Lifting default for ${\mathbb S}^1$-valued maps

Petru Mironescu (1)

1 EDPA - Équations aux dérivées partielles, analyse

Let $\varphi\in C^\infty ([0,1]^N ; {\mathbb R})$ and set $g=e^{\imath\varphi}$. When $0$<$s$<$1$, $p\ge 1$ and $1\le sp$<$N$, the $W^{s,p}$-semi-norm $|\varphi|_{W^{s,p}}$ of $\varphi$ is not controlled by $|g|_{W^{s,p}}$ (cf Bourgain, Brezis and the author, J. Anal. Math. 2000). (This question is related to the existence, for ${\mathbb S}^1$-valued maps $g$, of a lifting $\varphi$ as smooth as allowed by $g$.) In a subsequent work (Comm. Pure Appl. Math. 2005), the same authors suggested that $|g|_{W^{s,p}}$ does control a weaker quantity, namely $|\varphi|_{W^{s,p}+W^{sp,1}}$. Existence of such control is due to Bourgain and Brezis (J. Amer. Math. Soc. 2003) when $1$<$p\le 2$ and $s=1/p$ and to Nguyen (C. R. Acad. Sci. Paris 2008) when $N=1$, $p$>$1$ and $sp\ge 1$, or when $N\ge 2$, $p$>$1$ and $sp$>$1$. In this work, we establish the existence of a control in the full range $0$<$s$<$1$, $p\ge 1$ and $N\ge 1$. In particular, we do not require that $sp\ge 1$, as in the previous results.

Mots clés: lifting; unimodular maps; Sobolev spaces

Domaines: Analyse classique [math.CA]

Petru Mironescu. Lifting default for ${\mathbb S}^1$-valued maps. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2008, 346 (19-20), pp.1093-1044. ⟨hal-00747676⟩