[Restrictions de Fourier et courbes polynomiales ; une inégalité géométrique]
Le résultat que nous annonçons sur les restrictions de Fourier vaut pour des courbes polynomiales générales dans . Il permet de contrôler la norme de la transformée de Fourier relativement à la mesure d'arc affine (dont nous rappelons la définition) à la norme 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 . 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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@article{CRMATH_2008__346_1-2_45_0,
author = {Spyridon Dendrinos and James Wright},
title = {Fourier restriction, polynomial curves and a geometric inequality},
journal = {Comptes Rendus. Math\'ematique},
pages = {45--48},
publisher = {Elsevier},
volume = {346},
number = {1-2},
year = {2008},
doi = {10.1016/j.crma.2007.11.032},
language = {en},
}
Spyridon Dendrinos; James Wright. Fourier restriction, polynomial curves and a geometric inequality. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 45-48. doi : 10.1016/j.crma.2007.11.032. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.032/
[1] Fourier restriction to convex surfaces in , Publ. Mat., Volume 50 (2006), pp. 71-85
[2] Exponent of convergence of the singular integral in the Tarry problem, Dokl. Akad. Nauk SSSR, Volume 248 (1979) no. 2, pp. 268-272 (in Russian)
[3] J.G. Bak, D. Oberlin, A. Seeger, Restriction of Fourier transforms to curves and related oscillatory integrals, preprint
[4] Restriction for flat surfaces of revolution in , Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1905-1914
[5] Maximal functions and Hilbert transforms associated to polynomials, Rev. Mat. Iberoamericana, Volume 14 (1998) no. 1, pp. 117-144
[6] Restriction and decay for flat hypersurfaces, Publ. Mat., Volume 46 (2002), pp. 405-434
[7] On the restriction of the Fourier transform to curves: endpoint results and degenerate cases, Trans. Amer. Math. Soc., Volume 287 (1985), pp. 223-238
[8] Convolution operators with affine arclength measures on plane curves, J. Korean Math. Soc., Volume 36 (1999), pp. 193-207
[9] Restriction of Fourier transforms to curves, Ann. Inst. Fourier, Volume 35 (1985), pp. 117-123
[10] Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc., Volume 108 (1990), pp. 89-96
[11] Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc., Volume 97 (1985), pp. 111-125
[12] Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc., Volume 101 (1987), pp. 541-553
[13] Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 3591-3592
[14] Fourier restriction estimates for affine arclength measures in the plane, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 3303-3305
[15] A uniform Fourier restriction theorem for affine surfaces in , Proc. Amer. Math. Soc., Volume 132 (2004), pp. 1195-1199
[16] An affine restriction estimate in , Proc. Amer. Math. Soc., Volume 135 (2007) no. 4, pp. 1107-1113
[17] Fourier multipliers and estimates for the Fourier transform of measures carried by smooth curves in , Studia Math., Volume 51 (1974), pp. 169-182
[18] On some rules of Laguerre's and systems of equal sums of like powers, Rend. Math., Volume 6 (1971) no. 4, pp. 629-644
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