Localization principle for Triebel-Lizorkin spaces on spaces of homogeneous type
type: Item
instance of: scholarly article
title: Localization principle for Triebel-Lizorkin spaces on spaces of homogeneous type
zbMATH Open document ID: 1052.43015
published in: Revista Matemática Complutense
publication date: 2004-11-01T00:00:00Z
full work available at URL: https://eudml.org/doc/44520
review text: Let \((X,\rho,\mu)_{d,\theta}\) be a space of homogeneous type introduced by \textit{Y. Han} and \textit{D. Yang} [Dissertationes Math. (Rozprawy Mat.) 403, 1--102 (2002; Zbl 1019.43006)] and let \(F^s_{pq}(X)\) be a Triebel-Lizorkin space defined on \((X,d,\mu)\). Suppose that \(\{ \psi_{\alpha}\}_{\alpha}\) on \(X\) is a resolution of unity satisfying the conditions: (a) \(0\leq \psi_{\alpha}\leq 1\), \(\operatorname{supp} \psi_{\alpha} \subset B(z_{\alpha}, C)\)= \( \{x\in X: \rho(x,z_{\alpha})<C\}\), \(C>0\) is a constant and \[ B(z_{\alpha}, C/2)\cap B(z_{\beta}, C/2)=\emptyset \] if \(\alpha\neq \beta\); (b) if \(U= \cup_{\alpha} B(z_{\alpha}, C/2)\), then \(\mu(X\setminus U)=0\) and \(\psi_{\alpha}\in Lip_{\theta}(U)\) with a uniform \(Lip\) constant in \(\alpha\); (c) \(\sum_{\alpha}\psi_{\alpha}(x) =1\) on \(U\). The author shows that if \(s\in (-\theta,\theta)\), \(max(d/(d+\theta), d/(d+\theta+s))<p<\infty\) and \(max(d/(d+\theta), d/(d+\theta+s))<q\leq \infty\), then \[ \bigg\{ \sum_{\alpha} \| \psi_{\alpha}f\| ^{p}_{F^s_{pq}(X)}\bigg\}^{1/p} \] is an equivalent quasi-norm in \(F^s_{pq}(X)\).
reviewed by: Alexander Meskhi
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2113564
zbMATH Keywords: spaces of homogeneous type
zbMATH Keywords: Triebel-Lizorkin spaces
zbMATH Keywords: localization principle
zbMATH Keywords: atom
zbMATH Keywords: Littlewood-Paley \(S\)-function
zbMATH Keywords: Calderón-Zygmund reproducing formula
author: Dachun Yang
MaRDI profile type: Publication
DOI: 10.5209/REV_REMA.2004.V17.N1.16802
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description: scientific article; zbMATH DE number 2113564