{"title":"迪里希勒空间上的有界变化能力和索波列夫型不等式","authors":"Xiangyun Xie, Yu Liu, Pengtao Li, Jizheng Huang","doi":"10.1515/anona-2023-0119","DOIUrl":null,"url":null,"abstract":"\n In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak \n \n \n \n \n (\n \n 1\n ,\n 2\n \n )\n \n \n \\left(1,2)\n \n -Poincaré inequality instead of the weak \n \n \n \n \n (\n \n 1\n ,\n 1\n \n )\n \n \n \\left(1,1)\n \n -Poincaré inequality compared with the results in the previous references.","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":"37 9","pages":""},"PeriodicalIF":4.3000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces\",\"authors\":\"Xiangyun Xie, Yu Liu, Pengtao Li, Jizheng Huang\",\"doi\":\"10.1515/anona-2023-0119\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak \\n \\n \\n \\n \\n (\\n \\n 1\\n ,\\n 2\\n \\n )\\n \\n \\n \\\\left(1,2)\\n \\n -Poincaré inequality instead of the weak \\n \\n \\n \\n \\n (\\n \\n 1\\n ,\\n 1\\n \\n )\\n \\n \\n \\\\left(1,1)\\n \\n -Poincaré inequality compared with the results in the previous references.\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":\"37 9\",\"pages\":\"\"},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/anona-2023-0119\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0119","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak
(
1
,
2
)
\left(1,2)
-Poincaré inequality instead of the weak
(
1
,
1
)
\left(1,1)
-Poincaré inequality compared with the results in the previous references.
期刊介绍:
ACS Applied Electronic Materials is an interdisciplinary journal publishing original research covering all aspects of electronic materials. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials science, engineering, optics, physics, and chemistry into important applications of electronic materials. Sample research topics that span the journal's scope are inorganic, organic, ionic and polymeric materials with properties that include conducting, semiconducting, superconducting, insulating, dielectric, magnetic, optoelectronic, piezoelectric, ferroelectric and thermoelectric.
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