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

Published online by Cambridge University Press:  20 November 2018

Ciqiang Zhuo ,
Winfried Sickel ,
Dachun Yang  and
Wen Yuan
Ciqiang Zhuo
Affiliation:
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. e-mail: cqzhuo@mail.bnu.edu.cn
Winfried Sickel
Affiliation:
Mathematisches Institut, Friedrich-Schiller-Universität Jena, Jena 07743, Germany. e-mail: winfried.sickel@uni-jena.de
Dachun Yang
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. e-mail: dcyang@bnu.edu.cnwenyuan@bnu.edu.cn
Wen Yuan
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, P. R. China. e-mail: dcyang@bnu.edu.cnwenyuan@bnu.edu.cn
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Abstract

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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 lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$, which are of independent interest.

Keywords

42B2546E3542B35Besov spaceTriebel–Lizorkin spaceball averageCalderón reproducing formula
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 655 - 672
Copyright
Copyright © Canadian Mathematical Society 2017

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