Schatten and Sobolev Estimates for Green Operators on Compact Heisenberg Manifolds

Let $M = Γ\setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $Γ$. We give Schatten and Sobolev estimates for the Green operator $\mathcal{G}_α$ associated to a fixed element of a family of second order differential operators $\left\{ \mathcal{L}_α\right\}$ 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 $\mathcal{L}_α$.

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