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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.

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  • $\begingroup$ Try the scaling $x\mapsto\lambda x$, $t\mapsto\lambda^2 t$. $\endgroup$ – Fan Zheng Dec 3 '15 at 18:53
  • $\begingroup$ @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. $\endgroup$ – user83608 Dec 3 '15 at 19:12
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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.

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  • $\begingroup$ 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? $\endgroup$ – user83608 Dec 3 '15 at 19:35
  • $\begingroup$ @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. $\endgroup$ – Willie Wong Dec 3 '15 at 19:48
  • $\begingroup$ Actually, as regards the Euclidean case, there is also MR2547132. $\endgroup$ – user83608 Dec 3 '15 at 20:01
  • $\begingroup$ ... 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. $\endgroup$ – Willie Wong Dec 3 '15 at 20:13

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