Equivalent quasi-norms in Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth
type: Item
instance of: scholarly article
title: Equivalent quasi-norms in Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth
zbMATH Open document ID: 1465.46036
author: Qing Hong
author: Guorong Hu
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-02-25T00:00:00Z
review text: 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}^…
reviewed by: Eirik Berge
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 43A80
zbMATH DE Number: 7315657
zbMATH Keywords: Besov spaces
zbMATH Keywords: Triebel-Lizorkin spaces
zbMATH Keywords: quasi-norms
zbMATH Keywords: Lie groups of polynomial growth
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124769
OpenAlex ID: W3104483751
Wikidata QID: Q115345896
cites work: Besov and Triebel-Lizorkin spaces on Lie groups
cites work: Besov and Triebel-Lizorkin spaces associated to Hermite operators
cites work: Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces
cites work: Calderón reproducing formulas and new Besov spaces associated with operators
cites work: A T(b) theorem with remarks on analytic capacity and the Cauchy integral
cites work: Weighted subcoercive operators on Lie groups
cites work: Q3344703
cites work: A discrete transform and decompositions of distribution spaces
cites work: Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth
cites work: Besov spaces and frames on compact manifolds
cites work: Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators
cites work: Atomic and molecular decomposition of homogeneous spaces of distributions associated to non-negative self-adjoint operators
cites work: Spectral multipliers on spaces of distributions associated with non-negative self-adjoint operators
cites work: Modern Fourier Analysis
cites work: Croissance polynomiale et périodes des fonctions harmoniques
cites work: A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces
cites work: Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators
cites work: Besov and Triebel-Lizorkin spaces associated with non-negative self-adjoint operators
cites work: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces
cites work: Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions
cites work: Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Q4129323
cites work: Decomposition of spaces of distributions induced by Hermite expansions
cites work: Q4340096
cites work: Q3336200
cites work: Q4003552
cites work: Q4002250
DOI: 10.1016/J.JMAA.2020.124769
Recommended article: Besov and Triebel-Lizorkin spaces on Lie groups
Recommended article: Equivalent quasi-norms of Besov-Triebel-Lizorkin-type spaces via derivatives
Recommended article: Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth
Recommended article: Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth
Recommended article: Caffarelli–Kohn–Nirenberg inequalities on Lie groups of polynomial growth
Recommended article: Multilinear spectral multipliers on Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth
Recommended article: Triebel-Lizorkin spaces with variable smoothness and integrability on Lie groups of polynomial growth
Recommended article: Pointwise multipliers for Triebel-Lizorkin and Besov spaces on Lie groups
Recommended article: Besov algebras on Lie groups of polynomial growth
Recommended article: Multilinear and multiparameter spectral multipliers on homogeneous Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growth
description: scientific article; zbMATH DE number 7315657