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Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

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Abstract

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form

$${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$$

for arbitrary time t and arbitrary λ>0, where ψ is a smooth bump function supported in [−2,2] if λ<1 and supported in [1,2] if λ≥1. This generalizes previous results in Müller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.

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References

  1. Astengo, F.: The maximal ideal space of a heat algebra on solvable extensions of H-type groups. Boll. Unione Mat. Ital. A 9(7), 157–165 (1995)

    MathSciNet  MATH  Google Scholar 

  2. Astengo, F.: Multipliers for a distinguished Laplacean on solvable extensions of H-type groups. Monatsh. Math. 120, 179–188 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anker, J.P., Damek, E., Yacoub, C.: Spherical Analysis on harmonic AN groups, Ann. S. Norm. Super. Pisa Cl. Sci. 23(4), 643–679 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Cowling, M., Dooley, A.H., Korányi, A., Ricci, F.: H-type groups and Iwasawa decompositions. Adv. Math. 87, 1–41 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cowling, M., Giulini, S., Hulanicki, A., Mauceri, G.: Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. Studia Math. 111, 103–121 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Cowling, M., Dooley, A.H., Korányi, A., Ricci, F.: An approach to symmetric spaces of rank one via groups of Heisenberg type. J. Geom. Anal. 8, 199–237 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Cowling, M., Giulini, S., Meda, S.: Oscillatory multipliers related to the wave equation on noncompact symmetric spaces. J. Lond. Math. Soc. 66, 691–709 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Damek, E.: Curvature of a semidirect extension of a Heisenberg type nilpotent group. Colloq. Math. 53, 249–253 (1987)

    MathSciNet  MATH  Google Scholar 

  9. Damek, E.: Geometry of a semidirect extension of a Heisenberg type nilpotent group. Colloq. Math. 53, 255–268 (1987)

    MathSciNet  MATH  Google Scholar 

  10. Damek, E., Ricci, F.: A class of nonsymmetric harmonic Riemannian spaces. Bull. Am. Math. Soc. 27, 139–142 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Damek, E., Ricci, F.: Harmonic analysis on solvable extensions of H-type groups. J. Geom. Anal. 2, 213–248 (1992)

    MathSciNet  MATH  Google Scholar 

  12. Giulini, S., Meda, S.: Oscillating multipliers on noncompact symmetric spaces. J. Reine Angew. Math. 409, 93–105 (1990)

    MathSciNet  MATH  Google Scholar 

  13. Hebisch, W.: The subalgebra of L 1(AN) generated by the Laplacian. Proc. Am. Math. Soc. 117, 547–549 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hebisch, W., Steger, T.: Multipliers and singular integrals on exponential growth groups. Math. Z. 245, 37–61 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hunt, G.A.: Semigroups of measures on Lie groups. Trans. Am. Math. Soc. 81, 264–293 (1956)

    Article  MATH  Google Scholar 

  16. Ionescu, A.D.: Fourier integral operators on noncompact symmetric spaces of real rank one. J. Funct. Anal. 174, 274–300 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 145–159 (1975)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Müller, D.: Functional calculus on Lie groups and wave propagation, In: Proceedings of the International Congress of Mathematicians, vol. II, Berlin, 1998, number Extra vol. II, pp. 679–689 (1998)

  20. Müller, D., Thiele, C.: Wave equation and multiplier estimates on ax+b groups. Studia Math. 179, 117–148 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Peral, J.C.: L p-estimate for the wave equation. J. Funct. Anal. 36, 114–145 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vallarino, M.: Spectral multipliers on harmonic extensions of H-type groups. J. Lie Theory 17, 163–189 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Vallarino, M.: Spaces H 1 and BMO on ax+b-groups. Collect. Math. 60, 277–295 (2009)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Matematisches Seminar, Christian-Albrechts Universität, Ludewig-Meyn-Strasse 4, 24098, Kiel, Germany

    Detlef Müller

  2. Laboratoire MAPMO, Université d’Orléans, UFR Sciences, Bâtiment de Mathématique-Route de Chartres, B.P. 6759, 45067, Orléans cedex 2, France

    Maria Vallarino

Authors
  1. Detlef Müller
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  2. Maria Vallarino
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Correspondence to Maria Vallarino.

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Communicated by Fulvio Ricci.

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Müller, D., Vallarino, M. Wave Equation and Multiplier Estimates on Damek–Ricci Spaces. J Fourier Anal Appl 16, 204–232 (2010). https://doi.org/10.1007/s00041-009-9088-7

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  • DOI: https://doi.org/10.1007/s00041-009-9088-7

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