Fourier restriction, polynomial curves and a geometric inequality

Dendrinos, Spyridon; Wright, James

Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 45-48.

Le résultat que nous annonçons sur les restrictions de Fourier vaut pour des courbes polynomiales générales dans ${\mathbb{R}}^d$. Il permet de contrôler la norme $L^q$ de la transformée de Fourier relativement à la mesure d'arc affine (dont nous rappelons la définition) à la norme $L^p$ de la fonction, pour des p et q convenables. La borne est universelle pour toutes les courbes polynomiales de degré donné. Notre méthode repose sur une inégalité géométrique concernant les courbes polynomiales qui est intéressante en elle même, et s'applique à d'autres problèmes d'analyse harmonique euclidienne.

We announce a Fourier restriction result for general polynomial curves in ${\mathbb{R}}^d$. Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal bound for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. There are applications of this geometric inequality to other problems in euclidean harmonic analysis.

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