{"title":"测度空间中的Alberti秩一定理与拟共形映射","authors":"Panu Lahti","doi":"10.1016/j.jfa.2024.110758","DOIUrl":null,"url":null,"abstract":"<div><div>We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BV</mi></mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi>Y</mi><mo>)</mo></math></span> that at <span><math><msup><mrow><mo>‖</mo><mi>D</mi><mi>f</mi><mo>‖</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>-a.e. <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, the mapping <em>f</em> “behaves non-quasiconformally”.</div></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":"288 4","pages":"Article 110758"},"PeriodicalIF":1.7000,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Alberti's rank one theorem and quasiconformal mappings in metric measure spaces\",\"authors\":\"Panu Lahti\",\"doi\":\"10.1016/j.jfa.2024.110758\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for <span><math><mi>f</mi><mo>∈</mo><mrow><mi>BV</mi></mrow><mo>(</mo><mi>X</mi><mo>;</mo><mi>Y</mi><mo>)</mo></math></span> that at <span><math><msup><mrow><mo>‖</mo><mi>D</mi><mi>f</mi><mo>‖</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span>-a.e. <span><math><mi>x</mi><mo>∈</mo><mi>X</mi></math></span>, the mapping <em>f</em> “behaves non-quasiconformally”.</div></div>\",\"PeriodicalId\":15750,\"journal\":{\"name\":\"Journal of Functional Analysis\",\"volume\":\"288 4\",\"pages\":\"Article 110758\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2024-11-22\",\"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/S0022123624004464\",\"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/S0022123624004464","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Alberti's rank one theorem and quasiconformal mappings in metric measure spaces
We investigate a version of Alberti's rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for that at -a.e. , the mapping f “behaves non-quasiconformally”.
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
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