Let X be a Stein manifold of complex dimension n\geqslant 2 and \Omega \Subset X be a relatively compact domain with C^{2} smooth boundary in X. Assume that Ω is a weakly q-pseudoconvex domain in X. The purpose of this paper is to establish sufficient conditions for the closed range of \overline \partial on Ω. Moreover, we study the \overline \partial -problem on Ω. Specifically, we use the modified weight function method to study the weighted \overline \partial -problem with exact support in Ω. Our method relies on the L^{2} -estimates by Hörmander (1965) and by Kohn (1973)., Sayed Saber., and Obsahuje seznam literatury