[proofplan] The proof first separates the degenerate case in which the tuple $g=(g_1,\dots,g_p)$ vanishes identically on the chosen subdomain $\Omega'$; then the integral-closure estimate forces $f=0$ there and the zero coefficients solve the problem. In the nondegenerate case, the proof applies Skoda's $L^2$ division theorem to the [holomorphic function](/page/Holomorphic%20Function) $F=f^{q+1}$ on the relatively compact pseudoconvex subdomain $\Omega'$. The local integral-closure estimate gives enough control near the common zero set of $g$ to make Skoda's weighted integrability hypothesis finite for $\alpha>1$ close to $1$. Skoda's theorem then produces holomorphic coefficients $u_j$ and gives exactly the stated weighted estimate, and boundedness of $|g|$ on $\Omega'$ converts the weighted estimate into the unweighted $A^2(\Omega')$ membership. [/proofplan] [step:Fix the subdomain and introduce the tuple notation] Let $\Omega' \Subset \Omega$ be a relatively compact pseudoconvex domain. Define the holomorphic map \begin{align*} g:\Omega' &\to \mathbb{C}^p \\ z &\mapsto (g_1(z),\dots,g_p(z)), \end{align*} and define \begin{align*} F:\Omega' &\to \mathbb{C} \\ z &\mapsto f(z)^{q+1}. \end{align*} The function $F$ is holomorphic on $\Omega'$ because $f \in \mathcal{O}(\Omega)$ and $\Omega' \subset \Omega$. Since $\overline{\Omega'}$ is compact in $\Omega$, the local integral-closure hypothesis applied to $K=\overline{\Omega'}$ gives a constant $C_{\Omega'}>0$ such that \begin{align*} |f(z)| \leq C_{\Omega'} |g(z)| \qquad \text{for all } z \in \Omega'. \end{align*} Consequently, \begin{align*} |F(z)|^2 = |f(z)|^{2(q+1)} \leq C_{\Omega'}^{2(q+1)} |g(z)|^{2(q+1)} \qquad \text{for all } z \in \Omega'. \end{align*} [guided] We first restrict all objects to the pseudoconvex domain on which the division will be performed. The tuple of generators is treated as one holomorphic map \begin{align*} g:\Omega' &\to \mathbb{C}^p \\ z &\mapsto (g_1(z),\dots,g_p(z)), \end{align*} with squared norm \begin{align*} |g(z)|^2=\sum_{j=1}^p |g_j(z)|^2. \end{align*} We also define the [holomorphic function](/page/Holomorphic%20Function) to be divided by the tuple $g$: \begin{align*} F:\Omega' &\to \mathbb{C} \\ z &\mapsto f(z)^{q+1}. \end{align*} The integral-closure hypothesis is local on $\Omega$, so we must convert it into a uniform estimate on the chosen subdomain. Because $\Omega' \Subset \Omega$, its closure $\overline{\Omega'}$ is compact and contained in $\Omega$. Applying the hypothesis to $K=\overline{\Omega'}$ gives a constant $C_{\Omega'}>0$ such that \begin{align*} |f(z)| \leq C_{\Omega'} |g(z)| \qquad \text{for all } z \in \Omega'. \end{align*} Raising this inequality to the power $2(q+1)$ gives \begin{align*} |F(z)|^2 = |f(z)|^{2(q+1)} \leq C_{\Omega'}^{2(q+1)} |g(z)|^{2(q+1)} \qquad \text{for all } z \in \Omega'. \end{align*} This is the estimate that makes Skoda division applicable: it says that $F=f^{q+1}$ vanishes at least to the order required by the tuple $g$ near the common zero set of the $g_j$. [/guided] [/step] [step:Choose $\alpha>1$ so that Skoda's weighted integral is finite] Define \begin{align*} M_{\Omega'} := \sup_{z\in \Omega'} |g(z)|. \end{align*} If $M_{\Omega'}=0$, then $g_j=0$ on $\Omega'$ for every $j=1,\dots,p$. The estimate $|f(z)|\leq C_{\Omega'}|g(z)|$ then gives $f=0$ on $\Omega'$. In this case choose $u_j=0$ on $\Omega'$ for every $j=1,\dots,p$; the division identity holds and each $u_j$ belongs to $A^2(\Omega')$. The weighted estimate clause is not invoked in this degenerate case, because the ordinary Lebesgue integrands containing $|g|^{-2\alpha q}$ and $|g|^{-2(\alpha q+1)}$ are not defined when $|g|=0$ everywhere. Hence, for the rest of the proof, assume $M_{\Omega'}>0$, so the holomorphic tuple $g$ is not identically zero on $\Omega'$. Let $\alpha>1$. The weighted integrand appearing in Skoda's division theorem is \begin{align*} |F|^2 |g|^{-2(\alpha q+1)} = |f|^{2(q+1)} |g|^{-2(\alpha q+1)}. \end{align*} Using the estimate from the previous step, we obtain on $\Omega'$ \begin{align*} |F|^2 |g|^{-2(\alpha q+1)} \leq C_{\Omega'}^{2(q+1)} |g|^{2(q+1)-2(\alpha q+1)} = C_{\Omega'}^{2(q+1)} |g|^{-2q(\alpha-1)}. \end{align*} We use the local integrability theorem for small negative powers of a nonzero holomorphic tuple: if $h:V\to\mathbb{C}^r$ is holomorphic and not identically zero near a point of a complex domain $V$, then there exists $\varepsilon>0$ such that $|h|^{-2\varepsilon}\in L^1_{\mathrm{loc}}(V)$. Applying this result to $g:\Omega'\to\mathbb{C}^p$, for each point $a\in\overline{\Omega'}$ there are an open neighborhood $V_a\subset\Omega$ of $a$ and a number $\varepsilon_a>0$ such that $|g|^{-2\varepsilon_a}$ is integrable on $V_a\cap\Omega'$ with respect to $\mathcal L^{2m}$. Choose finitely many points $a_1,\dots,a_N\in\overline{\Omega'}$ with $\overline{\Omega'}\subset\bigcup_{\ell=1}^N V_{a_\ell}$ and set \begin{align*} \varepsilon_{\Omega'}:=\min_{1\leq \ell\leq N}\varepsilon_{a_\ell}>0. \end{align*} If $1<\alpha<1+\varepsilon_{\Omega'}/q$ when $q>0$, then $q(\alpha-1)<\varepsilon_{\Omega'}$, and the estimate above gives \begin{align*} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m} <\infty. \end{align*} If $q=0$, the same conclusion follows directly from the displayed integrand and the local integral-closure estimate, since the singular factor is then $|g|^{-2}$ and $|F|^2=|f|^2\leq C_{\Omega'}^2|g|^2$. [guided] Before discussing negative powers of $|g|$, we must exclude the one case where such powers are never locally integrable. Define \begin{align*} M_{\Omega'} := \sup_{z\in \Omega'} |g(z)|. \end{align*} If $M_{\Omega'}=0$, then $|g(z)|=0$ for every $z\in\Omega'$, so $g_j=0$ on $\Omega'$ for all $j=1,\dots,p$. The integral-closure estimate from the previous step gives $|f(z)|\leq C_{\Omega'}|g(z)|=0$, hence $f=0$ on $\Omega'$. Taking $u_j=0$ for every $j$ gives \begin{align*} f^{q+1}=0=\sum_{j=1}^p g_j u_j \qquad \text{on } \Omega', \end{align*} and each coefficient is in $A^2(\Omega')$. We do not write a weighted estimate in this case, because the expressions $|g|^{-2\alpha q}$ and $|g|^{-2(\alpha q+1)}$ are not ordinary [measurable functions](/page/Measurable%20Functions) when $|g|=0$ everywhere. Thus the weighted part of the theorem is handled only in the remaining case $M_{\Omega'}>0$, meaning the tuple $g$ is not identically zero on $\Omega'$. Skoda's theorem requires the finiteness of the weighted integral \begin{align*} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m}. \end{align*} After substituting $F=f^{q+1}$, the integrand is \begin{align*} |F|^2 |g|^{-2(\alpha q+1)} = |f|^{2(q+1)} |g|^{-2(\alpha q+1)}. \end{align*} The point of the integral-closure estimate is that it replaces the possible singularity of $f$ relative to $g$ by a controlled power of $|g|$. From the previous step, \begin{align*} |F|^2 \leq C_{\Omega'}^{2(q+1)} |g|^{2(q+1)}. \end{align*} Multiplying both sides by $|g|^{-2(\alpha q+1)}$ gives \begin{align*} |F|^2 |g|^{-2(\alpha q+1)} &\leq C_{\Omega'}^{2(q+1)} |g|^{2(q+1)} |g|^{-2(\alpha q+1)} \\ &= C_{\Omega'}^{2(q+1)} |g|^{2(q+1)-2(\alpha q+1)} \\ &= C_{\Omega'}^{2(q+1)} |g|^{-2q(\alpha-1)}. \end{align*} Thus the only remaining issue is whether the small negative power $|g|^{-2q(\alpha-1)}$ is integrable near the common zero set of $g_1,\dots,g_p$. The local integrability theorem for holomorphic tuples says the following: if $h:V\to\mathbb{C}^r$ is holomorphic and not identically zero near a point of a complex domain $V$, then some negative power $|h|^{-2\varepsilon}$ with $\varepsilon>0$ is locally integrable. We apply that theorem to \begin{align*} g:\Omega' &\to \mathbb{C}^p \\ z &\mapsto (g_1(z),\dots,g_p(z)). \end{align*} For every $a\in\overline{\Omega'}$ we obtain a neighborhood $V_a\subset\Omega$ and a number $\varepsilon_a>0$ such that $|g|^{-2\varepsilon_a}$ is integrable on $V_a\cap\Omega'$ with respect to $\mathcal L^{2m}$. Since $\overline{\Omega'}$ is compact, choose finitely many points $a_1,\dots,a_N$ with \begin{align*} \overline{\Omega'}\subset\bigcup_{\ell=1}^N V_{a_\ell}, \end{align*} and define \begin{align*} \varepsilon_{\Omega'}:=\min_{1\leq \ell\leq N}\varepsilon_{a_\ell}>0. \end{align*} If $q>0$ and $1<\alpha<1+\varepsilon_{\Omega'}/q$, then $q(\alpha-1)<\varepsilon_{\Omega'}$, so the preceding pointwise bound is integrable on each $V_{a_\ell}\cap\Omega'$ and hence on all of $\Omega'$. Therefore \begin{align*} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m} <\infty. \end{align*} If $q=0$, the singularity cancels directly from $|F|^2=|f|^2\leq C_{\Omega'}^2|g|^2$, so the same conclusion holds. The smallness condition on $\alpha-1$ is exactly why the statement only asserts the estimate for $\alpha>1$ chosen sufficiently close to $1$. [/guided] [/step] [step:Apply Skoda division to obtain holomorphic coefficients] Fix $\alpha>1$ sufficiently close to $1$ so that \begin{align*} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m}<\infty. \end{align*} We apply Skoda's $L^2$ division theorem in the following form: if $D\subset\mathbb{C}^m$ is pseudoconvex, $g_1,\dots,g_p\in\mathcal O(D)$, $Q=\min\{m,p-1\}$, $\alpha>1$, $\psi$ is plurisubharmonic on $D$, and $H\in\mathcal O(D)$ satisfies \begin{align*} \int_D |H|^2 |g|^{-2(\alpha Q+1)} e^{-\psi}\,d\mathcal L^{2m}<\infty, \end{align*} then there exist $v_1,\dots,v_p\in\mathcal O(D)$ with $H=\sum_{j=1}^p g_jv_j$ and \begin{align*} \int_D \sum_{j=1}^p |v_j|^2 |g|^{-2\alpha Q} e^{-\psi}\,d\mathcal L^{2m} \leq \frac{\alpha}{\alpha-1} \int_D |H|^2 |g|^{-2(\alpha Q+1)} e^{-\psi}\,d\mathcal L^{2m}. \end{align*} We use this theorem with $D=\Omega'$, $Q=q=\min\{m,p-1\}$, $H=F=f^{q+1}$, and $\psi=0$. The hypotheses are satisfied: $\Omega'$ is pseudoconvex by assumption, each $g_j$ is holomorphic on $\Omega'$, $F$ is holomorphic on $\Omega'$, the zero weight is plurisubharmonic, and the required weighted integral is finite by the previous step. Skoda's theorem yields holomorphic functions \begin{align*} u_1,\dots,u_p:\Omega' \to \mathbb{C} \end{align*} such that \begin{align*} F=\sum_{j=1}^p g_j u_j \qquad \text{on } \Omega', \end{align*} and \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} \le \frac{\alpha}{\alpha-1} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m}. \end{align*} Since $F=f^{q+1}$, this is precisely \begin{align*} f^{q+1}=\sum_{j=1}^p g_j u_j \qquad \text{on } \Omega', \end{align*} with \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} \le \frac{\alpha}{\alpha-1} \int_{\Omega'} |f|^{2(q+1)} |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m}. \end{align*} [guided] We now use the analytic division theorem that is designed for exactly this situation: Skoda's $L^2$ division theorem. In the form needed here, it says that if $D\subset\mathbb{C}^m$ is pseudoconvex, $g_1,\dots,g_p\in\mathcal O(D)$, $Q=\min\{m,p-1\}$, $\alpha>1$, $\psi$ is plurisubharmonic on $D$, and $H\in\mathcal O(D)$ satisfies \begin{align*} \int_D |H|^2 |g|^{-2(\alpha Q+1)} e^{-\psi}\,d\mathcal L^{2m}<\infty, \end{align*} then there are $v_1,\dots,v_p\in\mathcal O(D)$ such that \begin{align*} H=\sum_{j=1}^p g_jv_j \end{align*} and \begin{align*} \int_D \sum_{j=1}^p |v_j|^2 |g|^{-2\alpha Q} e^{-\psi}\,d\mathcal L^{2m} \leq \frac{\alpha}{\alpha-1} \int_D |H|^2 |g|^{-2(\alpha Q+1)} e^{-\psi}\,d\mathcal L^{2m}. \end{align*} We apply this theorem on $D=\Omega'$ with the holomorphic tuple $g_1,\dots,g_p$, the holomorphic dividend \begin{align*} F:\Omega' &\to \mathbb{C} \\ z &\mapsto f(z)^{q+1}, \end{align*} the parameter \begin{align*} q=\min\{m,p-1\}, \end{align*} and the plurisubharmonic weight $\psi=0$. Let us check the hypotheses. The domain $\Omega'$ is pseudoconvex by assumption. The functions $g_1,\dots,g_p$ are holomorphic on $\Omega'$ because they are holomorphic on the larger domain $\Omega$. The function $F=f^{q+1}$ is holomorphic on $\Omega'$ because powers of holomorphic functions are holomorphic. The weight $\psi=0$ is plurisubharmonic. Finally, the required weighted integral \begin{align*} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m} \end{align*} is finite by the preceding step. Skoda's theorem therefore produces holomorphic functions \begin{align*} u_1,\dots,u_p:\Omega' \to \mathbb{C} \end{align*} such that \begin{align*} F=\sum_{j=1}^p g_j u_j \qquad \text{on } \Omega', \end{align*} and such that the weighted $L^2$ estimate \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} \le \frac{\alpha}{\alpha-1} \int_{\Omega'} |F|^2 |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m} \end{align*} holds. Substituting back $F=f^{q+1}$ gives the division identity \begin{align*} f^{q+1}=\sum_{j=1}^p g_j u_j \qquad \text{on } \Omega', \end{align*} and the estimate becomes \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} \le \frac{\alpha}{\alpha-1} \int_{\Omega'} |f|^{2(q+1)} |g|^{-2(\alpha q+1)}\,d\mathcal L^{2m}. \end{align*} This is the effective part of the theorem: the coefficient functions are not merely shown to exist, but are produced with an explicit weighted norm bound. [/guided] [/step] [step:Deduce that the coefficients belong to $A^2(\Omega')$] The estimate from Skoda gives \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m}<\infty. \end{align*} Since each $g_j$ is holomorphic on a neighborhood of $\overline{\Omega'}$, the continuous function $|g|$ is bounded above on $\overline{\Omega'}$. Choose \begin{align*} M_{\Omega'} := \sup_{z\in \Omega'} |g(z)| < \infty. \end{align*} The case $M_{\Omega'}=0$ was already completed before the Skoda step, so here $M_{\Omega'}>0$. For every $z\in \Omega'$ with $|g(z)|>0$, \begin{align*} 1 \leq M_{\Omega'}^{2\alpha q}|g(z)|^{-2\alpha q}. \end{align*} At points where $|g(z)|=0$, the same inequality holds in the extended nonnegative sense. Therefore \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2\,d\mathcal L^{2m} \leq M_{\Omega'}^{2\alpha q} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} <\infty. \end{align*} Thus each $u_j$ is holomorphic and square-integrable on $\Omega'$, so \begin{align*} u_j \in A^2(\Omega') \qquad \text{for } j=1,\dots,p. \end{align*} The division identity and the weighted estimate have already been established, completing the proof. [guided] We still need to check that the holomorphic coefficients produced by Skoda are in the unweighted Bergman space $A^2(\Omega')$. Skoda gives the weighted estimate \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m}<\infty. \end{align*} Since each $g_j$ is holomorphic on a neighborhood of $\overline{\Omega'}$, the function \begin{align*} z\mapsto |g(z)|=\left(\sum_{j=1}^p |g_j(z)|^2\right)^{1/2} \end{align*} is continuous on $\overline{\Omega'}$. Compactness of $\overline{\Omega'}$ gives \begin{align*} M_{\Omega'}:=\sup_{z\in\Omega'}|g(z)|<\infty. \end{align*} We are in the nondegenerate case left after the earlier reduction, so $M_{\Omega'}>0$. For every point with $|g(z)|>0$, the bound $|g(z)|\leq M_{\Omega'}$ implies \begin{align*} 1\leq M_{\Omega'}^{2\alpha q}|g(z)|^{-2\alpha q}. \end{align*} At points where $|g(z)|=0$, the right-hand side is interpreted as $+\infty$, so the inequality holds in the extended nonnegative sense. Multiplying by the nonnegative function $\sum_{j=1}^p |u_j|^2$ and integrating with respect to $\mathcal L^{2m}$ gives \begin{align*} \int_{\Omega'} \sum_{j=1}^p |u_j|^2\,d\mathcal L^{2m} \leq M_{\Omega'}^{2\alpha q} \int_{\Omega'} \sum_{j=1}^p |u_j|^2 |g|^{-2\alpha q}\,d\mathcal L^{2m} <\infty. \end{align*} Therefore every $u_j$ is holomorphic and square-integrable on $\Omega'$, which is exactly \begin{align*} u_j\in A^2(\Omega') \qquad \text{for }j=1,\dots,p. \end{align*} Together with the division identity and Skoda's weighted estimate from the previous step, this completes the proof. [/guided] [/step]