Haarlet analysis of Lipschitz regularity in metric measure spaces
type: Item
instance of: scholarly article
title: Haarlet analysis of Lipschitz regularity in metric measure spaces
zbMATH Open document ID: 1261.42047
author: Hugo Aimar
author: A. L. Bernardis
author: Luis Nowak
published in: Science China. Mathematics
publication date: 2013-01-28T00:00:00Z
full work available at URL: http://cimec.org.ar/ojs/index.php/cmm/article/view/2999
review text: The authors give a characterization of Lipschitz spaces on spaces of homogeneous type via Haar coefficients. Let \((X,d,\mu)\) be a space of homogeneous type, where \(d\) is a distance on \(X\) and \(\mu\) satisfies the doubling condition. Lipschitz spaces were introduced by \textit{R. A. Macías} and \textit{C. Segovia} [Adv. Math. 33, 257--270 (1979; Zbl 0431.46018)]. It is said that a function \(f\) belongs to \(\operatorname{Lip}(\alpha), \;0 < \alpha <1\), if \[ | f(x) - f(y) | \leq C d(x,y)^{\alpha}. \] \textit{M. Christ} [Colloq. Math. 60/61, No. 2, 601--628 (1990; Zbl 0758.42009)] and \textit{H. Aimar} et al. [J. Geom. Anal. 21, No. 2, 288--304 (2011; Zbl 1219.42019)] introduced dyadic families of cubes \(\mathcal D\). The authors define some Haar system \({\mathcal H}_{\mathcal D}\) associated to \(\mathcal D\) and define the Carleson class \(C(\alpha, {\mathcal D})\) as follows. A function \(f\) is in \(C(\alpha, {\mathcal D})\) if and only if there exists a positive constant \(C\) such that \[ | \langle f,h \rangle | \leq C \operatorname{diam}(Q_h)^{\alpha} \mu (Q_h)^{1/2} \] holds for every \(h \in {\mathcal H}_{\mathcal D}\), where \(Q_h\) is the smallest dyadic cube containing the set \(\{ x \in X : h(x) \neq 0 \}\). They prove that \[ \operatorname{Lip}(\alpha) = C(\alpha, {\mathcal D}). \]
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 28C15
zbMATH DE Number: 6131970
zbMATH Keywords: Lipschitz spaces
zbMATH Keywords: Haar basis
zbMATH Keywords: spaces of homogeneous type
reviewed by: Yasuo Komori-Furuya
MaRDI profile type: Publication
OpenAlex ID: W1993683253
cites work: Q4351421
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cites work: Q3756738
DOI: 10.1007/S11425-012-4367-1
description: scientific article; zbMATH DE number 6131970