Schatten and Sobolev Estimates for Green Operators on Compact Heisenberg Manifolds

Let $M = \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We give Schatten and Sobolev estimates for the Green operator $\mathcal{G}_\alpha$ associated to a fixed element of a family of second order differential operators $\left\{ \mathcal{L}_\alpha \right\}$ on $M$. In particular, it follows that the Kohn Laplacian on functions on $M$ is subelliptic. Our main tool is Folland's description of the spectrum of $\mathcal{L}_\alpha$