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# Characterizations of Besov-Type and Triebel-Lizorkin-Type Spaces via Averages on Balls

Published:2016-12-09
Printed: Sep 2017
• Ciqiang Zhuo,
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
• Winfried Sickel,
Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena 07743, Germany
• Dachun Yang,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
• Wen Yuan,
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
 Format: LaTeX MathJax PDF

## Abstract

Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces via the sequence $\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between $f$ and the ball average $B_{\ell,2^{-k}}f$. These results give a way to introduce Besov-type spaces, Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any smoothness order on metric measure spaces. As special cases, the authors obtain a new characterization of Morrey-Sobolev spaces and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent interest.
 Keywords: Besov space, Triebel-Lizorkin space, ball average, Calderón reproducing formula
 MSC Classifications: 42B25 - Maximal functions, Littlewood-Paley theory 46E35 - Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems 42B35 - Function spaces arising in harmonic analysis

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