The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
Author:
Xie Xiangyun1, Liu Yu1, Li Pengtao2, Huang Jizheng3
Affiliation:
1. School of Mathematics and Physics, University of Science and Technology Beijing , Beijing 100083 , China 2. College of Mathematics, Qingdao University , Qingdao , Shandong 266071 , China 3. School of Science, Beijing University of Posts and Telecommunications , Beijing 100876 , P.R. China
Abstract
Abstract
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.
Publisher
Walter de Gruyter GmbH
Reference45 articles.
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