Annals of Global Analysis and Geometry

, Volume 38, Issue 4, pp 373–398 | Cite as

On the global boundedness of Fourier integral operators

Original Paper

Abstract

We consider a class of Fourier integral operators, globally defined on \({{\mathbb R}^{d}}\) , with symbols and phases satisfying product type estimates (the so-called SG or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces Mp. The minimal loss of derivatives is shown to be d|1/2 − 1/p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on Lp spaces are presented.

Keywords

SG-Fourier integral operators Modulation spaces Short-time Fourier transform 

Mathematics Subject Classification (2000)

35S30 47G30 42C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beurling A., Helson H.: Fourier transforms with bounded powers. Math. Scand. 1, 120–126 (1953)MATHMathSciNetGoogle Scholar
  2. 2.
    Bényi A., Gröchenig K., Okoudjou K.A., Rogers L.G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246(2), 366–384 (2007)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cappiello M.: Fourier integral operators of infinite order and applications to SG-hyperbolic equations. Tsukuba J. Math. 28, 311–361 (2004)MATHMathSciNetGoogle Scholar
  4. 4.
    Concetti, F., Toft, J.: Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols, I. Ark. Mat. (to appear)Google Scholar
  5. 5.
    Cordero E., Nicola F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254, 506–534 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cordero E., Nicola F.: Sharpness of some properties of Wiener amalgam and modulation spaces. Bull. Aust. Math. Soc. 80, 105–116 (2009)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cordero E., Nicola F., Rodino L.: Boundedness of Fourier integral operators on \({{\mathcal F} L^p}\) spaces. Trans. Amer. Math. Soc. 361, 6049–6071 (2009)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cordero E., Nicola F., Rodino L.: Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal. 9, 1–21 (2010)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cordes H.O.: The Technique of Pseudodifferential Operators. Cambridge University Press, Cambridge (1995)MATHCrossRefGoogle Scholar
  10. 10.
    Cordes H.O.: Precisely Predictable Dirac Observables. Fundamental Theories of Physics, vol. 154. Springer, Dordrecht (2007)Google Scholar
  11. 11.
    Coriasco S.: Fourier integral operators in SG classes II. Application to SG hyperbolic Cauchy problems. Ann. Univ. Ferrara Sez. VII (N.S.) XLIV, 81–122 (1998)MathSciNetGoogle Scholar
  12. 12.
    Coriasco S.: Fourier integral operators in SG classes I. Composition theorems and action on SG Sobolev spaces. Rend. Sem. Mat. Univ. Politec. Torino 57, 249–302 (1999)MATHMathSciNetGoogle Scholar
  13. 13.
    Coriasco S., Panarese P.: Fourier integral operators defined by classical symbols with exit behaviour. Math. Nachr. 242, 61–78 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Fefferman C.: The uncertainty principle. Bull. Amer. Math. Soc. 9, 129–205 (1983)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. Technical Report, University Vienna (1983) [also in Wavelets and Their Applications, M. Krishna, R. Radha, S. Thangavelu, editors, pp. 99–140. Allied Publishers, Bangalore (2003)]Google Scholar
  16. 16.
    Feichtinger H.G.: Atomic characterizations of modulation spaces through Gabor-type representations. Proc. Conf. Constr Funct Theory Rocky Mountain J. Math. 19, 113–126 (1989)MATHMathSciNetGoogle Scholar
  17. 17.
    Feichtinger H.G.: Generalized amalgams, with applications to Fourier transform. Canad. J. Math. 42(3), 395–409 (1990)MATHMathSciNetGoogle Scholar
  18. 18.
    Feichtinger H.G., Gröchenig K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146(2), 464–495 (1997)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Feichtinger H.G., Gröchenig K.: Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. Math. 108, 129–148 (1989)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Folland G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ (1989)MATHGoogle Scholar
  21. 21.
    Gröchenig K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston, MA (2001)Google Scholar
  22. 22.
    Gröchenig K., Leinert M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc. 17, 1–18 (2004)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hörmander L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hörmander L.: The Analysis of Linear Partial Differential Operators, vols. III, IV. Springer, Berlin (1985)Google Scholar
  25. 25.
    Krantz S.G., Parks H.R.: The Implicit Function Theorem. Birkhäuser Boston Inc, Boston (2002)MATHGoogle Scholar
  26. 26.
    Lebedev V., Olevskiĭ A.: C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers. Geom. Funct. Anal. 4(2), 213–235 (1994)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mascarello M., Rodino L.: Partial Differential Equations with Multiple Characteristics. Akad. Verl., Berlin (1997)MATHGoogle Scholar
  28. 28.
    Melrose R.B.: Spectral and scattering theory of the Laplacian on asymptotically Euclidean spaces. In: Ikawa, M. (eds) Spectral and Scattering Theory, pp. 85–130. Marcel Dekker, New York (1994)Google Scholar
  29. 29.
    Melrose R.B.: Geometric Scattering Theory. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
  30. 30.
    Okoudjou K.A.: A Beurling-Helson type theorem for modulation spaces. J. Funct. Spaces Appl. 7, 33–41 (2009)MATHMathSciNetGoogle Scholar
  31. 31.
    Parenti C.: Operatori pseudo-differenziali in \({{\mathbb R}\sp{n}}\) e applicazioni. Ann. Mat. Pura Appl. 93(4), 359–389 (1972)MATHMathSciNetGoogle Scholar
  32. 32.
    Phong D.H., Stein E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179(1), 105–152 (1997)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Rochberg R., Tachizawa K.: Pseudodifferential Operators, Gabor Frames, and Local Trigonometric Bases. Gabor Analysis and Algorithms. Applied and Numerical Harmonic Analysis, pp. 171–192. Birkhäuser Boston, Boston MA (1998)Google Scholar
  34. 34.
    Ruzhansky M.: On the sharpness of Seeger–Sogge–Stein orders. Hokkaido Math. J. 28, 357–362 (1999)MATHMathSciNetGoogle Scholar
  35. 35.
    Ruzhansky M.V.: Singularities of affine fibrations in the regularity theory of Fourier integral operators. Russian Math. Surveys 55, 93–161 (2000)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Ruzhansky M., Sugimoto M.: Global L 2-boundedness theorems for a class of Fourier integral operators. Comm. Partial Differential Equations 31(4-6), 547–569 (2006)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Schrohe E.: Spaces of Weighted Symbols and Weighted Sobolev Spaces on Manifolds. LNM, vol. 1256, pp. 360–377. Springer-Verlag, Berlin (1987)Google Scholar
  38. 38.
    Schulze B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Wiley, Chichester, New York (1998)MATHGoogle Scholar
  39. 39.
    Seeger A., Sogge C.D., Stein E.M.: Regularity properties of Fourier integral operators. Ann. Math. 134(2), 231–251 (1991)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Sogge C.D.: Fourier Integral in Classical Analysis. Cambridge Tracts in Mathematics, vol. 105. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  41. 41.
    Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  42. 42.
    Sugimoto M., Tomita N.: The dilation property of modulation spaces and their inclusion relation with Besov spaces. J. Funct. Anal. 248(1), 79–106 (2007)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Toft J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom. 26(1), 73–106 (2004)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Treves F.: Introduction to Pseudodifferential Operators and Fourier Integral Operators, vols I, II. Plenum Publishing Corporation, New York (1980)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorinoTorinoItaly
  2. 2.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

Personalised recommendations