From semiclassical Strichartz estimates to uniform $L^p$ resolvent estimates on compact manifolds

TL;DR

This paper establishes uniform $L^p$ resolvent estimates for the damped wave operator on compact manifolds, using semiclassical Strichartz estimates, extending previous results for the Laplacian to non-self-adjoint cases.

Contribution

It introduces an alternative proof method based on semiclassical Strichartz estimates, enabling analysis of non-self-adjoint perturbations of the Laplacian.

Findings

Proves uniform $L^p$ resolvent estimates for the damped wave operator.

Extends techniques to non-self-adjoint perturbations.

Provides a framework compatible with semiclassical spectral analysis.

Abstract

We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.