On $L^p$ resolvent estimates for Laplace-Beltrami operators on compact manifolds

TL;DR

This paper establishes $L^p$ resolvent estimates for Laplace-Beltrami operators on compact manifolds, extending Euclidean results and aiding in inverse problem solutions via complex geometrical optics.

Contribution

It generalizes Euclidean $L^p$ resolvent estimates to compact Riemannian manifolds using Hadamard's parametrix and oscillatory integral bounds.

Findings

Proves $L^p$ resolvent estimates on compact manifolds.

Links resolvent estimates to Carleman estimates for inverse problems.

Provides tools for constructing complex geometrical optics solutions.

Abstract

In this article we prove $L^p$ estimates for resolvents of Laplace-Beltrami operators on compact Riemannian manifolds, generalizing results of Kenig, Ruiz and Sogge in the Euclidean case and Shen for the torus. We follow Sogge and construct Hadamard's parametrix, then use classical boundedness results on integral operators with oscillatory kernels related to the Carleson and Sj\"olin condition. Our initial motivation was to obtain $L^p$ Carleman estimates with limiting Carleman weights generalizing those of Jerison and Kenig; we illustrate the pertinence of $L^p$ resolvent estimates by showing the relation with Carleman estimates. Such estimates are useful in the construction of complex geometrical optics solutions to the Schr\"odinger equation with unbounded potentials, an essential device for solving anisotropic inverse problems.