메뉴 건너뛰기




Volumn 20, Issue 3, 2010, Pages 627-656

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

Author keywords

hyperbolic trapped set; Schr dinger equation; Strichartz estimates

Indexed keywords


EID: 77956649223     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00039-010-0076-5     Document Type: Article
Times cited : (61)

References (44)
  • 1
    • 57849105880 scopus 로고    scopus 로고
    • Entropy and the localization of eigenfunctions
    • Anantharaman N.: Entropy and the localization of eigenfunctions. Annals of Math. 168(2), 435-475 (2008).
    • (2008) Annals of Math. , vol.168 , Issue.2 , pp. 435-475
    • Anantharaman, N.1
  • 2
    • 40949115555 scopus 로고    scopus 로고
    • Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
    • Anantharaman N., Nonnenmacher S.: Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier (Grenoble) 57(7), 1465-2523 (2007).
    • (2007) Ann. Inst. Fourier (Grenoble) , vol.57 , Issue.7 , pp. 1465-2523
    • Anantharaman, N.1    Nonnenmacher, S.2
  • 4
    • 0141511737 scopus 로고
    • Eigenfunctions which are concentrated in the neighborhood of a closed geodesic (in Russian)
    • Babič V. M.: Eigenfunctions which are concentrated in the neighborhood of a closed geodesic (in Russian). Zap. Nau cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 9, 15-63 (1968).
    • (1968) Zap. Nau Cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) , vol.9 , pp. 15-63
    • Babič, V.M.1
  • 5
    • 35648961455 scopus 로고    scopus 로고
    • The nonlinear Schrödinger equation on the hyperbolic space
    • Banica V.: The nonlinear Schrödinger equation on the hyperbolic space. Comm. PDE 32(10), 1643-1677 (2007).
    • (2007) Comm. PDE , vol.32 , Issue.10 , pp. 1643-1677
    • Banica, V.1
  • 6
    • 77956648227 scopus 로고    scopus 로고
    • p estimates, Arxiv: 0711. 358.
  • 7
    • 77956651279 scopus 로고    scopus 로고
    • Littlewood-Paley decompositions on manifolds with ends
    • Bouclet J.-M.: Littlewood-Paley decompositions on manifolds with ends. Bull. Soc. Math. Fr. 138(1), 1-37 (2010).
    • (2010) Bull. Soc. Math. Fr. , vol.138 , Issue.1 , pp. 1-37
    • Bouclet, J.-M.1
  • 8
    • 77956650458 scopus 로고    scopus 로고
    • J.-M. Bouclet, Strichartz estimates for asymptotically hyperbolic manifolds, Analysis and PDE, in press.
  • 9
    • 38049181248 scopus 로고    scopus 로고
    • Bouclet J.-M., Tzvetkov N.: Strichartz estimates for long range perturbations. Amer. J. Math. 129: 6, 1665-1609 (2007).
  • 10
    • 34250084341 scopus 로고
    • Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I
    • Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. Geom. Funct. Anal. 3, 107-156 (1993).
    • (1993) Schrödinger Equations. Geom. Funct. Anal. , vol.3 , pp. 107-156
    • Bourgain, J.1
  • 11
    • 3142714420 scopus 로고    scopus 로고
    • Smoothing effect for Schrödinger boundary value problems
    • Burq N.: Smoothing effect for Schrödinger boundary value problems. Duke Math. J. 123(2), 403-427 (2004).
    • (2004) Duke Math. J. , vol.123 , Issue.2 , pp. 403-427
    • Burq, N.1
  • 12
    • 2942564358 scopus 로고    scopus 로고
    • Strichartz inequalities and the non-linear Schrödinger equation on compact manifolds
    • Burq N., Gerard P., Tzvetkov N.: Strichartz inequalities and the non-linear Schrödinger equation on compact manifolds. Amer. J. Math 126, 569-605 (2004).
    • (2004) Amer. J. Math , vol.126 , pp. 569-605
    • Burq, N.1    Gerard, P.2    Tzvetkov, N.3
  • 13
    • 2642542715 scopus 로고    scopus 로고
    • On Nonlinear Schrödinger equations in exterior domains
    • Burq N., Gerard P., Tzvetkov N.: On Nonlinear Schrödinger equations in exterior domains. Ann. I. H. P. Ana. non lin. 21, 295-318 (2004).
    • (2004) Ann. I.H.P. Ana. Non Lin. , vol.21 , pp. 295-318
    • Burq, N.1    Gerard, P.2    Tzvetkov, N.3
  • 14
    • 1842843728 scopus 로고    scopus 로고
    • Geometric control in the presence of a black box
    • Burq N., Zworski M.: Geometric control in the presence of a black box. J. Amer. Math. Soc. 17(2), 443-471 (2004).
    • (2004) J. Amer. Math. Soc. , vol.17 , Issue.2 , pp. 443-471
    • Burq, N.1    Zworski, M.2
  • 15
    • 0036025586 scopus 로고    scopus 로고
    • Uniform estimates of the Laplace-Beltrami operator on infinite volume Riemannian manifolds II
    • Cardoso F., Vodev G.: Uniform estimates of the Laplace-Beltrami operator on infinite volume Riemannian manifolds II. Ann. H. Poincaré 3, 673-691 (2002).
    • (2002) Ann. H. Poincaré , vol.3 , pp. 673-691
    • Cardoso, F.1    Vodev, G.2
  • 16
    • 23844511635 scopus 로고    scopus 로고
    • Global existence results for nonlinear Schrödinger equations with quadratic potentials
    • Carles R.: Global existence results for nonlinear Schrödinger equations with quadratic potentials. Discrete Contin. Dyn. Syst. 13(2), 385-398 (2005).
    • (2005) Discrete Contin. Dyn. Syst. , vol.13 , Issue.2 , pp. 385-398
    • Carles, R.1
  • 17
    • 0035255579 scopus 로고    scopus 로고
    • Maximal functions associated to filtrations
    • Christ M., Kiselev A.: Maximal functions associated to filtrations. J. Funct. Anal. 179, 409-425 (2001).
    • (2001) J. Funct. Anal. , vol.179 , pp. 409-425
    • Christ, M.1    Kiselev, A.2
  • 18
    • 59449101954 scopus 로고    scopus 로고
    • Cutoff resolvent estimates and the semilinear Schrödinger equation, Proc
    • Christianson H.: Cutoff resolvent estimates and the semilinear Schrödinger equation, Proc. Amer. Math. Soc. 136(10), 3513-3520 (2008).
    • (2008) Amer. Math. Soc. , vol.136 , Issue.10 , pp. 3513-3520
    • Christianson, H.1
  • 19
    • 0040245853 scopus 로고
    • Effets régularisants locaux pour des équations dispersives générales
    • Constantin P., Saut J.-C.: Effets régularisants locaux pour des équations dispersives générales. C. R. AcadS ci. Paris Sér. I Math. 304(14), 407-410 (1987).
    • (1987) C. R. AcadS Ci. Paris Sér. I Math. , vol.304 , Issue.14 , pp. 407-410
    • Constantin, P.1    Saut, J.-C.2
  • 20
    • 59449094977 scopus 로고    scopus 로고
    • K. Datchev, Local smoothing for scattering manifolds with hyperbolic trapped sets, Comm. Math. Phys. 286: 3, 837-850.
  • 21
    • 0039250942 scopus 로고    scopus 로고
    • Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow
    • Doi S.-I.: Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow. Math. Ann. 318, 355-389 (2000).
    • (2000) Math. Ann. , vol.318 , pp. 355-389
    • Doi, S.-I.1
  • 22
    • 36549094784 scopus 로고
    • Semiclassical quantization of the scattering from a classically chaotic repellor
    • Gaspard P., Rice S. A.: Semiclassical quantization of the scattering from a classically chaotic repellor. J. Chem. Phys. 90, 2242-2254 (1989).
    • (1989) J. Chem. Phys. , vol.90 , pp. 2242-2254
    • Gaspard, P.1    Rice, S.A.2
  • 23
    • 77956649005 scopus 로고    scopus 로고
    • C. Guillarmou, S. Moroianu, J. Park, Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds., Advances in Math., to appear; Arxiv 0901. 4082.
  • 24
    • 77956652075 scopus 로고    scopus 로고
    • B. Hasselblatt, A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1997.
  • 25
    • 33748331204 scopus 로고    scopus 로고
    • Sharp Strichartz estimates on nontrapping asymptotically conic manifolds
    • Hassell A., Tao T., Wunsch J.: Sharp Strichartz estimates on nontrapping asymptotically conic manifolds. Amer. J. Math. 128(4), 963-1024 (2006).
    • (2006) Amer. J. Math. , vol.128 , Issue.4 , pp. 963-1024
    • Hassell, A.1    Tao, T.2    Wunsch, J.3
  • 26
    • 0000059859 scopus 로고    scopus 로고
    • Symbolic functional calculus and N-body resolvent estimates
    • Hassell A., Vasy A.: Symbolic functional calculus and N-body resolvent estimates. J. Funct. Anal. 173(2), 257-283 (2000).
    • (2000) J. Funct. Anal. , vol.173 , Issue.2 , pp. 257-283
    • Hassell, A.1    Vasy, A.2
  • 28
    • 0001138601 scopus 로고    scopus 로고
    • Endpoint Strichartz estimates
    • Keel M., Tao T.: Endpoint Strichartz estimates. Amer. J. Math. 15, 955-980 (1998).
    • (1998) Amer. J. Math. , vol.15 , pp. 955-980
    • Keel, M.1    Tao, T.2
  • 30
    • 0001189870 scopus 로고
    • Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature
    • Mazzeo R., Melrose R.: Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 75, 260-310 (1987).
    • (1987) J. Funct. Anal. , vol.75 , pp. 260-310
    • Mazzeo, R.1    Melrose, R.2
  • 31
    • 77956650664 scopus 로고    scopus 로고
    • R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, in "Spectral and Scattering Theory" (M. Ikawa, ed.), Marcel Dekker, 1994.
  • 32
    • 72249107608 scopus 로고    scopus 로고
    • Quantum decay rates in chaotic scattering
    • Nonnenmacher S., Zworski M.: Quantum decay rates in chaotic scattering. Acta Math. 203(2), 149-233 (2009).
    • (2009) Acta Math. , vol.203 , Issue.2 , pp. 149-233
    • Nonnenmacher, S.1    Zworski, M.2
  • 33
    • 77956652775 scopus 로고    scopus 로고
    • G. P. Paternain, Geodesic Flows. Progress in Mathematics 180, Birkhäuser Boston Inc., Boston MA (1999).
  • 34
    • 0000131067 scopus 로고
    • The limit set of a Fuchsian group
    • Patterson S. J.: The limit set of a Fuchsian group. Acta Math. 136, 241-273 (1976).
    • (1976) Acta Math. , vol.136 , pp. 241-273
    • Patterson, S.J.1
  • 35
    • 0035589894 scopus 로고    scopus 로고
    • Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume
    • Perry P.: Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume. Geom. Funct. Anal. 11, 132-141 (2001).
    • (2001) Geom. Funct. Anal. , vol.11 , pp. 132-141
    • Perry, P.1
  • 36
    • 77956648446 scopus 로고    scopus 로고
    • M. F. Py skina, The asymptotic behavior of eigenfunctions of the Helmholtz equation that are concentrated near a closed geodesic (in Russian), Zap. Nau cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 15 (1969), 154-160.
  • 37
    • 0000792579 scopus 로고
    • On the construction of quasimodes associated with stable periodic orbits
    • Ralston J. V.: On the construction of quasimodes associated with stable periodic orbits. Comm. Math. Phys. 51-3, 219-242 (1976).
    • (1976) Comm. Math. Phys. , vol.3-51 , pp. 219-242
    • Ralston, J.V.1
  • 38
    • 77956651243 scopus 로고    scopus 로고
    • J. Schmeling, R. Siegmund-Schultze, Hölder continuity of the holonomy maps for hyperbolic basic sets. I., Ergodic Theory and Related Topics, III (Güstrow, 1990), Springer Lecture Notes in Math. 1514 (1992), 174-191.
  • 39
    • 0036383355 scopus 로고    scopus 로고
    • Strichartz estimates for a Schrödinger operator with nonsmooth coefficients
    • Staffilani G., Tataru D.: Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Part. Diff. Eq. 27(7-8), 1337-1372 (2002).
    • (2002) Comm. Part. Diff. Eq. , vol.27 , Issue.7-8 , pp. 1337-1372
    • Staffilani, G.1    Tataru, D.2
  • 40
    • 84972553620 scopus 로고
    • Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
    • Strichartz R.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(3), 705-714 (1977).
    • (1977) Duke Math. J. , vol.44 , Issue.3 , pp. 705-714
    • Strichartz, R.1
  • 41
    • 51249182653 scopus 로고
    • The density at infinity of a discrete group of hyperbolic motions
    • Sullivan D.: The density at infinity of a discrete group of hyperbolic motions. Publ. IHES 50, 172-202 (1979).
    • (1979) Publ. IHES , vol.50 , pp. 172-202
    • Sullivan, D.1
  • 42
    • 0040970521 scopus 로고    scopus 로고
    • On 2D nonlinear Schrödinger equation on R × T
    • Takaoka H., Tzvetkov N.: On 2D nonlinear Schrödinger equation on R × T. J. Funct. Anal. 182(2), 427-442 (2001).
    • (2001) J. Funct. Anal. , vol.182 , Issue.2 , pp. 427-442
    • Takaoka, H.1    Tzvetkov, N.2
  • 43
    • 0040199721 scopus 로고    scopus 로고
    • Distribution of resonances for asymptotically Euclidean manifolds
    • Wunsch J., Zworski M.: Distribution of resonances for asymptotically Euclidean manifolds. J. Diff. Geom. 55, 43-82 (2000).
    • (2000) J. Diff. Geom. , vol.55 , pp. 43-82
    • Wunsch, J.1    Zworski, M.2
  • 44
    • 0033473946 scopus 로고    scopus 로고
    • Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces
    • Zworski M.: Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Invent. Math. 136(2), 353-409 (1999).
    • (1999) Invent. Math. , vol.136 , Issue.2 , pp. 353-409
    • Zworski, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.