A generalization of Hardy’s inequality to infinite tensors

Pub Date : 2024-02-20 DOI:10.1515/gmj-2024-2006

Morteza Saheli, Davoud Foroutannia, Sara Yusefian

{"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 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℭ</m:mi> </m:math> {\\mathfrak{C}} , and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form <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> \\|\\mathfrak{C}x^{k}\\|_{t,1}\\leq U\\|x\\|_{l_{p}}^{k} ( <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> k=1,2 ), where x is a sequence, <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> {\\mathfrak{C}x^{k}} is a tensor, and <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> {\\|\\cdot\\|_{t,1}} , <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> {\\|\\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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","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 ∥ ℭ ⁢ x k ∥ t , 1 ≤ U ⁢ ∥ x ∥ l p k

\|\mathfrak{C}x^{k}\|_{t,1}\leq U\|x\|_{l_{p}}^{k}

( k = 1 , 2

k=1,2

), where x is a sequence, ℭ ⁢ x k

{\mathfrak{C}x^{k}}

is a tensor, and ∥ ⋅ ∥ t , 1

{\|\cdot\|_{t,1}}

, ∥ ⋅ ∥ l p

{\|\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.

查看原文

哈代不等式对无限张量的推广

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。

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