New applications of Besov-type and Triebel-Lizorkin-type spaces
type: Item
instance of: scholarly article
title: New applications of Besov-type and Triebel-Lizorkin-type spaces
zbMATH Open document ID: 1185.42022
author: Yoshihiro Sawano
author: Wen Yuan
author: Dachun Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-12-10T00:00:00Z
review text: Given a bump function \(\psi\in \mathcal S(\mathbb R^n)\) with \(\chi_{B(0,2)}\leq \psi\leq \chi_{B(0,4)}\) and denoting \(\phi_j(x)=\psi(2^{-j}x)-\psi(2^{-j-1}), j\in \mathbb Z\), the Besov-Morrey spaces \(\mathcal N^s_{u,p,q}(\mathbb R^n)\) , for \(0<p\leq u<\infty, 0<q\leq\infty, s\in \mathbb R\), are defined as the set of elements in \(\mathcal S'_\infty\), where \(\mathcal S_\infty\) stands for the Schwarz functions with \(\int \phi(x)x^\gamma dx=0\) for \(\gamma\in (\mathbb N\cup \{0\})^n\), such that \[ \sum_{j=-\infty}^\infty \sup_P |P|^{q/u-q/p}\left(\int_P |2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^pdx\right)^{q/p}<\infty \] where \(P\) stands for a dyadic cube. Similarly the Triebel-Lizorkin-Morrey spaces \(\mathcal E^s_{u,p,q}(\mathbb R^n)\) are given by the condition \[ \sup_P |P|^{1/u-1/p} \int_P \left(\sum_{j=-\infty}^\infty|2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^q\right)^{p/q} dx <\infty. \] These spaces were considered by \textit{Y. Sawano} and \textit{H. Tanaka} [Math. Z. 257, No. 4, 871--905 (2007; Zbl 1133.42041)]. Also \textit{D. Yang} and \textit{W. Yuan} [J. Funct. Anal. 255, No. 10, 2760--2809 (2008; Zbl 1169.46016)] introduced, for \(0\leq \tau< \infty, 0<q\leq \infty, s\in \mathbb R\), the Besov-type spaces \(\mathcal B^{s,\tau}_{p,q}(\mathbb R^n)\) and the Triebel-Lizorkin spaces \(\mathcal F^{s,\tau}_{p,q}(\mathbb R^n)\), closely related to the previous ones, and given by the conditions \[ \sup_P |P|^{-\tau q} \sum_{j=-\log_2 \ell(P)}^\infty\left(\int_P |2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^p dx\right)^{q/p} <\infty \] and \[ \sup_P |P|^{-\tau q} \int_P \left(\sum_{j=-\log_2 \ell(P)}^\infty|2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^q\right)^{p/q} dx < \infty \] respectively. The authors show that for \(0<p<u<\infty\) and \(0<q<\infty\) then \(\mathcal N^s_{u,p,q}(\mathbb R^n)\) is a proper subspace of \(\mathcal B^{s,1/p-1/u}_{p,q}(\mathbb R^n)\) and that \(\mathcal N^s_{u,p,\infty}(\mathbb R^n)=\mathcal B^{s,1/p-1/u}_{p,\…
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5644906
zbMATH Keywords: Besov-type spaces
zbMATH Keywords: Trieblel-Lizorkin spaces
zbMATH Keywords: Atomic decomposition
zbMATH Keywords: trace
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.08.002
OpenAlex ID: W2078417726
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DOI: 10.1016/J.JMAA.2009.08.002
description: scientific article; zbMATH DE number 5644906