The following question references Forster's Lectures on Riemann Surfaces. It is shown that if $g \in \mathcal{E}(\mathbb{C})$ is of compact support, then there is a solution $f \in \mathcal{E}(\mathbb{C})$ of the equation $\partial f/\partial \bar{z} = g$ having compact support if and only if $$\int \int_{\mathbb{C}} z^n g(z)dz \wedge d\bar{z} = 0 \tag 1$$ for all $n \in \mathbb{N}$. Here, $\mathcal{E}(\mathbb{C})$ denotes the $\mathbb{C}$-algebra of functions differentiable with respect to the coordinates $x$ and $y$, for $z=x+iy.$ Let $\omega$ be a smooth one-form such that $\omega \in \Omega_c^{1}(X)$, where $\Omega_c^{1}(X)$ denotes the space of continuous one-forms with compact support. $X$ is a Riemann surface with boundary $\partial X$ consisting of piecewise smooth curves, for $\omega$ defined on a neighborhood of the closure of $X$.
If $\omega(z)$ has compact support on $X$, then is there a condition similar to $(1)$ that can be imposed, such that there is a solution $\sigma \in \Omega_c^{1}(X)$ of the equation $\partial \sigma/\partial \bar{z} = \omega$ having compact support if and only if this condition is satisfied?
Proposition: Let $\pi$ be a conformal diffeomorphism $\pi:X \to U,$ then for $\sigma=g(z)dz$, $$\int \int_{\mathbb{C}} z^n g(z)dz \wedge d\bar{z} =\int \int_{\pi^{-1}(\mathbb{C})=X} \pi^{*}(z^n \sigma \wedge d\bar{z}) \equiv \int \int_{X} \pi^{*}(z^n \sigma) \wedge \pi^{*}d\bar{z}.$$ It follows that if $\sigma \in \Omega_c^{1}(\mathbb{C})$ is of compact support, then there is a solution $\omega \in \Omega_c^{1}(\mathbb{C})$ of the equation $\partial \omega/ \partial \bar z=\sigma$ having compact support if and only if $$\int \int_{X} \pi^{*}(z^n \sigma) \wedge \pi^{*}d\bar{z}=0$$ for all $n \in \mathbb{N}.$
Thank you in advance.