Degenerate Poincaré-Sobolev inequalities via fractional integration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908690835046400 |
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| author | Claros, Alejandro |
| author_facet | Claros, Alejandro |
| contents | We present a local weighted estimate for the Riesz potential in $\mathbb{R}^n$, which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincaré-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by Pérez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the $A_1$ constant in the $(p^*,p)$ Poincaré-Sobolev inequality with $A_1$ weights. Our approach is versatile enough to prove Poincaré-Sobolev inequalities for high-order derivatives and fractional Poincaré-Sobolev inequalities with the BBM extra gain factor $(1-δ)^{1/p}$. In particular, we improve one of the main results from Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Int. Math. Res. Not. 20 (2023)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03651 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Degenerate Poincaré-Sobolev inequalities via fractional integration Claros, Alejandro Classical Analysis and ODEs Functional Analysis We present a local weighted estimate for the Riesz potential in $\mathbb{R}^n$, which improves the main theorem of Alberico, Cianchi, and Sbordone [C. R. Math. Acad. Sci. Paris \textbf{347} (2009)] in several ways. As a consequence, we derive weighted Poincaré-Sobolev inequalities with sharp dependence on the constants. We answer positively to a conjecture proposed by Pérez and Rela [Trans. Amer. Math. Soc. 372 (2019)] related to the sharp exponent in the $A_1$ constant in the $(p^*,p)$ Poincaré-Sobolev inequality with $A_1$ weights. Our approach is versatile enough to prove Poincaré-Sobolev inequalities for high-order derivatives and fractional Poincaré-Sobolev inequalities with the BBM extra gain factor $(1-δ)^{1/p}$. In particular, we improve one of the main results from Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Int. Math. Res. Not. 20 (2023)]. |
| title | Degenerate Poincaré-Sobolev inequalities via fractional integration |
| topic | Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2512.03651 |