Uncertainty Principles on weighted spheres, balls and simplexes

TL;DR

This paper extends the uncertainty principle to weighted spheres, balls, and simplexes using Dunkl operators, providing a generalized Heisenberg inequality for spherical $h$-harmonic expansions with reflection group invariance.

Contribution

It introduces a new decomposition of the Dunkl-Laplace-Beltrami operator to establish uncertainty principles on weighted geometric domains.

Findings

Established an uncertainty principle for weighted spheres and related domains.

Connected the principle to Dunkl harmonic analysis and reflection groups.

Provided a framework for further generalizations in harmonic analysis.

Abstract

This paper studies the uncertainty principle for spherical $h$-harmonic expansions on the unit sphere of $\mathbb{R}^d$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl-Laplace-Beltrami operator on the weighted sphere.