Propagation of smallness and spectral estimates

TL;DR

This paper demonstrates how spectral projector estimates for Laplace operators can be derived from propagation of smallness estimates for harmonic functions, linking local and global estimates and deriving consequences for heat equation control.

Contribution

It introduces a method to deduce spectral projector estimates from propagation of smallness, connecting local harmonic analysis to global spectral properties.

Findings

Spectral projector estimates can be obtained from propagation of smallness.

Global estimates are derived from local harmonic function estimates.

Applications to observability and control of heat equations are established.

Abstract

The purpose of this article is to show that the spectral projector estimates for Laplace operators can be deduced from Logunov-Malinnikova's Propagation of smallness estimates for harmonic functions [11, 10, 9]. The main point is to pass from the local estimates obtained in [4] (on a compact manifold) to global estimates. We also state classical consequences in terms of observability and control for heat equations, which are direct consequences of these spectral projector estimates..