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

Mironescu, Petru

Comptes Rendus. Mathématique, Tome 346 (2008) no. 19-20, pp. 1039-1044.

Soit $\phi \in C^{\infty }({[0,1]}^N,\mathbb{R})$. Si $0<s<1$, $p⩾1$ et $1⩽sp<N$, alors la semi-norme ${|\phi |}_{W^{s,p}}$ n'est pas contrôlée par ${|g|}_{W^{s,p}}$, où $g:=e^{ı\phi }$ [J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [Cette question est liée à l'existence, pour des g à valeurs dans ${\mathbb{S}}^1$, de relèvements φ aussi réguliers que g le permet.] Dans [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], il est conjecturé que ${|g|}_{W^{s,p}}$ contrôle une quantité plus faible que ${|\phi |}_{W^{s,p}}$, plus spécifiquement ${|\phi |}_{W^{s,p}+W^{1,sp}}$. L'existence d'un tel contrôle est due à J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation div $Y=f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] pour $1<p⩽2$ et $s=1/p$ et à H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] pour $N=1$, $p>1$ et $sp⩾1$ ou pour $N⩾2$, $p>1$ et $sp>1$. Dans cette Note, nous montrons l'existence d'un contrôle pour tout $s<1$, $p⩾1$ et N.

Let $\phi \in C^{\infty }({[0,1]}^N,\mathbb{R})$. When $0<s<1$, $p⩾1$ and $1⩽sp<N$, the $W^{s,p}$-semi-norm ${|\phi |}_{W^{s,p}}$ of φ is not controlled by ${|g|}_{W^{s,p}}$, where $g:=e^{ı\phi }$ [J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [This question is related to existence, for ${\mathbb{S}}^1$-valued maps g, of a lifting φ as smooth as allowed by g.] In [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], the authors suggested that ${|g|}_{W^{s,p}}$ does control a weaker quantity, namely ${|\phi |}_{W^{s,p}+W^{1,sp}}$. Existence of such control is due to J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation div $Y=f$ and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] when $1<p⩽2$, $s=1/p$ and to H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] when $N=1$, $p>1$ and $sp⩾1$ or when $N⩾2$, $p>1$ and $sp>1$. In this Note, we establish existence of control for all $s<1$, $p⩾1$ and N.

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