Lifting in Besov spaces. (April 2020) - Mironescu, Petru; Russ, Emmanuel; Sire, Yannick
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 .