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Littlewood–Paley characterizations of fractional Sobolev spaces via averages on balls

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

Published online by Cambridge University Press:  22 June 2018

Feng Dai ,
Jun Liu ,
Dachun Yang  and
Wen Yuan
Feng Dai
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada (fdai@ualberta.ca)
Jun Liu
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China (junliu@mail.bnu.edu.cn; dcyang@bnu.edu.cn; wenyuan@bnu.edu.cn)
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 (junliu@mail.bnu.edu.cn; dcyang@bnu.edu.cn; wenyuan@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 (junliu@mail.bnu.edu.cn; dcyang@bnu.edu.cn; wenyuan@bnu.edu.cn)
*
*Corresponding author.
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Abstract

By invoking some new ideas, we characterize Sobolev spaces Wα,p(ℝn) with the smoothness order α ∊ (0, 2] and p ∊ (max{1, 2n/(2α + n)},∞), via the Lusin area function and the Littlewood–Paley g*λ-function in terms of centred ball averages. We also show that the assumption p ∊ (max{1, 2n/(2α + n)},∞) is nearly sharp in the sense that these characterizations are no longer true when p ∊ (1, max{1, 2n/(2α + n)}). These characterizations provide a possible new way to introduce Sobolev spaces with smoothness order in (1, 2] on metric measure spaces.

Keywords

Sobolev spaceball averageLusin area functiong*λ-function

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 6 , December 2018 , pp. 1135 - 1163
Copyright
Copyright © Royal Society of Edinburgh 2018 

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