Restriction problems on the three-dimensional Heisenberg nilmanifold

In this paper, we prove a spectral restriction theorem on the three-dimensional Heisenberg nilmanifold. Since this manifold is an ${\mathbb{S}}^1$-bundle over the flat torus ${\mathbb{T}}^2$, the result provides a sub-elliptic counterpart of Zygmund’s restriction theorem on ${\mathbb{T}}^2$ \cite{zygmund}. We also establish its sharpness by means of the discrete short-time Fourier transform.