TITLE: $\partial \bar{\partial}$ lemma for contractible domains
QUESTION [10 upvotes]: Question. Is every $(p, \, p)$ closed form ($p\geq1$) in a contractible open set of $\mathbb{C}^n$ $\partial \bar{\partial}$ exact? 

We know that every $d$-exact $(p, \,p)$-form on a compact Kahler manifold is $\partial \bar{\partial}$ exact (by the Hodge theorem), but unfortunately, that can't be applied here...

REPLY [8 votes]: If in addition  you assume that your domain is pseudoconvex then by a theorem of
A.Aeppli what you want is true.The paper is titled :On the cohomology structure of Stein
manifolds 1965 (Proc.Conf.Complex Analysis(Minneapolis Minn 1964) pages 58 to 70 Springer,
Berlin.
  David's example indicates why one might need a hypothesis like pseudoconvexity.