[proofplan]
We prove the stated qualitative division theorem as a direct corollary of the established quantitative Skoda weighted $L^2$ division estimate, in the form proved by Skoda from the Hörmander $L^2$ existence theorem and the Skoda curvature inequality. The first step fixes the vector-valued notation and the convention for the singular powers of $|g|$. The quantitative estimate is then applied to the given [holomorphic function](/page/Holomorphic%20Function) $f$, and its finite weighted bound immediately gives holomorphic functions $h_1,\dots,h_m$ satisfying $f=\sum_{j=1}^m g_jh_j$ with the required integrability.
[/proofplan]
[step:Declare the vector-valued holomorphic data and the weighted Hilbert norm]
Let
\begin{align*}
g: \Omega &\to \mathbb{C}^m \\
z &\mapsto (g_1(z),\dots,g_m(z))
\end{align*}
be the holomorphic map determined by the given functions, and define the measurable function $|g|: \Omega \to [0,\infty)$ by
\begin{align*}
|g(z)|^2=\sum_{j=1}^m |g_j(z)|^2.
\end{align*}
For every real number $s>0$, the measurable function $|g|^{-2s}:\Omega\to[0,\infty]$ is understood with the convention that $|g(z)|^{-2s}=+\infty$ at points $z\in\Omega$ where $g_1(z)=\cdots=g_m(z)=0$.
For a holomorphic map
\begin{align*}
h: \Omega &\to \mathbb{C}^m \\
z &\mapsto (h_1(z),\dots,h_m(z)),
\end{align*}
write
\begin{align*}
|h(z)|^2=\sum_{j=1}^m |h_j(z)|^2.
\end{align*}
Set
\begin{align*}
I_f:=\int_\Omega |f|^2 |g|^{-2(\alpha q+1)} e^{-\psi}\,d\mathcal L^{2n}.
\end{align*}
The hypothesis says precisely that $I_f<\infty$.
[/step]
[step:Apply the established quantitative Skoda weighted $L^2$ estimate]
We invoke the quantitative Skoda weighted $L^2$ division estimate as an external analytic input; in Skoda's original notation this is Théorème 1 of H. Skoda, "Application des techniques $L^2$ à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids", Annales scientifiques de l'École Normale Supérieure 5 (1972), 545-579. The estimate says: if $\Omega\subset\mathbb{C}^n$ is pseudoconvex, $g_1,\dots,g_m\in\mathcal O(\Omega)$, $q=\min\{n,m-1\}$, $\alpha>1$, and $\psi$ is plurisubharmonic, then every $F\in\mathcal O(\Omega)$ satisfying
\begin{align*}
\int_\Omega |F|^2 |g|^{-2(\alpha q+1)}e^{-\psi}\,d\mathcal L^{2n}<\infty
\end{align*}
admits holomorphic functions $H_1,\dots,H_m\in\mathcal O(\Omega)$ such that
\begin{align*}
F=\sum_{j=1}^m g_jH_j
\end{align*}
and
\begin{align*}
\int_\Omega |H|^2 |g|^{-2\alpha q}e^{-\psi}\,d\mathcal L^{2n}
\leq \frac{\alpha}{\alpha-1}
\int_\Omega |F|^2 |g|^{-2(\alpha q+1)}e^{-\psi}\,d\mathcal L^{2n},
\end{align*}
where $H: \Omega\to\mathbb C^m$ is the holomorphic map $H(z)=(H_1(z),\dots,H_m(z))$.
All hypotheses of this estimate are exactly the hypotheses in the theorem: $\Omega$ is pseudoconvex, $g_1,\dots,g_m$ and $f$ are holomorphic on $\Omega$, $\psi$ is plurisubharmonic, $q=\min\{n,m-1\}$, $\alpha>1$, and the preceding step gives the required finite integral with $F=f$.
[guided]
The proof needs a division theorem with an estimate, not merely an algebraic assertion. We use the quantitative Skoda weighted $L^2$ division estimate as a previously established analytic result, specifically Théorème 1 of H. Skoda, "Application des techniques $L^2$ à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids", Annales scientifiques de l'École Normale Supérieure 5 (1972), 545-579. That result is stronger than the present statement because it gives an explicit weighted norm bound for the solution tuple.
Let $F:\Omega\to\mathbb C$ be the [holomorphic function](/page/Holomorphic%20Function) to which the estimate is applied. The estimate requires a pseudoconvex domain $\Omega\subset\mathbb C^n$, holomorphic functions $g_1,\dots,g_m\in\mathcal O(\Omega)$, the integer $q=\min\{n,m-1\}$, a real parameter $\alpha>1$, a plurisubharmonic function $\psi:\Omega\to[-\infty,\infty)$, and finiteness of
\begin{align*}
\int_\Omega |F|^2 |g|^{-2(\alpha q+1)}e^{-\psi}\,d\mathcal L^{2n}.
\end{align*}
We verify these inputs in the present theorem. The domain $\Omega$ is pseudoconvex by hypothesis. The functions $g_1,\dots,g_m$ and $f$ are holomorphic on $\Omega$ by hypothesis. The parameter satisfies $\alpha>1$ by hypothesis, the integer $q$ is exactly defined as $q=\min\{n,m-1\}$, and the weight $\psi$ is plurisubharmonic by hypothesis. Finally, the preceding step defined
\begin{align*}
I_f=\int_\Omega |f|^2 |g|^{-2(\alpha q+1)} e^{-\psi}\,d\mathcal L^{2n},
\end{align*}
and the theorem assumes $I_f<\infty$. Thus the quantitative Skoda estimate applies with $F=f$.
Applying the estimate with $F=f$ gives holomorphic functions $H_1,\dots,H_m\in\mathcal O(\Omega)$ satisfying
\begin{align*}
f=\sum_{j=1}^m g_jH_j
\end{align*}
and the quantitative bound
\begin{align*}
\int_\Omega |H|^2 |g|^{-2\alpha q}e^{-\psi}\,d\mathcal L^{2n}
\leq \frac{\alpha}{\alpha-1}
\int_\Omega |f|^2 |g|^{-2(\alpha q+1)}e^{-\psi}\,d\mathcal L^{2n},
\end{align*}
where $H:\Omega\to\mathbb C^m$ is the holomorphic map defined by $H(z)=(H_1(z),\dots,H_m(z))$. Since $\alpha>1$, we have
\begin{align*}
\frac{\alpha}{\alpha-1}<\infty.
\end{align*}
This is the point at which the strict inequality $\alpha>1$ is used quantitatively.
[/guided]
[/step]
[step:Rename the Skoda solution and verify the required conclusion]
Define $h_j:=H_j$ for each $j\in\{1,\dots,m\}$, and define
\begin{align*}
h: \Omega &\to \mathbb C^m \\
z &\mapsto (h_1(z),\dots,h_m(z)).
\end{align*}
Then $h_1,\dots,h_m\in\mathcal O(\Omega)$, and the division identity becomes
\begin{align*}
f=\sum_{j=1}^m g_jh_j.
\end{align*}
Moreover,
\begin{align*}
\int_\Omega |h|^2 |g|^{-2\alpha q}e^{-\psi}\,d\mathcal L^{2n}
&\leq \frac{\alpha}{\alpha-1} I_f \\
&<\infty,
\end{align*}
because $I_f<\infty$ and $\alpha/(\alpha-1)<\infty$. This is exactly the weighted $L^2$ integrability asserted in the theorem.
[/step]
