Heat-Semigroup-Based Besov Capacity on Dirichlet Spaces and Its Applications

In this paper, we investigate the Besov space and the Besov capacity and obtain several important capacitary inequalities in a strictly local Dirichlet space, which satisfies the doubling condition and the weak Bakry-emery condition. It is worth noting that the capacitary inequalities in this paper are proved if the Dirichlet space supports the weak (1,2)-Poincare inequality, which is weaker than the weak (1,1)-Poincare inequality investigated in the previous references. Moreover, we first consider the strong subadditivity and its equality condition for the Besov capacity in metric space.