{"title":"弱多孔集合和穆肯霍普 Ap 距离函数","authors":"","doi":"10.1016/j.jfa.2024.110558","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight <span><math><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>dist</mi><mspace></mspace><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> belongs to the Muckenhoupt class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, for some <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, if and only if <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of <em>E</em>. When <em>E</em> is weakly porous, we obtain a similar quantitative characterization of <span><math><mi>w</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, for <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span>, as well. At the end of the paper, we give an example of a set <span><math><mi>E</mi><mo>⊂</mo><mi>R</mi></math></span> which is not weakly porous but for which <span><math><mi>w</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> for every <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span>.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022123624002465/pdfft?md5=94e5160f264d3666ae1b2e160d10d5b3&pid=1-s2.0-S0022123624002465-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Weakly porous sets and Muckenhoupt Ap distance functions\",\"authors\":\"\",\"doi\":\"10.1016/j.jfa.2024.110558\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight <span><math><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>dist</mi><mspace></mspace><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> belongs to the Muckenhoupt class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, for some <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>, if and only if <span><math><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of <em>E</em>. When <em>E</em> is weakly porous, we obtain a similar quantitative characterization of <span><math><mi>w</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, for <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span>, as well. At the end of the paper, we give an example of a set <span><math><mi>E</mi><mo>⊂</mo><mi>R</mi></math></span> which is not weakly porous but for which <span><math><mi>w</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> for every <span><math><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mo>∞</mo></math></span>.</p></div>\",\"PeriodicalId\":15750,\"journal\":{\"name\":\"Journal of Functional Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-07-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022123624002465/pdfft?md5=94e5160f264d3666ae1b2e160d10d5b3&pid=1-s2.0-S0022123624002465-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/S0022123624002465\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123624002465","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Weakly porous sets and Muckenhoupt Ap distance functions
We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight belongs to the Muckenhoupt class , for some , if and only if is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of , for , as well. At the end of the paper, we give an example of a set which is not weakly porous but for which for every and .
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