Uniform Sobolev estimates for non-trapping metrics

TL;DR

This paper establishes uniform Sobolev estimates for the Laplacian on non-trapping asymptotically conic manifolds, extending previous results to more general geometries with non-constant coefficients.

Contribution

It generalizes uniform Sobolev estimates to non-constant coefficient Laplacians on non-trapping asymptotically conic manifolds, independent of the spectral parameter.

Findings

Proved uniform Sobolev estimates for Laplacians on non-trapping asymptotically conic manifolds.

Extended Kenig-Ruiz-Sogge results to more general geometric settings.

Estimates hold uniformly over all complex spectral parameters.

Abstract

We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(\Delta-\alpha)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $\Delta$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of $\alpha$ which ranges over all complex numbers. This generalizes to non-constant coefficient Laplacians a result of Kenig-Ruiz-Sogge.