Remarks on some dispersive estimates

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July 2011, 10(4): 1121-1128. doi: 10.3934/cpaa.2011.10.1121

Remarks on some dispersive estimates

Yonggeun Cho ^1, , Tohru Ozawa ^2, and Suxia Xia ^3,

1.	Department of Mathematics, and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea
2.	Department of Applied Physics, Waseda University, Tokyo, 169-8555
3.	School of Mathematics and System Sciences, Beihang University, Beijing 100191, China

Received March 2010 Revised October 2010 Published April 2011

Abstract
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In this paper we consider the initial value problem for $i\partial_t u + \omega(|\nabla|) u = 0$. Under suitable smoothness and growth conditions on $\omega$, we derive dispersive estimates which is the generalization of time decay and Strichartz estimates. We unify and also simplify dispersive estimates by utilizing the Bessel function. Another main ingredient of this paper is to revisit oscillatory integrals of [2].

Keywords: dispersive estimate, oscillatory integral, time decay, Dispersive equations, Bessel functions., Strichartz estimate.

Mathematics Subject Classification: Primary: 42B37; Secondary: 35Q4.

Citation: Yonggeun Cho, Tohru Ozawa, Suxia Xia. Remarks on some dispersive estimates. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1121-1128. doi: 10.3934/cpaa.2011.10.1121

References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, New York, 1976.
[2]	Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Disctrete Cont. Dynam. Syst., 17 (2007), 691-711. doi: 10.3934/dcds.2007.17.691.
[3]	S. Gustafson, K. Nakanishi and T.-P. Tsai, Scattering for the Gross-Pitaevskii equation, Math. Research Letters, 13 (2006), 273-285. doi: 10.1142/S0219199709003491.
[4]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[5]	E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Univ. Press, Princeton, N.J., 1993.

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References:

[1]	J. Bergh and J. Löfström, "Interpolation Spaces," Springer-Verlag, New York, 1976.
[2]	Y. Cho and T. Ozawa, On small amplitude solutions to the generalized Boussinesq equations, Disctrete Cont. Dynam. Syst., 17 (2007), 691-711. doi: 10.3934/dcds.2007.17.691.
[3]	S. Gustafson, K. Nakanishi and T.-P. Tsai, Scattering for the Gross-Pitaevskii equation, Math. Research Letters, 13 (2006), 273-285. doi: 10.1142/S0219199709003491.
[4]	M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.
[5]	E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Univ. Press, Princeton, N.J., 1993.

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