Low Frequency Estimates for Long Range Perturbations in Divergence Form

http://dx.doi.org/10.4153/CJM-2011-022-9
Canad. J. Math. 63(2011), 961-991

  Published:2011-04-25
 Printed: Oct 2011

We prove a uniform control as $ z \rightarrow 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 \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.