LIFTING OF S1 VALUED MAPS IN SUMS OF SOBOLEV SPACES

LIFTING OF S1-VALUED MAPS IN SUMS OF SOBOLEV SPACES PETRU MIRONESCU Abstract. We describe, in terms of lifting, the closure of smooth S1-valued maps in the space W s,p((?1, 1)N ;S1). (Here, 0 < s <∞ and 1 ≤ p <∞.) This description follows from an estimate for the phase of smooth maps: let 0 < s < 1, let ? ? C∞([?1, 1]N ;R) and set u = eı?. Then we may split ? = ?1 + ?2, where the smooth maps ?1 and ?2 satisfy (?) |?1|W s,p ≤ C|u|W s,p and ???2? sp Lsp ≤ C|u| p W s,p . (?) was proved for s = 1/2, p = 2 and arbitrary space dimension N by Bourgain and Brezis [3] and for N = 1, p > 1 and s = 1/p by Nguyen [14]. Our proof is a sort of continuous version of the Bourgain-Brezis approach (based on paraproducts). Estimate (?) answers (and generalizes) a question of Bourgain, Brezis, and the author [5]. 1. Introduction In [4], the authors addressed the problem of lifting of S1-valued maps in Sobolev spaces: (Ls,p) Given an arbitrary u ? W s,p(Q;S1), is there

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• s1-valued maps


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• bourgain- brezis argument


• gagliardo-nirenberg embedding


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