Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces
Résumé
We describe, in terms of lifting, the closure of smooth ${\mathbb S}^1$-valued maps in the space $W^{s,p} ((−1, 1)^N ; {\mathbb S}^1)$. Here, $0$<$s$<$\infty$ and $1\le p$<$\infty$. This description follows from an estimate for the phase of smooth maps: let $0$<$s$<$1$, $\varphi\in C^\infty ([-1,1]^N ; {\mathbb R})$ and set $u=e^{\imath\varphi}$. Then we may split $\varphi=\varphi_1+\varphi_2$, where the smooth maps $\varphi_1$ and $\varphi_2$ satisfy
(*) $|\varphi_1|_{W^{s,p}}\le C\, |u|_{W^{s,p}}$ and $\|\nabla\varphi_2\|_{L^{sp}}\le C\, |u|_{W^{s,p}}^{1/s}$.
Estimate (*) was obtained by Bourgain and Brezis (J. Amer. Math. Soc. 2003) when $s=1/2$ and $p=2$, and by Nguyen (C. R. Acad. Sci. 2008) when $N=1$, $p$>$1$ and $s=1/p$.
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 (Comm. Pure Appl. Math. 2005).
Domaines
Analyse classique [math.CA]Origine | Fichiers produits par l'(les) auteur(s) |
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