Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1661–1680 | Cite as

Analytic Smoothing Effect for the Nonlinear Schrödinger Equations Without Square Integrability

  • Gaku HoshinoEmail author
  • Ryosuke Hyakuna


In this study we consider the Cauchy problem for the nonlinear Schrödinger equations with data which belong to \(L^p,\)\(1<p<2.\) In particular, we discuss analytic smoothing effect with data which satisfy exponentially decaying condition at spatial infinity in \(L^p,\)\(1<p<2.\) We construct solutions in the function space of analytic vectors for the Galilei generator and the analytic Hardy space with the phase modulation operator based on \(L^{p}\).


Analytic smoothing effect Analytic Hardy space Nonlinear Schrödinger equations 

Mathematics Subject Classification




The authors would like to thank the anonymous referees for their helpful comments and suggestions.


  1. 1.
    Bergh, J., Löfström, J.: Interpolation Spaces an Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  2. 2.
    Carles, R., Mouzaoi, L.: On the Cauchy problem for the Hartree type equation in the Wiener algebra. Proc. Am. Math. Soc. 142(7), 2469–2482 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cazenave, T.: Semilinear Schrödinger equations. In: Courant Lecture Notes in Math., vol. 10. American Mathematical Society (2003)Google Scholar
  4. 4.
    Cazenave, T., Weissler, F.B.: Some Remarks on the Schrödinger Equation in the Critical Case. Lecture Notes in Mathematics, vol. 1394, pp. 18–29. Springer, Berlin (1989)zbMATHGoogle Scholar
  5. 5.
    de Bouard, A.: Analytic solution to non elliptic non linear Schrödinger equations. J. Differ. Equ. 104, 196–213 (1993)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ginibre, J., Velo, G.: On a class of nonlinear Schrodinger equations. I: The Cauchy problem. J. Funct. Anal. 32, 1–32 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grünrock, A.: Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS. Int. Math. Res. Not. 41, 2525–2558 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hayashi, N., Kato, K.: Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations. Commun. Math. Phys. 184, 273–300 (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hayashi, N., Saitoh, S.: Analyticity and smoothing effect for the Schrödinger equation. Ann. Inst. Henri Poincaré, Phys. Théor. 52, 163–173 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hayashi, N., Saitoh, S.: Analyticity and global existence of small solutions to some nonlinear Schrödinger equations. Commun. Math. Phys. 129, 27–41 (1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hayashi, N., Ozawa, T.: On the derivative nonlinear Schrödinger equation. Physica D 55, 14–36 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoshino, G., Hyakuna, R.: Trilinear \(L^p\) estimates with applications to the Cauchy problem for the Hartree-type equation (submitted)Google Scholar
  13. 13.
    Hoshino, G., Ozawa, T.: Analytic smoothing effect for a system of nonlinear Schrödinger equations. Differ. Equ. Appl. 5, 395–408 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions. Osaka J. Math. 51, 609–618 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hoshino, G., Ozawa, T.: Analytic smoothing effect form nonlinear Schrödinger equations with quintic nonlinearity. J. Math. Anal. Appl. 419, 285–297 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hyakuna, R.: Multilinear estimates with applications to the nonlinear Schrödinger and Hartree equation in \(\widehat{L^p}\)-spaces (preprint)Google Scholar
  17. 17.
    Hyakuna, R., Tanaka, T., Tsutsumi, M.: On the global well-posedness for the nonlinear Schrödinger equations with large initial data of infinite \(L^2\) norm. Nonlinear Anal. 74, 1304–1319 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hyakuna, R., Tsutsumi, M.: On the global wellposedness for the nonlinear Schrödinger equations with \(L^p\)-large initial data. Nonlinear Differ. Equ. Appl. 18, 309–327 (2011)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hyakuna, R., Tsutsumi, M.: On existence of global solutions of Schrödinger equations with subcritical nonlinearity for \(L^p\)-initial data. Proc. Am. Math. Soc. 140, 3905–3920 (2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    Kato, K., Ogawa, T.: Analytic smoothing effect and single point singularity for the nonlinear Schrödinger equations. J. Korean Math. Soc. 37, 1071–1084 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Phys. Théor. 46, 113–129 (1987)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kato, T., Masuda, K.: Nonlinear evolution equations and analyticity. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 3, 455–467 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nakamitsu, K.: Analytic finite energy solutions of the nonlinear Schrödinger equation. Commun. Math. Phys. 260, 117–130 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ozawa, T., Yamauchi, K.: Analytic smoothing effect for global solutions to nonlinear Schrödinger equation. J. Math. Anal. Appl. 364, 492–497 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sasaki, H.: Small analytic solutions to the Hartree equation. J. Funct. Anal. 270, 1064–1090 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  27. 27.
    Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse. Appl. Math. Sci., vol. 139. Springer (1999)Google Scholar
  28. 28.
    Tsutsumi, Y.: \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkc. Ekvac. 30, 115–125 (1987)zbMATHGoogle Scholar
  29. 29.
    Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, H.: Local well-posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions. Math. Methods Appl. Sci. 39, 4257–4267 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhou, Y.: Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space \(W^{s, p}\) for \(p<2,\) Trans. Am. Math. Soc. 362, 4683–4694 (2010)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Waseda UniversityTokyoJapan
  2. 2.Osaka UniversityToyonakaJapan

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