Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and motivate it. However, the question itself can be understood without reading the preceding paragraphs.
I was reading the chapter about Cousin problems in Krantz's "Function theory of several complex variables" and I was trying to understand the proof of the following fact, which according to the author contains the main ideas that are developed in the chapter.
Proposition. Let $\Omega\subseteq\mathbb{C}^n$ be pseudoconvex. Assume that $0\in\Omega$. If $f:\Omega\to\mathbb{C}$ is a holomorphic function that satisfies $f(0)=0$ , then there are holomorphic functions $f_1, f_2$ on $\Omega$ such that $f(z) = z_1f_1(z) + z_2f_2(z)$ on $\Omega$.
I think I can follow the proof proposed by the author (even though I find it unnecessarily convoluted, since it seems to me that the Cousin-like construction adopted in the proof is not essential and could be avoided), but I have some trouble understanding a crucial point.
The domain $\Omega$ is covered by open subsets $U_i$ and in the first part of the proof the author defines a family of smooth functions $h_1^i, h_2^i:U_i\to\mathbb{C}$ such that
- $z_1h_1^i(z) + z_2h_2^i(z) = 0$ on $U_i$;
- $\overline\partial h_1^j = \overline\partial h_1^i$ on $U_i\cap U_j$;
- $\overline\partial h_2^j = \overline\partial h_2^i$ on $U_i\cap U_j$.
Then the 1-forms $\alpha_1$ and $\alpha_2$ are defined on $\Omega$ imposing $\alpha_1 = \overline\partial h_1^i$ on each subset $U_i$ and similarly for $\alpha_2$. This makes sense because of the properties 2. and 3. above.
The following step of the proof, which is the one I would like to understand, consists in finding two smooth functions $h_1, h_2:\Omega\to\mathbb{C}$ such that $\overline\partial h_1 = \alpha_1$, $\overline\partial h_2 = \alpha_2$ and $z_1h_1 + z_2h_2 = 0$. I know that Hörmander's theorem implies the existence of $h_1$ and $h_2$ such that $\overline\partial h_1 = \alpha_1$ and $\overline\partial h_2 = \alpha_2$, but it is not necessarily true that $z_1h_1 + z_2h_2 = 0$. In fact, one has a lot of choices for such $h_1$ and $h_2$, and it is clear that for most of them $z_1h_1 + z_2h_2 = 0$ does not hold. So some clever strategy should be used to choose the right ones.
If I am reading the proof correctly, the author uses Hörmander's theorem to find $h_1$ and then defines $h_2(z) = -\frac{z_1h_1(z)}{z_2}$. But then I don't see why $h_2$ should be smooth; in fact I am pretty sure it is not, if $h_1$ is has not been chosen carefully.
I think the problem can be solved with the following plan:
- Note that $z_1\alpha_1 + z_2\alpha_2 = 0$ on $\Omega$.
- Write $\alpha_1 = z_2\beta_1$ and $\alpha_2 = z_1\beta_2$ for some smooth 1-forms $\beta_1$ and $\beta_2$.
- Apply Hörmander's theorem to $\beta_1$ and $\beta_2$.
However, I am not entirely sure on how to realize the second point. The fact that we are dealing with 1-forms instead of functions should be irrelevant. Also, the domain $\Omega$ does not play an important role. So the real question is:
Question. Let $f,g:\mathbb{C}^2\to\mathbb{C}$ be two smooth functions such that $z_1\ f(z_1, z_2) = z_2\ g(z_1, z_2)$ on $\mathbb{C}^2$. Is it true that there is a smooth function $h:\mathbb{C}^2\to\mathbb{C}$ such that $f(z_1,z_2) = z_2\ h(z_1, z_2)$?
Edit. Let me add that I have found a way around the issues in the proof of the proposition (basically, it is possible to make sure that $\alpha_1$ and $\alpha_2$ vanish in a neighborhood of $0$, and then the "plan" i have described becomes easier to realize), but I would like to know if the question has a positive answer in general.
