no. 2

Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces
Annales Mathématiques Blaise Pascal, Tome 17 (2010) no. 2, pp. 327-340.

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.

DOI : https://doi.org/10.5802/ambp.286
Classification : 43A80,  22E25
Mots clés : Wave equation, Damek–Ricci space 
@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},
     pages = {327--340},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {17},
     number = {2},
     year = {2010},
     doi = {10.5802/ambp.286},
     mrnumber = {2778917},
     zbl = {1207.43006},
     language = {en},
     url = {http://www.numdam.org/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) no. 2, pp. 327-340. doi : 10.5802/ambp.286. http://www.numdam.org/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., Volume 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, Volume 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., Volume 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., Volume 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., Volume 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, Volume 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., Volume 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., Volume 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., Volume 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., Volume 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., Volume 53 (1987), pp. 255-268 | MR 924070 | Zbl 0661.53033

[12] Damek, E. A Poisson kernel on Heisenberg type nilpotent groups, Colloq. Math., Volume 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., Volume 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, Volume 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, Math. Surveys and Monographs 39, American Mathematical Society, Providence RI, 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., Volume 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., Volume 8 (2002), pp. 101-113 | Article | MR 1923488 | Zbl 1018.43008

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

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