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Pointwise Characterizations of Even Order Sobolev Spaces via Derivatives of Ball Averages

Part of: Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  15 October 2018

Guangheng Xie ,
Dachun Yang  and
Wen Yuan
Guangheng Xie
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China Email: guanghengxie@mail.bnu.edu.cndcyang@bnu.edu.cnwenyuan@bnu.edu.cn
Dachun Yang
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China Email: guanghengxie@mail.bnu.edu.cndcyang@bnu.edu.cnwenyuan@bnu.edu.cn
Wen Yuan
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China Email: guanghengxie@mail.bnu.edu.cndcyang@bnu.edu.cnwenyuan@bnu.edu.cn
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Abstract

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Let $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$. In this article, the authors establish several equivalent characterizations of Sobolev spaces $W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any $\ell \in \mathbb{N}$ and $k\in \{0,\ldots ,\ell -1\}$, $\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$.

Keywords

Sobolev spacepointwise characterizationball average

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 681 - 699
Copyright
© Canadian Mathematical Society 2019 

Footnotes

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11571039, 11761131002, 11726621, 11671185 and 11871100). Dachun Yang is corresponding author.

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