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Inhomogeneous Strichartz estimates for the Schrödinger equation. (English) Zbl 1196.35074

The author considers the problem of determining the Lebesgue spaces containing the solution \(u(x,t)=-i\int_0^t e^{i(t-x)\triangle}F(\cdot,\tau)(x)\, d\tau\) to the Cauchy problem on \({\mathbb R}^{n+1}\) (\(n\geq 3\)) for the the inhomogeneous Schrödinger problem
\[ i\partial_t u+\triangle_x u=F(x,t),\qquad u(x,0)=f(x) \]
with initial data \(f=0\).
The Strichartz estimate with mixed norms \(\| u(x,t)\|_{L^q_t L^r_x} \leq c \| F\|_{L^{\bar q'}_t L^{\bar r'}_x}\) was known to hold for a range of exponents. The author improves the results of M. Keel and T. Tao and, following their ideas, in the main result it is shown that the Strichartz estimate holds when \(1/\bar r' -1/r=2/n\).
The paper also contains counterexamples which show that the estimates are false for some exponents.
A final note refers to a preprint by D. Foschi which contains related results.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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[1] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071
[2] Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in \?\textonesuperior , Manuscripta Math. 61 (1988), no. 4, 477 – 494. · Zbl 0696.35153 · doi:10.1007/BF01258601
[3] Thierry Cazenave and Fred B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75 – 100. · Zbl 0763.35085
[4] Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409 – 425. · Zbl 0974.47025 · doi:10.1006/jfan.2000.3687
[5] J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309 – 327 (English, with French summary). · Zbl 0586.35042
[6] Tosio Kato, An \?^{\?,\?}-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 223 – 238. · Zbl 0832.35131
[7] Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955 – 980. · Zbl 0922.35028
[8] S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations, Duke Math. J. 91 (1998), no. 2, 393 – 408. · Zbl 0955.35012 · doi:10.1215/S0012-7094-98-09117-7
[9] Irving Segal, Space-time decay for solutions of wave equations, Advances in Math. 22 (1976), no. 3, 305 – 311. · Zbl 0344.35058 · doi:10.1016/0001-8708(76)90097-9
[10] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[11] Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705 – 714. · Zbl 0372.35001
[12] Terence Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1471 – 1485. · Zbl 0966.35027 · doi:10.1080/03605300008821556
[13] Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477 – 478. · Zbl 0298.42011
[14] Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111 – 114.
[15] Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415 – 426. · Zbl 0638.35036
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