The optimal constant in the $L^2$ Folland-Stein inequality on the H-type group

We determine the optimal constant in the $L^2$ Folland-Stein inequality on the H-type group, which partially confirms the conjecture given by Garofalo and Vassilev (Duke Math. J., 2001). The proof is inspired by the work of Frank and Lieb (Ann. of Math., 2012) and Hang and Wang.