{"title":"Degenerate Poincaré-Sobolev inequalities via fractional integration","authors":"Alejandro Claros","doi":"10.1016/j.jfa.2025.111000","DOIUrl":null,"url":null,"abstract":"<div><div>We present a local weighted estimate for the Riesz potential in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, which improves the main theorem of Alberico et al. (2009) <span><span>[2]</span></span> 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 (2019) <span><span>[36]</span></span> related to the sharp exponent in the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> constant in the <span><math><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mi>p</mi><mo>)</mo></math></span> Poincaré-Sobolev inequality with <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> 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 <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup></math></span>. In particular, we improve one of the main results from Hurri-Syrjänen et al. (2023) <span><span>[24]</span></span>.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 111000"},"PeriodicalIF":1.6000,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002212362500182X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We present a local weighted estimate for the Riesz potential in , which improves the main theorem of Alberico et al. (2009) [2] 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 (2019) [36] related to the sharp exponent in the constant in the Poincaré-Sobolev inequality with 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 . In particular, we improve one of the main results from Hurri-Syrjänen et al. (2023) [24].
期刊介绍:
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis