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