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December 2009 On dilation operators in Triebel-Lizorkin spaces
Cornelia Schneider, Jan Vybíral
Funct. Approx. Comment. Math. 41(2): 139-162 (December 2009). DOI: 10.7169/facm/1261157806

Abstract

We consider dilation operators $T_k:f\rightarrow f(2^k\cdot)$ in the framework of Triebel-Lizorkin spaces $F^s_{p,q}(\mathbb{R}^n)$. If $s>n\max\big(\frac 1p -1,0\big)$, $T_k$ is a bounded linear operator from $F^s_{p,q}(\mathbb{R}{^n)$ into itself and there are optimal bounds for its norm. We study the situation on the line $s=n\max\big(\frac 1p -1,0\big)$, an open problem mentioned in [ET96, 2.3.1]. It turns out that the results shed new light upon the diversity of different approaches to Triebel-Lizorkin spaces on this line, associated to definitions by differences, Fourier-analytical methods and subatomic decompositions.

Citation

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Cornelia Schneider. Jan Vybíral. "On dilation operators in Triebel-Lizorkin spaces." Funct. Approx. Comment. Math. 41 (2) 139 - 162, December 2009. https://doi.org/10.7169/facm/1261157806

Information

Published: December 2009
First available in Project Euclid: 18 December 2009

zbMATH: 1194.46055
MathSciNet: MR2590330
Digital Object Identifier: 10.7169/facm/1261157806

Subjects:
Primary: 46E35

Rights: Copyright © 2009 Adam Mickiewicz University

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Vol.41 • No. 2 • December 2009
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