Sharp resolvent estimates on non-positively curved asymptotically hyperbolic manifolds

Abstract

We study the high energy estimate for the resolvent $R(\lambda )$ of the Laplacian on non-trapping asymptotically hyperbolic manifolds (AHM). In the literature, polynomial bound of the form $\Vert R(\lambda )\Vert =O(|\lambda {|}^N)$ for $|\lambda |$ large and $\lambda \in \mathbb{C}$ in strips where $R(\lambda )$ is holomorphic was established for some $N>-1$. We prove the optimal bound $O(|\lambda {|}^{-1})$ under the non-positive sectional curvature assumption by taking into account the oscillatory behavior of the Schwartz kernel of the resolvent.