The complex Green operator G on CR manifolds is the inverse of the Kohn-Laplacian □_b on the orthogonal complement of its kernel. In this note, we prove Schatten and Sobolev estimates for G on the unit sphere S^2n-1 in Cⁿ. We obtain these estimates by using the spectrum of □_b and the asymptotics of the eigenvalues of the usual Laplace-Beltrami operator.

This work is supported by the NSF (DMS-1659203). The work of the fourth author is also partially supported by a grant from the Simons Foundation (#353525).