# First Cousin Problem for Bergman spaces

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?