Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincaré-Sobolev inequalities
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| Format: | Preprint |
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2026
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| _version_ | 1866917444269899776 |
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| author | Claros, Alejandro Rela, Ezequiel |
| author_facet | Claros, Alejandro Rela, Ezequiel |
| contents | We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincaré-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25873 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincaré-Sobolev inequalities Claros, Alejandro Rela, Ezequiel Classical Analysis and ODEs Primary 42B35, Secondary 42B25, 46E35, 26D10 We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1 when the weight is nearly constant. As intermediate steps towards the result, we obtain quantitative estimates on the weighted and unweighted BMO norms of $\log w$ for an $A_\infty$ weight $w$. As a consequence, we show that a precise quantitative weighted Poincaré-Sobolev inequality can be proved for weights with small $[w]_{A_\infty}$ that recovers the classical Sobolev exponent $p^*=\frac{np}{n-p}$ when $[w]_{A_\infty}\to 1^+$. |
| title | Asymptotically sharp embedding of $A_\infty$ into $A_p$ for flat weights and applications to Poincaré-Sobolev inequalities |
| topic | Classical Analysis and ODEs Primary 42B35, Secondary 42B25, 46E35, 26D10 |
| url | https://arxiv.org/abs/2604.25873 |