Abstract
It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces ℳp,q, for 1≤p= q≤∞, provided their symbols belong to the Sjöstrand class M ∞,1. However, they generally fail to be bounded on ℳp,q for p≠q. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ℳp,q for p≠q, and between ℳp,q→ℳq,p, 1≤q<p≤∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asada, K., Fujiwara, D.: On some oscillatory transformation in L 2(ℝn). Jpn. J. Math. 4, 299–361 (1978)
Bényi, A., Gröchenig, K., Okoudjou, K.A., Rogers, L.G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246(2), 366–384 (2007)
Boulkhemair, A.: Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators. Math. Res. Lett. 4, 53–67 (1997)
Concetti, F., Garello, G., Toft, J.: Trace ideals for Fourier integral operators with non-smooth symbols II. Preprint. Available at arXiv:0710.3834
Cordero, E., Nicola, F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254, 506–534 (2008)
Cordero, E., Nicola, F.: Some new Strichartz estimates for the Schrödinger equation. J. Differ. Equ. 245, 1945–1974 (2008)
Cordero, E., Nicola, F., Rodino, L.: Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal. 9(1), 1–21 (2010). Available at arXiv:0710.3652v1
Cordero, E., Nicola, F., Rodino, L.: Sparsity of Gabor representation of Schrödinger propagators. Appl. Comput. Harmon. Anal. 26(3), 357–370 (2009)
Cordero, E., Nicola, F., Rodino, L.: Boundedness of Fourier integral operators on ℱL p spaces. Trans. Am. Math. Soc. 361, 6049–6071 (2009). Available at arXiv:0801.1444
Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups. Technical Report, University Vienna (1983). Also in Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, pp. 99–140. Allied Publishers (2003)
Feichtinger, H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. Rocky Mt. J. Math. 19, 113–126 (1989). Proc. Conf. Constructive Function Theory
Feichtinger, H.G.: Generalized amalgams, with applications to Fourier transform. Canad. J. Math. 42(3), 395–409 (1990)
Feichtinger, H.G., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108, 129–148 (1989)
Folland, G.B.: Harmonic Analysis in Phase Space. Princeton Univ. Press, Princeton (1989)
Fournier, J.J.F., Stewart, J.: Amalgams of L p and l q. Bull. Am. Math. Soc. (N.S.) 13(1), 1–21 (1985)
Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäuser, Basel (2001)
Helffer, B.: Théorie Spectrale pour des Operateurs Globalement Elliptiques. Astérisque. Société Mathématique de France, Paris (1984)
Helffer, B., Robert, D.: Comportement asymptotique precise du spectre d’operateurs globalement elliptiques dans ℝd. Sem. Goulaouic-Meyer-Schwartz 1980–1981, École Polytechnique, Exposé II (1980)
Hörmander, L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, vols. III, IV. Springer, Berlin (1985)
Krantz, S.G., Parks, H.R.: The Implicit Function Theorem. Birkhäuser Boston, Cambridge (2002)
Ruzhansky, M., Sugimoto, M.: Global L 2-boundedness theorems for a class of Fourier integral operators. Commun. Partial Differ. Equ. 31(4–6), 547–569 (2006)
Seeger, A., Sogge, C.D., Stein, E.M.: Regularity properties of Fourier integral operators. Ann. Math. (2) 134(2), 231–251 (1991)
Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom. 26(1), 73–106 (2004)
Triebel, H.: Modulation spaces on the Euclidean n-spaces. Z. Anal. Anwend. 2, 443–457 (1983)
Wolff, T.H.: Lectures on Harmonic Analysis. University Lecture Series. Am. Math. Soc., Providence (2003). Laba, I., Shubin, C. (eds.)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Karlheinz Gröchenig.
Rights and permissions
About this article
Cite this article
Cordero, E., Nicola, F. Boundedness of Schrödinger Type Propagators on Modulation Spaces. J Fourier Anal Appl 16, 311–339 (2010). https://doi.org/10.1007/s00041-009-9111-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9111-z
- Fourier integral operators
- Modulation spaces
- Wiener amalgam spaces
- Short-time Fourier transform
- Sjöstrand’s algebra