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

Feng Dai; Jun Liu; Dachun Yang; Wen Yuan

doi:10.1017/S0308210517000440

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

Show author details

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.

Article contents

Abstract

Rights & Permissions

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.