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Volumn 4, Issue 4, 2007, Pages 337-359

Weighted strichartz estimates for radial schrödinger equation on noncompact manifolds

Author keywords

Damek Ricci spaces; Noncompact manifold; Radial schr dinger equation; Strichartz estimates

Indexed keywords


EID: 39349085612     PISSN: 1548159X     EISSN: None     Source Type: Journal    
DOI: 10.4310/DPDE.2007.v4.n4.a3     Document Type: Article
Times cited : (33)

References (47)
  • 2
    • 85036944837 scopus 로고    scopus 로고
    • Ramona Anton. Strichartz Inequalities for Lipschitz Metrics on Manifolds and Nonlinear
    • Ramona Anton. Strichartz Inequalities for Lipschitz Metrics on Manifolds and Nonlinear
  • 3
    • 85036954053 scopus 로고    scopus 로고
    • Schrödinger Equation on Domains. to appear in Bulletin de la Société Mathématique de France.
    • Schrödinger Equation on Domains. to appear in Bulletin de la Société Mathématique de France.
  • 4
    • 0034355370 scopus 로고    scopus 로고
    • Besov spaces and Strichartz estimates on the Heisenberg group. (Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg.)
    • Hajer Bahouri, Patrick Gérard, and Chao-Jiang Xu. Besov spaces and Strichartz estimates on the Heisenberg group. (Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg.). J. Anal. Math., 82:93-118, 2000.
    • (2000) J. Anal. Math , vol.82 , pp. 93-118
    • Bahouri, H.1    Gérard, P.2    Xu, C.-J.3
  • 5
    • 85036939412 scopus 로고    scopus 로고
    • Valeria Banica. The nonlinear Schrödinger equation on hyperbolic space. to appear in Comm. P.D.E., 2005.
    • Valeria Banica. The nonlinear Schrödinger equation on hyperbolic space. to appear in Comm. P.D.E., 2005.
  • 6
    • 85036933125 scopus 로고    scopus 로고
    • Valeria Banica, Rémi Carles, and Gigliola Staffilani. Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space. to appear in G.A.F.A., 2006.
    • Valeria Banica, Rémi Carles, and Gigliola Staffilani. Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space. to appear in G.A.F.A., 2006.
  • 7
    • 0001240303 scopus 로고
    • Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation
    • Jacqueline E. Barab. Nonexistence of asymptotically free solutions for a nonlinear Schrödinger equation. J. Math. Phys., 25:3270-3273, 1984.
    • (1984) J. Math. Phys , vol.25 , pp. 3270-3273
    • Barab, J.E.1
  • 8
    • 33646121745 scopus 로고    scopus 로고
    • Some dispersive estimates for Schrödinger equations with repulsive potentials
    • Juan A. Barcelo, Alberto Ruiz, and Luis Vega. Some dispersive estimates for Schrödinger equations with repulsive potentials. J. Funct. Anal., 236(1):1-24, 2006.
    • (2006) J. Funct. Anal , vol.236 , Issue.1 , pp. 1-24
    • Barcelo, J.A.1    Ruiz, A.2    Vega, L.3
  • 9
    • 85036920335 scopus 로고    scopus 로고
    • F.A. Berezin and M.A. Shubin. The Schrödinger equation. Transl. from the Russian by Yu. Rajabov, D. A. Leites and N. A. Sakharova. Rev. and exp. version. Mathematics and Its Applications. Soviet Series. 66. Dordrecht etc.: Kluwer Academic Publishers. xvii, 555 p. 1991.
    • F.A. Berezin and M.A. Shubin. The Schrödinger equation. Transl. from the Russian by Yu. Rajabov, D. A. Leites and N. A. Sakharova. Rev. and exp. version. Mathematics and Its Applications. Soviet Series. 66. Dordrecht etc.: Kluwer Academic Publishers. xvii, 555 p. 1991.
  • 10
    • 0003207990 scopus 로고
    • Generalized Heisenberg groups and Damek-Ricci harmonic spaces
    • Berlin: Springer-Verlag. viii, 125 p
    • Jurgen Berndt, Franco Tricerri, and Lieven Vanhecke. Generalized Heisenberg groups and Damek-Ricci harmonic spaces. Lecture Notes in Mathematics. 1598. Berlin: Springer-Verlag. viii, 125 p., 1995.
    • (1995) Lecture Notes in Mathematics , vol.1598
    • Berndt, J.1    Tricerri, F.2    Vanhecke, L.3
  • 11
    • 85036935074 scopus 로고    scopus 로고
    • arXiv:math/0609455
    • Matthew Blair, Hart Smith, and Christopher Sogge. On Stricbartz; estimates for Schrödinger operators in compact manifolds with boundary. arXiv:math/0609455, 2006.
    • (2006)
    • Blair, M.1    Smith, H.2    Sogge, C.3
  • 12
    • 85036931777 scopus 로고    scopus 로고
    • Strichartz estimates on asymptotically hyperbolic manifolds
    • in preparation
    • Jean-Marc Bouclet. Strichartz estimates on asymptotically hyperbolic manifolds, in preparation.
    • Bouclet, J.-M.1
  • 13
    • 39349113667 scopus 로고    scopus 로고
    • arXiv:math/0611705
    • Jean-Marc Bouclet and Nikolay Tzvetkov. On global Strichartz estimates for non trapping metrics. arXiv:math/0611705, 2006.
    • (2006)
    • Bouclet, J.-M.1    Tzvetkov, N.2
  • 14
    • 34250084341 scopus 로고
    • Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations
    • J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations. Geom. Funct. Anal., 3(2):107-156, 1993.
    • (1993) Geom. Funct. Anal , vol.3 , Issue.2 , pp. 107-156
    • Bourgain, J.1
  • 17
    • 2942564358 scopus 로고    scopus 로고
    • Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds
    • N. Burq, P. Gérard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. Am. J. Math., 126(3):569-605, 2004.
    • (2004) Am. J. Math , vol.126 , Issue.3 , pp. 569-605
    • Burq, N.1    Gérard, P.2    Tzvetkov, N.3
  • 18
    • 12744280057 scopus 로고    scopus 로고
    • Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J., 53(6):1665-1680, 2004.
    • Nicolas Burq, Fabrice Planchon, John G. Stalker, and A. Shadi Tahvildar-Zadeh. Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J., 53(6):1665-1680, 2004.
  • 19
    • 85036950525 scopus 로고    scopus 로고
    • arXiv:math/0609391
    • Rémi Carles. On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential. arXiv:math/0609391, 2006.
    • (2006)
    • Carles, R.1
  • 21
    • 84967708598 scopus 로고
    • A class of nonsymmetric harmonic Riemannian spaces
    • Ewa Damek and Fulvio Ricci. A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc., New Ser., 27(1):139-142, 1992.
    • (1992) Bull. Am. Math. Soc., New Ser , vol.27 , Issue.1 , pp. 139-142
    • Damek, E.1    Ricci, F.2
  • 23
    • 33947363826 scopus 로고    scopus 로고
    • A singular critical potential for the Schrödinger operator
    • Thomas Duyckaerts. A singular critical potential for the Schrödinger operator. Canad. Math. Bull., 50(1):35-47, 2007.
    • (2007) Canad. Math. Bull , vol.50 , Issue.1 , pp. 35-47
    • Duyckaerts, T.1
  • 24
    • 51249179626 scopus 로고
    • A construction of the fundamental solution for the Schrödinger equation
    • Daisuke Fujiwara. A construction of the fundamental solution for the Schrödinger equation. J. Anal. Math., 35:41-96, 1979.
    • (1979) J. Anal. Math , vol.35 , pp. 41-96
    • Fujiwara, D.1
  • 25
    • 0346275418 scopus 로고    scopus 로고
    • Semilinear hyperbolic equations
    • Tokyo: Mathematical Society of Japan. viii, 208 p
    • Vladimir Georgiev. Semilinear hyperbolic equations. MSJ Memoirs. 7. Tokyo: Mathematical Society of Japan. viii, 208 p., 2000.
    • (2000) MSJ Memoirs , vol.7
    • Georgiev, V.1
  • 27
    • 85050669204 scopus 로고
    • The global Cauchy problem for the nonlinear Schrödinger equation revisited
    • J. Ginibre and G. Velo. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2(4):309-327, 1985.
    • (1985) Ann. Inst. H. Poincaré Anal. Non Linéaire , vol.2 , Issue.4 , pp. 309-327
    • Ginibre, J.1    Velo, G.2
  • 28
    • 33746746992 scopus 로고    scopus 로고
    • Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials
    • M. Goldberg. Dispersive bounds for the three-dimensional Schrödinger equation with almost critical potentials. Geom. Flunct. Anal., 16(3):517-536, 2006.
    • (2006) Geom. Flunct. Anal , vol.16 , Issue.3 , pp. 517-536
    • Goldberg, M.1
  • 29
    • 33645677003 scopus 로고    scopus 로고
    • Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials
    • DOI: 10.1155/IMRN/2006/13927
    • Goldberg, M. , Vega, L. , Visciglia, N., Counterexamples of Strichartz inequalities for Schrödinger equations with repulsive potentials, International Mathematics Research Notices(2006), VOL:2006, page: 1-16 , DOI: 10.1155/IMRN/2006/13927
    • (2006) International Mathematics Research Notices , vol.2006 , pp. 1-16
    • Goldberg, M.1    Vega, L.2    Visciglia, N.3
  • 30
    • 17444367698 scopus 로고    scopus 로고
    • A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds
    • Andrew Hassell, Terence Tao, and Jared Wunsch. A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds. Commun. Partial Differ. Equations, 30(2):157-205, 2005.
    • (2005) Commun. Partial Differ. Equations , vol.30 , Issue.2 , pp. 157-205
    • Hassell, A.1    Tao, T.2    Wunsch, J.3
  • 31
    • 0000570999 scopus 로고
    • Wave operators and similarity for some non-selfadjoint operators
    • Tosio Kato. Wave operators and similarity for some non-selfadjoint operators. Math. Ann., 162:258-279, 1965/1966.
    • (1965) Math. Ann , vol.162 , pp. 258-279
    • Kato, T.1
  • 32
    • 0001138601 scopus 로고    scopus 로고
    • Endpoint Strichartz estimates
    • Markus Keel and Terence Tao. Endpoint Strichartz estimates. Amer. J. Math., 120(5):955-980, 1998.
    • (1998) Amer. J. Math , vol.120 , Issue.5 , pp. 955-980
    • Keel, M.1    Tao, T.2
  • 33
    • 84972495996 scopus 로고
    • Convolution operators and L(p, q) spaces
    • R. O Neil. Convolution operators and L(p, q) spaces. Duke Math. J., 30:129-1427 1963.
    • (1963) Duke Math. J , vol.30 , pp. 129-1427
    • Neil, R.O.1
  • 34
    • 0003312578 scopus 로고    scopus 로고
    • Riemannian geometry
    • New York, NY: Springer
    • Peter Petersen. Riemannian geometry. Graduate Texts in Mathematics. 171. New York, NY: Springer. xvi, 432 p., 1998.
    • Graduate Texts in Mathematics , vol.171-16 , Issue.432 , pp. 1998
    • Petersen, P.1
  • 35
    • 39349093245 scopus 로고    scopus 로고
    • Weighted Strichartz estimates for the Schroedinger and wave equations on Damek-Ricci spaces
    • preprint
    • Vittoria Pierfelice. Weighted Strichartz estimates for the Schroedinger and wave equations on Damek-Ricci spaces. preprint, 2005.
    • (2005)
    • Pierfelice, V.1
  • 37
    • 85036919600 scopus 로고    scopus 로고
    • Luc Robbiano and Claude Zuily. Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials, arXiv:math/ 0610339 (2006).
    • Luc Robbiano and Claude Zuily. Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials, arXiv:math/ 0610339 (2006).
  • 38
    • 33845337712 scopus 로고    scopus 로고
    • Strichartz estimates for Schrödinger equations with variable coefficients
    • Mem. Soc. Math. Fr
    • Luc Robbiano and Claude Zuily. Strichartz estimates for Schrödinger equations with variable coefficients. Mem. Soc. Math. Fr., Nouv. Ser. 101-102, 2005.
    • (2005) Nouv. Ser , pp. 101-102
    • Robbiano, L.1    Zuily, C.2
  • 39
    • 1542508715 scopus 로고    scopus 로고
    • Time decay for solutions of Schrödinger equations with rough and time-dependent potentials
    • Igor Rodnianski and Wilhelm Schlag. Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math., 155(3):451-513, 2004.
    • (2004) Invent. Math , vol.155 , Issue.3 , pp. 451-513
    • Rodnianski, I.1    Schlag, W.2
  • 40
    • 0036383355 scopus 로고    scopus 로고
    • Strichartz estimates for a Schrödinger operator with nonsmooth coefficients
    • Gigliola Staffilani and Daniel Tataru. Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. Comm. Partial Differential Equations, 27(7-8):1337-1372, 2002.
    • (2002) Comm. Partial Differential Equations , vol.27 , Issue.7-8 , pp. 1337-1372
    • Staffilani, G.1    Tataru, D.2
  • 41
    • 85036910013 scopus 로고    scopus 로고
    • W.A. Strauss. Nonlinear scattering theory. Scattering Theory math. Phys., Proc. NATO advanced Study Inst., Denver 1973, 53-78 (1974)., 1974.
    • W.A. Strauss. Nonlinear scattering theory. Scattering Theory math. Phys., Proc. NATO advanced Study Inst., Denver 1973, 53-78 (1974)., 1974.
  • 42
    • 84972553620 scopus 로고
    • Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
    • Robert S. Strichartz. 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.S.1
  • 43
    • 0034355004 scopus 로고    scopus 로고
    • Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation
    • Terence Tao. Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Commun. Partial Differ. Equations, 25(7-8):1471-1485, 2000.
    • (2000) Commun. Partial Differ. Equations , vol.25 , Issue.7-8 , pp. 1471-1485
    • Tao, T.1
  • 44
    • 23044521817 scopus 로고    scopus 로고
    • Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation
    • Daniel Tataru. Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation. Trans. Am. Math. Soc., 353(2):795-807, 2001.
    • (2001) Trans. Am. Math. Soc , vol.353 , Issue.2 , pp. 795-807
    • Tataru, D.1
  • 45
    • 0035379526 scopus 로고    scopus 로고
    • Regularity of solutions to the free Schrödinger equation with radial initial data
    • M. C. Vilela. Regularity of solutions to the free Schrödinger equation with radial initial data. Ill. J. Math., 45(2):361-370, 2001.
    • (2001) Ill. J. Math , vol.45 , Issue.2 , pp. 361-370
    • Vilela, M.C.1
  • 46
    • 0000483491 scopus 로고
    • Existence of solutions for Schrödinger evolution equations
    • Kenji Yajima. Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys., 110:415-426, 1987.
    • (1987) Commun. Math. Phys , vol.110 , pp. 415-426
    • Yajima, K.1
  • 47
    • 3543151381 scopus 로고    scopus 로고
    • Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity
    • Kenji Yajima and Guoping Zhang. Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differ. Equations, 202(1):81-110, 2004.
    • (2004) J. Differ. Equations , vol.202 , Issue.1 , pp. 81-110
    • Yajima, K.1    Zhang, G.2


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