Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian
El Kamel, Jamel ; Yacoub, Chokri
Annales mathématiques Blaise Pascal, Tome 12 (2005), p. 147-160 / Harvested from Numdam
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In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens’ principle and the equipartition of energy hold if the inverse of the Harish-Chandra c-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.

@article{AMBP_2005__12_1_147_0,
     author = {El Kamel, Jamel and Yacoub, Chokri},
     title = {Huygens' principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {12},
     year = {2005},
     pages = {147-160},
     doi = {10.5802/ambp.199},
     zbl = {1088.35036},
     mrnumber = {2126445},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2005__12_1_147_0}
}

El Kamel, Jamel; Yacoub, Chokri. Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian. Annales mathématiques Blaise Pascal, Tome 12 (2005) pp. 147-160. doi : 10.5802/ambp.199. http://gdmltest.u-ga.fr/item/AMBP_2005__12_1_147_0/

[1] Anker, J.Ph.; Damek, E.; Yacoub, Ch. Spherical analysis on harmonic AN groups, Ann. Scuola Norm. Sup. Pisa, Tome XXIII (1996), pp. 643-679 | Numdam | MR 1469569 | Zbl 0881.22008

[2] Bloom, W.R.; Xu, Z. Fourier transforms of Schwartz functions on Chébli-Triméche hypergroups, Mh. Math., Tome 125 (1998), pp. 89-109 | Article | MR 1604930 | Zbl 0893.43001

[3] Branson, T.P.; Olafsson, G.; Schlichtkrull, H. Huygens’ principle in Riemannian symmetric spaces, Math. Ann., Tome 301 (1995), pp. 445-462 | Article | Zbl 0822.43002

[4] Branson, T.P.; Olafsson, G. Equipartition of energy for waves in symmetric spaces, J. Funct. Anal., Tome 97 (1991), pp. 403-416 | Article | MR 1111189 | Zbl 0796.43006

[5] Branson, T.P. Eventual partition of conserved quantities in wave motion, J. Math. Anal. Appl., Tome 96 (1983), pp. 54-62 | Article | MR 717494 | Zbl 0591.35036

[6] Chébli, H. Théorème de Paley-Wiener associé à un opérateur singulier sur (0,∞), J. Math. Pures Appl., Tome 58 (1979), pp. 1-19 | MR 533233 | Zbl 0394.34019

[7] Duffin, R.J. Equipartition of energy in wave motion, J. Math. Anal. Appl., Tome 32 (1970), pp. 386-391 | Article | MR 269190 | Zbl 0223.35055

[8] Helgason, S. Huygens’ principle for wave equation on symmetric spaces, J. Funct. Anal., Tome 107 (1992), pp. 279-288 | Article | Zbl 0757.58036

[9] Koornwinder, T.H.; Askey & Al., R.A. Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions : Group theoretical aspects and applications, Reidel (1984), pp. 1-85 | MR 774055 | Zbl 0584.43010

[10] Lax, P.D.; Phillips, R.S. Scattering theory, Academic Press, New York (1967) | MR 217440 | Zbl 0186.16301

[11] Lax, P.D.; Phillips, R.S. Translation representations for the solution of the non-euclidean wave equation, Comm. Pure Appl. Math., Tome 32 (1979), pp. 617-667 | Article | MR 533296 | Zbl 0425.35065

[12] Olafsson, G.; Schlichtkrull, H. Wave propagation on Riemannian symmetric spaces, J. Funct. Anal., Tome 107 (1992), pp. 270-278 | Article | MR 1172024 | Zbl 0757.58037

[13] Trimèche, K. Transformation intégrale de Weyl et thóréme de Paley-Wiener associés à un opérateur différentiel singulier sur (0,+∞), J. Math. Pures. Appl., Tome 60 (1981), pp. 51-98 | MR 616008 | Zbl 0416.44002

[14] Trimèche, K. Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comp. Harm. Anal., Tome 4 (1997), pp. 97-112 | Article | MR 1429682 | Zbl 0872.34059

[15] Yacoub, Ch. (1994) (Thèse, Université Paris-Sud)