Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908493867384832 |
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| author | Cai, Xiaxing |
| author_facet | Cai, Xiaxing |
| contents | In this paper, we consider two limiting cases ($α\rightarrow n$ and $α\rightarrow 0 $) of the recent affine HLS inequalities by Haddad and Ludwig. As $α\rightarrow n$, the affine logarithmic HLS inequality is established, which is stronger than the logarithmic HLS inequality by Carlen and Loss from 1992 and Beckner from 1993. As $α\rightarrow 0$, an affine version of Beckner's logarithmic Sobolev inequality is established, which is also a limiting case of the affine fractional $L^2$ Sobolev inequalities. The affine logarithmic Sobolev inequality is stronger than the original version by Beckner from 1995. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities Cai, Xiaxing Metric Geometry Functional Analysis In this paper, we consider two limiting cases ($α\rightarrow n$ and $α\rightarrow 0 $) of the recent affine HLS inequalities by Haddad and Ludwig. As $α\rightarrow n$, the affine logarithmic HLS inequality is established, which is stronger than the logarithmic HLS inequality by Carlen and Loss from 1992 and Beckner from 1993. As $α\rightarrow 0$, an affine version of Beckner's logarithmic Sobolev inequality is established, which is also a limiting case of the affine fractional $L^2$ Sobolev inequalities. The affine logarithmic Sobolev inequality is stronger than the original version by Beckner from 1995. |
| title | Affine Logarithmic HLS and Beckner-Type Logarithmic Sobolev Inequalities |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2504.09251 |