  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.
