Sharp Low Frequency Resolvent Estimates on Asymptotically Conical Manifolds

Abstract

On a class of asymptotically conical manifolds, we prove two types of low frequency estimates for the resolvent of the Laplace-Beltrami operator. The first result is a uniform ${L^2 \rightarrow L^2}$bound for ${\langle{r}\rangle^{-1} (- \Delta_G - z)^{-1}\langle{r}\rangle^{-1}}$when Re(z) is small, with the optimal weight ${\langle{r}\rangle^{-1}}$. The second one is about powers of the resolvent. For any integer N, we prove uniform ${L^2 \rightarrow L^2}$ bounds for ${ \langle{\epsilon r}\rangle^{-N} (-\epsilon^{-2} \Delta_G - Z)^{-N} \langle{\epsilon r}\rangle^{-N}}$ when Re(Z) belongs to a compact subset of (0, + ∞) and ${0 < \epsilon \ll 1}$. These results are obtained by proving similar estimates on a pure cone with a long range perturbation of the metric at infinity.