The union problem on complex manifolds (Q1608212)

Let \(X\) be an \(n\)-dimensional complex manifold with a \(C^\infty\) Hermitian metric. Let \(\Omega_1\Subset \Omega_2\Subset\cdots\) be an increasing sequence of Stein open sets in \(X\) such that their union \(\Omega= \bigcup^\infty_{j=1} \Omega_j\) is relatively compact in \(X\). The following statement is the main result of the paper. Theorem 1. The union \(\Omega\) is Stein if and only if given an \(f\in L^2_{(p,q)}(\Omega)\) \(\overline\partial\)-closed, there is a \(u\in L^2_{(p,q-1)}(\Omega)\) with \(\overline\partial u= f\) and such that \[ \|u\|_{L^2(p,q-1)(\Omega)}\leq K\|f\|_{L^2(p,q)(\Omega)}, \] where \(K\) depends only on \(\Omega\), but not on \(f\).