Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces
Astengo, Francesca ; Di Blasio, Bianca
Annales mathématiques Blaise Pascal, Tome 17 (2010), p. 327-340 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We prove that Huygens’ principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek–Ricci spaces. We also prove a Paley–Wiener type theorem for the inverse of the Helgason Fourier transform on Damek–Ricci spaces.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/ambp.286
Classification:  43A80,  22E25 
@article{AMBP_2010__17_2_327_0,
     author = {Astengo, Francesca and Di Blasio, Bianca},
     title = {Huygens' principle and a Paley--Wiener type theorem on Damek--Ricci spaces},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     year = {2010},
     pages = {327-340},
     doi = {10.5802/ambp.286},
     zbl = {1207.43006},
     mrnumber = {2778917},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2010__17_2_327_0}
}

Astengo, Francesca; Di Blasio, Bianca. Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces. Annales mathématiques Blaise Pascal, Tome 17 (2010) pp. 327-340. doi : 10.5802/ambp.286. http://gdmltest.u-ga.fr/item/AMBP_2010__17_2_327_0/

[1] Andersen, N. B. Real Paley–Wiener theorem for the inverse Fourier transform on a Riemannian symmetric space, Pacific J. Math., Tome 213 (2004), pp. 1-13 | Article | MR 2040247 | Zbl 1049.43004

[2] Anker, J. Ph.; Damek, E.; Yacoub, C. Spherical analysis on harmonic AN groups, Ann. Scuola Norm. Sup. Pisa, Tome 23 (1996), pp. 643-679 | Numdam | MR 1469569 | Zbl 0881.22008

[3] Astengo, F.; Blasio, B. Di A Paley-Wiener theorem on NA harmonic spaces, Colloq. Math., Tome 80 (1999), pp. 211-233 | MR 1703838 | Zbl 0938.43003

[4] Astengo, F.; Blasio, B. Di Some properties of horocycles on Damek–Ricci spaces, Diff. Geo. Appl., Tome 26 (2008), pp. 676-682 | Article | MR 2474430 | Zbl 1156.43004

[5] Astengo, F.; Camporesi, R.; Di Blasio, B. The Helgason Fourier transform on a class of nonsymmetric harmonic spaces, Bull. Austral. Math. Soc., Tome 55 (1997), pp. 405-424 | Article | MR 1456271 | Zbl 0894.43003

[6] Ayadi, F. Equipartition of energy for the wave equation associated to the Dunkl-Cherednik Laplacian, J. Lie Theory, Tome 18 (2008), pp. 747-755 | MR 2523134 | Zbl 1171.35421

[7] Ben Saïd, S. Huygens’ principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators, Math. Res. Lett., Tome 13 (2006), pp. 43-58 | MR 2199565 | Zbl 1088.39018

[8] Branson, T.; Ólafsson, G.; Pasquale, A. The Paley-Wiener Theorem for the Jacobi transform and the local Huygens’ principle for root systems with even multiplicities, Indag. Mathem., Tome 16 (2005), pp. 429-442 | Article | MR 2313632 | Zbl 1168.43302

[9] Branson, T.; Ólafsson, G.; Schlichtkrull, H. Huygens’ principle in Riemannian symmetric spaces, Math. Ann., Tome 301 (1995), pp. 445-462 | Article | MR 1324519 | Zbl 0822.43002

[10] Cowling, M.; Dooley, A. H.; Korányi, A.; Ricci, F. H-type groups and Iwasawa decompositions, Adv. Math., Tome 87 (1991), pp. 1-41 | Article | MR 1102963 | Zbl 0761.22010

[11] Damek, E. The geometry of a semidirect extension of a Heisenberg type nilpotent group, Colloq. Math., Tome 53 (1987), pp. 255-268 | MR 924070 | Zbl 0661.53033

[12] Damek, E. A Poisson kernel on Heisenberg type nilpotent groups, Colloq. Math., Tome 53 (1987), pp. 239-247 | MR 924068 | Zbl 0661.53035

[13] Damek, E.; Ricci, F. Harmonic analysis on solvable extensions of H–type groups, J. Geom. Anal., Tome 2 (1992), pp. 213-248 | MR 1164603 | Zbl 0788.43008

[14] El Kamel, J.; Yacoub, C. Huygens’ priciple and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian, Ann. Math. Blaise Pascal, Tome 12 (2005), pp. 147-160 | Article | Numdam | MR 2126445 | Zbl 1088.35036

[15] Hadamard, J. Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, New Haven (1923)

[16] Helgason, S. Geometric Analysis on Symmetric Spaces, American Mathematical Society, Providence RI, Math. Surveys and Monographs 39 (1994) | MR 1280714 | Zbl 0809.53057

[17] Kaplan, A. Fundamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc., Tome 258 (1980), pp. 147-153 | Article | MR 554324 | Zbl 0393.35015

[18] Noguchi, M. The Solution of the Shifted Wave equation on Damek–Ricci Space, Interdiscip. Inform. Sci., Tome 8 (2002), pp. 101-113 | Article | MR 1923488 | Zbl 1018.43008

[19] Taylor, M. E. Partial Differential Equations, Springer-Verlag, New York, Texts in Applied Mathematics 23 (1996) | MR 1395147 | Zbl 0869.35001

[20] Thangavelu, S. On Paley–Wiener and Hardy theorems for NA groups, Math. Z., Tome 245 (2003), pp. 483-502 | Article | MR 2021567 | Zbl 1045.22008