Abstract
We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣs(ℝn). We control the weightedL 2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular solutions of the free Schrödinger equation.
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J.A. Barceló, J.M. Bennett, A. Carbery, A. Ruiz, and M.C. Vilela, Some special solutions of the Schrödinger equation,Indiana Univ. Math. J. 56 (2007), 1581–1593.
J.A. Barceló, J.M. Bennett, A. Carbery, A. Ruiz, and M.C. Vilela, A note on weighted estimates for the Schrödinger operator,Rev. Mat. Complut. 21 (2008), 481–488.
J.A. Barceló, A. Ruiz, and L. Vega, Weighted estimates for the Helmholtz equation and some applications,J. Funct. Anal. 150 (1997), 356–382.
J.M. Bennett, A. Carbery, F. Soria, and A. Vargas, A Stein conjecture for the circle,Math. Ann. 336 (2006), 671–695.
O. Blasco, A. Ruiz, and L. Vega, Non-interpolation in Morrey-Campanato and block spaces,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 31–40.
J. Bergh and J. Löfström,Interpolation Spaces, An Introduction, Springer-Verlag, Berlin-New York, 1976.
T. Cazenave, L. Vega, and M.C. Vilela, A note on the nonlinear Schrödinger equation in weakL p spaces,Commun. Contemp. Math. 3 (2001), 153–162.
T. Cazenave and F.B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation inH 1,Manuscripta Math. 61 (1988), 477–494.
S. Chanillo and E. Sawyer, Unique continuation for Δ +V and the C. Fefferman-Phong class,Trans. Amer. Math. Soc. 318 (1990), 275–300.
C. Fefferman and D.H. Phong, Lower bounds for Schrödinger equations,Conference on Partial Differential Equations (Saint Jean de Monts), Conf. No. 7, Soc. Math. France, Paris, 1982.
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.
A. Jeffrey (Ed.)Table of Integrals, Series and Products, Academic Press, New York-London, 1965.
M. Keel and T. Tao, Endpoint Strichartz estimates,Amer. J. Math. 120 (1998), 955–980.
A. Ruiz and L. Vega, Unique continuation for Schrödinger operators with potential in Morrey spaces, Conference on Mathematical Analysis (El Escorial, 1989),Publ. Mat. 35 (1991), 291–298.
A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials,Duke Math. J. 76 (1994), 913–940.
R.S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,Duke Math. J. 44 (1977), 705–714.
T.H. Wolff, Unique continuation for ⋎Δu⋎ ≤V ⋎∇u⋎ and related problems,Rev. Mat. Iberoamericana 6 (1990), 155–200.
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The first and fourth authors were supported by Spanish Grant MTM2008-02568, the fifth by Spanish Grant MTM2007-62186, the second by EPSRC Grant EP/E022340/1 and the third by EC project “Pythagoras II” and a Leverhulme Study Abroad Fellowship.
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Barceló, J.A., Bennett, J.M., Carbery, A. et al. Strichartz inequalities with weights in Morrey-Campanato classes. Collect. Math. 61, 49–56 (2010). https://doi.org/10.1007/BF03191225
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DOI: https://doi.org/10.1007/BF03191225