High Frequency Weighted Resolvent Estimates for the Dirichlet Laplacian in the Exterior Domain

In this paper, we want to present several resolvent estimates for the Dirichlet Laplacian in exterior domain. The estimates evaluate a weighted \(L^2\) norm with a weight measured by a negative power of the distance from the boundary. We consider an exterior domain \(\varOmega \) , that is the complementary of a compact in \(\textbf{R}^n\) , and the inhomogeneous Helmotz equation on it. If the exterior domain is non-trapping, there are cut-off resolvent estimates without weights. Our main result is that we can improve the estimates putting the weights. The main idea is the polar change of coordinates, where \(r=d(x,\partial \varOmega )\) , that allows us to use the Hardy inequality close to the boundary of the domain. Kato smoothing estimate is obtained as a consequence of the weighted cut-off resolvent estimates.