An equivalence theorem for some integral conditions with general measures related to Hardy's inequality
type: Item
instance of: scholarly article
title: An equivalence theorem for some integral conditions with general measures related to Hardy's inequality
zbMATH Open document ID: 1113.26024
author: Christopher A. Okpoti
author: Lars-Erik Persson
author: Gord Sinnamon
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Let \(M(x):= \int_{[x,\infty)} d\mu<\infty\) and \(\Lambda(x):= \int_{(-\infty,x]} d\lambda<\infty\), where \(\lambda\) and \(\mu\) are measures on \(\mathbb{R}\). The authors define five functions as follows \[ \begin{aligned} A_1(x;\alpha,\beta)&= M(x)^\alpha\Lambda(x)^\beta,\\ A_2(x;\alpha,\beta,s)&= \Biggl(\int_{[x,\infty)}\Lambda^{(\beta- s)/\alpha} \,d\mu\Biggr)^\alpha\Lambda(x)^s,\\ A_3(x;\alpha,\beta, s)&= \Biggl(\int_{(-\infty, x]} M^{(\alpha- s)/\beta} \,d\lambda\Biggr)^\beta M(x)^s,\\ A_4(x;\alpha,\beta, s)&= \Biggl(\int_{(-\infty, x]}\Lambda^{(\beta+ s)/\alpha} \,d\mu\Biggr)^\alpha\Lambda(x)^{-s},\\ A_5(x;\alpha,\beta, s)&= \Biggl(\int_{[-x,\infty)} M^{(\alpha+ s)/\beta} \,d\lambda\Biggr)^\beta M(x)^{-s}, \end{aligned} \] where \(\alpha\), \(\beta\) and \(s\) are positive fixed numbers. They prove as their main result that the numbers \(\sup_{x\in \mathbb{R}} A_s(x;\alpha,\beta)\) and \(\sup_{x\in \mathbb{R}} A_i(x;\alpha, \beta,s)\), \(i= 2,3,4,5\), are mutually equivalent. This is a general equivalence theorem of independent interest. In particular it can be applied for characterizing the Hardy type inequality with general measures for the case \(1< p\leq q<\infty\) (see the Hardy inequality).
reviewed by: Borislav Crstici
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5083574
zbMATH Keywords: discrete Hardy's inequality
zbMATH Keywords: Hardy operator
zbMATH Keywords: Hardy's inequality with general measures
zbMATH Keywords: scales of characterizations
zbMATH Keywords: weight functions
zbMATH Keywords: weight sequences
zbMATH Keywords: equivalent integral conditions
zbMATH Keywords: comparisons
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.015
OpenAlex ID: W4251346663
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DOI: 10.1016/J.JMAA.2006.03.015
description: scientific article; zbMATH DE number 5083574