Almost sharp global wellposedness and scattering for the defocusing conformal wave equation on the hyperbolic space

In this paper we prove a global well-posedness and scattering result for the defocusing conformal nonlinear wave equation in the hyperbolic space $\mathbb{H}^d, d \geq 3$. We take advantage of the hyperbolic geometry which yields stronger Morawetz and Strichartz estimates. We show that the solution is globally wellposed and scatters if the initial data is radially symmetric and lies in $H^{\frac{1}{2}+\epsilon}(\mathbb{H}^d)\times H^{-\frac{1}{2}+\epsilon}(\mathbb{H}^d)$, $\epsilon>0$.