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Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group

Published online by Cambridge University Press:  20 November 2018

Giulia Furioli ,
Camillo Melzi  and
Alessandro Veneruso
Giulia Furioli
Affiliation:
Dipartimento di Ingegneria Gestionale e dell'Informazione, Università di Bergamo, Viale Marconi 5, I-24044 Dalmine (BG), Italy email: [email protected]
Camillo Melzi
Affiliation:
Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Università dell'Insubria, Via Valleggio 11, I-22100 Como, Italy email: [email protected]
Alessandro Veneruso
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I-16146 Genova, Italy email: [email protected]
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Abstract

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We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

Keywords

22E2535B65nilpotent and solvable Lie groupssmoothness and regularity of solutions of PDEs
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1301 - 1322
Copyright
Copyright © Canadian Mathematical Society 2007

References

[A] Alexopoulos, G., Spectral multipliers on Lie groups of polynomial growth. Proc. Amer. Math. Soc. 120(1994), no. 3, 973–979.Google Scholar
[BG] Bahouri, H. and Gallagher, I., Paraproduit sur le groupe de Heisenberg et applications. Rev. Mat. Iberoamericana 17(2001), no. 1, 69–105.Google Scholar
[BGX] Bahouri, H., Gérard, P., and Xu, C.-J., Espaces de Besov et estimations de Strichartz généralisées sur le groupe de Heisenberg. J. Anal. Math. 82(2000), 93–118.Google Scholar
[BJR] Benson, C., Jenkins, J., and Ratcliff, G., The spherical transform of a Schwartz function on the Heisenberg group. J. Funct. Anal. 154(1998), no. 2, 379–423.Google Scholar
[BJRW] Benson, C., Jenkins, J., Ratcliff, G., and Worku, T., Spectra for Gelfand pairs associated with the Heisenberg group. Colloq. Math. 71(1996), no. 2, 305–328.Google Scholar
[C] Christ, M., Lp bounds for spectral multipliers on nilpotent groups. Trans. Amer. Math. Soc. 328(1991), no. 1, 73–81.Google Scholar
[D] Del Hierro, M., Dispersive and Strichartz estimates on H-type groups. Studia Math. 169(2005), no. 1, 1–20.Google Scholar
[EMOT] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher transcendental functions. Vol. 2. McGraw-Hill, New York, 1953.Google Scholar
[F] Faraut, J., Analyse harmonique et fonctions spéciales. In: Deux cours d’analyse harmonique. Progr. Math. 69, Birkhäuser, Boston, 1987, pp. 1–151.Google Scholar
[FMV] Furioli, G., Melzi, C., and Veneruso, A., Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279(2006), no. 9-10, 1028–1040.Google Scholar
[FV] Furioli, G. and Veneruso, A., Strichartz inequalities for the Schrödinger equation with the full Laplacian on the Heisenberg group. Studia Math. 160(2004), no. 2, 157–178.Google Scholar
[Ge1] Geller, D., Fourier analysis on the Heisenberg group. Proc. Natl. Acad. Sci. U.S.A. 74(1977), no. 4, 1328–1331.Google Scholar
[Ge2] Geller, D., Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 36(1980), no. 2, 205–254.Google Scholar
[Gi] Ginibre, J., An introduction to nonlinear Schrödinger equations. In: Nonlinear Waves. GAKUTO Internat. Ser. Math. Sci. Appl. 10, 1997, pp. 85–133.Google Scholar
[GV] Ginibre, J. and Velo, G., Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133(1995), no. 1, 50–68.Google Scholar
[He] Hebisch, W., Multiplier theorem on generalized Heisenberg groups. Colloq. Math. 65(1993), no. 2, 231–239.Google Scholar
[Hu] Hulanicki, A., A functional calculus for Rockland operators on nilpotent Lie groups. Studia Math. 78(1984), no. 3, 253–266.Google Scholar
[KPV] Kenig, C., Ponce, G., and Vega, L., Oscillatory integrals and regularity of dispersive equations. Indiana Univ. Math. J. 40(1991), no. 1, 33–69.Google Scholar
[KVW] Korányi, A., Vági, S., and Welland, G. V., Remarks on the Cauchy integral and the conjugate function in generalized half-planes. J. Math. Mech. 19(1969/1970), 1069–1081.Google Scholar
[M] Mauceri, G., Maximal operators and Riesz means on stratified groups. Sympos. Math. 29, Academic Press, New York, 1987, pp. 47–62.Google Scholar
[MRS1] Müller, D., Ricci, F., and Stein, E. M., Marcinkiewicz multipliers and multi-parameter structure on the Heisenberg(-type) groups. I. Invent.Math. 119(1995), no. 2, 199–233.Google Scholar
[MRS2] Müller, D., Marcinkiewicz multipliers and multi-parameter structure on the Heisenberg(-type) groups. II. Math. Z. 221(1996), no. 2, 267–291.Google Scholar
[MS] Müller, D. and Stein, E. M., On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. 73(1994), no. 4, 413–440.Google Scholar
[N] Nachman, A. I., The wave equation on the Heisenberg group. Comm. Partial Differential Equations 7(1982), no. 6, 675–714.Google Scholar
[P] Peetre, J., New thoughts on Besov spaces. Duke University Mathematics Series 1, Mathematics Department, Duke University, Durham, NC, 1976.Google Scholar
[S1] Skrzypczak, L., Atomic decompositions on manifolds with bounded geometry. Forum Math. 10(1998), no. 1, 19–38.Google Scholar
[S2] Skrzypczak, L., Besov spaces and Hausdorff dimension for some Carnot–Carathéodory metric spaces. Canad. J. Math. 54(2002), no. 6, 1280–1304.Google Scholar
[St] Stein, E. M., Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals.. Princeton Mathematical Series 43, Princeton University Press, Princeton, 1993.Google Scholar
[S] Strichartz, R. S., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44(1977), no. 3, 705–714.Google Scholar
[V] Veneruso, A., Schwartz kernels on the Heisenberg group. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 6(2003), 657–666.Google Scholar