Log-Sobolev and Beckner Inequalities and Stability of Poincaré Inequality with Weighted Gaussian Measures

We employ a Markov semigroup approach combined with the Γ-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincaré inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and L^2 stability estimates of the Poincaré inequality. Furthermore, we formulate a scale-dependent version of the Poincaré inequality for homogeneous Gaussian-type measures and apply it to analyze the stability of the Heisenberg Uncertainty Principle with homogeneous weights. Finally, we establish a Logarithmic Sobolev inequality for weighted Gaussian measures and utilize it to derive the Euclidean Logarithmic Sobolev inequality with homogeneous log-concave weights.