The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces

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.

Keywords

• bv space
• dirichlet space
• heat kernel
• capacity
• sobolev inequality
• 28a12
• 31e05
• 26b30