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 $L^2\to L^2$ bound for $⟨r{⟩}^{-1}(-{\Delta }_G-z{)}^{-1}⟨r{⟩}^{-1}$ when $ \mbox{Re}(z) $ is small, with the optimal weight $⟨r{⟩}^{-1}$. The second one is about powers of the resolvent. For any integer $N$, we prove uniform $L^2\to L^2$ bounds for $⟨\epsilon r{⟩}^{-N}(-{\epsilon }^{-2}{\Delta }_G-Z{)}^{-N}⟨\epsilon r{⟩}^{-N}$ 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.

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References in corpus (1)

• Local Energy Decay for the Damped Wave Equation

Cited by in corpus (4)

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