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

Sobolev and Schatten Estimates for the Complex Green Operator on Spheres