no. 6
Semiclassical resolvent estimates at trapped sets
[Estimations de résolvantes semi-classiques et ensembles captifs]
Datchev, Kiril1 ; Vasy, András2
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.

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 : ∥χR h (E+i0)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, χ ˜, est à support dans l’ensemble captif, obtenant ∥χR h (E+i0)χ ˜∥=𝒪(a(h)h -1 ) lorsque la borne a priori est ∥χ ˜R h (E+i0)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: ∥χR h (E+i0)χ∥=𝒪(h -1 ), a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜, to be supported at the trapped set, giving ∥χR h (E+i0)χ ˜∥=𝒪(a(h)h -1 ) when the a priori bound is ∥χ ˜R h (E+i0)χ ˜∥=𝒪(a(h)h -1 ).

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

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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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2752/

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