Abstract
This article discusses some smoothing estimates of the initial value problem for dispersive equations with constant coefficients. In particular, it is shown that a certain condition for the principal part of the symbol (see the assumption (1.3) below, which is equivalent to the one “of principal type” in the paper by Ben-Artzi and Devinatz [2]) is necessary and sufficient for the maximal smoothing in space-time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Agmon and L. Hörmander,Asymptotic properties of solutions of differential equations with simple characteristics, J. Analyse Math.30 (1976), 1–38.
M. Ben-Artzi and A. Devinatz,Local smoothing and convergence properties of Schrödinger type equations, J. Funct. Anal.101 (1991), 231–254.
M. Ben-Artzi and S. Klainerman,Decay and regularity for the Schrödinger equation, J. Analyse Math.58 (1992), 25–37.
H. Chihara,Smoothing effects of dispersive pseudodifferential equations, Comm. Partial Differential Equations27 (2002), 1953–2005.
P. Constantin and J. C. Saut,Local smoothing properties of dispersive equations, J. Amer. Math. Soc.1 (1988), 413–439.
W. Craig, T. Kappeler and W. Strauss,Microlocal dispersive smoothing for the Schrödinger equation, Comm. Pure Appl. Math.48 (1995), 769–860.
H. Cycon and P. Perry,Local time-decay of high energy scattering states for the Schrödinger equation, Math. Z.188 (1984), 125–142.
S. Doi,Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J.82 (1996), 679–706.
J. Ginibre and G. Velo,Smoothing properties and retarded estimates for some dispersive equations, Comm. Math. Phys.144 (1992), 163–188.
T. Hoshiro,Mourre's method and smoothing properties of dispersive equations, Comm. Math. Phys.202 (1999), 255–265.
W. Ichinose,Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math.21 (1984), 565–581.
A. Jensen and P. Perry,Commutator method and Besov space estiamtes for Schrödinger operators, J. Operator Theory14 (1985), 181–188.
T. Kato and K. Yajima,Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys.1 (1989), 481–496.
C. Kenig, G. Ponce and L. Vega,Well-posedness and scattering results for the generalized Koteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math.46 (1993), 527–620.
S. Mizohata,Some remarks on the Cauchy problem, J. Math. Kyoto Univ.1 (1961), 109–127.
E. Mourre,Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys.78 (1978), 391–408.
E. Mourre,Operateur conjugués et propriétés de propagation, Comm. Math. Phys.91 (1983), 279–300.
P. Sjölin,Regularity of solutions to the Schrödinger equation, Duke Math. J.55 (1987), 699–715.
L. Vega,The Schrödinger equation: pointwise convergence to the initial data, Proc. Amer. Math. Soc.102 (1988), 874–878.
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Norio Shimakura
The author was supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 13640187).
Rights and permissions
About this article
Cite this article
Hoshiro, T. Decay and regularity for dispersive equations with constant coefficients. J. Anal. Math. 91, 211–230 (2003). https://doi.org/10.1007/BF02788788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02788788