Uncertainty Principles on Harmonic Manifolds of Rank One

TL;DR

This paper establishes several uncertainty principles for the Fourier transform on harmonic manifolds of rank one, extending classical Euclidean results to a broader geometric setting.

Contribution

It introduces new uncertainty principles and generalizes the Hausdorff-Young inequality to harmonic manifolds of rank one, expanding harmonic analysis in non-Euclidean spaces.

Findings

Derived a Heisenberg uncertainty principle for harmonic manifolds.

Proved a Morgen theorem in this geometric context.

Extended the Hausdorff-Young inequality to harmonic manifolds.

Abstract

We show various uncertainty principles for the Fourier transform on harmonic manifolds of rank one. In particular, we derive a Heisenberg uncertainty principle, a Morgen theorem, an uncertainty principle for the Schr\"odinger equation and a version of H\"omanders theorem. Furthermore, we generalise the in the Euclidean case well-known Hausdorff-Young inequality for the Fourier transform, to harmonic manifolds of rank one.