Semi-classical functional calculus on manifolds with ends and weighted L p estimates
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 1181-1223

For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related L p boundedness properties of these operators and show in particular that, although they are not bounded on L p in general, they are always bounded on suitable weighted L p spaces.

Pour une classe de variétés riemanniennes à bouts, nous donnons des développements semi-classiques de fonctions bornées du Laplacien. Nous étudions ensuite des propriétés de continuité L p de ces opérateurs et montrons en particulier que, bien qu’ils ne soient en général pas bornés sur L p , ils le sont toujours sur des espaces L p à poids convenables.

DOI : https://doi.org/10.5802/aif.2638
Classification:  58J40
Keywords: Manifold with ends, L p estimates, h-pseudodifferential operators
@article{AIF_2011__61_3_1181_0,
     author = {Bouclet, Jean-Marc},
     title = {Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {1181-1223},
     doi = {10.5802/aif.2638},
     mrnumber = {2918727},
     zbl = {1236.58033},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_1181_0}
}
Bouclet, Jean-Marc. Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1181-1223. doi : 10.5802/aif.2638. http://www.numdam.org/item/AIF_2011__61_3_1181_0/

[1] Ammann, B.; Lauter, R.; Nistor, V.; Vasy, A. Complex powers and non compact manifolds, Comm. PDE, Tome 29 (2004), pp. 671-705 | Article | MR 2059145 | Zbl 1071.58022

[2] Beals, R. Characterization of pseudo-differential operators and applications, Duke Math. J., Tome 44 (1977) no. 1, pp. 45-57 (and Correction, Duke Math. J. 46, no. 1, 215, (1979)) | Article | MR 435933 | Zbl 0353.35088

[3] Bony, J. M. Caractérisation des opérateurs pseudo-différentiels, Séminaire X-EDP, exp. XXIII (1996-1997) | Numdam | MR 1482829 | Zbl 1061.35531

[4] Bouclet, J. M. Strichartz estimates on asymptotically hyperbolic manifolds (Analysis and PDE (to appear)) | MR 2783305

[5] Bouclet, J. M. Littlewood-Paley decompositions on manifolds with ends, Bulletin de la SMF, Tome 138, fascicule 1 (2010), pp. 1-37 | Numdam | MR 2638890 | Zbl 1198.42013

[6] Bouclet, J. M.; Tzvetkov, N. Strichartz estimates for long range perturbations, Amer. J. Math., Tome 129 (2007) no. 6, pp. 1565-1609 | Article | MR 2369889 | Zbl 1154.35077

[7] Burq, N.; Gérard, P.; Tzvetkov, N. Strichartz inequalities and the non linear Schrödinger equation on compact manifolds, Amer. J. Math., Tome 126 (2004), pp. 569-605 | Article | MR 2058384 | Zbl 1067.58027

[8] Cheeger, J.; Gromov, M.; Taylor, M. Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds, J. Diff. Geom., Tome 17 (1982), pp. 15-53 | MR 658471 | Zbl 0493.53035

[9] Clerc, J. L.; Stein, E. M. L p -multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A., Tome 71 (1974), p. 3911-3912 | Article | MR 367561 | Zbl 0296.43004

[10] Davies, E. B. Spectral theory and differential operators, Cambridge University Press (1995) | MR 1349825 | Zbl 0893.47004

[11] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, Cambridge University Press, London Mathematical Society Lecture Note Series, Tome 268 (1999) | MR 1735654 | Zbl 0926.35002

[12] Grubb, G. Functionnal calculus of pseudo-differential boundary problems, Birkhäuser, Boston Tome 65 (1986) | Zbl 0622.35001

[13] Hassel, A.; Tao, T.; Wunsch, J. A Strichartz inequality for the Schrödinger equation on non-trapping asymptotically conic manifolds, Comm. PDE, Tome 30 (2004), pp. 157-205 | Article | MR 2131050 | Zbl 1068.35119

[14] Helffer, B.; Robert, D. Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Analysis, Tome 53 (1983), pp. 246-268 | Article | MR 724029 | Zbl 0524.35103

[15] Kordyukov, Y. A. L p estimates for functions of elliptic operators on manifolds of bounded geometry, Russian J. Math. Phys., Tome 7 (2000) no. 2, pp. 216-229 | MR 1836640 | Zbl 1065.58021

[16] Melrose, R. B. Geometric scattering theory, Cambridge Univ. Press, Stanford lecture (1995) | MR 1350074 | Zbl 0849.58071

[17] Robert, D. Autour de l’approximation semi-classique, Birkhaüser, Progress in mathematics, Tome 68 (1987) | MR 897108 | Zbl 0621.35001

[18] Schulze, B. W. Pseudo-differential operators on manifolds with singularities, North-Holland, Amsterdam (1991) | MR 1142574 | Zbl 0747.58003

[19] Seeley, R. T. Complex powers of an elliptic operator, Proc. Symp. in Pure Math., Tome 10 (1967), pp. 288-307 | MR 237943 | Zbl 0159.15504

[20] Seeley, R. T. The resolvent of an elliptic boundary problem, Amer. J. Math., Tome 91 (1969), pp. 889-920 | Article | MR 265764 | Zbl 0191.11801

[21] Stein, E.M. Singular integrals and differentiability properties of functions, Princeton Univ. Press (1970) | MR 290095 | Zbl 0207.13501

[22] Taylor, M. L p estimates on functions of the Laplace operator, Duke Math. J., Tome 58 (1989) no. 3, pp. 773-793 | Article | MR 1016445 | Zbl 0691.58043

[23] Taylor, M. Partial Differential Equations II, Linear Equations, Springer, Appl. Math. Sci., Tome 116 (1996) | MR 1395149 | Zbl 0869.35003

[24] Taylor, M. Partial Differential Equations III, Nonlinear Equations, Springer, Appl. Math. Sci., Tome 117 (1996) | MR 1477408 | Zbl 1206.35004