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15 June 2007 Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds
N. Burq, P. Gérard, N. Tzvetkov
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Duke Math. J. 138(3): 445-486 (15 June 2007). DOI: 10.1215/S0012-7094-07-13834-1

Abstract

We estimate the Lp-norm (2p+) of the restriction to a curve of the eigenfunctions of the Laplace-Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere, these estimates are sharp. If the curve has nonvanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the sphere. Moreover, we sketch their extension to higher dimensions.

On prouve une estimation de la norme Lp (2p+) de la restriction à une courbe des fonctions propres de l'opérateur de Laplace-Beltrami sur une surface riemannienne. Si la courbe est une géodésique de la sphère, on montre que nos estimations sont optimales. En revanche, si la courbe possède une courbure géodésique non nulle, on améliore le résultat. Toutes nos estimées sont optimales sur la sphère. Nous en esquissons par ailleurs des généralisations aux dimensions supérieures

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N. Burq. P. Gérard. N. Tzvetkov. "Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds." Duke Math. J. 138 (3) 445 - 486, 15 June 2007. https://doi.org/10.1215/S0012-7094-07-13834-1

Information

Published: 15 June 2007
First available in Project Euclid: 18 June 2007

zbMATH: 1131.35053
MathSciNet: MR2322684
Digital Object Identifier: 10.1215/S0012-7094-07-13834-1

Subjects:
Primary: 35P20
Secondary: 35J15 , 53C21

Rights: Copyright © 2007 Duke University Press

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Vol.138 • No. 3 • 15 June 2007
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