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Stability of Gaussian Poincar\'{e} inequalities and Heisenberg Uncertainty Principle with monimial weights
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abstract
We use the Bakry-\'{E}mery curvature-dimension criterion and $\Gamma$-calculus to establish the Poincar\'{e} 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 Poincar\'{e} 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 Poincar\'{e} 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 Poincar\'{e} 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].
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Quantitative stability for the Brascamp-Lieb inequality and moment measures
For any convex potential, the L1 distance from a function to the Brascamp-Lieb optimizer manifold is controlled by the square root of its deficit, with a dimension-only constant independent of the potential.
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