Stability of Gaussian Poincaré inequalities and Heisenberg Uncertainty Principle with monimial weights
Nguyen Lam, Guozhen Lu, Andrey Russanov
TL;DR
The paper extends the stability theory of Poincaré and Heisenberg Uncertainty inequalities to monomial-weighted Gaussian measures by employing the Bakry-Émery curvature-dimension criterion and Γ-calculus. It proves a base Poincaré inequality with constant 1 for monomial Gaussian weights and derives gradient-stability via a duality method, then obtains sharp, improved stability results for the classical Gaussian case through Hermite spectral analysis. The authors introduce scale-dependent Poincaré inequalities and show how these yield scale-aware stability results for the Heisenberg Uncertainty Principle with monomial weights, including refined stability bounds involving advanced optimizer families. Collectively, the results provide explicit optimal constants, attainability by nonlinear functions, and affirmative stability statements in weighted Gaussian spaces, thereby broadening quantitative stability insights for uncertainty principles in multi-dimensional settings.
Abstract
We use the Bakry-Émery curvature-dimension criterion and $Γ$-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 inspect the stability 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 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, which is of independent interest. Moreover, we also establish the stability of the sharp stability inequality of the classical Heisenberg Uncertainty Principle proved in [15].