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Characterization of the SG-Wave Front Set in Terms of the FBI-Transform

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Abstract

The \(\mathrm {SG}\)-wave front set, which measures microlocally the deviation of a tempered distribution from being rapidly decaying and smooth, is studied using a Fourier–Bros–Iagolnitzer transform. This generalizes the established characterization of the classical Hörmander \({\mathscr {C}^\infty }\)-wave front set. In particular, the transform used is capable of identifying singularities both at finite arguments as well as such arising at infinity.

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Notes

  1. Such a function may be constructed as follows: take any smooth positive function f supported in \([-1,0]\) with \(\int |f(x)|\ dx=C\). Then set, for \(x\in [-1,0]\), \(\phi ^0(x)=C^{-1}\int _{-1}^x f(x)\ dx\) and \(\phi ^0(x)=1-\phi ^0(x-1)\) for \(x\in [0,1]\).

References

  1. Bony, J.M.: Equivalence des diverses notions de spectre singulier analytique. Séminaire Goulaouic-Schwartz exp. n. 3:1–12 (1976–1977)

  2. Bros, J., Iagolnitzer, D.: Support essentiel et structure analytique des distributions. Séminaire Goulaouic-Meyer-Schwartz exp. 18 (1975–1976)

  3. Chung, S.Y., Kim, D.: A quasianalytic singular spectrum with res pect to the Denjoy-Carleman class. Nagoya Math. J. 148, 137–149 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Commun. Partial Differ. Equ. 3(11), 979–1005 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coriasco, S., Johansson, K., Toft, J.: Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators. J. Differ. Equ. 254(8), 3228–3258 (2013). doi:10.1016/j.jde.2013.01.014

    Article  MathSciNet  MATH  Google Scholar 

  7. Coriasco, S., Johansson, K., Toft, J.: Global wave-front sets of intersection and union type. Fourier Analysis, Trends in Mathematics 2014, vol. 9, pp. 1–106. Springer International Publishing (2014)

  8. Coriasco, S., Johansson, K., Toft, J.: Global wave-front properties for Fourier integral operators and hyperbolic problems. J. Fourier. Anal. Appl. (2015). doi:10.1007/s00041-015-9422-1

  9. Coriasco, S., Maniccia, L.: Wave front set at infinity and hyperbolic linear operators with multiple characteristics. Ann. Global Anal. Geom. 24, 375–400 (2003)

  10. Coriasco, S., Schulz, R.: The global wave front set of tempered oscillatory integrals with inhomogeneous phase functions. J. Fourier Ana l. Appl. 19(5), 1093–1121 (2013). doi:10.1007/s00041-013-9283-4

    Article  MathSciNet  MATH  Google Scholar 

  11. Delort, J.-M.: FBI Transformation: Second Microlocalization and SeMilinear Caustics, vol. 1522. Springer, Berlin (1992)

  12. Egorov, Y.V., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  14. Gérard, P.: Moyennisation et régularité deux-microlocale. Ann. scient. Ec. Norm. Sup. 4ème série 23, 89–121 (1990)

    MATH  Google Scholar 

  15. Hassel, A., Wunsch, J.: The semiclassical resolvent and the prop agator for non-trapping scattering metrics. Adv. Math. 217(2), 586–682 (2008)

    Article  MathSciNet  Google Scholar 

  16. Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  17. Hörmander, L.: Fourier Integral operators I. Acta Math. 127(1), 79–183 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaneko, A.: On the global existence of real analytic solutions of linear partial differential equations on unbounded domain. J. Fac. Sci. Univ. Tokyo Sec. IA 32, 319–372 (1985)

    MathSciNet  MATH  Google Scholar 

  19. Kato, K., Kobayashi, M., Ito, S.: Remark on characterization of wave front set by wave packet transform, preprint. arXiv:1408.1370v1 (2014)

  20. Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Universitext. Springer, New York (2002)

  21. Melrose, R.: Spectral and scattering theory for the Laplacian on a symptotically Euclidian spaces. In: Spectral and Scattering Theory”, Sanda 1992. Lecture Notes in Pure and Appl. Math., vol. 161, pp. 85–130. Dekker, New York (1994)

  22. Melrose, R.: Geometric Scattering Theory, Stanford Lectures. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  23. Melrose, R.: Lecture Notes for Graduate Analysis, 2004, available online at http://www-math.mit.edu/~rbm/18.155-F04-notes/Lecture-notes.pdf, last download 17/11/2015

  24. Nicola, F., Rodino, L.: Global Pseudo-differential Calculus on Euclidean Spaces. Birkhäuser, Basel (2010)

    Book  MATH  Google Scholar 

  25. Parenti, C.: Operatori pseudo-differentiali in \({\mathbb{R}}^n\) e applicazioni. Annali Mat. Pura Appl. 93, 359–389 (1972)

  26. Pilipovic, S., Teofanov, N., Toft, J.: Micro-local analysis with Fourier Lebesgue spaces, Part I. J. Fourier Anal. Appl. 17:374 -407

  27. Schulz, R.: Microlocal Analysis of Tempered Distributions. Diss. Niedersächsische Staats-und Universitätsbibliothek Göttingen (2014)

  28. Schulz, R., Wahlberg, P.: The equality of the homogeneous and the Gabor wave front set, preprint. arXiv:1304.7608 (2013)

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

  30. Wunsch, J., Zworski, M.: The FBI transform on compact \(C^{\infty }\)-manifolds. Trans. Am. Math. Soc. 353(3), 1151–1167 (2001)

  31. Zworski, M.: Semiclassical Analysis, Graduate Studies in Mathemat ics 138. AMS (2012)

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Acknowledgments

The author would like to thank Patrik Wahlberg, as well as the anonymous reviewers, for their remarks and corrections which led to numerous improvements of the manuscript. The present article elaborates on certain pieces of the author’s thesis [27], which was completed under supervision of Dorothea Bahns at the University of Göttingen. During this time, the author was supported by the Studienstiftung des deutschen Volkes. The author would further like to thank the Leibniz Universität Hannover as well as the Georg-August Universität Göttingen for institutional support.

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  1. Institut für Analysis, Leibniz Universität Hannover, Welfenplatz 1, 30167, Hannover, Germany

    René M. Schulz

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  1. René M. Schulz
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Correspondence to René M. Schulz.

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Communicated by Hans G. Feichtinger.

Appendix: The FBI Transform Visualized

Appendix: The FBI Transform Visualized

Explicitly computable FBI-transforms of certain model distributions may serve as a visualization for Theorem 3.5 as well as the results building up to it. We have (cf. [9])

Fig. 1
figure 1

The transformed chirp for various parameters \(\lambda \) and \(\mu \)

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Fig. 2
figure 2

The transformed Gaussian peak for various parameters \(\lambda \) and \(\mu \)

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The first two identities may be directly read off from

$$\begin{aligned} \left| {\mathscr {F}_{\lambda ,\mu }}\delta _{x_0}(x,\xi )\right| =(2\pi ^{3/2})^{-d/2} \mu ^{d/4}\lambda ^{3d/4} e^{-\frac{\lambda }{2\mu }|x_0-\mu x|^2} \end{aligned}$$

and Fourier symmetry (3.4). The third and fourth FBI-transforms may be calculated explicitly as well, and the graphs of absolute value of their transforms are depicted in Figs. 1 and 2 (including level set lines and suitably scaled). These underline how the time-frequency plane is shifted under the action of the generalized FBI transform for various parameters.

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Schulz, R.M. Characterization of the SG-Wave Front Set in Terms of the FBI-Transform. J Fourier Anal Appl 22, 1141–1156 (2016). https://doi.org/10.1007/s00041-015-9451-9

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