Refined Strichartz Estimates for sub-Laplacians in Heisenberg and $H$-type groups

We obtain refined Strichartz estimates for the sub-Riemannian Schr\"{o}dinger equation on $H$-type Carnot groups using Fourier restriction techniques. In particular, we extend the previously known Strichartz estimates previously obtained for the Heisenberg group also to non radial initial data. The same arguments permits to obtain refined Strichartz estimates for the wave equation on $H$-type groups. Our proof is based on estimates for the spectral projectors for sub-Laplacians and reinterprets Strichartz estimates as Fourier restriction theorems for nilpotent groups in the context of trace-class operator valued measures.