Stochastic analysis of Beckner's and related functional inequalities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908982047670272 |
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| author | Hariya, Yuu |
| author_facet | Hariya, Yuu |
| contents | Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincaré's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincaré's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a Hölder-type inequality that holds among the entropy, variance and related functionals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12462 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic analysis of Beckner's and related functional inequalities Hariya, Yuu Probability Functional Analysis 39B62 (Primary) 60H30 (Secondary) Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincaré's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincaré's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a Hölder-type inequality that holds among the entropy, variance and related functionals. |
| title | Stochastic analysis of Beckner's and related functional inequalities |
| topic | Probability Functional Analysis 39B62 (Primary) 60H30 (Secondary) |
| url | https://arxiv.org/abs/2604.12462 |