Schatten and Sobolev Estimates for Green Operators on Compact Heisenberg Manifolds

Let $M=\mathrm{\Gamma }\setminus {\mathbb{H}}_d$ be a compact quotient of the $d$-dimensional Heisenberg group ${\mathbb{H}}_d$ by a lattice subgroup $\mathrm{\Gamma }$. 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 ${\mathcal{L}}_{\alpha }$.