Pointwise multipliers of Triebel-Lizorkin spaces on Carnot-Carathéodory spaces
Summary: Let \((\mathcal X, d, \mu)\) be a Carnot-Carathéodory space, namely, \(\mathcal X\) is a smooth manifold, \(d\) is a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. \(\mu\) is a nonnegative Borel regular measure on \(\mathcal X\) satisfying that there exists a constant \(C_0 \in [1, \infty)\) such that for all \(x \in \mathcal X\) and \(0 < r < \mathrm{diam}\mathcal X\), \(\mu(B(x, 2r)) := \mu ({y \in \mathcal X : d(x, y) < 2r}) \leq C_0 \mu(B(x, r)) < \infty\) (doubling property). Using the discrete Calderón reproducing formula and the Plancherel-Pólya characterization of the inhomogeneous Triebel-Lizorkin spaces developed by \textit{Y.-S. Han} et al. [Trans. Am. Math. Soc. 365, No. 1, 319--360 (2013; Zbl 1275.42035); Abstr. Appl. Anal. 2008, Article ID 893409 (2008; Zbl 1193.46018)], pointwise multipliers of inhomogeneous Triebel-Lizorkin spaces are obtained.
scientific article; zbMATH DE number 6134535