Sobolev and Schatten Estimates for the Complex Green Operator on Spheres

The complex Green operator $\mathcal{G}$ on CR manifolds is the inverse of the Kohn-Laplacian ${\mathrm{\square }}_b$ 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 ${\mathrm{\square }}_b$ and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator.