Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-13T19:27:11.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Oscillatory Integrals with Nonhomogeneous Phase Functions Related to Schrödinger Equations

Published online by Cambridge University Press:  20 November 2018

Lawrence A. Kolasa
Lawrence A. Kolasa*
Affiliation:
Ryerson Polytechnic University, e-mail: lkolasa@acs.ryerson.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider solutions to the free Schrödinger equation in $n+1$ dimensions. When we restrict the last variable to be a smooth function of the first $n$ variables we find that the solution, so restricted, is locally in ${{L}^{2}}$, when the initial data is in an appropriate Sobolev space.

Keywords

42A2542B25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 3 , 01 September 1998 , pp. 306 - 317
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Bourgain, J., A Remark on Schrödinger Operators. Israel J. Math. 77 (1992), 1–16.Google Scholar
2. Carleson, L., Some Analytical Problems Related to Statistical Mechanics. In: Euclidean Harmonic Analysis. Lecture Notes in Math. 779 (1979), 5–45.Google Scholar
3. Dahlberg, B. and Kenig, C., A Note on the Almost EverywhereConvergence of Solutions of the Schrödinger Equation. In: Harmonic Analysis. Lecture Notes in Math. 908 (1982), 205–209.Google Scholar
4. Greenleaf, A. and Seeger, A., Fourier Integral Operators With Fold Singularities. J. Reine Angew. Math. 455 (1994), 35–56.Google Scholar
5. Hörmander, L., The Analysis of Linear Partial Differential Operators I. Grundlehren Math. Wiss. 256, Springer-Verlag, Berlin-New York, 1990.Google Scholar
6. Hörmander, L., Oscillatory Integrals and Multipliers on FLp. Ark. Mat. 11 (1971), 1–11.Google Scholar
7. Kolasa, L., Oscillatory Integrals and Schrödinger Maximal Operators. Pacific J. Math. 177 (1997), 77–102.Google Scholar
8. Kenig, C. and Ruiz, A., A Strong Type (2,2) Estimate for aMaximal Operator Associated to the Schrödinger Equation. Trans. Amer.Math. Soc. 280 (1983), 239–246.Google Scholar
9. Pan, Y. and Sogge, C., Oscillatory Integrals Associated to Folding Canonical Relations. Colloq. Math. 60 (1990), 413–419.Google Scholar
10. Sjölin, P., Regularity of Solutions to the Schrödinger Equation. DukeMath. J. 55 (1987), 669–715.Google Scholar
11. Stein, E. M., Harmonic Analysis. Princeton University Press, Princeton, NJ, 1993.Google Scholar
12. Vega, L., Schrödinger Equations: Pointwise Convergence to the Initial Data. Proc. Amer. Math. Soc. 102 (1988), 874–878.Google Scholar