Fourier Uncertainty Principles on Riemannian Manifolds

TL;DR

This paper extends Fourier uncertainty principles to compact Riemannian manifolds, addressing challenges posed by eigenfunction growth and relating findings to Bourgain's $ extstyle ext{ extLambda}_q$ theorem.

Contribution

It introduces a Fourier uncertainty principle for Riemannian manifolds, contrasting it with abelian group cases and exploring eigenfunction growth implications.

Findings

Established a Fourier uncertainty principle on compact Riemannian manifolds.

Connected eigenfunction growth to Bourgain's $ extstyle ext{ extLambda}_q$ theorem.

Highlighted differences between manifold and abelian group Fourier analysis.

Abstract

The purpose of this paper is to develop a Fourier uncertainty principle on compact Riemannian manifolds and contrast the underlying ideas with those arising in the setting of locally compact abelian groups. The key obstacle is the growth of eigenfunctions, and connections to Bourgain's celebrated $\Lambda_q$ theorem are discussed in this context.