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 $ \mbox{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 $ \mbox{Re}(Z) $ belongs to a compact subset of $ (0,+\infty) $ 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.