On the compactness of the product of Hankel operators on the sphere

Consider Hankel operators H phi and H psi on the unit sphere in C-n. If n = 1, then a necessary condition for H-phi*H-psi to be compact is lim(vertical bar z vertical bar up arrow 1)parallel to H(phi)k(z)parallel to parallel to H(psi)k(z)parallel to = 0. We show that when n >= 2, this condition is no longer necessary for H-phi*H psi to be compact.