Uncertainty principle on weighted spheres, balls and simplexes

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.