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...

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Detalles Bibliográficos
Autor: Claros, A.
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
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spelling Degenerate Poincaré-Sobolev inequalities via fractional integrationClaros, A.Riesz potentialWeighted inequalitiesWeighted Poincaré-Sobolev inequalitiesWe 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)].PRE2021-099091202520252025info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1979reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://doi.org/10.1016/j.jfa.2025.111000info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/CEX2021-001142-Sinfo:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/19792026-06-19T12:47:47Z
dc.title.none.fl_str_mv Degenerate Poincaré-Sobolev inequalities via fractional integration
title Degenerate Poincaré-Sobolev inequalities via fractional integration
spellingShingle Degenerate Poincaré-Sobolev inequalities via fractional integration
Claros, A.
Riesz potential
Weighted inequalities
Weighted Poincaré-Sobolev inequalities
title_short Degenerate Poincaré-Sobolev inequalities via fractional integration
title_full Degenerate Poincaré-Sobolev inequalities via fractional integration
title_fullStr Degenerate Poincaré-Sobolev inequalities via fractional integration
title_full_unstemmed Degenerate Poincaré-Sobolev inequalities via fractional integration
title_sort Degenerate Poincaré-Sobolev inequalities via fractional integration
dc.creator.none.fl_str_mv Claros, A.
author Claros, A.
author_facet Claros, A.
author_role author
dc.subject.none.fl_str_mv Riesz potential
Weighted inequalities
Weighted Poincaré-Sobolev inequalities
topic Riesz potential
Weighted inequalities
Weighted Poincaré-Sobolev inequalities
description 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)].
publishDate 2025
dc.date.none.fl_str_mv 2025
2025
2025
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info:eu-repo/semantics/acceptedVersion
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dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1979
url http://hdl.handle.net/20.500.11824/1979
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://doi.org/10.1016/j.jfa.2025.111000
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/CEX2021-001142-S
info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2022-2025
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
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