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2013 A new characterization of Triebel-Lizorkin spaces on $\mathbb R^n$
Dachun Yang, Wen Yuan, Yuan Zhou
Publ. Mat. 57(1): 57-82 (2013).


In this paper, the authors characterize the Triebel-Lizorkin space $\dot F^\alpha_{p,q}(\mathbb{R}^n)$ via a new square function

$$S_{\alpha,q}(f)(x)=\left\{\sum_{k\in\mathbb{Z}} 2^{k\alpha q}\left|\frac1{|B(x,2^{-k})|}\int_{B(x,2^{-k})}[f(x)-f(y)]\,dy \right|^q \right\}^{1/q}$$

where $f\in L^1_{\operatorname{loc}}({\mathbb R}^n)\cap \mathcal{S}'({\mathbb R}^n)$, $x\in{\mathbb R}^n$, $\alpha\in(0,2)$ and $p, q\in(1,\infty]$. Similar characterizations are also established for Triebel-Lizorkin spaces $\dot F^\alpha_{p,q}(\mathbb{R}^n)$ with $\alpha\in(0,\infty)\setminus 2{\mathbb N}$ and $p,q\in(1,\,\infty]$, and for Besov spaces $\dot B^\alpha_{p,q}(\mathbb{R}^n)$ with $\alpha\in(0,\infty)\setminus 2{\mathbb N}$, $p\in(1,\infty]$ and $q\in(0,\infty]$.


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Dachun Yang. Wen Yuan. Yuan Zhou. "A new characterization of Triebel-Lizorkin spaces on $\mathbb R^n$." Publ. Mat. 57 (1) 57 - 82, 2013.


Published: 2013
First available in Project Euclid: 18 December 2012

zbMATH: 1291.46036
MathSciNet: MR3058927

Primary: 46E35
Secondary: 42B25 , 42B35

Keywords: Besov space , Calderó reproducing formula , square function , Triebel-Lizorkin space

Rights: Copyright © 2013 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.57 • No. 1 • 2013
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