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CONTACT:   

Abdulloh Jaelani,           abdjae@fst.unair.ac.id           Department of Mathematics, Faculty of Science and 

Technology, Universitas Airlangga, Surabaya, Indonesia  

 

       The article can be accessed here. https://doi.org/10.15642/mantik.2022.8.1.36-44 

Relation of Morrey Sequence Spaces, Weak Type Morrey 

Sequence Spaces, and Sequence Spaces   
 

 

Pragasto Aji Hendro Puadi1*, Eridani1, Abdulloh Jaelani1* 

1Department of Mathematics, Universitas Airlangga, Surabaya, Indonesia 

 

 

Article history: 

Received Apr 26, 2022 

Revised, Jun 5, 2022 

Accepted, Jun 28, 2022 

 

Kata Kunci:  

Ruang Barisan, 

Ruang Morrey 

Ruang Morrey Tipe 

Lemah 

 

Abstrak. Ruang Morrey merupakan ruang yang cukup penting dan 

banyak dikaji dalam banyak cabang matematika. Ruang Morrey tipe 

lemah merupakan ruang berorde semu dan memiliki sifat dasar yang 

sama dengan ruang barisan Morrey.Pada artikel ini diselidiki sifat 

elementer antara ruang barisan Morrey dan ruang barisan Morrey 

tipe lemah. Selanjutnya ditunjukkan hubungannya antara ruang 

barisan Morrey dan ruang barisan Morrey tipe lemah dengan ruang 

barisan. Berdasarkan hasil pembahasan diperoleh sifat elementer 

ruang barisan Morrey ℓ𝑝 ⊂ ℓ𝑞
𝑝

 dengan 1 ≤ 𝑝 ≤ 𝑞 < ∞ dan ℓ𝑞
𝑝2 ⊆

ℓ𝑞
𝑝1  dengan 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. Sifat elementer dari ruang 

barisan Morrey tipe lemah adalah ℓ𝑞
𝑝

⊆ 𝜔ℓ𝑞
𝑝

 dengan 1 ≤ 𝑝 ≤ 𝑞 <

∞, 𝜔ℓ𝑞
𝑝2 ⊆ 𝜔ℓ𝑞

𝑝1  dengan 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ dan ruang barisan 

Morrey tipe lemah merupakan ruang quasi norma. Lebih lanjut 

hubungan ruang barisan Morrey, ruang barisan Morrey lemah dan 

ruang barisan adalah ℓ𝑝 ⊂ ℓ𝑞
𝑝

⊆ 𝜔ℓ𝑞
𝑝
. 

 

 

Keywords:  

Sequence Spaces, 

Morrey Spaces,  

Weak Type Morrey 

Spaces,  

 

 

 

Abstract. Morrey space is a space that important spaces and widely 

studied in many branches of mathematics. Weak type Morrey spaces 

is a quasinormed spaces and have alike elementary properties with 

Morrey sequence spaces. This article investigates some properties 

the elementary properties of Morrey sequence spaces and weak type 

Morrey sequence space. Next is show relation Morrey sequence 

spaces and weak type Morrey sequence space with sequence spaces. 

Based on the results of the discussion, it was obtained that the 

elementary properties of Morrey sequence spaces is ℓ𝑝 ⊂ ℓ𝑞
𝑝
 with 

1 ≤ 𝑝 ≤ 𝑞 < ∞ and ℓ𝑞
𝑝2 ⊆ ℓ𝑞

𝑝1  with 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. The 

elementary properties of weak type Morrey sequence spaces 𝜔ℓ𝑞
𝑝
 is 

ℓ𝑞
𝑝

⊆ 𝜔ℓ𝑞
𝑝
 with 1 ≤ 𝑝 ≤ 𝑞 < ∞, 𝜔ℓ𝑞

𝑝2 ⊆ 𝜔ℓ𝑞
𝑝1  with 1 ≤ 𝑝1 ≤ 𝑝2 ≤

𝑞 < ∞  and  weak type Morrey sequence spaces is quasinormed 
space. Furthermore, the relation of Morrey sequence spaces, weak 

type Morrey sequence spaces and sequence spaces is ℓ𝑝 ⊂ ℓ𝑞
𝑝

⊆

𝜔ℓ𝑞
𝑝
 

  

How to cite: 

P. A. H. Puadi, Eridani, & A. Jaelani, “Relation of Morrey Sequence Spaces, Weak Type Morrey 

Sequence Spaces, and Sequence Spaces”. J. Mat Mantik, vol. 8, no. 1, pp. 36-44, Jun. 2022. 

  

Jurnal Matematika MANTIK 

Vol. 8, No. 1, June 2022, pp. 36-44 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:abdjae@fst.unair.ac.id
https://doi.org/10.15642/mantik.2021.7.1.9-19
http://u.lipi.go.id/1458103791


P. A. H. Puadi, Eridani, & A. Jaelani 

Relation of Morrey Sequence Spaces, Weak Type Morrey Sequence Spaces, and Sequence Spaces 

37 

1. Introduction 

Morrey spaces were first introduced by Charles Bardfield Morrey Jr. (1907-1984) on 

1938. Morrey spaces became an important space in many branches of mathematics even 

though it was first discovered to solve partial differential equations, now there are hundreds 

of articles and journals that discuss Morrey space to take up the recent development of 

Morrey Spaces [1], [2].  

On 2016, it is defined that Morrey sequence space is denoted by ℓ𝑞
𝑝

 with 1 ≤ 𝑝 ≤ 𝑞 <

 ∞ is a set of all real sequence 𝑥 =  〈𝑥𝑘 〉𝑘∈ℤ that holds 

 ‖𝑥‖ ℓ𝑞
𝑝 =  sup 

𝑁
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

<  ∞ 

with 𝑁 ∈  ℕ, 𝑆𝑁 = {−𝑁, −(𝑁 − 1), … , 0, … , 𝑁 − 1, 𝑁} and |𝑆𝑁 | denote the cardinality of 
𝑆𝑁 [3], [4]. Sequence spaces denote by ℓ

𝑝 = ℓ𝑝
𝑝

 with 𝑝 = 𝑞 is a set of all sequences that 

holds   

 ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ ℤ < ∞ and  𝑝 as parameter [5]–[7] . 
Morrey sequence space ℓ𝑞

𝑝
 has a very close relation with sequence space ℓ𝑝. One of 

the Morrey sequence space’s elementary properties is  ‖𝑥‖ ℓ𝑞
𝑝

 
≤ ‖𝑥‖ ℓ 𝑝  

for all 𝑥 ∈ ℓ𝑝, hence  ℓ𝑝 ⊆  ℓ𝑞
𝑝

 for 1 ≤ 𝑝 ≤ 𝑞 < ∞. In other words, Morrey sequence space 

ℓ𝑞
𝑝
 is an extension of sequence space ℓ𝑝.  Morrey sequence space ℓ𝑞

𝑝
  can be more extend 

to weak type Morrey sequence space ωℓ𝑞
𝑝
 [8]–[10]. Weak type Morrey sequence spaces is 

a set of all real sequence 𝑥 =  〈𝑥𝑘 〉𝑘∈ℤ that holds 

‖𝑥‖ω ℓ𝑞
𝑝 =  sup 

𝑁∈ℕ,𝛾>0
|𝑆𝑁 |

1
𝑞

 − 
1
𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|

1/𝑝
 

<  ∞. 

Weak type Morrey sequence spaces is a quasinormed spaces and have alike 

elementary properties with Morrey sequence spaces. Based on the description above, we 

will discuss about Morrey sequence spaces’ elementary properties and weak type Morrey 

sequence spaces and also the relation between sequence space ℓp, Morrey sequence space 
ℓ𝑞

𝑝
 dan weak type Morrey sequence space ωℓ𝑞

𝑝
.  

 

Definition [11]  

Quasinorm ‖∙ ‖ on a vector space 𝑉 over a field ℝ is a function from 𝑉 to [0, ∞) and holds 
(i) ‖𝑢‖ = 0 if and only if 𝑢 = 0 
(ii) ‖𝑟𝑢‖ =  |𝑟|‖𝑢‖ for every 𝑟 ∈ ℝ and 𝑢 ∈ 𝑉 
(iii) There exists 𝐶 ≥ 1 so that if 𝑢, 𝑣 ∈ 𝑉 then ‖𝑢 + 𝑣‖ ≤  𝐶(‖𝑢‖ +  ‖𝑣‖) 

if ‖   ‖ is a quasinorm and (𝑉, ‖∙ ‖) is a quasinormed space. 
 

Theorem [12] (Minkowski inequality) 

If 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 𝑝 ≥ 1 then 
 (∑ |𝑥𝑘 +  𝑦𝑘 |

𝑝𝑛
𝑘=1 )

1/𝑝 ≤  (∑ |𝑥𝑘 |
𝑝𝑛

𝑘=1 )
1/𝑝 +  (∑ |𝑦𝑘 |

𝑝𝑛
𝑘=1 )

1/𝑝.  
 

Theorem [12] 

If 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 0 < 𝑝 < 1 then  
 ∑ |𝑥𝑘 + 𝑦𝑘 |

𝑝𝑛
𝑘=1 ≤  ∑ |𝑥𝑘 |

𝑝𝑛
𝑘=1 +  ∑ |𝑦𝑘 |

𝑝𝑛
𝑘=1 . 

 

Theorem [12] (Hölder inequality)  

If 𝑥1, 𝑥2, … , 𝑥𝑛 , 𝑦1, 𝑦2, … , 𝑦𝑛 ∈ ℝ and 𝑝, 𝑞 is an exponent conjugation then  
 ∑ |𝑥𝑘 𝑦𝑘 |

𝑛
𝑘=1 ≤  (∑ |𝑥𝑘 |

𝑝𝑛
𝑘=1 )

1/𝑝(∑ |𝑦𝑘 |
𝑞𝑛

𝑘=1 )
1/𝑞 . 

 

 

 



Jurnal Matematika MANTIK 

Vol. 8, No. 1, June 2022, pp.36-44 
 

 

38 

Definition [12]  

Suppose that 𝑝 ∈ (0, ∞). ℓ𝑝 is a set of all sequence 𝑥 ∶ ℕ →  ℝ that holds ∑ |𝑥𝑘 |
𝑝∞

𝑖=1  

convergent or 

                                    ℓ𝑝 =  {𝑥 = 〈𝑥𝑛 〉  ∶  ∑ |𝑥𝑘 |
𝑝∞

𝑘=1  <  ∞}. 
But on this article, ℓ𝑝 sequence space is a set of all sequence 𝑥 ∶ ℤ →  ℝ that holds 
                                       ∑ |𝑥𝑘 |

𝑝
𝑘∈ℤ =  ∑ |𝑥𝑘 |

𝑝∞
−∞ <  ∞. 

 

Definition [3]  

Suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞, 𝑁 ∈ ℕ, 𝑆𝑁 =  {−𝑁, −(𝑁 − 1), … , 0, … , 𝑁 − 1, 𝑁} and 
|𝑆𝑁 | = 2𝑁 + 1 denote the cardinality of 𝑆𝑁. Let ℓ𝑞

𝑝
 denote Morrey sequence space is a set 

of all sequence 〈𝑥𝑘 〉𝑘∈ℤ that holds 

                                       ‖𝑥‖ ℓ𝑞
𝑝 =  sup 

𝑁
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

<  ∞. 

 

Definition [3]  

Suppose that  1 ≤ 𝑝 ≤ 𝑞 < ∞. Let 𝜔ℓ𝑞
𝑝
 denote weak type Morrey sequence space is a set 

of all sequence 〈𝑥𝑘 〉𝑘∈ℤ that holds 

                      ‖𝑥‖ω ℓ𝑞
𝑝 =  sup 

𝑁∈ℕ,𝛾>0
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝

 

<  ∞. 

  

2. Methods 

We used the properties of definition quasinorm, Minkowski Inequality, Hölder 

inequality, and characteristics of norm on the set of real [4], [13]–[15] to obtain properties 

of Morrey sequence spaces, weak type Morrey sequence space, and relation Morrey 

sequence spaces and weak type Morrey sequence space with sequence spaces.  

 

3. Results and Discussion 

Proposition 

Morrey Sequence Space ℓ𝑞
𝑝

 is a vector space over ℝ. 

Proof : 

Suppose that 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ, 𝑦 = 〈𝑦𝑘 〉𝑘∈ℤ is an element of ℓ𝑞
𝑝
. 

For every 𝑎 ∈ ℝ, 

                                   ‖𝑎𝑥‖ ℓ𝑞
𝑝 = sup 

𝑁
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 (∑ |𝑎𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

                                                   = |𝑎| sup 
𝑁

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

                                                 = |𝑎|‖𝑥‖ ℓ𝑞
𝑝 < ∞              

Then 𝑎𝑥 ∈ ℓ𝑞
𝑝

. 

With Minkowski Inequality we got 

 (∑ |𝑥𝑘 + 𝑦𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
≤  (∑ |𝑥𝑘 |

𝑝
𝑘 ∈ 𝑆𝑁

)
1/𝑝

+ (∑ |𝑦𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

                                           ≤  |𝑆𝑁 |
1

𝑞
 − 

1

𝑝(∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
+  |𝑆𝑁 |

1

𝑞
 − 

1

𝑝(∑ |𝑦𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
 

                     ‖𝑥 + 𝑦‖ ℓ𝑞
𝑝 ≤  ‖𝑥‖ ℓ𝑞

𝑝 +  ‖𝑦‖ ℓ𝑞
𝑝 . 

If ‖𝑥‖ ℓ𝑞
𝑝 < ∞ and ‖𝑦‖ ℓ𝑞

𝑝 < ∞, then ‖𝑥 + 𝑦‖ ℓ𝑞
𝑝 < ∞ so that 𝑥 + 𝑦 ∈ ℓ𝑞

𝑝
. ℓ𝑞

𝑝
 is closed 

under the operation (+). 

Let 𝑧 ∈  ℓ𝑞
𝑝
 then  

(i) 𝑥 + 𝑦 =  〈𝑥𝑘 + 𝑦𝑘 〉𝑘∈ℤ =  〈𝑦𝑘 + 𝑥𝑘 〉𝑘∈ℤ = 𝑦 + 𝑥 for every 𝑥, 𝑦 ∈  ℓ𝑞
𝑝

 



P. A. H. Puadi, Eridani, & A. Jaelani 

Relation of Morrey Sequence Spaces, Weak Type Morrey Sequence Spaces, and Sequence Spaces 

39 

(ii) 𝑥 + (𝑦 + 𝑧) = (𝑥 + 𝑦) + 𝑧 for every 𝑥, 𝑦, 𝑧 ∈  ℓ𝑞
𝑝

 

(iii)  There exists 0 = 〈… ,0,0,0, … . 〉, then 0 + 𝑥 = 𝑥 + 0 for every  𝑥 ∈ ℓ𝑞
𝑝

 

(iv)  There exists −𝑥 = 〈−𝑥𝑘 〉𝑘∈ℤ, then (−𝑥) + 𝑥 = 𝑥 + (−𝑥) = 0 for every 𝑥 ∈ ℓ𝑞
𝑝
 

(v) Let 𝑏 ∈ ℝ. 
𝑎(𝑥 + 𝑦) = 𝑎𝑥 + 𝑎𝑦 
(𝑎 + 𝑏)𝑥 = 𝑎𝑥 + 𝑏𝑥 

(𝑎𝑏)𝑥 = 𝑎(𝑏𝑥) 
for every 𝑥 ∈ ℓ𝑞

𝑝
 

(vi)  There exists 1 ∈ ℝ so that 1𝑥 = 𝑥 for every 𝑥 ∈ ℓ𝑞
𝑝

 

thus ℓ𝑞
𝑝

 is a vector space over ℝ.∎ 

 

Example:  

Suppose that 1 ≤ 𝑝 < 𝑞 < ∞. A sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 =  |𝑘|
−𝑞/𝑝 for 𝑘 ≠ 0 and 

𝑥𝑘 = 0 for 𝑘 = 0. 
 

 

For every 𝑁 ∈ ℕ, 

                                           ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
= 2 ∑

1

𝑘𝑞
<𝑁𝑘=1 2 ∑

1

𝑘𝑞
∞
𝑘=1 .                                     (1) 

 

                  ∑
1

𝑘𝑞
∞
𝑘=1 = 1 + (

1

2𝑞
+

1

3𝑞
) + (

1

4𝑞
+

1

5𝑞
+

1

6𝑞
+

1

7𝑞
) + ⋯ 

                                    < 1 +
2

2𝑞
+

4

4𝑞
+

8

8𝑞
+ ⋯ = 1 +  

1

2𝑞−1
+

1

4𝑞−1
+

1

8𝑞−1
+ ⋯ 

If 𝑞 > 1 then 0 <
1

2𝑞−1
< 1 implies that 

                                                              ∑
1

𝑘𝑞
∞
𝑘=1 <

1

1−
1

2𝑞−1

.                                                       (2) 

From (1) and (2) obtained 

    |𝑆𝑁 |
𝑝

𝑞
−1

∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
= (

1

|𝑆𝑁|
1−

𝑝
𝑞

) ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
< ∑ |𝑥𝑘 |

𝑝
𝑘 ∈ 𝑆𝑁

<
2

1−
1

2𝑞−1

 

               (|𝑆𝑁|
𝑝

𝑞
−1

∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝

< (
2𝑞

2𝑞−1−1
)

1/𝑞

 

     sup 
𝑁

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
≤ (

2𝑞

2𝑞−1−1
)

1/𝑞

. 

Thus 𝑥 ∈ ℓ𝑞
𝑝

. ∎ 

 

Theorem   

Suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞. ‖𝑥‖ ℓ𝑞
𝑝 ≤  ‖𝑥‖ℓp  for every 𝑥 ∈ ℓ

𝑝. 

Proof : 

For all 𝑁 ∈ ℕ, it’s easy to see that 0 < |𝑆𝑁 |
1

𝑞
−

1

𝑝 ≤ 1 then 

                               |𝑆𝑁 |
1

𝑞
−

1

𝑝(∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
≤  (∑ |𝑥𝑘 |

𝑝
𝑘 ∈ 𝑆𝑁

)
1/𝑝

 

and 

                   sup 
𝑁

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
≤ sup

𝑁
 (∑ |𝑥𝑘 |

𝑝
𝑘 ∈ 𝑆𝑁

)
1/𝑝

 

 

                                                             ‖𝑥‖ ℓ𝑞
𝑝 ≤  ‖𝑥‖ℓp . ∎ 

  



Jurnal Matematika MANTIK 

Vol. 8, No. 1, June 2022, pp.36-44 
 

 

40 

Example : 

Let 1 ≤ 𝑝 < 𝑞 < ∞ and a sequence 𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 =  |𝑘|
−1/𝑞 for 𝑘 ≠ 0 and 𝑥𝑘 = 0 

for 𝑘 = 0. 

If  0 <
𝑝

𝑞
< 1 then 

                                    ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ ℤ = 2 ∑
1

𝑘𝑝/𝑞
∞
𝑘=1 ≥ 2 ∑

1

𝑘
∞
𝑘=1  . 

 

We know that ∑
1

𝑘
∞
𝑘=1  is harmonic series then it’s divergent and  ∑ |𝑥𝑘 |

𝑝
𝑘 ∈ 𝑆𝑁

 divergent 

thus  𝑥 ∉ ℓ𝑝. 

For all 𝑁 ∈ ℕ,    ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
= ∑

1

|𝑘|𝑝/𝑞
 
𝑘 ∈ 𝑆𝑁,𝑘≠0

= 2 ∑
1

𝑘𝑝/𝑞
𝑁
𝑘=1 . 

For every 𝑝 and 𝑞 with 1 ≤ 𝑝 < 𝑞 < ∞, function 𝑦 = 𝑥−𝑝/𝑞 has 𝑦′ < 0 and 𝑦′′ > 0 on 
𝑥 ≥ 1. 

 
Figure 1. Area of partition below 𝑦 = 𝑥−𝑝/𝑞  

 

We know that the sum of areas of partitions equal to 

  ∑
1

𝑘𝑝/𝑞
𝑁
𝑘=2  

And  

        ∑
1

𝑘𝑝/𝑞
𝑁
𝑘=2 ≤ ∑

1

𝑘𝑝/𝑞
𝑁
𝑘=1 = 1 + ∑

1

𝑘𝑝/𝑞
𝑁
𝑘=2 ≤  1 + ∫

1

𝑥𝑝/𝑞
 𝑑𝑥

𝑁

1
 

                                                                                  ≤  1 −
𝑞

𝑞−𝑝
+  

𝑞

𝑞−𝑝
𝑁

1−
𝑝

𝑞 . 

Hence, 

                                       |𝑆𝑁 |
𝑝

𝑞
−1

∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
≤ 2 (

 1−
𝑞

𝑞−𝑝
+ 

𝑞

𝑞−𝑝
𝑁

1−
𝑝
𝑞

(2𝑁+1)
1−

𝑝
𝑞

) 

 

and 

                          lim
𝑁→∞

2 (
 1−

𝑞

𝑞−𝑝
+ 

𝑞

𝑞−𝑝
𝑁

1−
𝑝
𝑞

(2𝑁+1)
1−

𝑝
𝑞

) = 2 (
𝑞

𝑞−𝑝
2

𝑝

𝑞
−1

) =
𝑞

𝑞−𝑝
2

𝑝

𝑞   

 

                              |𝑆𝑁 |
1

𝑞
−

1

𝑝(∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝

≤ (2 (
 1−

𝑞

𝑞−𝑝
+ 

𝑞

𝑞−𝑝
𝑁

1−
𝑝
𝑞

(2𝑁+1)
1−

𝑝
𝑞

))

1/𝑝

 

                   sup 
𝑁

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘 ∈ 𝑆𝑁
)

1/𝑝
≤ 21/𝑝 (

𝑞

𝑞−𝑝
)

1/𝑝
 

thus 𝑥 ∈ ℓ𝑞
𝑝

.∎ 

 

Theorem  

Suppose that 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞. For all 𝑥 ∈ ℓ𝑞
𝑝2  applies that 



P. A. H. Puadi, Eridani, & A. Jaelani 

Relation of Morrey Sequence Spaces, Weak Type Morrey Sequence Spaces, and Sequence Spaces 

41 

‖𝑥‖
 ℓ𝑞

𝑝1 ≤  ‖𝑥‖ ℓq
p2 . 

Proof : 

By Hölder inequality, we have  

                                       ∑ |𝑥𝑘 |
𝑝1

𝑘∈𝑆𝑁
≤  (∑ |𝑥𝑘 |

𝑝2
𝑘∈𝑆𝑁

)
𝑝1
𝑝2 (∑ 1𝑘∈𝑆𝑁 )

1−
𝑝1
𝑝2       

                             (
1

|𝑆𝑁|
∑ |𝑥𝑘 |

𝑝1
𝑘∈𝑆𝑁

)

1

𝑝1 ≤ (
1

|𝑆𝑁|
∑ |𝑥𝑘 |

𝑝2
𝑘∈𝑆𝑁

)

1

𝑝2                                          

                     |𝑆𝑁 |
1

𝑞
−

1

𝑝1 (∑ |𝑥𝑘 |
𝑝1

𝑘∈𝑆𝑁
)

1

𝑝1 ≤  |𝑆𝑁 |
1

𝑞
−

1

𝑝2 (∑ |𝑥𝑘 |
𝑝2

𝑘∈𝑆𝑁
)

1

𝑝2                            

  
                                                ‖𝑥‖

 ℓ𝑞
𝑝1 ≤  ‖𝑥‖ ℓq

p2 . ∎    

From the above theorem, we have ℓ𝑞
𝑝2 ⊆ ℓ𝑞

𝑝1 on 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ but we can’t 

approve ℓ𝑞
𝑝2 ⊂ ℓ𝑞

𝑝1  yet [3]. 

 

Theorem  

If 1 ≤ 𝑝 ≤ 𝑞 < ∞ then ‖𝑥‖𝜔ℓ𝑞
𝑝 ≤ ‖𝑥‖ℓ𝑞

𝑝 

For every 𝑥 ∈ ℓ𝑞
𝑝

. 

Proof: 

 |𝑆𝑁 |
1

𝑞
−

1

𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 =  |𝑆𝑁 |

1

𝑞
−

1

𝑝|𝛾𝑝{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 

                                                              = |𝑆𝑁 |
1

𝑞
−

1

𝑝(∑ 𝛾𝑝𝑘∈𝑆𝑁 ,|𝑥𝑘|>𝛾 )
1/𝑝

 

                                                                ≤ |𝑆𝑁 |
1

𝑞
−

1

𝑝(∑ |𝑥𝑘 |
𝑝

𝑘∈𝑆𝑁,|𝑥𝑘|>𝛾 
)

1/𝑝
 

 |𝑆𝑁 |
1

𝑞
−

1

𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 ≤ |𝑆𝑁 |

1

𝑞
−

1

𝑝(∑ |𝑥𝑘 |
𝑝

𝑘∈𝑆𝑁  
)

1/𝑝
                     

supremum over 𝑁 ∈ ℕ and 𝛾 > 0 on the above inequality, we have 

 sup 
𝑁∈ℕ,𝛾>0

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 ≤ sup 

𝑁
|𝑆𝑁 |

1

𝑞
−

1

𝑝 (∑ |𝑥𝑘 |
𝑝

𝑘∈𝑆𝑁
)

1/𝑝
  

                                                                  ‖𝑥‖𝜔ℓ𝑞
𝑝 ≤ ‖𝑥‖ℓ𝑞

𝑝 . 

 

Thus ℓ𝑞
𝑝

⊆ 𝜔ℓ𝑞
𝑝
 ∎. 

 

Example : 

Suppose that 1 ≤ 𝑝 ≤ 𝑞 < ∞. A sequence  𝑥 = 〈𝑥𝑘 〉𝑘∈ℤ with 𝑥𝑘 =  |𝑘|
−1/𝑝 for 𝑘 ≠ 0 and 

𝑥𝑘 = 0 for 𝑘 = 0. 

   ∑ |𝑥𝑘 |
𝑝

𝑘 ∈ ℤ = 2 ∑
1

𝑘
∞
𝑘=1 . 

We know that ∑
1

𝑘
∞
𝑘=1  is divergent then 𝑥 ∉ ℓ

𝑝 but for any 𝛾 > 0, 

          𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑘|
−1/𝑝 > 𝛾}|

1/𝑝
= 2𝛾|{𝑘 ∈ ℕ ∶ 1 ≤ 𝑘 < 𝑁, 𝑘−1/𝑝 > 𝛾}|

1/𝑝
 

and  𝑘−1/𝑝 > 𝛾 ⟹ 𝑘−1 > 𝛾𝑝 ⟹ 𝑘 <
1

𝛾𝑝
 then 

     𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑘|
−1/𝑝 > 𝛾}|

1/𝑝
< 2𝛾 (

1

𝛾𝑝
)

1/𝑝
 

  𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑘|
−1/𝑝 > 𝛾}|

1/𝑝
< 2𝛾 (

1

𝛾
) = 2 

then 𝑥 ∈ 𝜔ℓ𝑝
𝑝

  and ℓ𝑝 ⊂ 𝜔ℓ𝑝
𝑝

= 𝜔ℓ𝑝. 

 

Theorem 

If 1 ≤ 𝑝 ≤ 𝑞 < ∞ then ‖   ‖𝜔ℓ𝑞
𝑝 is a  quasinorm and (𝜔ℓ𝑞

𝑝
, ‖   ‖𝜔ℓ𝑞

𝑝  ) is a quasinormed 

space. 



Jurnal Matematika MANTIK 

Vol. 8, No. 1, June 2022, pp.36-44 
 

 

42 

Proof: 

From the definition, we know that ‖𝑥‖𝜔ℓ𝑞
𝑝 ≥ 0 for all 𝑥 ∈ 𝜔ℓ𝑞

𝑝
. 

If 𝑥 = 0 (𝑥𝑘 = 0 for all 𝑘 ∈ ℤ) then {𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾} is an empty space and 
|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}| = 0, 

     |𝑆𝑁 |
1

𝑞
 − 

1

𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 = 0 

          sup 
𝑁∈ℕ,𝛾>0

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}|
1/𝑝 = 0               

           ‖𝑥‖𝜔ℓ𝑞
𝑝 = 0. 

If ‖𝑥‖𝜔ℓ𝑞
𝑝 = 0 then |{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝛾}| = 0 for all 𝑁 ∈ ℕ and 𝛾 > 0, hence for every 

𝑘 ∈ ℤ applies that 
0 ≤ |𝑥𝑘 | ≤  𝛾 

⟹𝑥𝑘 = 0 untuk setiap 𝑘 ∈ ℤ (𝑥 = 0). 
Let 𝑥 ∈ 𝜔ℓ𝑞

𝑝
 and 𝑟 = 0 then ‖𝑟𝑥‖𝜔ℓ𝑞

𝑝 = |𝑟|‖𝑥‖𝜔ℓ𝑞
𝑝 = 0. If 𝑟 ≠ 0 then applies that  

  ‖𝑟𝑥‖𝜔ℓ𝑞
𝑝 = sup 

𝑁∈ℕ,𝛾>0
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑟𝑥𝑘 | > 𝛾}|
1/𝑝 

                                = sup 
𝑁∈ℕ,𝛾>0

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 𝛾 |{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | >
𝛾

|𝑟|
}|

1/𝑝
 

let  
𝛾

|𝑟|
= 𝑎 ⟹ 𝛾 = |𝑟|𝑎 then 

  ‖𝑟𝑥‖𝜔ℓ𝑞
𝑝  = sup 

𝑁∈ℕ,𝑎>0
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝 |𝑟|𝑎|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝑎}|
1/𝑝 

                                = |𝑟| sup 
𝑁∈ℕ,𝑎>0

|𝑆𝑁 |
1

𝑞
 − 

1

𝑝 𝑎|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | > 𝑎}|
1/𝑝 

                  = |𝑟|‖𝑥‖𝜔ℓ𝑞
𝑝 .                                               

Let  𝑥, 𝑦 ∈ 𝜔ℓ𝑞
𝑝

. For any 𝑁 ∈ ℕ, if 𝑘 ∈ 𝑆𝑁 dan |𝑥𝑘 | ≤ |𝑦𝑘 | then 

|𝑥𝑘 + 𝑦𝑘 | ≤ |𝑥𝑘 | + |𝑦𝑘 | ≤ 2|𝑦𝑘 | 
else if 𝑘 ∈ 𝑆𝑁 dan |𝑥𝑘 | > |𝑦𝑘 | then 

 |𝑥𝑘 + 𝑦𝑘 | ≤ |𝑥𝑘 | + |𝑦𝑘 | ≤ 2|𝑥𝑘 |. 
Thus  

{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 + 𝑦𝑘 | > 𝛾} ⊆ {𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | + |𝑦𝑘 | > 𝛾}                               
                                                      ⊆ {𝑘 ∈ 𝑆𝑁 ∶  2|𝑥𝑘 | > 𝛾} ∪ {𝑘 ∈ 𝑆𝑁 ∶  2|𝑦𝑘 | > 𝛾}. 

For any 𝛾 > 0 dan 𝑁 ∈ ℕ, 
|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 + 𝑦𝑘 | > 𝛾}| ≤  |{𝑘 ∈ 𝑆𝑁 ∶  2|𝑥𝑘 | > 𝛾}| + |{𝑘 ∈ 𝑆𝑁 ∶  2|𝑦𝑘 | > 𝛾}| 

Multiply both sides by  (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝛾)
𝑝

 applies that 

  (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝛾)
𝑝

|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 + 𝑦𝑘 | > 𝛾}|                                                         

               ≤ (|𝑆𝑁|
1

𝑞
−

1

𝑝𝛾)
𝑝

|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 | >
𝛾

2
}| + (|𝑆𝑁 |

1

𝑞
−

1

𝑝𝛾)
𝑝

|{𝑘 ∈ 𝑆𝑁 ∶  |𝑦𝑘 | >
𝛾

2
}| 

Let 
𝛾

2
= 𝜎 ⟹ 𝛾 = 2𝜎  

  |𝑆𝑁 |
1

𝑞
−

1

𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 + 𝑦𝑘 | > 𝛾}|
1/𝑝                                                                     

    ≤ 2 ((|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎)
𝑝

|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}| + (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎)
𝑝

|{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|)

1/𝑝

. 

If 𝑝 > 1 ⟹  
1

𝑝
< 1then  

 |𝑆𝑁 |
1
𝑞

−
1
𝑝𝛾|{𝑘 ∈ 𝑆𝑁 ∶  |𝑥𝑘 + 𝑦𝑘 | > 𝛾}|

1/𝑝                                                                         



P. A. H. Puadi, Eridani, & A. Jaelani 

Relation of Morrey Sequence Spaces, Weak Type Morrey Sequence Spaces, and Sequence Spaces 

43 

  ≤ 2 ((|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎)
𝑝

|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}| + (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎)
𝑝

|{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|)

1/𝑝

  

  ≤ 2 (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝜎}|)
1/𝑝

+ 2 (|𝑆𝑁 |
1

𝑞
−

1

𝑝𝜎|{𝑘 ∈ 𝑆𝑁 : |𝑦𝑘 | > 𝜎}|)
1/𝑝

   

taking supremum  over 𝛾 > 0 dan 𝑁 ∈ ℕ from the above inequality, we get 
‖𝑥 + 𝑦‖𝜔ℓ𝑞

𝑝 ≤ 2‖𝑥‖𝜔ℓ𝑞
𝑝 + 2‖𝑦‖𝜔ℓ𝑞

𝑝  

                           = 2 (‖𝑥‖𝜔ℓ𝑞
𝑝 + ‖𝑦‖𝜔ℓ𝑞

𝑝 ). 

Thus  ‖    ‖𝜔ℓ𝑞
𝑝  is a quasinorm and (𝜔ℓ𝑞

𝑝
, ‖   ‖𝜔ℓ𝑞

𝑝  ) is a quasinormed space.∎ 

 

Theorem  

If  1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 then ‖𝑥‖𝜔ℓ𝑞
𝑝1 ≤ ‖𝑥‖𝜔ℓ𝑞

𝑝2  

for all 𝑥 ∈ 𝜔ℓ𝑞
𝑝2 . 

Proof : 

From the definition, for all 𝑥 ∈ ‖𝑥‖
𝜔ℓ𝑞

𝑝2  and any 𝑁 ∈ ℕ. 

 |𝑆𝑁 |
1

𝑞
−

1

𝑝2 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}|
1/𝑝2 ≤ sup 

𝑁∈ℕ,𝛾>0
|𝑆𝑁 |

1

𝑞
 − 

1

𝑝2 𝛾|{𝑘 ∈ 𝑆𝑁 : |𝑥𝑘 | > 𝛾}|
1/𝑝2  

                       = ‖𝑥‖
𝜔ℓ𝑞

𝑝2 . 

And it is equivalent to 

                                                              𝛾 ≤
|𝑆𝑁|

1
𝑝2

−
1
𝑞

|{𝑘∈𝑆𝑁∶|𝑥𝑘|>𝛾}|
1/𝑝2

‖𝑥‖
𝜔ℓ𝑞

𝑝2  

 |𝑆𝑁 |
1

𝑞
−

1

𝑝1 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}|
1/𝑝1 ≤

|𝑆𝑁 |
1

𝑝2
−

1
𝑝1

|{𝑘∈𝑆𝑁 ∶|𝑥𝑘|>𝛾}|
1

𝑝2
−

1
𝑝1

‖𝑥‖
𝜔ℓ𝑞

𝑝2  

 |𝑆𝑁 |
1

𝑞
−

1

𝑝1 𝛾|{𝑘 ∈ 𝑆𝑁 ∶ |𝑥𝑘 | > 𝛾}|
1/𝑝1 ≤ (

|{𝑘∈𝑆𝑁∶|𝑥𝑘|>𝛾}|

|𝑆𝑁|
)

1

𝑝1
−

1

𝑝2 ‖𝑥‖
𝜔ℓ𝑞

𝑝2  

                          ≤ ‖𝑥‖
𝜔ℓ𝑞

𝑝2  

taking supremum over  𝑁 ∈ ℕ and 𝛾 > 0 on the above inequality, we get 
‖𝑥‖

𝜔ℓ𝑞
𝑝1 ≤ ‖𝑥‖𝜔ℓ𝑞

𝑝2 . ∎ 

From the above inequality we have 𝜔ℓ𝑞
𝑝2 ⊆ 𝜔ℓ𝑞

𝑝1 . 
 

 

4. Conclusions 

Based on the results and discussion, we obtained some conclusions:  

1) There are some Morrey sequence space’s elementary properties : 

i. If 1 ≤ 𝑝 ≤ 𝑞 < ∞ then for all 𝑥 ∈ ℓ𝑝, we have ‖𝑥‖ ℓ𝑞
𝑝 ≤  ‖𝑥‖ℓp   and ℓ

𝑝 ⊂ ℓ𝑞
𝑝
. 

ii. If 1 ≤ 𝑝1 ≤ 𝑝2 ≤ 𝑞 < ∞ then for all 𝑥 ∈ ℓ𝑞
𝑝2 , we have  ‖𝑥‖

 ℓ𝑞
𝑝1 ≤  ‖𝑥‖ ℓq

p2  and 

ℓ𝑞
𝑝2 ⊆ ℓ𝑞

𝑝1 . 

2) There are some weak type Morrey sequence space’s elementary properties : 

i. If  1 ≤ p ≤ q < ∞ then for all x ∈ ℓq
p
, we have ‖x‖ωℓq

p ≤ ‖x‖ℓq
p  and ℓq

p
⊆ ωℓq

p
. 

ii. If  1 ≤ p ≤ q < ∞ then ‖   ‖ωℓq
p  is a quasinorm and (ωℓq

p
, ‖   ‖ωℓq

p  ) is a 

quasinormed space. 

iii. If 1 ≤ p1 ≤ p2 ≤ q < ∞ then for all x ∈ ωℓq
p2, we have ‖x‖

ωℓq
p1 ≤ ‖x‖ωℓq

p2              

and 𝜔ℓ𝑞
𝑝2 ⊆ 𝜔ℓ𝑞

𝑝1 . 



Jurnal Matematika MANTIK 

Vol. 8, No. 1, June 2022, pp.36-44 
 

 

44 

3) Based on the results in points 1 and 2 it can be concluded that ℓ𝑝 sequence space is a 
subset of Morrey Sequence Space and Morrey Sequence Space is a subset of weak 

type Morrey Sequence Space or equivalent to ℓ𝑝 ⊂ ℓ𝑞
𝑝

⊆ 𝜔ℓ𝑞
𝑝

. 

 

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