement to Theorem 2.13. The analogous result in the case of Morrey spaces can be found in [2, 5].
Proof of Theorem 3.5.
Assume to the contrary that . Let . It follows from (3.1) that . Since , we have , so that . Combining this with (3.2), we get . Consequently, . This contradicts the fact that . Thus, we must have . ∎
Acknowledgement. The first and second authors are supported by P3MI–ITB Research and Innovation Program 2018.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 5[5] H. Gunawan, D.I. Hakim, E. Nakai, and Y. Sawano, “On inclusion relation between weak Morrey spaces and Morrey spaces”, Nonlinear Anal. 168 (2018), 27-31.
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