[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 $F=f$.[/step]

[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: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]