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# Remarks on some dispersive estimates

• 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].
Mathematics Subject Classification: Primary: 42B37; Secondary: 35Q40.

 Citation:

•  [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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