Closed range property of on nonpseudoconvex domain

The problem on strictly pseudoconvex domains was solved by Kohn Later, H6rmander 6 solved the problem on pseudoconvex domains by introducing weighted norms, which bypassed the question of boundary regularity. On domains that are not necessarily pseudoconvex the 0 problem on (p, q) forms can still be solved if the Levi-form has n -q positive eigenvalues at every point on the boundary (2, 4, 6, 8 and others). There remains the question of solving the problem when the n q eigenvalues of the Levi-form are allowed to be zero. In a previous paper The above condition of weak q-convexity on the one hand makes the improvement of allowing the eigenvalues of the Levi-form to be zero, but on the other hand it requires the sum of the eigenvalues instead of the individual eigenvalue to be nonnegative. Hence, by considering the case of (p, n 1) forms, we can see that the weak q-convexity is not the optimal condition imposed on the Levi-form that we can solve .I n this paper we try to further improve the requirement on the Levi-form. For (p, n 1) forms we only require that at every point on the boundary there is one holomorphic vector field whose Levi-form is nonnegative. This is the expected minimal condition. However, we can only prove that 0 has closed range. We also attempt to extend the result to (p, q) forms, with a partial success. We assume that the Levi-form can be split into blocks with an (n q) minor being positive semi-definite. Adding some extra assumptions we prove similarly that 0 has closed range.