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On the Wave Equation Associated to the Hermite and the Twisted Laplacian

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Abstract

The dispersive properties of the wave equation u tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|2 or the twisted Laplacian −(∇ x −iy)2/2−(∇ y +ix)2/2. In both cases we prove optimal L 1−L ∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ν) associated to fractional powers of the twisted Laplacian for 0<ν<1.

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References

  1. Beals, R.: L p boundedness of Fourier integral operators. Mem. Am. Math. Soc. 38(264), 1–57 (1982)

    MathSciNet  Google Scholar 

  2. Dziubański, J.: Triebel-Lizorkin spaces associated with Laguerre and Hermite expansions. Proc. Am. Math. Soc. 125(12), 3547–3554 (1997)

    Article  MATH  Google Scholar 

  3. Epperson, J.: Triebel-Lizorkin spaces for Hermite expansions. Studia Math. 114(1), 87–103 (1995)

    MATH  MathSciNet  Google Scholar 

  4. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vols. I, II. McGraw-Hill, New York (1953). Based, in part, on notes left by Harry Bateman

    Google Scholar 

  5. Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 2(4), 309–327 (1985)

    MATH  MathSciNet  Google Scholar 

  6. Hardy, G.H., Riesz, M.: The General Theory of Dirichlet Series. Cambridge University Press, London (1915)

    Google Scholar 

  7. Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120(5), 955–980 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Koch, H., Ricci, F.: Spectral projections for the twisted Laplacian. Studia Math. 180(2), 103–110 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Koch, H., Tataru, D.: L p eigenfunction bounds for the Hermite operator. Duke Math. J. 128(2), 369–392 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Miyachi, A.: On some estimates for the wave equation in L p and H p. J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 27(2), 331–354 (1980)

    MATH  MathSciNet  Google Scholar 

  11. Müller, D., Ricci, F.: Analysis of second order differential operators on Heisenberg groups. I. Invent. Math. 101(3), 545–582 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Müller, D., Seeger, A.: Sharp l p-estimates for the wave equation on Heisenberg type groups. (2008, in preparation)

  13. Müller, D., Stein, E.M.: L p-estimates for the wave equation on the Heisenberg group. Rev. Mat. Iberoam. 15(2), 297–334 (1999)

    MATH  Google Scholar 

  14. Ólafsson, G., Zheng, S.: Function spaces associated with Schrödinger operators: the Pöschl-Teller potential. J. Fourier Anal. Appl. 12(6), 653–674 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Peral, J.C.: L p estimates for the wave equation. J. Funct. Anal. 36(1), 114–145 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Shatah, J., Struwe, M.: Geometric Wave Equations. Courant Lecture Notes in Mathematics, vol. 2. New York University Courant Institute of Mathematical Sciences, New York (1998)

    MATH  Google Scholar 

  18. Thangavelu, S.: Riesz transforms and the wave equation for the Hermite operator. Commun. Part. Differ. Equ. 15(8), 1199–1215 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, vol. 159. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  20. Tie, J., Wong, M.W.: Wave kernels of the twisted Laplacian. In: Modern Trends in Pseudo-Differential Operators. Oper. Theory Adv. Appl., vol. 172, pp. 107–115. Birkhäuser, Basel (2007)

    Chapter  Google Scholar 

  21. Yajima, K.: Schrödinger evolution equations with magnetic fields. J. Anal. Math. 56, 29–76 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Piazzale A. Moro 2, 00185, Rome, Italy

    Piero D’Ancona

  2. Bâtiment de Matématiques, Unversité d’Orléans, Route de Chartres, B.P. 6759, 45067, Orléans Cedex 2, France

    Vittoria Pierfelice

  3. Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri, 56100, Pisa, Italy

    Fulvio Ricci

Authors
  1. Piero D’Ancona
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  2. Vittoria Pierfelice
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  3. Fulvio Ricci
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Correspondence to Piero D’Ancona.

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Communicated by Hans Feichtinger.

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D’Ancona, P., Pierfelice, V. & Ricci, F. On the Wave Equation Associated to the Hermite and the Twisted Laplacian. J Fourier Anal Appl 16, 294–310 (2010). https://doi.org/10.1007/s00041-009-9104-y

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  • DOI: https://doi.org/10.1007/s00041-009-9104-y

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