Besov Space via Heat Semigroup on Carnot Group and Its Capacity

In this paper, we first introduce a class of Besov spaces via the heat semigroup on Carnot groups, and provide some basic properties about these spaces. Especially, we reveal the relationship between the Besov space and the known Sobolev space on Carnot groups. Then some Sobolev type inequalities for Besov spaces are established. Finally, we introduce the Besov capacity associated with Besov spaces and investigate some measure theories and geometric properties for this capacity. Furthermore, several applications of the Besov capacity are also obtained.