[proofplan] We reduce the statement to the sharp divisor form of the [Ohsawa-Takegoshi extension theorem](/theorems/3697), then translate its residue measure into the hypersurface measure appearing in the statement. The hypotheses $|s|\le 1$, pseudoconvexity of $\Omega$, and plurisubharmonicity of $\varphi$ are exactly the analytic hypotheses needed for the [extension theorem](/theorems/59). The lower and upper bounds on $\|ds_z\|_{\mathrm{op}}$ make the residue measure and $\mathcal H^{2n-2}$ uniformly comparable on $D$, so the resulting constant depends only on the stated data. [/proofplan] [step:Apply the divisor form of the Ohsawa-Takegoshi extension theorem] Let $\bar\partial$ denote the Cauchy-Riemann operator acting on smooth complex-valued functions on open subsets of $\mathbb C^n$. We use the divisor form of the [Ohsawa-Takegoshi $L^2$ Extension Theorem](/theorems/???): if $\Omega\subset\mathbb C^n$ is bounded and pseudoconvex, $s\in\mathcal O(\Omega)$ satisfies $|s|\le 1$, $D=\{z\in\Omega:s(z)=0\}$ is a smooth divisor, and $\varphi$ is plurisubharmonic, then every $f\in\mathcal O(D)$ with \begin{align*} \int_D |f|^2 e^{-\varphi}\,\frac{d\mathcal H^{2n-2}(z)}{\|ds_z\|_{\mathrm{op}}^2}<\infty \end{align*} has an extension $F\in\mathcal O(\Omega)$ satisfying $F|_D=f$ and \begin{align*} \int_\Omega |F|^2e^{-\varphi}\,d\mathcal L^{2n}(z) \le C_{\mathrm{OT}}(\Omega)\int_D |f|^2 e^{-\varphi}\,\frac{d\mathcal H^{2n-2}(z)}{\|ds_z\|_{\mathrm{op}}^2}, \end{align*} where $C_{\mathrm{OT}}(\Omega)>0$ depends only on $\Omega$ and on the fixed normalisations of $\mathcal L^{2n}$ and $\mathcal H^{2n-2}$. All hypotheses of this theorem are hypotheses of the present theorem: $\Omega$ is bounded and pseudoconvex, $s$ is holomorphic, $D$ is the smooth zero divisor of $s$, $|s|\le 1$ on $\Omega$, and $\varphi$ is plurisubharmonic. [/step] [step:Verify that the residue norm is finite] Define the residue-density function $r:D\to(0,\infty)$ by \begin{align*} r(z)=\|ds_z\|_{\mathrm{op}}^{-2}. \end{align*} The assumed lower bound $m\le \|ds_z\|_{\mathrm{op}}$ for $z\in D$ gives \begin{align*} 0<r(z)\le m^{-2}\qquad\text{for every }z\in D. \end{align*} Since the hypothesis on $f$ is \begin{align*} \int_D |f|^2e^{-\varphi}\,d\mathcal H^{2n-2}<\infty, \end{align*} we obtain \begin{align*} \int_D |f|^2 e^{-\varphi}\,\frac{d\mathcal H^{2n-2}(z)}{\|ds_z\|_{\mathrm{op}}^2} \le m^{-2}\int_D |f|^2e^{-\varphi}\,d\mathcal H^{2n-2}(z)<\infty. \end{align*} Thus $f$ satisfies the integrability hypothesis required in the divisor [extension theorem](/theorems/59). [/step] [step:Convert the Ohsawa-Takegoshi estimate to the stated hypersurface estimate] Applying the [extension theorem](/theorems/59) gives $F\in\mathcal O(\Omega)$ with $F|_D=f$ and \begin{align*} \int_\Omega |F|^2e^{-\varphi}\,d\mathcal L^{2n}(z) &\le C_{\mathrm{OT}}(\Omega)\int_D |f|^2 e^{-\varphi}\,\frac{d\mathcal H^{2n-2}(z)}{\|ds_z\|_{\mathrm{op}}^2} \\ &\le C_{\mathrm{OT}}(\Omega)m^{-2}\int_D |f|^2e^{-\varphi}\,d\mathcal H^{2n-2}(z). \end{align*} Set \begin{align*} C=C_{\mathrm{OT}}(\Omega)m^{-2}. \end{align*} Then $C$ depends only on $\Omega$, $m$, and the chosen measure normalisations. This dependence is allowed by the theorem statement, and it is independent of $f$. The constructed [holomorphic function](/page/Holomorphic%20Function) $F$ satisfies both $F|_D=f$ and the asserted weighted $L^2$ estimate, completing the proof. [/step]