Schatten and Sobolev Estimates for Green Operators on Compact Heisenberg Manifolds

TL;DR

This paper establishes Schatten and Sobolev estimates for Green operators on compact Heisenberg manifolds, demonstrating subellipticity of the Kohn Laplacian using spectral analysis.

Contribution

It provides new Schatten and Sobolev estimates for Green operators on compact Heisenberg manifolds, leveraging Folland's spectral description.

Findings

Green operators satisfy Schatten and Sobolev estimates

Kohn Laplacian is subelliptic on these manifolds

Spectral analysis via Folland's description is key

Abstract

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 $\mathcal{L}_\alpha$.