On $L^p$ resolvent estimates for elliptic operators on compact manifolds

TL;DR

This paper establishes uniform $L^p$ resolvent estimates for higher order elliptic operators on compact manifolds, extending previous results for Laplace--Beltrami operators and demonstrating the optimality of spectral regions.

Contribution

It generalizes existing $L^p$ resolvent estimates from Laplace--Beltrami operators to higher order elliptic operators on compact manifolds, showing spectral region optimality.

Findings

Uniform $L^p$ resolvent estimates proved for higher order elliptic operators.

Spectral regions in the estimates are shown to be optimal.

Methodology closely follows previous approaches for Laplace--Beltrami operators.

Abstract

We prove uniform $L^p$ estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding resul of [3] in the case of Laplace-- Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1] very closely. We also show that spectral regions in our $L^p$ resolvent estimates are optimal.