Equivalent quasi-norms in Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth
Two classes of well-behaved function spaces are the \textit{Besov spaces} \(B_{p,q}^{s}(\mathbb{R}^{n})\) and the \textit{Triebel-Lizorkin spaces} \(F_{p,q}^{s}(\mathbb{R}^{n})\) on \(\mathbb{R}^{n}\). These spaces have been generalized in recent years to various non-Euclidean settings. Of particular importance is the work [\textit{G.~Kerkyacharian} and \textit{P.~Petrushev}, Trans. Am. Math. Soc. 367, No.~1, 121--189 (2015; Zbl 1321.58017)], where the Besov and Triebel-Lizorkin spaces are developed on a metric measure space derived from a non-negative self-adjoint operator. In the classical setting, the Besov and Triebel-Lizorkin spaces can be characterized by derivatives; one has the quasi-norm characterizations \[ \|f\|_{B_{p,q}^{s}(\mathbb{R}^{n})} \sim \sum_{|\alpha| \leq m}\|D^{\alpha}f\|_{B_{p,q}^{s - m}(\mathbb{R}^{n})} \sim \|f\|_{B_{p,q}^{s - m}(\mathbb{R}^{n})} + \sum_{k = 1}^{n}\left\|\frac{\partial^{m}f}{\partial x_{k}^{m}}\right\|_{B_{p,q}^{s - m}(\mathbb{R}^{n})}, \] \[ \|f\|_{F_{p,q}^{s}(\mathbb{R}^{n})} \sim \sum_{|\alpha| \leq m}\|D^{\alpha}f\|_{F_{p,q}^{s - m}(\mathbb{R}^{n})} \sim \|f\|_{F_{p,q}^{s - m}(\mathbb{R}^{n})} + \sum_{k = 1}^{n}\left\|\frac{\partial^{m}f}{\partial x_{k}^{m}}\right\|_{F_{p,q}^{s - m}(\mathbb{R}^{n})}. \] The paper aims to extend these equivalent quasi-norm characterizations to the setting of Lie groups with polynomial growth. More precisely, let \(\mathbb{G}\) be a Lie group with polynomial growth and let \[ \mathbb{X} = \{X_{1}, \dots, X_{n}\} \] be a family of left-invariant vector fields on \(\mathbb{G}\) satisfying the Hörmander condition (also called the bracket generating condition). The paper shows, by utilizing smooth atomic decompositions, that one can characterize the Besov spaces \(B_{p,q}^{s}(\mathbb{G})\) and the Tribel-Lizorkin spaces \(F_{p,q}^{s}(\mathbb{G})\) by the quasi-norms \[ \|f\|_{B_{p,q}^{s}(\mathbb{G})} \sim \sum_{|\alpha| \leq m}\|X^{\alpha}f\|_{B_{p,q}^{s - m}(\mathbb{G})} \sim \|f\|_{B_{p,q}^{s - m}(\mathbb{G})} + \sum_{|\alpha| = m}\|X^{\alpha}f\|_{B_{p,q}^{s - m}(\mathbb{G})}, \] \[ \|f\|_{F_{p,q}^{s}(\mathbb{G})} \sim \sum_{|\alpha| \leq m}\|X^{\alpha}f\|_{F_{p,q}^{s - m}(\mathbb{G})} \sim \|f\|_{F_{p,q}^{s - m}(\mathbb{G})} + \sum_{|\alpha| = m}\|X^{\alpha}f\|_{F_{p,q}^{s - m}(\mathbb{G})}. \] It should be noted that, after the completion of the paper, the paper [\textit{T.~Bruno} et al., Math. Ann. 377, No.~1--2, 335--377 (2020; Zbl 1453.46025)] developed a theory of Besov and Triebel-Lizorkin spaces on general non-compact, connected Lie groups equipped with a sub-Riemannian structure. For certain values of the parameters \(p,q, s\), the paper [loc. cit.] independently shows the quasi-norm characterizations for the Besov and Triebel-Lizorkin spaces in a more general setting.
scientific article; zbMATH DE number 7315657
scientific article