Restriction of toral eigenfunctions to hypersurfaces

Jean Bourgain; Zeév Rudnick

doi:10.1016/j.crma.2009.08.008

Partial Differential Equations

Restriction of toral eigenfunctions to hypersurfaces
[Sur la restriction de fonctions propre du tore à des hyper-surfaces]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Zeév Rudnick ^1,²

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United States
² Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1249-1253.

Résumé
Abstract

Soit $T^{d} = R^{d} / Z^{d}$ le tore plat d-dimensionnel. Pour $d = 2$ et $d = 3$ , on établit des bornes supérieures et inférieures uniformes sur les restrictions des fonctions propres de l'opérateur de Laplace–Beltrami à des surfaces lisses de courbure non nulle.

Let $T^{d} = R^{d} / Z^{d}$ be the d-dimensional flat torus. We establish for $d = 2, 3$ uniform upper and lower bounds on the restrictions of the eigenfunctions of the Laplacian to smooth hyper-surfaces with non-vanishing curvature.

Reçu le : 2009-07-28
Accepté le : 2009-08-06
Publié le : 2009-10-01

DOI : 10.1016/j.crma.2009.08.008

Affiliations des auteurs :

Jean Bourgain ¹ ; Zeév Rudnick ^{1, 2}

¹ School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, United States
² Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

@article{CRMATH_2009__347_21-22_1249_0,
     author = {Jean Bourgain and Ze\'ev Rudnick},
     title = {Restriction of toral eigenfunctions to hypersurfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1249--1253},
     publisher = {Elsevier},
     volume = {347},
     number = {21-22},
     year = {2009},
     doi = {10.1016/j.crma.2009.08.008},
     language = {en},
}

TY  - JOUR
AU  - Jean Bourgain
AU  - Zeév Rudnick
TI  - Restriction of toral eigenfunctions to hypersurfaces
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 1249
EP  - 1253
VL  - 347
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2009.08.008
LA  - en
ID  - CRMATH_2009__347_21-22_1249_0
ER  -

%0 Journal Article
%A Jean Bourgain
%A Zeév Rudnick
%T Restriction of toral eigenfunctions to hypersurfaces
%J Comptes Rendus. Mathématique
%D 2009
%P 1249-1253
%V 347
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2009.08.008
%G en
%F CRMATH_2009__347_21-22_1249_0

Jean Bourgain; Zeév Rudnick. Restriction of toral eigenfunctions to hypersurfaces. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1249-1253. doi : 10.1016/j.crma.2009.08.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.08.008/

Bibliographie
Cité par

[1] N. Burq; P. Gérard; N. Tzvetkov Restrictions of the Laplace–Beltrami eigenfunctions to submanifolds, Duke Math. J., Volume 138 (2007) no. 3, pp. 445-486

[2] J. Bourgain, Z. Rudnick, P. Sarnak, in preparation

[3] J. Cilleruelo; A. Córdoba Trigonometric polynomials and lattice points, Proc. Amer. Math. Soc., Volume 115 (1992) no. 4, pp. 899-905

[4] J. Cilleruelo; A. Granville Lattice points on circles, squares in arithmetic progressions and sumsets of squares, Additive Combinatorics, CRM Proc. Lecture Notes, vol. 43, Amer. Math. Soc., Providence, RI, 2007, pp. 241-262

[5] C.S. Herz Fourier transforms related to convex sets, Ann. of Math. (2), Volume 75 (1962), pp. 81-92

[6] R. Hu, $L^{p}$ norm estimates of eigenfunctions restricted to submanifolds, Forum Math., in press

[7] V. Jarnik Über die Gitterpunkte auf konvexen Kurven, Math. Z., Volume 24 (1926) no. 1, pp. 500-518

[8] P. Sarnak Letter to Reznikov, June 2008 http://www.math.princeton.edu/sarnak/ (see)

Cité par Sources :

Commentaires - Politique