Stability Estimates for the Heisenberg Uncertainty Principle with Dunkl Weights

Abstract

We study the stability of the Heisenberg Uncertainty Principle with Dunkl weights, inspired by recent progress on stability in functional and geometric inequalities. First, using the Bakry-Émery \(\varGamma _2\)-calculus, we establish several versions of the Poincaré inequality for measures involving Dunkl-Gaussian weights. We then prove a Heisenberg-type uncertainty principle adapted to Dunkl weights through suitable identities. By combining these results, we obtain several stability estimates for the Dunkl-weighted Heisenberg uncertainty principle. Our work extends existing methods developed for monomial weights to the more complex Dunkl setting.