Endpoint resolvent estimates for compact Riemannian manifolds

TL;DR

This paper establishes endpoint resolvent estimates for the Laplace-Beltrami operator on compact Riemannian manifolds, matching Euclidean behavior and identifying optimal spectral regions.

Contribution

It extends Euclidean resolvent bounds to compact manifolds at the critical endpoint, clarifying the spectral region where estimates hold.

Findings

Resolves endpoint $L^p$ bounds for Laplace-Beltrami resolvent

Identifies optimal spectral region for estimates on spheres

Matches Euclidean resolvent behavior in the manifold setting

Abstract

We prove $L^p\to L^{p'}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.