Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.