On the Hardy inequality with measures
type: Item
instance of: scholarly article
title: On the Hardy inequality with measures
zbMATH Open document ID: 1222.26031
DOI: 10.1134/S1064562408060100
author: Yong-Cai Geng
author: Sumit K. Garg
published in: Doklady Mathematics
publication date: 2011-06-08T00:00:00Z
review text: The author presents four theorems (without proofs) concerning necessary and sufficient conditions for the fulfillment of the general Hardy inequality of the form \[ \Biggl(\int_{[a,b]} v(x)\biggl(\int_{[a,x]} k(x,y)f(y)u(y)\,d\lambda(y)\biggr)^q \,d\mu(x)\Biggr)^{1/q} \leq C\Biggl(\int_{[a,b]}f^p w\,dv\Biggr)^{1/p} \] for all \(f\in\{\mathfrak{M}_{\lambda}\}^+\), where \(0<p<1\); \(0<q<+\infty\); \(\lambda,\mu\) and \(v\) are \(\sigma\)-finite measures on \([a,b]\subset\mathbb R\); \(\lambda\) and \(v\) are defined on the same \(\sigma\)-algebra \(\mathfrak{M}\) of subsets of the set \([a,b]\) containing all Borel subsets of \([a,b];\) \(u,v,w\) and \(k\) are the respective measurable functions. Those theorems generalize significantly previous results obtained by other mathematicians, including the author himself (seven papers cited in the list of references). Especially interesting in this paper is the substitution of Oinarov's condition for the kernel \(k\) by some monotonic-type conditions for \(k\).
reviewed by: Roman Wituła
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5905453
MaRDI profile type: Publication
cites work: Q3997067
cites work: Q4784073
cites work: Weighted Norm Inequalities of Hardy Type for a Class of Integral Operators
cites work: Hardy's inequality with three measures
cites work: Inequalities of Hardy type for a class of integral operators with measures
cites work: SOME ELEMENTARY INEQUALITIES, III
description: scientific article; zbMATH DE number 5905453