Degenerate Poincaré-Sobolev inequalities via fractional integration
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 depende...
| Autor: | |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1979 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/1979 |
| Access Level: | acceso abierto |
| Palabra clave: | Riesz potential Weighted inequalities Weighted Poincaré-Sobolev inequalities |
| Sumario: | 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. \textbf{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 $(1-\delta)^\frac{1}{p}$. In particular, we improve one of the main results from Hurri-Syrjänen, Martínez-Perales, Pérez, and Vähäkangas [Int. Math. Res. Not. \textbf{20} (2023)]. |
|---|