On a Hardy type general weighted inequality in spaces \(L ^{p(\cdot )}\)
type: Item
instance of: scholarly article
title: On a Hardy type general weighted inequality in spaces \(L ^{p(\cdot )}\)
zbMATH Open document ID: 1203.26027
author: Farman I. Mamedov
author: Aziz Harman
published in: Integral Equations and Operator Theory
publication date: 2010-08-23T00:00:00Z
review text: A Hardy type two-weighted inequality \[ \|v^{1/q(x)}Hf(x)\|_{q(\cdot)}\leq C\|\omega^{1/p(x)}f(x)\|_{p(\cdot)} \] is studied in the norms of generalized Lebesgue spaces \(L^{p(\cdot)}(\mathbb R^n)\) for the multidimensional Hardy operator \(Hf(x)=\int_{\{t\in \mathbb R^n;|t|<|x|\}} f(t) \,dt\). Equivalent necessary and sufficient conditions are found for the \({L^{p(\cdot)} \rightarrow L^{q(\cdot)}}\) boundedness of the Hardy operator when the exponents satisfy \(q(0) < p(0), q(\infty ) < p(\infty )\). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.
reviewed by: László Losonczi
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 5774005
zbMATH Keywords: Hardy operator
zbMATH Keywords: Hardy inequality
zbMATH Keywords: variable exponent
zbMATH Keywords: weighted inequality
MaRDI profile type: Publication
full work available at URL: s00020-010-1765-z
OpenAlex ID: W2029928705
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DOI: 10.1007/S00020-010-1765-Z
description: scientific article; zbMATH DE number 5774005