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Low Frequency Estimates for Long Range Perturbations in Divergence Form

Published online by Cambridge University Press:  20 November 2018

Jean-Marc Bouclet
Jean-Marc Bouclet*
Affiliation:
Institut de Mathématiques de Toulouse, Université de Toulouse, Toulouse, France, F-31062 email: jean-marc.bouclet@math.univ-toulouse.fr
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Abstract

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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.

Keywords

35P25resolvent estimatesthresholdsscattering theoryRiesz transform
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 5 , 18 October 2011 , pp. 961 - 991
Copyright
Copyright © Canadian Mathematical Society 2011

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