Sharp fractional Sobolev and related inequalities on H-type groups

We determine the sharp constants for the fractional Sobolev inequalities associated with the conformally invariant fractional powers ${\mathcal{L}}_s(0<s<1)$ of the sublaplacian on H-type groups. From these inequalities we derive a sharp log-Sobolev inequality by considering a limiting case and a sharp Sobolev trace inequality. The later extends to this context the result of Frank, González, Monticelli and Tan (Adv. Math, 2015).