A note on weighted norm inequalities for fractional maximal operators with non-doubling measures
type: Item
instance of: scholarly article
title: A note on weighted norm inequalities for fractional maximal operators with non-doubling measures
zbMATH Open document ID: 1266.42050
author: Weihong Wang
author: Chaoqiang Tan
author: Zengjian Lou
published in: Taiwanese Journal of Mathematics
publication date: 2013-02-21T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/2225
review text: The aim of the present paper is to study two-weight norm inequalities for fractional maximal functions associated to a non-negative ``\(n-\)dimensional'' Borel measure \(\mu\) on \(\mathbb{R}^d\), which only satisfies the growth condition \(\mu(Q)\leq l(Q)^n\) for any cube \(Q\subset \mathbb{R}^d\), where \(l(Q)\) stands for the side length of \(Q\). The non-centered maximal function \(\displaystyle M_{\alpha}\) is defined as \[ \displaystyle M_{\alpha}f(x)=\sup_{x\in Q}\frac{1}{\mu(5Q)^{1-\alpha}}\int_{Q}|f(y)|\,\text{d}\mu(y) \] for \(\alpha\in[0,1)\) and the supremum taken over all cubes \(Q\) that contain \(x\). The authors investigate the pairs of weights (i.e non-negative and locally integrable functions) for which \(M_{\alpha}\) satisfies a weak or a strong type inequality. More precisely, there are two main results. The first one explores the pairs of weights for which the maximal operator \(M_{\alpha}\) satisfies a weak type inequality. The second one characterizes those pairs of weights for which \(M_{\alpha}\) satisfies a strong type inequality, which can be achieved by certain Orlicz norms localized in cubes.
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 6137975
zbMATH Keywords: non-homogeneous space
zbMATH Keywords: fractional maximal operator
zbMATH Keywords: Muckenhoupt weight
reviewed by: Anna Savvopoulou
MaRDI profile type: Publication
OpenAlex ID: W1763402742
DOI: 10.11650/TWJM/1500406741
description: scientific article; zbMATH DE number 6137975