Uncertainty principle on weighted spheres, balls and simplexes

TL;DR

This paper establishes uncertainty principles for functions on weighted spheres, balls, and simplexes invariant under reflection groups, extending classical results to weighted and symmetric settings with new inequalities.

Contribution

It introduces uncertainty principles for weighted spaces on spheres, balls, and simplexes, including invariance under reflection groups and broader classes of functions.

Findings

Uncertainty inequality on weighted spheres involving the spherical gradient.

Extension of uncertainty principles to the unit ball and simplex.

Some results hold for all functions, not just invariant ones.

Abstract

For a family of weight functions $h_\kappa$ that are invariant under a reflection group, the uncertainty principle on the unit sphere in the form of $$ \min_{1 \le i \le d} \int_{\mathbb{S}^{d-1}} (1- x_i) |f(x)|^2 h_\kappa^2(x) d\sigma \int_{\mathbb{S}^{d-1}}\left |\nabla_0 f(x)\right |^2 h_\kappa^2(x) d\sigma \ge c $$ is established for invariant functions $f$ that have unit norm and zero mean, where $\nabla_0$ is the spherical gradient. In the same spirit, uncertainty principles for weighted spaces on the unit ball and on the standard simplex are established, some of them hold for all admissible functions instead of invariant functions.