Josep M. Gallegos , Mihalis Mourgoglou , Xavier Tolsa
{"title":"Extrapolation of solvability of the regularity and the Poisson regularity problems in rough domains","authors":"Josep M. Gallegos , Mihalis Mourgoglou , Xavier Tolsa","doi":"10.1016/j.jfa.2024.110672","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, be an open set satisfying the corkscrew condition with <em>n</em>-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span> and the weak-<span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span> is equivalent to the solvability of the regularity problem in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span> for some <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>. We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space <span><math><msup><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></math></span> is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 1","pages":"Article 110672"},"PeriodicalIF":1.7000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624003604/pdfft?md5=559d2fce88142d22708d8e9a462e2ff0&pid=1-s2.0-S0022123624003604-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123624003604","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
Let , , be an open set satisfying the corkscrew condition with n-Ahlfors regular boundary ∂Ω, but without any connectivity assumption. We study the connection between solvability of the regularity problem for divergence form elliptic operators with boundary data in the Hajłasz-Sobolev space and the weak- property of the associated elliptic measure. In particular, we show that solvability of the regularity problem in is equivalent to the solvability of the regularity problem in for some . We also prove analogous extrapolation results for the Poisson regularity problem defined on tent spaces. Moreover, under the hypothesis that ∂Ω supports a weak -Poincaré inequality, we show that the solvability of the regularity problem in the Hajłasz-Sobolev space is equivalent to a stronger solvability in a Hardy-Sobolev space of tangential derivatives.
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