{"title":"A generalization of Hardy’s inequality to infinite tensors","authors":"Morteza Saheli, Davoud Foroutannia, Sara Yusefian","doi":"10.1515/gmj-2024-2006","DOIUrl":null,"url":null,"abstract":"In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℭ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0150.png\" /> <jats:tex-math>{\\mathfrak{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>x</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>k</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0106.png\" /> <jats:tex-math>\\|\\mathfrak{C}x^{k}\\|_{t,1}\\leq U\\|x\\|_{l_{p}}^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0107.png\" /> <jats:tex-math>k=1,2</jats:tex-math> </jats:alternatives> </jats:inline-formula>), where <jats:italic>x</jats:italic> is a sequence, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0149.png\" /> <jats:tex-math>{\\mathfrak{C}x^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a tensor, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0155.png\" /> <jats:tex-math>{\\|\\cdot\\|_{t,1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0153.png\" /> <jats:tex-math>{\\|\\cdot\\|_{l_{p}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are the tensor and sequence norms, respectively. The constant <jats:italic>U</jats:italic> is independent of <jats:italic>x</jats:italic>, and we seek the smallest possible value of <jats:italic>U</jats:italic>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"81 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Georgian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2006","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ{\mathfrak{C}}, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ℭxk∥t,1≤U∥x∥lpk\|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} (k=1,2k=1,2), where x is a sequence, ℭxk{\mathfrak{C}x^{k}} is a tensor, and ∥⋅∥t,1{\|\cdot\|_{t,1}}, ∥⋅∥lp{\|\cdot\|_{l_{p}}} are the tensor and sequence norms, respectively. The constant U is independent of x, and we seek the smallest possible value of U.
在本文中，我们将哈代不等式扩展到无限张量。为此，我们引入 Cesàro 张量 ℭ {\mathfrak{C}} ，并将其视为从序列空间到张量空间的张量映射。 ，并将它们视为从序列空间到张量空间的张量映射。事实上，我们证明了形式为 ∥ ℭ x k ∥ t , 1 ≤ U ∥ x ∥ l p k\|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k} 的不等式。 ( k = 1 , 2 k=1,2 ), 其中 x 是一个序列，ℭ x k {\mathfrak{C}x^{k}} 是一个张量，并且 ∥ ⋅ ∥ t , 1 {\|\cdot\|_{t,1}} , ∥ ⋅ ∥ l p {\|\cdot\|_{l_{p}}} 分别是张量规范和序列规范。常数 U 与 x 无关，我们寻求 U 的最小值。
期刊介绍：
The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter. The concern of this international journal is the publication of research articles of best scientific standard in pure and applied mathematics. Special emphasis is put on the presentation of results obtained by Georgian mathematicians.
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