ON REGULARITY OF (sic)partial derivative-SOLUTIONS ON aq DOMAINS WITH C2 BOUNDARY IN COMPLEX MANIFOLDS

We study regularity of solutions u to (sic)partial derivative u = f on a relatively compact C (2) domain D in a complex manifold of dimension n , where f is a (0, q ) form. Assume that there are either (q + 1) negative or (n - q ) positive Levi eigenvalues at each point of boundary partial derivative D . Under the necessary condition that a locally L (2) solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain 1/2 derivative when q = 1 and f is in the H & ouml;lder-Zygmund space Lambda (R)(D) with r > 1. For q > 1, the same regularity for the solutions is achieved when partial derivative D is either sufficiently smooth or of (n - q ) positive Levi eigenvalues everywhere on partial derivative D .