\(T1\) theorems on Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications
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)]).
scientific article; zbMATH DE number 1895114
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