Linear Schrödinger equation on $\mathbb{H}^{d}$

Question

Consider the linear Schrödinger equation $i\partial_t u = -\Delta u$, where $\Delta$ is the Laplacian on the hyperbolic space $\mathbb{H}^d$. What are the admissible pairs $(p, q)$ such that we have Strichartz estimates of the form $$ \Vert u\Vert_{L^p_tL^q_x(\mathbb{R}\times\mathbb{H}^d)} \leq C_{p, q}\Vert u_0\Vert_{L^2(\mathbb{H}^d)}?$$ Is the theory similar to that on $\mathbb{R}^d$? Also, If we replace the Laplacian in the above equation with the fractional Laplacian $(-\Delta)^{\alpha/2}$, where $\alpha \in (0, 2)$, do we know about the admissible pairs? This is mainly a reference request.

Answer 1

For the standard Schroedinger, the result is due to Anker and Pierfelice http://www.sciencedirect.com/science/article/pii/S0294144909000250 and separately Ionescu and Staffilani http://link.springer.com/article/10.1007%2Fs00208-009-0344-6

Their Strichartz estimate reads:

Let $u$ solve $i\partial_t u + \triangle u = F$ on $\mathbb{H}^n \times\mathbb{R}$, and let $(p^{-1}, q^{-1})$ and $(\bar{p}^{-1}, \bar{q}^{-1})$ both belong to the triangle $$ T_n = \{(x,y) \in (0,1/2]\times (0,1/2): x + ny \geq n/2\} \cup \{ (0,1/2)\} $$ then the estimate $$ \|u\|_{L^p_t L^q_x} \lesssim \|u_0\|_{L^2_x} + \|F\|_{L^{\bar{p}'}_tL^{\bar{q}'}_x} $$

Where $\prime$ denote the Holder conjugate.