Inclusion Properties of The Homogeneous Herz-Morrey y aces CAUCHY – JURNAL MATEMATIKA MURNI DAN APLIKASI Volume 6(3) (2020), Pages 117-121 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: August 18, 2020 Reviewed: October 05, 2020 Accepted: November 05, 2020 DOI: http://dx.doi.org/10.18860/ca.v6i3.10114 Hairur Rahman Department of Mathematics, Islamic State University of Maulana Malik Ibrahim Malang Email: hairur@mat.uin-malang.ac.id ABSTRACT In this paper, we have discussed the inclusion properties of the homogeneous Herz-Morrey spaces and the homogeneous weak homogeneous spaces. We also studied the inclusion relation between those spaces. Keywords: homogeneous Herz-Morrey spaces; homogeneous weak Herz-Morrey spaces; inclusion properties. INTRODUCTION The subject discussion about inclusion properties of any spaces or inclusion relation between spaces has interested to study. Some authors have studied about inclusion relation in some spaces (see [1], [2], [3], [4] and [5]). It guided us to discuss the inclusion properties of other spaces. Regarding C.B. Morrey in [6] who introduced Morrey spaces, many authors have defined the generalization of Morrey spaces and combined with other spaces. Lu and Xu [7] introduce one of the homogeneous Herz-Morrey spaces. These spaces are the generalization of Morrey spaces and Herz spaces. Let 𝛼 ∈ ℝ, 0 < 𝑝 ≀ ∞, 0 < π‘ž < ∞, and 0 ≀ πœ† < ∞, the homogeneous Herz-Morrey spaces ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛) are defined by ℳ�̇�𝑝,π‘ž 𝛼,πœ†(𝑅𝑛) ∢= {𝑓 ∈ πΏπ‘™π‘œπ‘ π‘ž (ℝ𝑛/{0}) ∢ ‖𝑓‖ ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛) < ∞}, where ‖𝑓‖ ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛) = sup πΏβˆˆβ„€ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝 with π΅π‘˜ = 𝐡(0,2 π‘˜ ) = {π‘₯ ∈ ℝ𝑛: |π‘₯| ≀ 2π‘˜ }, π΄π‘˜ = π΅π‘˜ /π΅π‘˜βˆ’1 for π‘˜ ∈ β„€ and πœ’π‘˜ = πœ’π΄π‘˜ for π‘˜ ∈ β„€ be the characteristic function of the set π΄π‘˜ . Lu and Xu also defined the homogeneous weak Herz-Morrey spaces. For 𝛼 ∈ ℝ, let 0 < 𝑝 ≀ ∞, πœ† β‰₯ 0 and 0 < π‘ž < ∞, the homogeneous weak Herz-Morrey spaces (π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(ℝ𝑛)) is a set of measurable 𝑓 ∈ πΏπ‘™π‘œπ‘ π‘ž (ℝ𝑛/{0}) which completed by norm such that ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(ℝ𝑛) = sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘π‘šπ‘˜ (𝛾, 𝑓) 𝑝 π‘ž 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝 < ∞, where π‘šπ‘˜ (𝛾, 𝑓) = |{π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾}|. Some authors have studied those spaces in different term of discussion (see [7], [8], [9], [10]). Meanwhile, in this article, the authors would like to discuss the inclusion properties and inclusion relation of the homogeneous Herz-Morrey spaces and the homogeneous weak Herz-Morrey spaces. Inclusion Properties of The Homogeneous Herz-Morrey y aces http://dx.doi.org/10.18860/ca.v6i3.10114 mailto:hairur@mat.uin-malang.ac.id Inclusion Properties of The Homogeneous Herz-Morrey Hairur Rahman 118 RESULTS AND DISCUSSION Now, we formulate our main results of this paper as follows: Theorem 1.1. Let 1 ≀ 𝑝1 ≀ 𝑝2 < π‘ž < ∞, then the following inclusion holds: ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛). Generally, by Theorem 1.1, we have the following inclusions of the homogeneous Herz-Morrey spaces. Theorem 1.2. Let 1 ≀ 𝑝1 ≀ 𝑝2 < π‘ž < ∞, then the following inclusion holds: πΏπ‘ž (𝑅𝑛) = β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) βŠ† ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛). Besides, we have the inclusion property of the homogeneous weak Herz-Morrey spaces, also the inclusion relation of the homogeneous Herz-Morrey spaces. Theorem 1.3. Let 1 ≀ 𝑝1 ≀ 𝑝2 ≀ π‘ž < ∞, the following inclusion holds: π‘Šβ„³οΏ½Μ‡οΏ½π‘2,π‘ž 𝛼,πœ† (ℝ𝑛) βŠ† π‘Šβ„³οΏ½Μ‡οΏ½π‘1,π‘ž 𝛼,πœ† (ℝ𝑛). Theorem 1.4. Let 1 ≀ 𝑝 ≀ π‘ž. Then the inclusion ℳ�̇�𝑝,π‘ž 𝛼,πœ† (ℝ𝑛) βŠ† π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(𝑅𝑛) is proper. The proof of each theorem will be described in the following section. THE PROOF OF THEOREM 1.1. For proofing Theorem 1.1., we shall show that ‖𝑓‖ ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛) ≀ ‖𝑓‖ ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) by applying HΓΆlder inequality. Proof of Theorem 1.1. Let we first take for any 𝑓 ∈ ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛), then by using HΓΆlder inequality and 𝑝1 ≀ 𝑝2 we obtain that ‖𝑓‖ ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛) = sup πΏβˆˆπ‘ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘1 𝐿 π‘˜=βˆ’βˆž β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1 ) 1 𝑝1 ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† (( βˆ‘ (2π‘˜π›Όπ‘1 ) 𝑝2 𝑝1 𝐿 π‘˜=βˆ’βˆž ) 𝑝1 𝑝2 ( βˆ‘ (β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1 ) 𝑝2 𝑝2βˆ’π‘1 𝐿 π‘˜=βˆ’βˆž ) 1βˆ’ 𝑝1 𝑝2 ) 1 𝑝1 ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† (( βˆ‘ 2π‘˜π›Όπ‘2 𝐿 π‘˜=βˆ’βˆž ) 𝑝1 𝑝2 ( βˆ‘ β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1𝑝2 𝑝2βˆ’π‘1 𝐿 π‘˜=βˆ’βˆž ) 1βˆ’ 𝑝1 𝑝2 ) 1 𝑝1 ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘2 𝐿 π‘˜=βˆ’βˆž ( βˆ‘ β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝1𝑝2 𝑝2βˆ’π‘1 𝐿 π‘˜=βˆ’βˆž ) 𝑝2βˆ’π‘1 𝑝1 ) 1 𝑝2 ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘2 𝐿 π‘˜=βˆ’βˆž β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝2 ) 1 𝑝2 ≀ ‖𝑓‖ ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) . By this observation, we know that 𝑓 ∈ ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛). Hence it concludes that ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛). Inclusion Properties of The Homogeneous Herz-Morrey Hairur Rahman 119 THE PROOF OF THEOREM 1.2. Since it has been stated in Theorem 1.1 that ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝1,π‘ž 𝛼,πœ† (𝑅𝑛), therefore in proving Theorem 1.2, we need to prove that πΏπ‘ž (𝑅𝑛) = β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛). Proof of Theorem 1.2. To prove that πΏπ‘ž (𝑅𝑛) = β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛), we need to show that ‖𝑓‖𝑳𝒒(𝑹𝒏) = β€–π‘“β€–π‘€οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) . Let take for any 𝑓 ∈ π‘€οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛), by applying HΓΆlder inequality for the norm. We obtain that ‖𝑓‖ π‘€οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘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‘ 2βˆ’πΏπœ† βˆ‘ 2π‘˜π›Ό 𝐿 π‘˜=βˆ’βˆž (∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž (2π‘˜π‘‘ ) 1 π‘ž ≀ sup πΏβˆˆπ‘ 2βˆ’πΏπœ† βˆ‘ 2 π‘˜π›Ό+ π‘˜π‘‘ π‘ž 𝐿 π‘˜=βˆ’βˆž (∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž ≀ 𝐢 (∫ |𝑓(π‘₯)|π‘ž 𝑑𝑦 𝐡(0,2π‘˜) ) 1 π‘ž ≀ ‖𝒇‖𝑳𝒒(𝑹𝒏), it means that 𝑓 ∈ πΏπ‘ž (𝑅𝑛). Then πΏπ‘ž (𝑅𝑛) βŠ† β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ† (𝑅𝑛). Meanwhile, for any 𝑓 ∈ πΏπ‘ž (𝑅𝑛), we can find any constant 𝐢 such that 𝐢 = sup πΏβˆˆπ‘ 2βˆ’πΏπœ† βˆ‘ 2 π‘˜π›Ό+ π‘˜π‘‘ π‘žπΏ π‘˜=βˆ’βˆž , then it shows that 𝑓 ∈ β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) which means β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) βŠ† πΏπ‘ž (𝑅𝑛). Hence, it concludes that πΏπ‘ž (𝑅𝑛) = β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛). Next, we have to prove that β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ† (𝑅𝑛) βŠ† ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) by showing ‖𝑓‖ ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) ≀ ‖𝑓‖ β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) . By using a similar way for proving Theorem 1.1., and since π‘ž > 𝑝2, it is clear that ‖𝑓‖ℳ�̇�𝑝2,π‘ž 𝛼,πœ† (𝑅𝑛) ≀ ‖𝑓‖ β„³οΏ½Μ‡οΏ½π‘ž,π‘ž 𝛼,πœ†(𝑅𝑛) . Therefore, the proof is complete. THE PROOF OF THEOREM 1.3. One way for proving Theorem 1.3. is showed that ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘1,π‘ž 𝛼,πœ† (𝑅𝑛) ≀ ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘2,π‘ž 𝛼,πœ† (𝑅𝑛) . Proof of Theorem 1.3. Let take for any 𝑓 ∈ ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘1,π‘ž 𝛼,πœ† (𝑅𝑛) , then by observing the norm of 𝑓 we obtain that ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘1,π‘ž 𝛼,πœ† (ℝ𝑛) = sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘1 π‘šπ‘˜ (𝛾, 𝑓) 𝑝1 π‘ž 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝1 ≀ sup 𝛾>0 𝛾 sup πΏβˆˆβ„€ 2βˆ’πΏπœ† ( βˆ‘ 2π‘˜π›Όπ‘2 π‘šπ‘˜ (𝛾, 𝑓) 𝑝2 π‘ž 𝐿 π‘˜=βˆ’βˆž ) 1 𝑝2 ≀ ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘2,π‘ž 𝛼,πœ† (𝑅𝑛) . By the observation, it concludes that π‘Šβ„³οΏ½Μ‡οΏ½π‘2,π‘ž 𝛼,πœ† (ℝ𝑛) βŠ† π‘Šβ„³οΏ½Μ‡οΏ½π‘1,π‘ž 𝛼,πœ† (ℝ𝑛). Inclusion Properties of The Homogeneous Herz-Morrey Hairur Rahman 120 THE PROOF OF THEOREM 1.4. Proving Theorem 1.4 is used a similar idea as previous theorems which shall show that ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(ℝ𝑛) ≀ ‖𝑓‖ ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛) . Proof of Theorem 1.4. Let 𝑓 ∈ ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛 ), π‘Ž ∈ ℝ𝑛, and 𝛾 > 0. We observe that |{π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾}| 𝑝 π‘ž ≀ (∫ |𝑓(π‘₯)πœ’π‘˜ | π‘ž 𝑑π‘₯ 𝐡(0,2π‘˜) ) 𝑝 π‘ž = β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝 Multiplying both side by βˆ‘ 2π‘˜π›Όπ‘πΏπ‘˜=βˆ’βˆž , then we obtain that βˆ‘ 2π‘˜π›Όπ‘|{π‘₯ ∈ π΄π‘˜ : |𝑓(π‘₯)| > 𝛾}| 𝑝 π‘ž 𝐿 π‘˜=βˆ’βˆž ≀ βˆ‘ 2π‘˜π›Όπ‘β€–π‘“πœ’π‘˜ β€–πΏπ‘ž(ℝ𝑛) 𝑝 𝐿 π‘˜=βˆ’βˆž . It says merely that ‖𝑓‖ π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(ℝ𝑛) ≀ ‖𝑓‖ ℳ�̇�𝑝,π‘ž 𝛼,πœ†(ℝ𝑛) , therefore 𝑓 ∈ π‘Šβ„³οΏ½Μ‡οΏ½π‘,π‘ž 𝛼,πœ†(ℝ𝑛). 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