  The purpose of this paper is to establish local regularity of the solution
operator to the Kohn-Laplace equation, called the complex Green operator, on
abstract CR manifolds of hypersurface type. For a cut-off function $\sigma$, we
introduce the $\sigma$-superlogarithmic property, a potential theoretical
condition on CR manifolds. We prove that if the given datum is smooth on an
open set containing the support of $\sigma$ then the solution is smooth on the
interior of $\{x\in M:\sigma(x)=1\}$. Furthermore, we also study the smoothness
of the integral kernel of the complex Green operator.
