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

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.

• Try the scaling $x\mapsto\lambda x$, $t\mapsto\lambda^2 t$. Dec 3, 2015 at 18:53
• @FanZheng Scaling would only give you the possible pairs. But it is not clear to me that the Strichartz estimates would hold for all possible pairs. Dec 3, 2015 at 19:12

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.
• Exactly what I wanted. One question, if I may: $\lesssim (....)$ here means $\leq C_{p, q, p', q'}(....)$. Is it known what the optimal constants $C_{p, q, p', q'}$ are and if they are attained? Dec 3, 2015 at 19:35
• @user83608: that I don't know. As far as I know even the case for Euclidean space is not entirely resolved: Foschi proved for $n = 1,2$ but only estimates of the best constants are available in general ejde.math.txstate.edu/Volumes/2015/270/selvitella.pdf ; that there exists maximizers however is known [Shao, arXiv: 0809.0153]. I'll be surprised if the general hyperbolic space case is solved. Dec 3, 2015 at 19:48
• ... which has also only been successfully used to compute cases when $n = 1$ or $2$. My point is that given so much is still not known about the Euclidean case, I am doubtful whether anyone seriously looked at the best constants issue for hyperbolic space. Dec 3, 2015 at 20:13