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Almost Diagonalization of Pseudodifferential Operators

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this review we focus on the almost diagonalization of pseudodifferential operators and highlight the advantages that time-frequency techniques provide here. In particular, we retrace the steps of an insightful paper by Gröchenig, who succeeded in characterizing a class of symbols previously investigated by Sjöstrand by noticing that Gabor frames almost diagonalize the corresponding Weyl operators. This approach also allows to give new and more natural proofs of related results such as boundedness of operators or algebra and Wiener properties of the symbol class. Then, we discuss some recent developments on the theme, namely an extension of these results to a more general family of pseudodifferential operators and similar outcomes for a symbol class closely related to Sjöstrand’s one.

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References

  1. Boggiatto, P., De Donno, G., Oliaro, A.: Time-frequency representations of Wigner type and pseudo-differential operators. Trans. Amer. Math. Soc. 362(9), 4955–4981 (2010). https://doi.org/10.1090/S0002-9947-10-05089-0

  2. Cordero, E., Gröchenig, K., Nicola, F., Rodino, L.: Wiener algebras of fourier integral operators. Journal de mathématiques pures et appliquées 99(2), 219–233 (2013)

    Google Scholar 

  3. Cordero, E., Gröchenig, K., Nicola, F., Rodino, L.: Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class. J. Math. Phys. 55(8), 081,506, 17 (2014). https://doi.org/10.1063/1.4892459

  4. Cordero, E., Nicola, F.: Some new Strichartz estimates for the Schrödinger equation. J. Differential Equations 245(7), 1945–1974 (2008). https://doi.org/10.1016/j.jde.2008.07.009

  5. Cordero, E., Nicola, F., Trapasso, S.I.: Almost diagonalization of \(\tau \)-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces. J. Fourier Anal. Appl. https://doi.org/10.1007/s00041-018-09651-z

  6. Dias, N.C., de Gosson, M.A., Prata, J.a.N.: Maximal covariance group of Wigner transforms and pseudo-differential operators. Proc. Amer. Math. Soc. 142(9), 3183–3192 (2014). https://doi.org/10.1090/S0002-9939-2014-12311-2

  7. Feichtinger, H.G.: On a new Segal algebra. Monatsh. Math. 92(4), 269–289 (1981). https://doi.org/10.1007/BF01320058

  8. Feichtinger, H.G.: Modulation spaces on locally compact abelian groups. Universität Wien. Mathematisches Institut (1983)

    Google Scholar 

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

    Google Scholar 

  10. Feichtinger, H.G.: Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process. 5(2), 109–140 (2006)

    Google Scholar 

  11. de Gosson, M.: The Wigner transform. Advanced Textbooks in Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2017). https://doi.org/10.1142/q0089

  12. de Gosson, M.A.: Symplectic methods in harmonic analysis and in mathematical physics, Pseudo-Differential Operators. Theory and Applications, vol. 7. Birkhäuser/Springer Basel AG, Basel (2011). https://doi.org/10.1007/978-3-7643-9992-4

  13. de Gosson, M.A.: Symplectic covariance properties for Shubin and Born-Jordan pseudo-differential operators. Trans. Amer. Math. Soc. 365(6), 3287–3307 (2013). https://doi.org/10.1090/S0002-9947-2012-05742-4

  14. de Gosson, M.A., Gröchenig, K., Romero, J.L.: Stability of Gabor frames under small time Hamiltonian evolutions. Lett. Math. Phys. 106(6), 799–809 (2016). https://doi.org/10.1007/s11005-016-0846-6

  15. Gröchenig, K.: Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA (2001). https://doi.org/10.1007/978-1-4612-0003-1

  16. Gröchenig, K.: Composition and spectral invariance of pseudodifferential operators on modulation spaces. Journal d’Analyse Mathmatique 98(1), 65–82 (2006)

    Google Scholar 

  17. Gröchenig, K.: Time-frequency analysis of Sjöstrand’s class. Rev. Mat. Iberoam. 22(2), 703–724 (2006). https://doi.org/10.4171/RMI/471

  18. Gröchenig, K.: Four Short Courses on Harmonic Analysis, chap. Wieners Lemma: Theme and Variations. An Introduction to Spectral Invariance and Its Applications. Birkhuser Basel (2010)

    Google Scholar 

  19. Gröchenig, K., Rzeszotnik, Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. In: Annales de l’institut Fourier, vol. 58, pp. 2279–2314 (2008)

    Google Scholar 

  20. Heil, C.: An introduction to weighted Wiener amalgams. In: Wavelets and their Applications, pp. 183–216. Allied Publishers (2003).

    Google Scholar 

  21. Guo, K., Labate, D.: Representation of Fourier integral operators using shearlets. J. Fourier Anal. Appl. 14(3), 327–371 (2008). https://doi.org/10.1007/s00041-008-9018-0

  22. Hörmander, L.: The analysis of linear partial differential operators. III, volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1985)

    Google Scholar 

  23. Meyer, Y.: Ondelettes et operateurs ii: Operateurs de calderon-zygmund.(wavelets and operators ii: Calderon-zygmund operators). Hermann, Editeurs des Sciences et des Arts, Paris (1990)

    Google Scholar 

  24. Rochberg, R., Tachizawa, K.: Pseudodifferential operators, gabor frames, and local trigonometric bases. In: Gabor analysis and algorithms, pp. 171–192. Springer (1998)

    Google Scholar 

  25. Ruzhansky, M., Wang, B., Zhang, H.: Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces. J. Math. Pures Appl. (9) 105(1), 31–65 (2016). https://doi.org/10.1016/j.matpur.2015.09.005

  26. Sjöstrand, J.: An algebra of pseudodifferential operators. Math. Res. Lett. 1(2), 185–192 (1994)

    Google Scholar 

  27. Sjöstrand, J.: Wiener type algebras of pseudodifferential operators. In: Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, pp. Exp. No. IV, 21. École Polytech., Palaiseau (1995)

    Google Scholar 

  28. Sugimoto, M., Tomita, N., Wang, B.: Remarks on nonlinear operations on modulation spaces. Integral Transforms Spec. Funct. 22(4-5), 351–358 (2011). https://doi.org/10.1080/10652469.2010.541054

  29. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal. 207(2), 399–429 (2004). https://doi.org/10.1016/j.jfa.2003.10.003

  30. Toft, J.: Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom. 26(1), 73–106 (2004). https://doi.org/10.1023/B:AGAG.0000023261.94488.f4

  31. Wang, B., Huo, Z., Hao, C., Guo, Z.: Harmonic analysis method for nonlinear evolution equations. I. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2011). https://doi.org/10.1142/9789814360746

  32. Wong, M.W.: Weyl transforms. Universitext. Springer-Verlag, New York (1998)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    S. Ivan Trapasso

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  1. S. Ivan Trapasso
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Correspondence to S. Ivan Trapasso .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Elena Cordero

  3. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Maurice de Gosson

  4. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Fabio Nicola

  6. Dipartimento di Matematica “G. Peano”, University of Torino, Turin, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Torino, Turin, Italy

    Anita Tabacco

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Trapasso, S.I. (2019). Almost Diagonalization of Pseudodifferential Operators. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-05210-2_14

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