{"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":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":3.2000,"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\":51301,\"journal\":{\"name\":\"Advances in Nonlinear Analysis\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Nonlinear Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/anona-2023-0119\",\"RegionNum\":1,\"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":"Advances in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/anona-2023-0119","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","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:
Advances in Nonlinear Analysis (ANONA) aims to publish selected research contributions devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The Journal focuses on papers that address significant problems in pure and applied nonlinear analysis. ANONA seeks to present the most significant advances in this field to a wide readership, including researchers and graduate students in mathematics, physics, and engineering.