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A new characterization of Triebel-Lizorkin spaces on $\mathbb R^n$

Dachun Yang, Wen Yuan, and Yuan Zhou

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Abstract

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 $q\in(0,\infty]$.

Article information

Source
Publ. Mat., Volume 57, Number 1 (2013), 57-82.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.pm/1355854298

Mathematical Reviews number (MathSciNet)
MR3058927

Zentralblatt MATH identifier
1291.46036

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis

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

Citation

Yang, Dachun; Yuan, Wen; Zhou, Yuan. A new characterization of Triebel-Lizorkin spaces on $\mathbb R^n$. Publ. Mat. 57 (2013), no. 1, 57--82. https://projecteuclid.org/euclid.pm/1355854298


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