Several functional capacities and Carleson type embeddings of fractional Sobolev sapces on stratified Lie groups

In this paper, we focus on the functional and geometrical aspects of the fractional Sobolev capacity, the Besov capacity and the Riesz capacity on stratified lie groups, respectively. Firstly, we provide a new Carleson characterization of the extension of fractional Sobolev spaces to $L^q(\text{X}\times {\mathbb{R}}_{+},\mu )$ with $q\in {\mathbb{R}}_{+}$ using the fractional heat semigroup and the Caffarelli-Silvestre type extension on stratified Lie groups $\text{X}$. Secondly, a characterization of $\nu$ on $\text{X}$ which ensures the continuity of the fractional Sobolev space belonging to $L^q(\text{X},\nu )$ is also obtained via taking $t\to 0$. Finally, with the help of inequalities related to the Besov capacity and its properties, we also obtain a characterization of $\nu$ on $\text{X}$ which ensures the continuity of the Besov type space belonging to $L^q(\text{X},\nu )$.