Mathematische Annalen

, Volume 335, Issue 3, pp 645–673 | Cite as

A smoothing property of Schrödinger equations in the critical case

  • Michael Ruzhansky
  • Mitsuru SugimotoEmail author


This paper deals with the critical case of the global smoothing estimates for the Schrödinger equation. Although such estimates fail for critical orders of weights and smoothing, it is shown that they are still valid if one works with operators with symbols vanishing on a certain set. The geometric meaning of this set is clarified in terms of the Hamiltonian flow of the Laplacian. The corresponding critical case of the limiting absorption principle for the resolvent is also established. Obtained results are extended to dispersive equations of Schrödinger type, to hyperbolic equations and to equations of other orders.


Dispersive Equation Hyperbolic Equation Geometric Meaning Critical Case Smoothing Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ben-Artzi, M., Klainerman, S.: Decay and regularity for the Schrödinger equation. J. Analyse Math. 58, 25–37 (1992)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Chihara, H.: Smoothing effects of dispersive pseudodifferential equations. Comm. Partial Differential Equations 27, 1953–2005 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cordes, H.O.: The technique of pseudodifferential operators. Cambridge Univ. Press, Cambridge, 1995Google Scholar
  4. 4.
    Coriasco, S.: Fourier integral operators in SG classes I: composition theorems and action on SG Sobolev spaces. Rend. Sem. Mat. Univ. Pol. Torino 57, 249–302 (1999)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Hardy, G.H., Littlewood, J.E.: Some Properties of Fractional Integrals, I. Math. Zeit. 27, 565–606 (1928)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Springer-Verlag, Berlin-New York, 1983Google Scholar
  7. 7.
    Kato, T., Yajima, T.: Some examples of smooth operators and the associated smoothing effect. Rev. Math. Phys. 1, 481–496 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry. II., Interscience, New York-London-Sydney, 1969Google Scholar
  9. 9.
    Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Amer. Math. Soc. 255, 343–362 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Matsumura, M.: Asymptotic behavior at infinity for Green's functions of first order systems with characteristics of nonuniform multiplicity. Publ. Res. Inst. Math. Sci. 12, 317–377 (1976)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Miyachi, A.: On some estimates for the wave equation in L p and H p. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 331–354 (1980)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Ruzhansky, M., Sugimoto, M.: A new proof of global smoothing estimates for dispersive equations. Operator Theory: Advances and Applications 155, 65–75 (2004)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Ruzhansky, M., Sugimoto, M.: Global L 2-boundedness theorems for a class of Fourier integral operators. Comm. Partial Differential Equations 31, 547–569 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Ruzhansky, M., Sugimoto, M.: Structural properties of derivative nonlinear Schrödinger equations, (preprint)Google Scholar
  15. 15.
    Ruzhansky, M., Sugimoto, M.: Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations. Operator Theory: Advances and Applications 164, 65–78 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ruzhansky, M., Sugimoto, M.: Weighted L 2 estimates for a class of Fourier integral operators, (preprint)Google Scholar
  17. 17.
    Sjölin, P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Stein, E.M., Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7, 503–514 (1958)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Sugimoto, M.: Global smoothing properties of generalized Schrödinger equations. J. Anal. Math. 76, 191–204 (1998)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Sugimoto, M.: A Smoothing property of Schrödinger equations along the sphere. J. Anal. Math. 89, 15–30 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Sugimoto, M., Tsujimoto, K.: A resolvent estimate and a smoothing property of inhomogeneous Schrödinger equations. Proc. Japan Acad. Ser. A Math. Sci. 74, 74–76 (1998)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Watanabe, K.: Smooth perturbations of the selfadjoint operator |Δ|α/2. Tokyo J. Math. 14, 239–250 (1991)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan

Personalised recommendations