Characterizations of Sobolev spaces via averages on balls. (November 2015)

• Record Type: Journal Article
• Authors: Dai, Feng; Gogatishvili, Amiran; Yang, Dachun; Yuan, Wen
• Is Part Of: Nonlinear analysis. Volume 128(2015)
• Page Start: 86
• Page End: 99
• Publication Date: 2015-11
• Subjects: primary 46E35 -- secondary 42B25 42B35; Sobolev space -- Average on ball -- Difference -- Euclidean space -- Space of homogeneous type; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.07.024
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, the authors prove several equivalent characterizations of Sobolev spaces of even integer orders on R n, using the average B t f ( x ) ≔ 1 | B ( x, t ) | ∫ B ( x, t ) f ( y ) d y of a function f over the ball B ( x, t ) ≔ { y ∈ R n : | y − x | < t } with x ∈ R n and t ∈ ( 0, ∞ ) . These characterizations rely only on the metric and the Lebesgue measure on R n and are simpler than those obtained recently by Alabern et al. (2012). Moreover, these results may shed new light on the theory of high order Sobolev spaces on spaces of homogeneous type.