Low Frequency Estimates for Long Range Perturbations in Divergence Form

We prove a uniformcontrol as z → 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 ≥ 3 when P is defined on ℝd and in dimension d ≥ 2 when P is defined outside a compact obstacle with Dirichlet boundary conditions.