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Dispersive estimates for manifolds with one trapped orbit. (English) Zbl 1152.58024
For a large class of complete, non-compact Riemannian manifolds \((M, g)\) with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator:
\[ \int^T_0\|\rho_se^{it(\Delta_g-V)u_0}\|^2_{H^{1/2-\varepsilon}(M)}\,dt\leq C_T\|u_0\|^2_{L^2(M)}, \]
where \(\rho_s(x)\in{\mathcal C}^\infty(M)\) satisfies \(\rho_s =\langle \text{dist}_g(x,x_0)\rangle^{-s}\), \(s>\frac12\), and \(V\in{\mathcal C}^\infty(M)\), \(0\leq V\leq C\) satisfies \(|\nabla V|\leq C\langle\text{dist}(x,x_0)\rangle^{-1-\delta}\) for some \(\delta > 0\). From the local smoothing estimate, we deduce a family of Strichartz-type estimates, which are used to prove two well-posedness results for the nonlinear Schrödinger equation.
As a second application, we prove the following sub-exponential local energy decay estimate for solutions to the wave equation when \(\dim M = n\geq 3\) is odd and \(M\) is equal to \(\mathbb R^n\) outside a compact set: \[ \int_M|\psi\partial_tu|^2+|\psi \nabla u|^2\,dx\leq Ce^{-T^{1/2}/c}(\|U(x,0)\|^2_{H^{1+\varepsilon(M)}}+\|D_tu(x,0)\|^2_{H^\varepsilon(M)}), \]
where \(\psi\in{\mathcal C}^\infty(M)\), \(\psi\equiv e^{-|x|2}\) outside a compact set.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
35B99 Qualitative properties of solutions to partial differential equations
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