Low frequency resolvent estimates for long range perturbations of the euclidean Laplacian

Let $P$ be a long range metric perturbation of the Euclidean Laplacian on $\R^d,\, d\ge 3$. We prove that the following resolvent estimate holds: \begin{equation*} \Vert\<x\ >^{-\alpha}(P-z)^{-1}\<x\ >^{-\beta}\Vert\lesssim 1\quad \forall z\in \C\setminus \R,\,\vert z\vert<1, \end{equation*} if $\alpha,\beta > 1/2$ and $\alpha+\beta > 2$. The above estimate is false for the Euclidean Laplacian in dimension $3$ if $\alpha\le 1/2$ or $\beta\le 1/2$ or $\alpha+\beta<2$.

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Published 1 January 2010