Harmonic Analysis/Mathematical Analysis
Fourier restriction, polynomial curves and a geometric inequality
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 45-48.

We announce a Fourier restriction result for general polynomial curves in Rd. 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.

Le résultat que nous annonçons sur les restrictions de Fourier vaut pour des courbes polynomiales générales dans Rd. Il permet de contrôler la norme Lq de la transformée de Fourier relativement à la mesure d'arc affine (dont nous rappelons la définition) à la norme Lp 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.11.032
Dendrinos, Spyridon 1; Wright, James 2

1 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
2 School of Mathematics, University of Edinburgh, JCMB, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
@article{CRMATH_2008__346_1-2_45_0,
     author = {Dendrinos, Spyridon and Wright, James},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.11.032/}
}
TY  - JOUR
AU  - Dendrinos, Spyridon
AU  - Wright, James
TI  - Fourier restriction, polynomial curves and a geometric inequality
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 45
EP  - 48
VL  - 346
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.11.032/
DO  - 10.1016/j.crma.2007.11.032
LA  - en
ID  - CRMATH_2008__346_1-2_45_0
ER  - 
%0 Journal Article
%A Dendrinos, Spyridon
%A Wright, James
%T Fourier restriction, polynomial curves and a geometric inequality
%J Comptes Rendus. Mathématique
%D 2008
%P 45-48
%V 346
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.11.032/
%R 10.1016/j.crma.2007.11.032
%G en
%F CRMATH_2008__346_1-2_45_0
Dendrinos, Spyridon; Wright, James. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.11.032/

[1] Abi-Khuzam, F.; Shayya, B. Fourier restriction to convex surfaces in R3, Publ. Mat., Volume 50 (2006), pp. 71-85

[2] Arkipov, G.I.; Karatsuba, A.A.; Chubarikov, V.N. 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] Carbery, A.; Kenig, C.; Ziesler, S. Restriction for flat surfaces of revolution in R3, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1905-1914

[5] Carbery, A.; Ricci, F.; Wright, J. Maximal functions and Hilbert transforms associated to polynomials, Rev. Mat. Iberoamericana, Volume 14 (1998) no. 1, pp. 117-144

[6] Carbery, A.; Ziesler, S. Restriction and decay for flat hypersurfaces, Publ. Mat., Volume 46 (2002), pp. 405-434

[7] Christ, M. 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] Choi, Y. Convolution operators with affine arclength measures on plane curves, J. Korean Math. Soc., Volume 36 (1999), pp. 193-207

[9] Drury, S.W. Restriction of Fourier transforms to curves, Ann. Inst. Fourier, Volume 35 (1985), pp. 117-123

[10] Drury, S.W. Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc., Volume 108 (1990), pp. 89-96

[11] Drury, S.W.; Marshall, B. Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc., Volume 97 (1985), pp. 111-125

[12] Drury, S.W.; Marshall, B. Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc., Volume 101 (1987), pp. 541-553

[13] Oberlin, D. Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 3591-3592

[14] Oberlin, D. Fourier restriction estimates for affine arclength measures in the plane, Proc. Amer. Math. Soc., Volume 129 (2001), pp. 3303-3305

[15] Oberlin, D. A uniform Fourier restriction theorem for affine surfaces in R3, Proc. Amer. Math. Soc., Volume 132 (2004), pp. 1195-1199

[16] Shayya, B. An affine restriction estimate in R3, Proc. Amer. Math. Soc., Volume 135 (2007) no. 4, pp. 1107-1113

[17] Sjölin, P. Fourier multipliers and estimates for the Fourier transform of measures carried by smooth curves in R2, Studia Math., Volume 51 (1974), pp. 169-182

[18] Steinig, J. On some rules of Laguerre's and systems of equal sums of like powers, Rend. Math., Volume 6 (1971) no. 4, pp. 629-644

Cited by Sources: