Sharp resolvent estimates on non-positively curved asymptotically hyperbolic manifolds

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 $\|R(\lambda)\| = O(|\lambda|^{N})$ 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.