A necessary and sufficient condition for Hardy's operator in the variable Lebesgue space
type: Item
instance of: scholarly article
title: A necessary and sufficient condition for Hardy's operator in the variable Lebesgue space
zbMATH Open document ID: 1470.42041
DOI: 10.1155/2014/342910
author: Yusuf Zeren
author: Farman I. Mamedov
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: The variable exponent Hardy inequality \(\|x^{\beta(x) - 1} \int_0^x f(t) d t\|_{L^{p(.)}(0, l)} \leq C \|x^{\beta(x)} f\|_{L^{p(.)}(0, l)}\), \(f \geq 0\) is proved assuming that the exponents \(p :(0, l) \rightarrow(1, \infty)\), \(\beta :(0, l) \rightarrow \mathbb{R}\) not rapidly oscilate near origin and \(1 / p'(0) - \beta > 0\). The main result is a necessary and sufficient condition on \(p\), \(\beta\) generalizing known results on this inequality.
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7022194
Wikidata QID: Q59036292
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/342910
OpenAlex ID: W2081514441
cites work: Q4169289
cites work: Q5439977
cites work: Q5634306
cites work: Q3994467
cites work: Electrorheological fluids: Modeling and mathematical theory
cites work: On some variational problems
cites work: Regularity for elliptic equations with general growth conditions
cites work: Regularity results for a class of functionals with non-standard growth
cites work: Lebesgue and Sobolev spaces with variable exponents
cites work: Variable Lebesgue spaces. Foundations and harmonic analysis
cites work: Q5691788
cites work: Convolution type operators inlp(x)
cites work: Maximal and fractional operators in weighted \(L^{p(x)}\) spaces
cites work: Q3374004
cites work: Q4676261
cites work: Hardy type inequality in variable Lebesgue spaces
cites work: Q5440922
cites work: Hardy's inequality in power-type weighted \(L^{p(\cdot)}(0,\infty )\) spaces
cites work: On a weighted inequality of Hardy type in spaces \(L^{p(\cdot)}\)
cites work: On a Hardy type general weighted inequality in spaces \(L ^{p(\cdot )}\)
cites work: On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces
cites work: On equivalent conditions for the general weighted Hardy type inequality in space \(L^{p(\cdot)}\)
cites work: On boundedness of weighted Hardy operator in \(L^{p(\cdot)}\) and regularity condition
cites work: A necessary and sufficient condition for Hardy's operator in
cites work: On the spaces \(L^{p(x)}(\Omega)\) and \(W^{m,p(x)}(\Omega)\)
cites work: Q4021390
cites work: Q3237284
description: scientific article; zbMATH DE number 7022194