We use the Bakry–Émery curvature-dimension criterion and \(\Gamma \) -calculus to establish the Poincaré inequality with monomial Gaussian measure, and then apply the duality approach to study its improvements and its gradient stability. We also set up the scale-dependent Poincaré inequality with monomial Gaussian type measure and use it to establish the stability with exact sharp constants of the Heisenberg Uncertainty Principle with monomial weight. Finally, we apply the improved versions of the monomial Gaussian Poincaré inequality to investigate the improved stability with exact optimal constants of the Heisenberg Uncertainty Principle with monomial weight. As special cases of our main results, we obtain the gradient stability of the classical Gaussian Poincaré inequality with exact sharp constant, which is of independent interest and surprisingly absent in the literature. Moreover, we also establish the stability of the sharp stability inequality of the classical Heisenberg Uncertainty Principle proved in [15].
