Semiclassical resolvent estimates at trapped sets  [ Estimations de résolvantes semi-classiques et ensembles captifs ]
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, p. 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 : https://doi.org/10.5802/aif.2752
Classification:  58J47,  35L05
Mots clés: Estimations de résolvantes, ensembles captifs, propagation de singularités
@article{AIF_2012__62_6_2379_0,
     author = {Datchev, Kiril and Vasy, Andr\'as},
     title = {Semiclassical resolvent estimates at~trapped sets},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2379-2384},
     doi = {10.5802/aif.2752},
     mrnumber = {3060761},
     zbl = {1271.58015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_6_2379_0}
}
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/item/AIF_2012__62_6_2379_0/

[1] Burq, Nicolas Lower bounds for shape resonances widths of long range Schrödinger operators, Amer. J. Math., Tome 124 (2002) no. 4, pp. 677-755 | MR 1914456 | Zbl 1013.35019

[2] Burq, Nicolas; Zworski, Maciej Geometric control in the presence of a black box, J. Amer. Math. Soc., Tome 17:2 (2004) no. 4, pp. 443-471 | MR 2051618 | Zbl 1050.35058

[3] Cardoso, Fernando; Vodev, Georgi Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II, Ann. Henri Poincaré, Tome 3 (2002) no. 4, pp. 673-691 | MR 1933365 | Zbl 1021.58016

[4] Christianson, Hans Semiclassical non-concentration near hyperbolic orbits, J. Funct. Anal., Tome 246 (2007) no. 2, pp. 145-195 (Corrigendum, J. Funct. Anal., 258 (2010), no. 3 p. 1060-1065) | MR 2321040 | Zbl 1119.58018

[5] Christianson, Hans; Schenck, Emmanuel; Vasy, András; Wunsch, Jared From resolvent estimates to damped waves (To appear in J. Anal. Math. Preprint available at arXiv:1206.1565, 2012)

[6] Datchev, Kiril; Vasy, András Propagation through trapped sets and semiclassical resolvent estimates, Annales de l’Institut Fourier, Tome 62.6 (2012), pp. 2345-2375

[7] Hörmander, Lars The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Springer Verlag (1994) | MR 1313500 | Zbl 0601.35001