Bounded eigenfunctions in the real Hyperbolic space

Sandrine Grellier (1) , Jean-Pierre Otal (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We characterize the distributions on the boundary of the hyperbolic space whose Poisson-Helgason transforms are bounded λ-eigenfunctions of the Laplace operator. Our main result states that these distributions are exactly the derivatives of Holder functions on the unit sphere, whose smoothness order can be precisely expressed in terms of the eigenvalue λ; this extends the results obtained in the case n= 2 by the second author.

Mots clés: Eigenfunctions; Hyperbolic space; Laplace-Beltrami operator; Poisson-Helgason Transform

Domaines: Analyse classique [math.CA]

Sandrine Grellier, Jean-Pierre Otal. Bounded eigenfunctions in the real Hyperbolic space. international mathematical research notices, 2005, 62, pp.3867-3897. ⟨hal-00022000⟩