Decomposition of analytic functions Given two open overlapping sets $\Omega_1$, $\Omega_2$ and an analytic function $f$ on $\Omega_1\cap\Omega_2$, how does one prove that there are analytic functions $g_1$ on $\Omega_1$ and $g_2$ on $\Omega_2$ such that $f=g_1+g_2$ on $\Omega_1\cap\Omega_2$ ?
 A: Take a smooth function $\rho:\Omega_1\cup\Omega_2\to[0,1]$ such that $\rho\equiv 0$ outside $\Omega_1$ and $\rho\equiv 1$ outside $\Omega_2$.
Now, the function $f\rho$ extends to $\Omega_2$ and the function $f(1-\rho)$ extends to $\Omega_1$, both as smooth functions; we would like to modify them so that the sum remains unchanged but they both become holomorphic.
The first condition implies that we want to define
$$g_1=u+f(1-\rho)$$
$$g_2=-u+f\rho$$
for a smooth function $u$ defined on $\Omega_1\cup\Omega_2$.
The second condition implies that $\overline{\partial}g_1=\overline{\partial}g_2=0$; i.e.
$$\overline{\partial}u=f\overline{\partial}\rho$$
we note that $f\overline{\partial\rho}$ is a well define $(0,1)$-form on $\Omega_1\cup\Omega_2$.
Solving the $\overline{\partial}$-equation, we get a smooth function $u$ which solves our problem, so that $g_1$ and $g_2$ become holomorphic.
NB: This is just an adaptation of the solution of the Cousin I problem to the case with only $2$ open sets in $\mathbb{C}$ (where on every open domain we can solve the Cauchy-Riemann equation). So, if you assume to know that Cousin I problem has a holomorphic solution, the whole exercise becomes trivial.
