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arXiv · 2604.16791

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

Nguyen Lam·Guozhen Lu·Andrey Russanov
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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.

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BibTeXRIS

Nguyen Lam, Guozhen Lu, Andrey Russanov. 2026-04-18. Log-Sobolev and Beckner inequalities and stability of Poincar\'e inequality with weighted Gaussian measures. https://arxiv.org/abs/2604.16791

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