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Pointwise characterizations of Besov and Triebel-Lizorkin spaces in terms of averages on balls


Authors: Dachun Yang and Wen Yuan
Journal: Trans. Amer. Math. Soc. 369 (2017), 7631-7655
MSC (2010): Primary 46E35; Secondary 42B25, 42B35
DOI: https://doi.org/10.1090/tran/6871
Published electronically: April 11, 2017
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Abstract: In this article, the authors characterize Besov spaces $ \dot B^{\alpha }_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin spaces $ \dot F^{\alpha }_{p,q}(\mathbb{R}^n)$ with $ \alpha \in (0,\infty )$, $ p\in (1,\infty ]$ and $ q\in (0,\infty ]$ ( $ q\in (1,\infty ]$ for $ F$ spaces), as well as their inhomogeneous versions, via some pointwise inequalities involving ball averages. These pointwise characterizations provide a way to introduce Besov and Triebel-Lizorkin spaces with arbitrary positive smoothness on metric measure spaces.


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Additional Information

Dachun Yang
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: dcyang@bnu.edu.cn

Wen Yuan
Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
Email: wenyuan@bnu.edu.cn

DOI: https://doi.org/10.1090/tran/6871
Keywords: Besov space, Triebel-Lizorkin space, ball average, Calder\'on reproducing formula, pointwise characterization
Received by editor(s): June 6, 2015
Received by editor(s) in revised form: October 22, 2015
Published electronically: April 11, 2017
Additional Notes: This project was supported by the National Natural Science Foundation of China (Grant Nos. 11571039 and 11471042), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20120003110003) and the Fundamental Research Funds for Central Universities of China (Grant No. 2014KJJCA10).
The second author is the corresponding author.
Article copyright: © Copyright 2017 American Mathematical Society

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