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.