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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 orderdifferential operators $\left\{ \mathcal{L}_\alpha \right\}$ on $M$. Inparticular, it follows that the Kohn Laplacian on functions on $M$ issubelliptic. Our main tool is Folland's description of the spectrum of$\mathcal{L}_\alpha$.

Schatten and Sobolev Estimates for Green Operators on Compact Heisenberg Manifolds