Inclusion Properties of Herz-Morrey Spaces With Variable Exponent CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(1) (2021), Pages 22-27 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: May 02, 2021 Reviewed: August 24, 2021 Accepted: October 06, 2021 DOI: https://doi.org/10.18860/ca.v7i1.12141 Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman Departement of Mathematics, Islamic State University of Maulana Malik Ibrahim Malang Email: hairur@mat.uin-malang.ac.id ABSTRACT The inclusion properties in Herz-Morrey spaces has proved by Rahman in 2020. This paper aims to discuss the inclusion of the homogeneous Herz-Morrey spaces and homogeneous weak Herz- Morrey spaces with variable exponent. We also investigated the inclusion between both spaces. This result will be useful to prove fractional integral on the homogeneous Herz-Morrey spaces with variable exponent. Keywords: Herz-Morrey spaces; inclusion properties; variable exponent. INTRODUCTION Inclusion properties or inclusion relation between spaces has received a lot of attention from researchers. It seems that many authors have studied this issue in some spaces (see [1]-[5]). Thus, this lead the author for discussing the inclusion properties especially in Herz-Morrey spaces. Herz spaces can be traced back to the work of Beurling. Beurling [6] introduced a space π’œπ‘, which is the original version of non homogeneous Herz spaces. Lu et al [7] has given the inclusion properties in homogeneous Herz spaces, as a proposition below. Proposition 1.1. Let 𝛼 ∈ ℝ, 𝑝 > 0, and π‘ž ≀ ∞. The following inclusions are valid. a. If 𝑝1 ≀ 𝑝2, then πΎπ‘ž 𝛼,𝑝1 (ℝ𝑛) βŠ‚ πΎπ‘ž 𝛼,𝑝2 (ℝ𝑛) b. If π‘ž2 ≀ π‘ž1, then πΎπ‘ž1 𝛼,𝑝 (ℝ𝑛) βŠ‚ πΎπ‘ž2 π›Όβˆ’π‘›( 1 π‘ž1 βˆ’ 1 π‘ž2 ),𝑝 (ℝ𝑛). This proposition can be proved by simply computation. In fact, if 0 < π‘Ÿ < 1, (a) is a consequence of the inequality (βˆ‘|π‘Žπ‘˜ | ∞ π‘˜=1 ) π‘Ÿ ≀ βˆ‘|π‘Žπ‘˜ | π‘Ÿ ∞ π‘˜=1 . While, (b) can be deduced directly from the HΓΆlder inequality. In 2016, Gunawan et al. (see [1] [2]) have proved the inclusion of Morrey spaces and generalized Morrey spaces. Recently, Rahman [8] also has proved the inclusion properties in Herz-Morrey spaces. These result have been motivated the author to study more about inclusion in homogenous Herz-Morrey spaces, but in this case the author uses variable exponent. Since 1991, the research of Kovacik and Rakosnik [9] motivated many researchers to study about function spaces with variable exponent in several discussion. Suppose that Ξ© βŠ‚ ℝ𝑛 is an open set, 𝑝(β‹…): Ξ© β†’ [1, ∞) is a measurable https://doi.org/10.18860/ca.v7i1.12141 mailto:hairur@mat.uin-malang.ac.id Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman 23 function and 𝐿𝑝(β‹…)(Ξ©) is denoted the set of measurable functions 𝑓 on Ξ©, such that for some positive πœ† satisfied ∫ ( | 𝑓(π‘₯) | πœ† ) 𝑝( π‘₯ ) 𝑑π‘₯ Ξ© < ∞. If 𝐿𝑝(β‹…)(Ξ©) equipped by the Luxemburg-Nakano norm β€– 𝑓 β€– 𝐿 𝑝(β‹…)(Ξ©) = inf { πœ† > 0 ∢ ∫ ( | 𝑓(π‘₯) | πœ† ) 𝑝(π‘₯) 𝑑π‘₯ Ξ© ≀ 1}, then 𝐿𝑝(β‹…)(Ξ©) becomes a Banach function spaces. Since these spaces generalize the standard 𝐿𝑝 spaces, they are also referred to as variable 𝐿𝑝 spaces. 𝐿𝑝(β‹…)(Ξ©) is isometrically isomorphic to 𝐿𝑝(Ξ©), when 𝑝(π‘₯) = 𝑝 is a constant. In 2010, the boundedness of sublinear operators on Herz-Morrey space with variable exponent ℳ�̇�𝑝(β‹…) 𝛼,π‘ž and ℳ�̇�𝑝(β‹…) οΏ½Μ…οΏ½,π‘ž was proved by Izuki [10]. Then, Xu and Yang [11] developed the definition of Herz-Morrey spaces with variabel exponents. Let 𝑝(β‹…) ∈ 𝒫(ℝ𝑛), 0 < π‘ž < ∞, 0 ≀ πœ† < ∞, and 𝛼(β‹…) is a bounded real-valued measurable function on ℝ𝑛 , the homogeneous Herz-Morrey spaces with variable exponent β„³οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) consists all functions 𝑓 ∈ πΏπ‘™π‘œπ‘ π‘ž ( ℝ𝑛 /{0} ) such that ‖𝑓‖ β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) = sup πΏβˆˆβ„€ 1 2πΏπœ† (βˆ‘ 2π‘˜π›Ό(β‹…)𝑝‖ π‘“πœ’π‘˜ β€–πΏπ‘ž (ℝ𝑛) 𝑝𝐿 π‘˜=βˆ’βˆž ) 1 𝑝(β‹…) < ∞, , where π΅π‘˜ = { π‘₯ ∈ ℝ 𝑛 : |π‘₯| ≀ 2π‘˜ }, π΄π‘˜ = π΅π‘˜ /π΅π‘˜βˆ’1 and πœ’π‘˜ = πœ’π΄π‘˜ is the characteristic function of the set π΄π‘˜ for π‘˜ ∈ β„€. As another spaces which have their weak type spaces, Herz-Morrey spaces also have their weak type spaces. For 𝛼(β‹…) ∈ ℝ𝑛 , 𝑝(β‹…) ∈ 𝒫(ℝ𝑛), 0 ≀ πœ† ≀ ∞ and 0 < π‘ž ≀ ∞, the homogeneous weak Herz-Morrey spaces with variabel exponent ( π‘Š β„³οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛)) is a set of measurable 𝑓 ∈ πΏπ‘™π‘œπ‘ π‘ž (ℝ𝑛 /{0}) which is equipped with norm such that β€– 𝑓 β€– π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) = sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝(β‹…)π‘šπ‘˜ (𝛾, 𝑓) 𝑝(β‹…) π‘ž 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝(β‹…) < ∞, where π‘šπ‘˜ ( 𝛾, 𝑓 ) = |{ π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾 }|. Some authors have investigated those spaces in various terms of discussion (see [12] - [15]). Meanwhile, this article aims to discuss in terms inclusion properties and inclusion relation of the homogeneous Herz-Morrey spaces and homogeneous weak Herz-Morrey spaces with variable exponent. RESULT AND DISCUSSION Our main results are the following: Theorem 2.1. Let 1 ≀ 𝑝1(β‹…) ≀ 𝑝2(β‹…) < π‘ž < ∞, and 𝛼(β‹…) is a bounded real-valued measurable fuction on ℝ𝑛 . Then, the inclusion β„³οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† β„³οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛), Is valid. Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman 24 Proof. We may take any 𝑓 ∈ β„³οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). Then, by using HΓΆlder inequality and 𝑝1 ≀ 𝑝2 we have ‖𝑓‖ β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) = sup πΏβˆˆπ‘ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝1(β‹…) 𝐿 π‘˜=βˆ’βˆž β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1(β‹…) ) 1 𝑝1(β‹…) ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† (( βˆ‘ (2π‘˜π›Ό(β‹…)𝑝1(β‹…)) 𝑝2(β‹…) 𝑝1(β‹…) 𝐿 π‘˜=βˆ’βˆž ) 𝑝1(β‹…) 𝑝2(β‹…) ( βˆ‘ (β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1(β‹…) ) 𝑝2(β‹…) 𝑝2(β‹…)βˆ’π‘1(β‹…) 𝐿 π‘˜=βˆ’βˆž ) 1βˆ’ 𝑝1(β‹…) 𝑝2(β‹…) ) 1 𝑝1(β‹…) ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† (( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝2(β‹…) 𝐿 π‘˜=βˆ’βˆž ) 𝑝1(β‹…) 𝑝2(β‹…) ( βˆ‘ β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1(β‹…)𝑝2(β‹…) 𝑝2(β‹…)βˆ’π‘1(β‹…) 𝐿 π‘˜=βˆ’βˆž ) 1βˆ’ 𝑝1(β‹…) 𝑝2(β‹…) ) 1 𝑝1(β‹…) ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝2(β‹…) 𝐿 π‘˜=βˆ’βˆž ( βˆ‘ β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1(β‹…)𝑝2(β‹…) 𝑝2(β‹…)βˆ’π‘1(β‹…) 𝐿 π‘˜=βˆ’βˆž ) 𝑝2(β‹…)βˆ’π‘1(β‹…) 𝑝1(β‹…) ) 1 𝑝2(β‹…) ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝2(β‹…) 𝐿 π‘˜=βˆ’βˆž β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝2(β‹…) ) 1 𝑝2(β‹…) ≀ ‖𝑓‖ β„³οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) . It is easy to know that 𝑓 ∈ β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛), where 𝛼(β‹…) ∈ (ℝ𝑛) and 𝑝(β‹…) ∈ 𝒫(ℝ𝑛). Then, we have β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† ( ℝ𝑛). By the previous theorem, the author established the following inclusions. Theorem 2.2. Let 1 ≀ 𝑝1(β‹…) ≀ 𝑝2(β‹…) < π‘ž < ∞, and 𝛼(β‹…) is a bounded real-valued measurable fuction on ℝ𝑛 , then the following inclusion is valid. πΏπ‘ž ( 𝑅𝑛) = β„³ οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼(β‹…),πœ† ( ℝ𝑛 ) βŠ† β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† ( ℝ𝑛 ) βŠ† β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† ( ℝ𝑛). Proof. Theorem 2.1 has stated that β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). Then, we only prove that πΏπ‘ž (ℝ𝑛) = β„³οΏ½Μ‡οΏ½ π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† ℳ𝐾 Μ‡ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). Let 𝑓 ∈ π‘€οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ), by using similar method as before, we get β€– 𝑓 β€– 𝑀�̇� π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)π‘ž 𝐿 π‘˜=βˆ’βˆž ((∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž (∫ |πœ’π‘˜ | π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž ) π‘ž ) 1 π‘ž ≀ sup πΏβˆˆπ‘ 1 2πΏπœ† βˆ‘ 2 π‘˜π›Ό(β‹…) 𝐿 π‘˜=βˆ’βˆž (∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž ( 2 π‘˜π‘‘ ) 1 π‘ž ≀ 𝐢 (∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman 25 ≀ β€– 𝒇 β€– 𝑳𝒒(ℝ𝒏). Hence, 𝑓 ∈ πΏπ‘ž (ℝ𝑛) and πΏπ‘ž (ℝ𝑛) βŠ† β„³ οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ). In the other hand, for any 𝑓 ∈ πΏπ‘ž (ℝ𝑛), there exist any constant 𝐢 such that 𝐢 = sup πΏβˆˆπ‘ 1 2πΏπœ† βˆ‘ 2 π‘˜π›Ό(β‹…) + π‘˜π‘‘ π‘žπΏ π‘˜=βˆ’βˆž . Consequently, we have 𝑓 ∈ β„³ οΏ½Μ‡οΏ½ π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) and β„³οΏ½Μ‡οΏ½ π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† πΏπ‘ž (ℝ𝑛 ). It gives conclusion that πΏπ‘ž (ℝ𝑛) = β„³ οΏ½Μ‡οΏ½ π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ), where 𝛼(β‹…) ∈ (ℝ𝑛 ). Furthermore, we will prove that β„³ οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). By using similar method as the proof of Theorem 2.1, we have β€– 𝑓 β€– β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) ≀ β€– 𝑓 β€– β„³ οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) , where 𝛼(β‹…) ∈ (ℝ𝑛 ). The author also added the inclusion of the homogeneous weak Herz-Morrey spaces with variable exponent by the following theorem. Theorem 2.3. Let 1 ≀ 𝑝1(β‹…) ≀ 𝑝2(β‹…) ≀ π‘ž < ∞, and 𝛼(β‹…) is a bounded real-valued measurable fuction on ℝ𝑛 , the following inclusion holds: π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). Proof. Let 𝑓 ∈ ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) , we have β€– 𝑓 β€– π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) = sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 1 2πΏπœ† ( βˆ‘ 2π‘˜π›Ό(β‹…)𝑝1(β‹…)π‘šπ‘˜ (𝛾, 𝑓) 𝑝1(β‹…) π‘ž 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝1(β‹…) ≀ sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 1 2πΏπœ† (βˆ‘ 2π‘˜π›Ό(β‹…)𝑝2(β‹…)π‘šπ‘˜ (𝛾, 𝑓) 𝑝2(β‹…) π‘žπΏ π‘˜=βˆ’βˆž ) 1 𝑝2(β‹…) ≀ ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) . The above inequality has shown that π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝2(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) βŠ† π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝1(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). Now, we state the inclusion relation between both spaces. Theorem 2.4. Let 1 ≀ 𝑝(β‹…) ≀ π‘ž, and 𝛼(β‹…) is a bounded real-valued measurable function on ℝ𝑛 . Then, the inclusion β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ) βŠ† π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) is proper. Proof. We use similar idea as before to prove this theorem. Let 𝑓 ∈ β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ), π‘Ž(β‹…) ∈ ℝ𝑛 , 𝑝(β‹…) ∈ 𝒫(ℝ𝑛) and 𝛾 > 0. We have observed that | {π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾} | 𝑝(β‹…) π‘ž ≀ (∫ |𝑓(π‘₯)πœ’π‘˜ | π‘ž 𝑑π‘₯ 𝐡(0,2π‘˜) ) 𝑝(β‹…) π‘ž = β€– π‘“πœ’π‘˜ β€– πΏπ‘ž(ℝ𝑛) 𝑝(β‹…) . Multiplying both sides by βˆ‘ 2π‘˜π›Ό(β‹…)𝑝(β‹…)πΏπ‘˜=βˆ’βˆž , we get Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman 26 βˆ‘ 2π‘˜π›Ό(β‹…)𝑝(β‹…)|{ π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾 }| 𝑝(β‹…) π‘ž 𝐿 π‘˜=βˆ’βˆž ≀ βˆ‘ 2π‘˜π›Ό(β‹…)𝑝(β‹…) β€– π‘“πœ’π‘˜ β€– πΏπ‘ž(ℝ𝑛) 𝑝(β‹…) 𝐿 π‘˜=βˆ’βˆž . Clearly, we see that β€– 𝑓 β€– π‘Šβ„³οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) ≀ β€– 𝑓 β€– ℳ𝐾 Μ‡ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛) and 𝑓 ∈ π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛), which implies that β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛 ) βŠ† π‘Š β„³ οΏ½Μ‡οΏ½ 𝑝(β‹…),π‘ž 𝛼(β‹…),πœ† (ℝ𝑛). CONCLUSION By this result, the author can conclude that the homogeneous Herz-Morrey spaces with variable exponent have inclusion properties ... . This result will be useful to be used in proving fractional integral on the homogeneous Herz-Morrey spaces with variable exponent. ACKNOWLEDGMENT This paper is partially supported by UIN Maulana Malik Ibrahim Malang Research and Innovation Program 2020. REFERENCES [1] H. Gunawan, D. I. Hakim, K. M. Limanta and A. A. Masta, "Inclusion property of generalized Morrey spaces," Math. Nachr., pp. 1-9, 2016. [2] H. Gunawan, D. I. Hakim and M. Idris, "Proper inclusions of Morrey spaces," Glasnik Matematicki, vol. 53, no. 1, 2017. [3] H. Gunawan, D. I. Hakim, E. Nakai and Y. Sawano, "On inclusion relation between weak Morrey spaces and Morrey spaces," Nonlineae Analysis, vol. 168, pp. 27-31, 2018. [4] H. Gunawan, E. Kikianty and C. Schwanke, "Discrete Morrey spaces and their inclusion properties," Math. Nachr., pp. 1-14, 2017. [5] A. A. Masta, H. Gunawan and W. Setya-Budhi, "An Inclusion Property of Orlicz- Morrey Spaces," J. Phys.: Conf. Ser, vol. 893, pp. 1-7, 2017. [6] A. Beurling, "Construction and analysis of some convolution algebras," Annales de L'Institut Fourier Grenoble, vol. 14, pp. 1-32, 1964. [7] S. Lu, D. Yang and H. Guoen, Herz Type Spaces and Their Applications, Beijing: Science Press, 2008. [8] H. Rahman, "Inclusion properties of the homogeneous Herz-Morrey," Cauchy, vol. 6, no. 3, pp. 117-121, 2020. [9] O. Kovacik and J. Rakosnik, "On space and," Czchoslovak Math. J., vol. 41, pp. 592- 618, 1991. [10] M. Izuki, "Boundedness of Sublinear Operators on Herz Spaces with Variable Exponent and Application to Wavelet Characterization," vol. 36 (1), no. Analysis Mathematics, pp. 33-50, 2010. [11] J. Yang and J. Xu, "Herz-Morrey-Hardy Spaces with Variable Exponents and Their Applications," no. Journal of Function Spaces, pp. 1-19, 2015. [12] S. Lu and L. Xu, "Boundedness of Rough Singular Integral Operators on The Homogeneous Morrey-Herz Spaces," Hokkaido Math. Journal, vol. 34, pp. 299-314, Inclusion Properties of Herz-Morrey Spaces With Variable Exponent Hairur Rahman 27 2005. [13] M. Izuki, "Fractional Integral on Herz-Morrey spaces with variable exponent," Hiroshima Math. J., vol. 40, pp. 343-355, 2010. [14] Y. Mizuta and T. Ohno, "Herz-Morrey spaces of variable exponent, Riesz potential operator and duality," Complex Variable and Elliptic Equations, vol. 60, no. 2, pp. 211-240, 2015. [15] Y. Shi, X. Tao and T. Zheng, "Multilinier Riesz potential on Morrey-Herz spaces with non-doubling measure," Journal of Inequality and Applications, vol. 10, 2010.