Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces

Astengo, Francesca; Di Blasio, Bianca

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.

Classification : 43A80, 22E25
Mots clés : Wave equation, Damek–Ricci space

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

[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 | DOI | Numdam | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

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

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

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