 
Summary: NONLINEAR SCHR¨ODINGER EQUATION
ON REAL HYPERBOLIC SPACES
JEANPHILIPPE ANKER & VITTORIA PIERFELICE
Abstract. We consider the Schr¨odinger equation with no radial assumption on real
hyperbolic spaces Hn
. We obtain in all dimensions n2 sharp dispersive and Strichartz
estimates for a large family of admissible pairs. As a first consequence, we obtain strong
wellposedness results for NLS. Specifically, for small initial data, we prove L2
and H1
global wellposedness for any subcritical power (in contrast with the Euclidean case)
and with no gauge invariance assumption on the nonlinearity F. On the other hand,
if F is gauge invariant, L2
charge is conserved and hence, as in the Euclidean case, it
is possible to extend local L2
solutions to global ones. The corresponding argument in
H1
requires conservation of energy, which holds under the stronger condition that F is
defocusing. Recall that global wellposedness in the gauge invariant case was already
proved by Banica, Carles and Staffilani [4], for small radial L2
data or for large radial
