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A Strichartz inequality for the Schrödinger equation on nontrapping asymptotically conic manifolds. (English) Zbl 1068.35119
The aim of the article is to prove for a solution $$u$$ of the Schrödinger equation $$iu_t+\frac12\Delta_Mu=0$$ the $$L^4$$ space-time Strichartz inequality $$\int_0^1\int_M|u(t,z)|^4\,dg(z)\,dt\leq C\|u(0)\|^4_{H^{1/4}(M)}$$ on any smooth three-dimensional asymptotically conic Riemannian manifold $$M$$ obeying a nontrapping condition. The proof is based on the interaction Morawetz inequality as a positive commutator inequality for the tensor product $$U(t,z',z'')\equiv u(t,z')u(t,z'')$$ of the solution with itself and smoothing estimates for Schrödinger solutions. It is shown that the exponent in $$H^{1/4}(M)$$ is sharp.

MSC:
 35Q40 PDEs in connection with quantum mechanics 35Q55 NLS equations (nonlinear Schrödinger equations) 58J47 Propagation of singularities; initial value problems on manifolds
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