Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces

Petru Mironescu (1)

1 EDPA - Équations aux dérivées partielles, analyse

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).

Mots clés: lifting; unimodular maps; Sobolev spaces

Domaines: Analyse classique [math.CA]

Petru Mironescu. Lifting of ${\mathbb S}^1$-valued maps in sums of Sobolev spaces. 2008. ⟨hal-00747663⟩