Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T11:32:47.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform Sobolev estimates for non-trapping metrics

Published online by Cambridge University Press:  07 October 2013

Colin Guillarmou  and
Andrew Hassell
Colin Guillarmou
Affiliation:
DMA, U.M.R. 8553 CNRS, Ecole Normale Supérieure, 45 rue d’Ulm, F 75230 Paris cedex 05, France ([email protected])
Andrew Hassell
Affiliation:
Department of Mathematics, Australian National University, Canberra ACT 0200, Australia ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove uniform Sobolev estimates $\Vert u\Vert _{{L}^{p\prime } } \leq C\Vert (\Delta - \alpha )u\Vert _{{L}^{p} } $ for $\alpha \in \mathbb{C} $ and $p= 2n/ (n+ 2), {p}^{\prime } = 2n/ (n- 2)$ on non-trapping asymptotically conic manifolds of dimension $n\geq 3$, generalizing to non-constant coefficient Laplacians a result of Kenig, Ruiz and Sogge [13].

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 3 , July 2014 , pp. 599 - 632
Copyright
©Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bourgain, J., Shao, P., Sogge, C. D. and Yao, X., On ${L}^{p} $ -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds (arXiv:1204.3927).Google Scholar
Cheng, S. Y, Li, P. and Yau, S.-T., On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), 1021–1063.CrossRefGoogle Scholar
Datchev, K., Local smoothing for scattering manifolds with hyperbolic trapped sets, Commun. Math. Phys. 286 (3) (2009), 837–850.CrossRefGoogle Scholar
Dimassi, M. and Sjöstrand, J., Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, Volume 268, 1999), xii+227 pp.CrossRefGoogle Scholar
Dos Santos Ferreira, D., Kenig, C. E. and Salo, M., On ${L}^{p} $ resolvent estimates for Laplace–Beltrami operators on compact manifolds, Forum Math. to appear.Google Scholar
García-Cuerva, J. and Gatto, A. E., Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (2004), 245–261.CrossRefGoogle Scholar
Guillarmou, C. and Hassell, A., Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds, I. Math. Ann. 341 (4) (2008), 859–896.CrossRefGoogle Scholar
Guillarmou, C., Hassell, A. and Sikora, A., Resolvent at low energy III: the spectral measure, Trans. Amer. Math. Soc. 365 (2013), 6103–6148.Google Scholar
Guillarmou, C., Hassell, A. and Sikora, A., Restriction and spectral multiplier theorems on asymptotically conic manifolds, Anal. PDE 6 (4) (2013), 893–950.CrossRefGoogle Scholar
Hassell, A. and Vasy, A., The resolvent for Laplace-type operators on asymptotically conic spaces, Ann. Inst. Fourier (Grenoble) 51 (5) (2001), 1299–1346.CrossRefGoogle Scholar
Hassell, A. and Wunsch, J., The semiclassical resolvent and the propagator for non-trapping scattering metrics, Adv. Math. 217 (2) (2008), 586–682.CrossRefGoogle Scholar
Ionescu, A. D. and Jerison, D., On the absence of positive eigenvalues of Schrödinger operators with rough potentials, Geom. Funct. Anal. 13 (5) (2003), 1029–1081.Google Scholar
Kenig, C. E., Ruiz, A. and Sogge, C. D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), 329–347.CrossRefGoogle Scholar
Koch, H. and Tataru, D., Carleman estimates and absence of embedded eigenvalues, Commun. Math. Phys. 267 (2) (2006), 419–449.CrossRefGoogle Scholar
Melrose, R. B., The Atiyah–Patodi–Singer index theorem. (AK Peters, Wellesley, 1993).CrossRefGoogle Scholar
Melrose, R. B., Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, in Spectral and scattering theory (Sanda, 1992), Lecture Notes in Pure and Appl. Math., Volume 161, pp. 85–130 (Dekker, New York, 1994).Google Scholar
Melrose, R. B. and Zworski, M., Scattering metrics and geodesic flow at infinity, Invent. Math. 124 (1–3) (1996), 389–436.Google Scholar
Rodnianski, I. and Tao, T., Effective limiting absorption principles, and applications (arXiv:1105.0873).Google Scholar
Seeley, R. T., Complex powers of an elliptic operator, in Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307 (Amer. Math. Soc., Providence, RI, 1967).Google Scholar
Shen, Z., On absolute continuity of the periodic Schrödinger operators, Internat. Math. Res. Not. 1 (2001), 1–31.CrossRefGoogle Scholar
Stein, E., Harmonic analysis, in Real variable methods, orthogonality and oscillatory integrals. (Princeton University Press, 1993).Google Scholar
Varopoulos, N., Hardy–Littlewood Sobolev for semigroups, J. Funct. Anal. 63 (2) (1985), 240–260.CrossRefGoogle Scholar
Vasy, A. and Zworski, M., Semi-classical estimates in asymptotically Euclidean scattering, Commun. Math. Phys. 212 (2000), 205–217.Google Scholar
Wunsch, J. and Zworski, M., Distribution of resonances for asymptotically Euclidean manifolds, J. Differential Geom. 55 (1) (2000), 43–82.CrossRefGoogle Scholar
Zworski, M., Semiclassical analysis, Graduate Studies in Mathematics, Volume 138 (AMS, 2012).CrossRefGoogle Scholar