{"title":"On a Hardy–Morrey inequality","authors":"Ryan Hynd , Simon Larson , Erik Lindgren","doi":"10.1016/j.jfa.2025.111002","DOIUrl":null,"url":null,"abstract":"<div><div>Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality<span><span><span><math><mi>λ</mi><msubsup><mrow><mo>‖</mo><mfrac><mrow><mi>u</mi></mrow><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>n</mi><mo>/</mo><mi>p</mi></mrow></msubsup></mrow></mfrac><mo>‖</mo></mrow><mrow><mo>∞</mo></mrow><mrow><mi>p</mi></mrow></msubsup><mo>≤</mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><mo>|</mo><mi>D</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></math></span></span></span> for any open set <span><math><mi>Ω</mi><mo>⊊</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. This inequality is valid for functions supported in Ω and with <em>λ</em> a positive constant independent of <em>u</em>. The crucial hypothesis is that the exponent <em>p</em> exceeds the dimension <em>n</em>. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of Ω, sharp constants, and the existence of a nontrivial <em>u</em> which saturates the inequality.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"289 6","pages":"Article 111002"},"PeriodicalIF":1.7000,"publicationDate":"2025-04-18","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/S0022123625001843","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
Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality for any open set . This inequality is valid for functions supported in Ω and with λ a positive constant independent of u. The crucial hypothesis is that the exponent p exceeds the dimension n. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of Ω, sharp constants, and the existence of a nontrivial u which saturates the inequality.
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