A characterization of some weighted norm inequalities for maximal operators
type: Item
instance of: scholarly article
title: A characterization of some weighted norm inequalities for maximal operators
zbMATH Open document ID: 1205.42017
DOI: 10.4171/ZAA/1414
author: Tengiz Kopaliani
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2010-11-17T00:00:00Z
review text: The main result of the paper concerns the boundedness of the generalized maximal operator \[ M_\varphi f(x)=\sup\limits_Q \frac{1}{\varphi(|Q|)}\int\limits_Q |f(y)|\,dy \] within the frameworks of the variable exponent Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n),\) where the function \(\varphi\) is non-negative, continuous, increasing, with \(\varpi(0)=0\) and satisfying the subadditivity condition of the type \(\varphi\left(\sum t_i\right)\leq C \sum\varphi\left(t_i\right), \;\;\;t_i\in \mathbb{R}^1_+\). The main result is given in Theorem 4.7 which provides a certain necessary and sufficient condition on the exponent functions \(p(x), q(x) \) and \(\varphi(x)\) for the operator \(M_\varphi\) to be bounded from \(L^{p(\cdot)}(\mathbb{R}^n)\) to \(L^{q(\cdot)}(\mathbb{R}^n), \;\;p(x)\leq q(x)\). The proof is based on the previous results of the author, E.~Berezhnoi and L.~Diening and the results of the first sections of the paper on the validity of the generalized Minkowsky inequality for the variable exponent Lebesgue spaces which are of interest in themselves.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5816828
zbMATH Keywords: generalized maximal operator
zbMATH Keywords: variable \(L^{p}\) space
zbMATH Keywords: Muckenhoupt type weights
zbMATH Keywords: Minkowsky inequality
reviewed by: S. G. Samko
MaRDI profile type: Publication
cites work: Regularity results for a class of functionals with non-standard growth
cites work: Q4871609
cites work: Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces
cites work: Q3886302
cites work: Q4458796
full work available at URL: https://doi.org/10.4171/zaa/1414
OpenAlex ID: W2075029450
description: scientific article; zbMATH DE number 5816828