Localization principle for Triebel-Lizorkin spaces on spaces of homogeneous type
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 \(F^s_{pq}(X)\).
scientific article; zbMATH DE number 2113564