Lorentz space extension of Strichartz estimates

Ahn, Cheonghee; Cho, Yonggeun

doi:10.1090/S0002-9939-05-07891-3

Lorentz space extension of Strichartz estimates
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by Cheonghee Ahn and Yonggeun Cho PDF

Proc. Amer. Math. Soc. 133 (2005), 3497-3503 Request permission

Abstract:

In this paper, Strichartz estimates for the solution of the Schrödinger evolution equation are considered on a mixed normed space with Lorentz norm with respect to the time variable.

References

Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
N. Burq, F. Planchon, J.G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the Wave and Schrödinger Equations with Potentials of Critical Decay, in preprint.
Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. MR 1055532, DOI 10.1016/0362-546X(90)90023-A
Yonggeun Cho, Eunhee Koh, and Sanghyuk Lee, A maximal inequality for filtration on some function spaces, Osaka J. Math. 41 (2004), no. 2, 267–276. MR 2069087
Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
Michael Christ and Alexander Kiselev, Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: some optimal results, J. Amer. Math. Soc. 11 (1998), no. 4, 771–797. MR 1621882, DOI 10.1090/S0894-0347-98-00276-8
F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), no. 1, 87–109. MR 1124294, DOI 10.1016/0022-1236(91)90103-C
Michael Cwikel, On $(L^{po}(A_{o}),\,\ L^{p_{1}}(A_{1}))_{\theta },\,_{q}$, Proc. Amer. Math. Soc. 44 (1974), 286–292. MR 358326, DOI 10.1090/S0002-9939-1974-0358326-0
Tosio Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281–306. MR 1383498, DOI 10.1007/BF02787794
Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sci. Publ. Math. 19 (1964), 5–68 (French). MR 165343, DOI 10.1007/BF02684796
Kenji Nakanishi, Asymptotically-free solutions for the short-range nonlinear Schrödinger equation, SIAM J. Math. Anal. 32 (2001), no. 6, 1265–1271. MR 1856248, DOI 10.1137/S0036141000369083
Kenji Nakanishi and Tohru Ozawa, Remarks on scattering for nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 45–68. MR 1891695, DOI 10.1007/s00030-002-8118-9
F. Nier and A. Soffer, Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator, J. Funct. Anal. 198 (2003), no. 2, 511–535. MR 1964550, DOI 10.1016/S0022-1236(02)00099-X
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
Hartmut Pecher, $L^{p}$-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. 150 (1976), no. 2, 159–183. MR 435604, DOI 10.1007/BF01215233
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451–513. MR 2038194, DOI 10.1007/s00222-003-0325-4
Hart F. Smith and Christopher D. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian, Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171–2183. MR 1789924, DOI 10.1080/03605300008821581
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Terence Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1471–1485. MR 1765155, DOI 10.1080/03605300008821556