Completeness of Muckenhoupt classes
type: Item
instance of: scholarly article
title: Completeness of Muckenhoupt classes
zbMATH Open document ID: 1180.28002
author: Hugo Aimar
author: Marilina Carena
author: Bibiana Iaffei
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-11-10T00:00:00Z
full work available at URL: http://cimec.org.ar/ojs/index.php/cmm/article/view/1385
review text: The authors show that the Hausdorff distance between compact sets and the Kantorovich distance between measures provide an adequate setting to describe the convergence in the class of Muckenhoupt weights. They first recall that the Cantor set and its standard approximation can be described as a convergence result of \((Y_n,\mu_n)\to (C,\mu)\) where one considers the distance \(d_{HK}((Y_n,\mu_n), (C,\mu))\) given by the sum of the Hausdorff distance \(d_H(Y_n,C)\) and the Kantorovich-Rubinshtein distance \(d_k(\mu_n,\mu)\). Then they study in this particular case the weighted situation \(w_n(y)=|y|^\alpha, y\in Y_n\) showing that, for \(s=\frac{\log 2}{\log 3}\) and \(-s<\alpha<s(p_1)\), the terms \(w_n\) become \(A_p\)-weights with a uniform constant. The aim of the note is actually to show that in the general setting of doubling metric compact spaces \((X, d)\), given a sequence of compact probabilistic subspaces \((Y_n,\mu_n)\) satisfying the doubling condition \(0<\mu_n(B_d(y,2r))\leq A\mu_n(B_d(y,r))\) for all \(y\in Y_n, r>0\) and converging to \((Y,\mu)\) in the above distance \(d_{HK}\) and given a sequence of weights \(w_n\in A_p(Y_n,\mu_n)\) with uniform Muckhenhoupt constant \(C\), normalized to have integral \(1\), and weak*-converging to a measure \(\nu\), one has that the limit measure becomes an \(A_p\)-weight, that is \(d\nu=wd\mu\) with \(w\in A_p(Y,\mu)\).
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 5629195
zbMATH Keywords: Muckenhoupt weights
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: Kantorovich distance
MaRDI profile type: Publication
OpenAlex ID: W2080739944
cites work: Weighted Norm Inequalities for the Hardy-Littlewood Maximal Operator on Spaces of Homogeneous Type
cites work: Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type
cites work: Discrete approximation of spaces of homogeneous type
cites work: Q4667206
cites work: Q4269108
cites work: Inequalities for the maximal function relative to a metric
cites work: A T(b) theorem with remarks on analytic capacity and the Cauchy integral
cites work: A Note on Weighted Norm Inequalities for the Hardy-Littlewood Maximal Operator
cites work: Weighted norm inequalities for maximal functions and singular integrals
cites work: Q5565773
cites work: Q3700300
cites work: Q5293718
cites work: Lipschitz functions on spaces of homogeneous type
cites work: Q4538028
cites work: Q5607484
cites work: Q3794674
cites work: Hausdorff dimension and doubling measures on metric spaces
DOI: 10.1016/J.JMAA.2009.07.027
description: scientific article; zbMATH DE number 5629195