Sobolev and Schatten Estimates for the Complex Green Operator on Spheres

TL;DR

This paper establishes Schatten and Sobolev estimates for the complex Green operator on spheres, linking spectral properties of the Kohn-Laplacian and Laplace-Beltrami eigenvalues to analyze regularity and compactness.

Contribution

It provides the first known Schatten and Sobolev estimates for the Green operator on CR spheres, connecting spectral theory with geometric analysis.

Findings

Schatten estimates for the Green operator on spheres

Sobolev estimates derived from eigenvalue asymptotics

Spectral analysis links to regularity properties

Abstract

The complex Green operator $\mathcal{G}$ on CR manifolds is the inverse of the Kohn-Laplacian $\square_b$ on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for $\mathcal{G}$ on the unit sphere $\mathbb{S}^{2n-1}\subset \mathbb{C}^n$. We obtain these estimates by using the spectrum of $\square_b$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator.