Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913042186371072 |
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| author | Lam, Nguyen Lu, Guozhen Russanov, Andrey |
| author_facet | Lam, Nguyen Lu, Guozhen Russanov, Andrey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16791 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures Lam, Nguyen Lu, Guozhen Russanov, Andrey Functional Analysis Analysis of PDEs Probability 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. |
| title | Log-Sobolev and Beckner inequalities and stability of Poincaré inequality with weighted Gaussian measures |
| topic | Functional Analysis Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2604.16791 |