Published online by Cambridge University Press: 20 November 2018
We prove a uniformcontrol as $z\,\to \,0$ for the resolvent
${{(P-z)}^{-1}}$ of long range perturbations
$P$ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension
$d\,\ge \,3$ when
$P$ is defined on
${{\mathbb{R}}^{d}}$ and in dimension
$d\,\ge \,2$ when
$P$ is defined outside a compact obstacle with Dirichlet boundary conditions.