Wave equation on general noncompact symmetric spaces

Équation des ondes sur les espaces symétriques non compacts généraux

Jean-Philippe Anker (1) , Hong-Wei Zhang (1)

1 IDP - Institut Denis Poisson

We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. As consequences, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on hyperbolic spaces.

Mots clés: Noncompact symmetric space of higher rank; Semilinear wave equation; Dispersive property; Strichartz inequality; Global well-posedness

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Philippe Anker, Hong-Wei Zhang. Wave equation on general noncompact symmetric spaces. American Journal of Mathematics, In press. ⟨hal-02969730⟩