Uniform Sobolev estimates for non-trapping metrics

Abstract

We prove uniform Sobolev estimates $||u|{|}_{L^{p^{\prime }}}\le C||(\Delta -\alpha )u|{|}_{L^p}$, where $p=2n/(n+2),p^{\prime }=2n/(n-2)$, for the Laplacian $\Delta$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of $\alpha$ which ranges over all complex numbers. This generalizes to non-constant coefficient Laplacians a result of Kenig-Ruiz-Sogge.

28 pages

References in corpus (3)

• Local smoothing for scattering manifolds with hyperbolic trapped sets
• Effective limiting absorption principles, and applications
• On $L^p$-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds

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