Haarlet analysis of Lipschitz regularity in metric measure spaces
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 \[ \operatorname{Lip}(\alpha) = C(\alpha, {\mathcal D}). \]
scientific article; zbMATH DE number 6131970