{"title":"卡诺群上有限畸变映射的开性和离散性","authors":"S. G. Basalaev, S. K. Vodopyanov","doi":"10.1134/s0037446623060046","DOIUrl":null,"url":null,"abstract":"<p>We prove that a mapping of finite distortion\n<span>\\( f:\\Omega\\to 𝔾 \\)</span> in a domain <span>\\( \\Omega \\)</span>\nof an <span>\\( H \\)</span>-type Carnot group <span>\\( 𝔾 \\)</span>\nis continuous, open, and discrete provided that\nthe distortion function <span>\\( K(x) \\)</span> of <span>\\( f \\)</span> belongs to <span>\\( L_{p,\\operatorname{loc}}(\\Omega) \\)</span>\nfor some <span>\\( p>\\nu-1 \\)</span>.\nIn fact, the proof is suitable for each Carnot group\nprovided it has a <span>\\( \\nu \\)</span>-harmonic function of the\nform <span>\\( \\log\\rho \\)</span>, where the homogeneous norm\n<span>\\( \\rho \\)</span> is <span>\\( C^{2} \\)</span>-smooth.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":"29 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Openness and Discreteness of Mappings of Finite Distortion on Carnot Groups\",\"authors\":\"S. G. Basalaev, S. K. Vodopyanov\",\"doi\":\"10.1134/s0037446623060046\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that a mapping of finite distortion\\n<span>\\\\( f:\\\\Omega\\\\to 𝔾 \\\\)</span> in a domain <span>\\\\( \\\\Omega \\\\)</span>\\nof an <span>\\\\( H \\\\)</span>-type Carnot group <span>\\\\( 𝔾 \\\\)</span>\\nis continuous, open, and discrete provided that\\nthe distortion function <span>\\\\( K(x) \\\\)</span> of <span>\\\\( f \\\\)</span> belongs to <span>\\\\( L_{p,\\\\operatorname{loc}}(\\\\Omega) \\\\)</span>\\nfor some <span>\\\\( p>\\\\nu-1 \\\\)</span>.\\nIn fact, the proof is suitable for each Carnot group\\nprovided it has a <span>\\\\( \\\\nu \\\\)</span>-harmonic function of the\\nform <span>\\\\( \\\\log\\\\rho \\\\)</span>, where the homogeneous norm\\n<span>\\\\( \\\\rho \\\\)</span> is <span>\\\\( C^{2} \\\\)</span>-smooth.</p>\",\"PeriodicalId\":49533,\"journal\":{\"name\":\"Siberian Mathematical Journal\",\"volume\":\"29 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-11-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Siberian Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s0037446623060046\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0037446623060046","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Openness and Discreteness of Mappings of Finite Distortion on Carnot Groups
We prove that a mapping of finite distortion
\( f:\Omega\to 𝔾 \) in a domain \( \Omega \)
of an \( H \)-type Carnot group \( 𝔾 \)
is continuous, open, and discrete provided that
the distortion function \( K(x) \) of \( f \) belongs to \( L_{p,\operatorname{loc}}(\Omega) \)
for some \( p>\nu-1 \).
In fact, the proof is suitable for each Carnot group
provided it has a \( \nu \)-harmonic function of the
form \( \log\rho \), where the homogeneous norm
\( \rho \) is \( C^{2} \)-smooth.
期刊介绍:
Siberian Mathematical Journal is journal published in collaboration with the Sobolev Institute of Mathematics in Novosibirsk. The journal publishes the results of studies in various branches of mathematics.