Abstract
This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we define a condition which is called (H-q) condition which is related to the Levi form on the complex manifold. Under the (H-q) condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the L-2(partial derivative) over bar Cauchy problems on domains in C-n, Kahler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted L-2 (partial derivative) over bar Cauchy problem is studied under the same condition in a Kahler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the L-2 theory for the (partial derivative) over bar -operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy (Hn-q-1) and an outer domain which satisfy (H-q).