\(T1\) theorems on Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications
type: Item
instance of: scholarly article
title: \(T1\) theorems on Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications
zbMATH Open document ID: 1041.43006
DOI: 10.4171/ZAA/1132
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2003-01-01T00:00:00Z
review text: The aim of this paper is to establish the reduced \(T1\) theorems for the (inhomogeneous) Besov spaces \(B^ s_{p,q}(X)\) (when \(p\in(p_0,\infty]\), \(q\in(0,\infty]\) with some \(p_0\in(0,1)\)) and for the (inhomogeneous) Triebel-Lizorkin spaces \(F^ s_{p,q}(X)\) (when \(p\in(p_0,\infty)\), \(q\in(p_0,\infty]\) with some \(p_0\in(0,1)\)) on a space \(X\) of homogeneous type. (The spaces \(B^ s_{p,q}(X)\) and \(F^ s_{p,q}(X)\) were introduced in \textit{Y. Han} and \textit{D. Yang} [Stud. Math. 156, No. 1, 67--97 (2003; Zbl 1032.42025)].) The \(T1\) theorems for the homogeneous Besov spaces \(\dot{B}^ s_{p,q}(X)\) and Triebel-Lizorkin spaces \(\dot{F}^ s_{p,q}(X)\) are also stated (some partial results were obtained by \textit{D. Dong} and \textit{Y. Han} [\(T1\) theorems for Besov and Triebel-Lizorkin spaces] and by the author [Real interpolation for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type]). As an application of the \(T1\) theorems it is shown that an operator of Bessel potential type can be used as the lifting operator of these spaces. It generalizes the corresponding results from \textit{Y. Han} and \textit{D. Yang} [Diss. Math. 403, 1--102 (2002; Zbl 1019.43006)] for \(p, q>1\). Recall that the spaces \(X\) of homogeneous type cover (when \(\text{diam\,}X<\infty\)) the boundaries of bounded Lipschitz domains in \(\mathbb R^ n\), the \(n\)-torus in \(\mathbb R^ n\), \(C^\infty\)-compact Riemann manifolds, compact \(d\)-sets (which include various kinds of fractals) and (when \(\text{diam\,}X=\infty\)) Euclidean spaces, the boundaries of unbounded Lipschitz domains in \(\mathbb R^ n\) and Lipschitz manifolds of the non-compact case (from \textit{H. Triebel} [Rev. Mat. Comput. 15, 475--524 (2002; Zbl 1034.46033)]).
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1895114
zbMATH Keywords: spaces of homogeneous type
zbMATH Keywords: Besov spaces
zbMATH Keywords: Triebel-Lizorkin spaces
zbMATH Keywords: \(T1\) theorems
zbMATH Keywords: Calderón-Zygmund operators
zbMATH Keywords: Bessel potentials
author: Dachun Yang
reviewed by: Petr Gurka
MaRDI profile type: Publication
cites work: A T(b) theorem with remarks on analytic capacity and the Cauchy integral
cites work: Q2548398
cites work: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (Calderón-Zygmund operators, para-accretive functions and interpolation)
cites work: INHOMOGENEOUS PLANCHEREL–PÔLYA INEQUALITIES ON SPACES OF HOMOGENEOUS TYPE AND THEIR APPLICATIONS
cites work: Some Maximal Inequalities
cites work: Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications
cites work: Inhomogeneous Calderón reproducing formula on spaces of homogeneous type
cites work: Discrete Calderón-type reproducing formula
cites work: Q4719275
cites work: Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type
cites work: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces
cites work: New characterizations and applications of inhomogeneous Besov and Triebel–Lizorkin spaces on homogeneous type spaces and fractals
cites work: Some new spaces of Besov and Triebel–Lizorkin type on homogeneous spaces
cites work: Lipschitz functions on spaces of homogeneous type
cites work: Generalized uncertainty principles on spaces of homogeneous type
cites work: Q5689215
cites work: Q3336200
cites work: Q4357054
cites work: Q2731784
cites work: Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers.
cites work: Real interpolations for Besov and Triebel‐Lizorkin spaces on spaces of homogeneous type
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description: scientific article; zbMATH DE number 1895114