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I recall (an easy case of) the first Cousin problem :

Let $\Omega_1, \Omega_2$ be two open subsets of the complex plane $\mathbb{C}$ with non-empty intersection and $f$ be holomorphic on $\Omega_1 \cap \Omega_2$.Are there two functions $f_1, f_2$ such that $f_1$ holomorphic on $\Omega_1$, $f_2$ holomorphic on $\Omega_2$ and $f=f_1-f_2$ on $\Omega_1 \cap \Omega_2$?

The answer is yes.

Now, let me state the same problem for the Bergman spaces. I recall that if $\Omega \subset \mathbb{C}$ is open, we say that $f$ belongs to the Bergman space $A^2(\Omega)$ if $f$ is holomorphic on $\Omega$ and $\int_\Omega |f(x+iy)|^2 dx dy <\infty$. So, the problem becomes :

Let $\Omega_1, \Omega_2$ be two open subsets of the complex plane $\mathbb{C}$ with non-empty intersection and $f \in A^2\left(\Omega_1 \cap \Omega_2 \right)$. Are there two functions $f_1, f_2$ such that $f_1 \in A^2\left(\Omega_1\right)$, $f_2 \in A^2\left(\Omega_2 \right)$ and $f=f_1-f_2$ on $\Omega_1 \cap \Omega_2$?

My question is : Is there any work on this topic?

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