I am studying complex analysis. In Krantz's book : "Function theory in several complex variables " page 209 : the author notices that $H^p(\mathcal U,\mathcal E^q) = 0$ for every $p>0$ in the proof of theorem 6.3.1. A hint is to use the following theorem :
Let $X$ be a paracompact manifold, $\mathcal F$ is a sheaf over $X$ and $\mathcal U = $ {$U_i$}$_{i\in I}$ an open covering on $X$ such that $H^p(U_{i_0}\cap...\cap U_{i_p},\mathcal F) = 0$ for every $p \geq 1$ and every choice of $i_0,...,i_p$. Then $H^p(\mathcal U,\mathcal F) = H^p(\mathcal X,\mathcal F)$ for all $p\geq 0$.
Here, $\Omega \subseteq \mathbb C^n$ is an open set and $\mathcal U$ is an open covering of $\Omega$ by domains of holomorphy (I also do not understand this mean all domains of holomorphy that have intersection with $\Omega$ or an arbitrary open covering by domains of holomorphy). $\mathcal E^q$ is the sheaf of germs of $(0,q)$ forms with $C^{\infty}$ coefficents.