Lifting in Besov spaces. (April 2020)

• Record Type: Journal Article
• Authors: Mironescu, Petru; Russ, Emmanuel; Sire, Yannick
• Is Part Of: Nonlinear analysis. Volume 193(2020)
• Publication Date: 2020-04
• Subjects: 46E35; Besov spaces -- Lifting -- Weighted Sobolev spaces -- VMO -- Jacobian -- Trace -- Restriction; 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.2019.03.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let Ω be a smooth bounded (simply connected) domain in R n and let u be a complex-valued measurable function on Ω such that | u ( x ) | = 1 a.e. Assume that u belongs to a Besov space B p, q s ( Ω ; ℂ ) . We investigate whether there exists a real-valued function φ ∈ B p, q s ( Ω ; R ) such that u = e ı φ . This complements the corresponding study in Sobolev spaces due to Bourgain, Brezis and the first author. The microscopic parameter q turns out to play an important role in some limiting situations. The analysis of this lifting problem relies on some interesting new properties of Besov spaces, in particular a non-restriction property when q > p .