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On the global boundedness of Fourier integral operators

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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 M p. 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 L p spaces are presented.

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References

  1. Beurling A., Helson H.: Fourier transforms with bounded powers. Math. Scand. 1, 120–126 (1953)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cappiello M.: Fourier integral operators of infinite order and applications to SG-hyperbolic equations. Tsukuba J. Math. 28, 311–361 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Concetti, F., Toft, J.: Schatten-von Neumann properties for Fourier integral operators with non-smooth symbols, I. Ark. Mat. (to appear)

  5. Cordero E., Nicola F.: Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation. J. Funct. Anal. 254, 506–534 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cordero E., Nicola F.: Sharpness of some properties of Wiener amalgam and modulation spaces. Bull. Aust. Math. Soc. 80, 105–116 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cordero E., Nicola F., Rodino L.: Time-frequency analysis of Fourier integral operators. Commun. Pure Appl. Anal. 9, 1–21 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cordes H.O.: The Technique of Pseudodifferential Operators. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  10. Cordes H.O.: Precisely Predictable Dirac Observables. Fundamental Theories of Physics, vol. 154. Springer, Dordrecht (2007)

    Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  13. Coriasco S., Panarese P.: Fourier integral operators defined by classical symbols with exit behaviour. Math. Nachr. 242, 61–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fefferman C.: The uncertainty principle. Bull. Amer. Math. Soc. 9, 129–205 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  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)]

  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)

    MATH  MathSciNet  Google Scholar 

  17. Feichtinger H.G.: Generalized amalgams, with applications to Fourier transform. Canad. J. Math. 42(3), 395–409 (1990)

    MATH  MathSciNet  Google Scholar 

  18. Feichtinger H.G., Gröchenig K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146(2), 464–495 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Feichtinger H.G., Gröchenig K.: Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. Math. 108, 129–148 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Folland G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton, NJ (1989)

    MATH  Google Scholar 

  21. Gröchenig K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston, MA (2001)

    Google Scholar 

  22. Gröchenig K., Leinert M.: Wiener’s lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc. 17, 1–18 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hörmander L.: Fourier integral operators I. Acta Math. 127, 79–183 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hörmander L.: The Analysis of Linear Partial Differential Operators, vols. III, IV. Springer, Berlin (1985)

    Google Scholar 

  25. Krantz S.G., Parks H.R.: The Implicit Function Theorem. Birkhäuser Boston Inc, Boston (2002)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mascarello M., Rodino L.: Partial Differential Equations with Multiple Characteristics. Akad. Verl., Berlin (1997)

    MATH  Google Scholar 

  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. Melrose R.B.: Geometric Scattering Theory. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  30. Okoudjou K.A.: A Beurling-Helson type theorem for modulation spaces. J. Funct. Spaces Appl. 7, 33–41 (2009)

    MATH  MathSciNet  Google Scholar 

  31. Parenti C.: Operatori pseudo-differenziali in \({{\mathbb R}\sp{n}}\) e applicazioni. Ann. Mat. Pura Appl. 93(4), 359–389 (1972)

    MATH  MathSciNet  Google Scholar 

  32. Phong D.H., Stein E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179(1), 105–152 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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. Ruzhansky M.: On the sharpness of Seeger–Sogge–Stein orders. Hokkaido Math. J. 28, 357–362 (1999)

    MATH  MathSciNet  Google Scholar 

  35. Ruzhansky M.V.: Singularities of affine fibrations in the regularity theory of Fourier integral operators. Russian Math. Surveys 55, 93–161 (2000)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Schulze B.-W.: Boundary Value Problems and Singular Pseudo-Differential Operators. Wiley, Chichester, New York (1998)

    MATH  Google Scholar 

  39. Seeger A., Sogge C.D., Stein E.M.: Regularity properties of Fourier integral operators. Ann. Math. 134(2), 231–251 (1991)

    Article  MathSciNet  Google Scholar 

  40. Sogge C.D.: Fourier Integral in Classical Analysis. Cambridge Tracts in Mathematics, vol. 105. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  41. Stein E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  43. Toft J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom. 26(1), 73–106 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  44. Treves F.: Introduction to Pseudodifferential Operators and Fourier Integral Operators, vols I, II. Plenum Publishing Corporation, New York (1980)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Torino, via Carlo Alberto 10, 10123, Torino, Italy

    Elena Cordero & Luigi Rodino

  2. Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Torino, Italy

    Fabio Nicola

Authors
  1. Elena Cordero
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  2. Fabio Nicola
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  3. Luigi Rodino
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Correspondence to Fabio Nicola.

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Cordero, E., Nicola, F. & Rodino, L. On the global boundedness of Fourier integral operators. Ann Glob Anal Geom 38, 373–398 (2010). https://doi.org/10.1007/s10455-010-9219-z

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  • DOI: https://doi.org/10.1007/s10455-010-9219-z

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