Semiclassical resolvent estimates at trapped sets

Datchev, Kiril; Vasy, András

doi:10.5802/aif.2752

Semiclassical resolvent estimates at trapped sets
[Estimations de résolvantes semi-classiques et ensembles captifs]

Datchev, Kiril ¹ ; Vasy, András ²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4397, U.S.A.
² Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.

Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2379-2384.

Résumé
Abstract

Nous étendons nos résultats récents sur la propagation d’estimations de résolvantes semi-classiques à travers des ensembles captifs sous des bornes a priori de type polynomial. Précédemment, nous obtenions des estimations non-captives dans des situations captives quand la résolvante est contrôlée par au dessus et en dessous par des fonctions cutoff $χ$ dont le support microlocal est situé loin de l’ensemble captif : $∥ χ R_{h} (E + i 0) χ ∥ = 𝒪 (h^{- 1})$ (version microlocale d’un résultat de Burq et Cardoso-Vodev). Nous considérons maintenant le cas où l’une des deux fonctions cutoff, $\tilde{χ}$ , est à support dans l’ensemble captif, obtenant $∥ χ R_{h} (E + i 0) \tilde{χ} ∥ = 𝒪 (\sqrt{a (h)} h^{- 1})$ lorsque la borne a priori est $∥ \tilde{χ} R_{h} (E + i 0) \tilde{χ} ∥ = 𝒪 (a (h) h^{- 1})$ .

We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs $χ$ microlocally supported away from the trapping: $∥ χ R_{h} (E + i 0) χ ∥ = 𝒪 (h^{- 1})$ , a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, $\tilde{χ}$ , to be supported at the trapped set, giving $∥ χ R_{h} (E + i 0) \tilde{χ} ∥ = 𝒪 (\sqrt{a (h)} h^{- 1})$ when the a priori bound is $∥ \tilde{χ} R_{h} (E + i 0) \tilde{χ} ∥ = 𝒪 (a (h) h^{- 1})$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2752

Classification : 58J47, 35L05
Keywords: Resolvent estimates, trapping, propagation of singularities
Mot clés : Estimations de résolvantes, ensembles captifs, propagation de singularités

Affiliations des auteurs :

Datchev, Kiril ¹ ; Vasy, András ²

¹ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4397, U.S.A.
² Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.

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     author = {Datchev, Kiril and Vasy, Andr\'as},
     title = {Semiclassical resolvent estimates at~trapped sets},
     journal = {Annales de l'Institut Fourier},
     pages = {2379--2384},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     doi = {10.5802/aif.2752},
     zbl = {1271.58015},
     mrnumber = {3060761},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2752/}
}

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Datchev, Kiril; Vasy, András. Semiclassical resolvent estimates at trapped sets. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2379-2384. doi : 10.5802/aif.2752. http://www.numdam.org/articles/10.5802/aif.2752/

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