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On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications
1. | Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, South Korea |
2. | School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, South Korea |
3. | Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea |
References:
[1] |
J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[2] |
T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[3] |
Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.
doi: 10.1512/iumj.2013.62.4970. |
[4] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[5] |
E. Cordero and F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr., 281 (2008), 25-41.
doi: 10.1002/mana.200610585. |
[6] |
E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974.
doi: 10.1016/j.jde.2008.07.009. |
[7] |
P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290.
doi: 10.1007/s00208-005-0672-0. |
[8] |
D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[9] |
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327. |
[10] |
L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008. |
[11] |
Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[12] |
T. Kato, An $L^{q,r}$ -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223-238. |
[13] |
Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861.
doi: 10.1016/j.jmaa.2011.09.039. |
[14] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[15] |
Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160.
doi: 10.1016/j.jmaa.2010.06.019. |
[16] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[17] |
S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468.
doi: 10.1016/j.jmaa.2011.11.067. |
[18] |
S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726.
doi: 10.4171/RMI/797. |
[19] |
V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338.
doi: 10.1007/s00208-005-0720-9. |
[20] |
I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227.
doi: 10.1512/iumj.2011.60.4824. |
[21] |
S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168.
doi: 10.4171/RMI/591. |
[22] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993. |
[23] |
R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[24] |
M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
show all references
References:
[1] |
J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976. |
[2] |
T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys., 147 (1992), 75-100.
doi: 10.1007/BF02099529. |
[3] |
Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.
doi: 10.1512/iumj.2013.62.4970. |
[4] |
M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409-425.
doi: 10.1006/jfan.2000.3687. |
[5] |
E. Cordero and F. Nicola, Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation, Math. Nachr., 281 (2008), 25-41.
doi: 10.1002/mana.200610585. |
[6] |
E. Cordero and F. Nicola, Some new Strichartz estimates for the Schrödinger equation, J. Differential equations., 245 (2008), 1945-1974.
doi: 10.1016/j.jde.2008.07.009. |
[7] |
P. D'Ancona, V. Pierfelice and N. Visciglia, Some remarks on the Schrödinger equation with a potential in $L_t^rL_x^s$, Math. Ann., 333 (2005), 271-290.
doi: 10.1007/s00208-005-0672-0. |
[8] |
D. Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24.
doi: 10.1142/S0219891605000361. |
[9] |
J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéare, 2 (1985), 309-327. |
[10] |
L. Grafakos, Classical Fourier Analysis, $2^{nd}$ edition, Graduate Texts in Mathematics, 249. Springer, New York, 2008. |
[11] |
Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations, J. Anal. Math., 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[12] |
T. Kato, An $L^{q,r}$ -theory for nonlinear Schrödinger equations, in Spectral and scattering theory and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, 23 (1994), 223-238. |
[13] |
Y. Ke, Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861.
doi: 10.1016/j.jmaa.2011.09.039. |
[14] |
M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
doi: 10.1353/ajm.1998.0039. |
[15] |
Y. Koh, Improved inhomogeneous Strichartz estimates for the Schrödinger equation, J. Math. Anal. Appl., 373 (2011), 147-160.
doi: 10.1016/j.jmaa.2010.06.019. |
[16] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[17] |
S. Lee and I. Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl., 389 (2012), 461-468.
doi: 10.1016/j.jmaa.2011.11.067. |
[18] |
S. Lee and I. Seo, On inhomogeneous Strichartz estimates for the Schrödinger equation, Rev. Mat. Iberoam., 30 (2014), 711-726.
doi: 10.4171/RMI/797. |
[19] |
V. Naibo and A. Stefanov, On some Schrödinger and wave equations with time dependent potentials, Math. Ann., 334 (2006), 325-338.
doi: 10.1007/s00208-005-0720-9. |
[20] |
I. Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J., 60 (2011), 1203-1227.
doi: 10.1512/iumj.2011.60.4824. |
[21] |
S. Shao, Sharp linear and bilinear restriction estimate for paraboloids in the cylinderically symmetric case, Rev. Mat. Iberoam., 25 (2009), 1127-1168.
doi: 10.4171/RMI/591. |
[22] |
E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. 1993. |
[23] |
R. S. Strichartz, Restriction of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), 705-714.
doi: 10.1215/S0012-7094-77-04430-1. |
[24] |
M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136.
doi: 10.1090/S0002-9947-06-04099-2. |
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