I'm following Otto Forster's notation in his book Lectures on Riemann Surfaces. Let $X$ be a noncompact Riemann surface. $\mathcal{E}(X)$ is the sheaf of infinitely differentiable functions on $X$, $\mathcal{E}^{1}(X)$ is the sheaf of smooth 1-forms and $\mathcal{E}^{2}(X)$ is the sheaf of smooth 2-forms on $X$. I'm trying to do problem 26.2(a) in the book. Given any $\omega \in \mathcal{E}^{2}(X)$ I want to show that there exists $f \in \mathcal{E}(X)$ such that $d'd''f = \omega$.
I already know that if $\gamma$ is an antiholomorphic 1-form then there exists $g \in \mathcal{E}(X)$ such that $d''g = \gamma$ and I want to use that, but I'm not sure how.
Edit: $d'': \mathcal{E}(X) \to \mathcal{E}^{1}(X)$ and $d': \mathcal{E}^{1}(X) \to \mathcal{E}^{2}(X)$. I'll mention what they do locally on charts even though they are global. If $(U,z)$ is a chart then $d''f = \frac{\partial f}{\partial \overline{z}} d\bar{z}$ on $U$ and $d'(fdz + g d\bar{z}) = \frac{\partial g}{\partial z} dz \wedge d \bar{z}$. This is what they do locally and I think that's what's important (and easier to state).