Estimates for sums of eigenfunctions of elliptic pseudo-differential operators on compact Lie groups

TL;DR

This paper generalizes eigenfunction sum estimates from Laplacians to positive elliptic pseudo-differential operators on compact Lie groups, with applications in control theory for diffusion models.

Contribution

It extends existing eigenfunction estimates to a broader class of operators on compact Lie groups using symbol positivity criteria.

Findings

Established eigenfunction sum estimates for elliptic pseudo-differential operators.

Proved null-controllability for diffusion models on compact Lie groups.

Connected spectral estimates with control theory applications.

Abstract

We extend the estimates proved by Donnelly and Fefferman and by Lebeau and Robbiano for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in the control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.