Log-Sobolev and Beckner inequalities and stability of Poincar\'e inequality with weighted Gaussian measures

TL;DR

This paper develops new inequalities and stability results for weighted Gaussian measures using semigroup and $\Gamma$-calculus methods, extending classical inequalities to weighted and homogeneous settings.

Contribution

It introduces generalized Beckner and Logarithmic Sobolev inequalities for weighted Gaussian measures and analyzes their stability and scale-dependent versions.

Findings

Derived a generalized Beckner inequality for weighted Gaussian measures.

Established stability estimates for the Poincaré inequality in weighted settings.

Formulated a scale-dependent Poincaré inequality and applied it to the Heisenberg Uncertainty Principle.

Abstract

We employ a Markov semigroup approach combined with the $\Gamma$-calculus to establish a generalized Beckner inequality associated with weighted Gaussian measures. As a direct consequence, we derive the corresponding Poincar\'e inequality in the same setting. Subsequently, by means of a duality argument, we investigate gradient and $L^2$ stability estimates of the Poincar\'e inequality. Furthermore, we formulate a scale-dependent version of the Poincar\'e 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.