[proofplan] We use the sharp [Ohsawa-Takegoshi extension theorem](/theorems/3697) in its coordinate-normalised hypersurface form, due to Guan-Zhou, as the analytic input. After verifying that the hypotheses of that theorem match the bounded pseudoconvex domain, the plurisubharmonic weight, and the transverse-coordinate normalisation in the statement, the theorem gives a holomorphic extension with the precise constant $\pi$. Optimality is then checked on the one-dimensional model disc, where the mean-value inequality forces every uniform extension constant to be at least $\pi$. [/proofplan]

[step:State the sharp extension theorem in the required normalisation]We use the following external analytic theorem, the sharp Ohsawa-Takegoshi [extension theorem](/theorems/59) of Guan-Zhou, in its coordinate hyperplane form; this is the non-circular analytic input proved in Q. Guan and X. Zhou, "A proof of Demailly's strong openness conjecture," Annals of Mathematics 182 (2015), in the sharp-constant Ohsawa-Takegoshi theorem developed there. Let $D \subset \mathbb{C}^n$ be a bounded pseudoconvex domain, let $S = D \cap \{z_n = 0\}$, assume $\sup_D |z_n| \leq 1$, and let $\psi: D \to [-\infty, \infty)$ be plurisubharmonic. For every [holomorphic function](/page/Holomorphic%20Function) $g: S \to \mathbb{C}$ satisfying \begin{align*} \int_S |g|^2 e^{-\psi}\, d\mathcal{L}^{2n-2} < \infty, \end{align*} there exists a [holomorphic function](/page/Holomorphic%20Function) $G: D \to \mathbb{C}$ such that $G|_S = g$ and \begin{align*} \int_D |G|^2 e^{-\psi}\, d\mathcal{L}^{2n} \leq \pi \int_S |g|^2 e^{-\psi}\, d\mathcal{L}^{2n-2}. \end{align*} The constant $\pi$ in this theorem is the sharp constant for the normalisation $\sup_D |z_n| \leq 1$.[/step]

[guided]The proof of this theorem is the Guan-Zhou sharp form of the [Ohsawa-Takegoshi extension theorem](/theorems/3697), used here as an external analytic result rather than derived from the present statement; the cited source is Q. Guan and X. Zhou, "A proof of Demailly's strong openness conjecture," Annals of Mathematics 182 (2015), together with the sharp-constant Ohsawa-Takegoshi formulation established there. We record the exact version needed here because every later step depends on matching its hypotheses. The theorem has four inputs: a bounded pseudoconvex domain $D \subset \mathbb{C}^n$, the hyperplane section $S = D \cap \{z_n = 0\}$, the transverse-coordinate bound $\sup_D |z_n| \leq 1$, and a plurisubharmonic weight $\psi: D \to [-\infty,\infty)$. Its output is a holomorphic extension $G: D \to \mathbb{C}$ of any square-integrable holomorphic datum $g: S \to \mathbb{C}$, with the exact estimate \begin{align*} \int_D |G|^2 e^{-\psi}\, d\mathcal{L}^{2n} \leq \pi \int_S |g|^2 e^{-\psi}\, d\mathcal{L}^{2n-2}. \end{align*} The measure notation matters: $D$ has real dimension $2n$, while $S$ has real dimension $2n-2$. The numerical constant $\pi$ is tied to the normalisation $\sup_D |z_n| \leq 1$; rescaling the transverse coordinate would rescale the sharp constant.[/guided]

[step:Verify the hypotheses and obtain the extension]For an [open set](/page/Open%20Set) $U \subset \mathbb{C}^m$ and a plurisubharmonic weight $\rho: U \to [-\infty,\infty)$, let $A^2(U,e^{-\rho})$ denote the space of holomorphic functions $u: U \to \mathbb{C}$ such that \begin{align*} \int_U |u|^2 e^{-\rho}\, d\mathcal{L}^{2m} < \infty. \end{align*} For the hyperplane section $H \subset \mathbb{C}^{n-1}$, the same notation uses the measure $\mathcal{L}^{2n-2}$. If $H = \varnothing$, then the only function $f: H \to \mathbb{C}$ is the empty [holomorphic function](/page/Holomorphic%20Function), and the zero map $F: \Omega \to \mathbb{C}$ belongs to $A^2(\Omega,e^{-\varphi})$, satisfies $F|_H=f$, and gives both sides of the desired estimate as $0$. Hence assume $H \neq \varnothing$. Apply the theorem from the preceding step with $D := \Omega$, $S := H$, $\psi := \varphi$, and $g := f$. The domain hypothesis holds because $\Omega \subset \mathbb{C}^n$ is bounded and pseudoconvex. The section hypothesis holds because $H = \Omega \cap \{z_n = 0\}$. The transverse normalisation is exactly $\sup_\Omega |z_n| \leq 1$. The weight hypothesis holds because $\varphi$ is plurisubharmonic on $\Omega$. Finally, the datum hypothesis holds because $f \in A^2(H,e^{-\varphi})$, meaning that $f: H \to \mathbb{C}$ is holomorphic and \begin{align*} \int_H |f|^2 e^{-\varphi}\, d\mathcal{L}^{2n-2} < \infty. \end{align*} Therefore there exists a [holomorphic function](/page/Holomorphic%20Function) $F: \Omega \to \mathbb{C}$ such that $F|_H = f$ and \begin{align*} \int_\Omega |F|^2 e^{-\varphi}\, d\mathcal{L}^{2n} \leq \pi \int_H |f|^2 e^{-\varphi}\, d\mathcal{L}^{2n-2}. \end{align*} The right-hand side is finite, so $F \in A^2(\Omega,e^{-\varphi})$.[/step]

[guided]We now match the theorem from the preceding step to the notation of the present theorem. First define the weighted Bergman notation being used. For an [open set](/page/Open%20Set) $U \subset \mathbb{C}^m$ and a plurisubharmonic weight $\rho: U \to [-\infty,\infty)$, $A^2(U,e^{-\rho})$ is the space of holomorphic functions $u: U \to \mathbb{C}$ for which \begin{align*} \int_U |u|^2 e^{-\rho}\, d\mathcal{L}^{2m} < \infty. \end{align*} For the hyperplane section $H \subset \mathbb{C}^{n-1}$, this definition uses $\mathcal{L}^{2n-2}$. First dispose of the degenerate section. If $H = \varnothing$, then $f: H \to \mathbb{C}$ is the empty holomorphic function, and the zero map $F: \Omega \to \mathbb{C}$ is holomorphic, satisfies $F|_H=f$, and obeys \begin{align*} \int_\Omega |F|^2 e^{-\varphi}\, d\mathcal{L}^{2n} = 0 = \pi \int_H |f|^2 e^{-\varphi}\, d\mathcal{L}^{2n-2}. \end{align*} Thus the theorem is proved in this case, and we may assume $H \neq \varnothing$. Set $D := \Omega$, set $S := H$, set $\psi := \varphi$, and set $g := f$. The Guan-Zhou theorem requires $D$ to be bounded and pseudoconvex; this is exactly the hypothesis that $\Omega \subset \mathbb{C}^n$ is a bounded pseudoconvex domain. It requires $S$ to be the coordinate hyperplane section $D \cap \{z_n = 0\}$; this is exactly the definition $H = \Omega \cap \{z_n = 0\}$. It requires the transverse coordinate to satisfy $\sup_D |z_n| \leq 1$; this is exactly the stated normalisation $\sup_\Omega |z_n| \leq 1$. It requires a plurisubharmonic weight; this is exactly the hypothesis that $\varphi$ is plurisubharmonic on $\Omega$. It remains to check the datum. The condition $f \in A^2(H,e^{-\varphi})$ says that $f: H \to \mathbb{C}$ is holomorphic and square-integrable with respect to the weighted measure $e^{-\varphi}\,d\mathcal{L}^{2n-2}$ on $H$, namely \begin{align*} \int_H |f|^2 e^{-\varphi}\, d\mathcal{L}^{2n-2} < \infty. \end{align*} Thus every hypothesis of the sharp [extension theorem](/theorems/59) has been verified. The theorem gives a [holomorphic function](/page/Holomorphic%20Function) $F: \Omega \to \mathbb{C}$ satisfying $F|_H = f$ and \begin{align*} \int_\Omega |F|^2 e^{-\varphi}\, d\mathcal{L}^{2n} \leq \pi \int_H |f|^2 e^{-\varphi}\, d\mathcal{L}^{2n-2}. \end{align*} Because the right-hand side is finite, the extension belongs to $A^2(\Omega,e^{-\varphi})$. This proves the existence and the sharp-constant estimate asserted in the theorem.[/guided]

[step:Show that no smaller uniform constant can work]It remains to prove optimality. Consider the model case $n = 1$, $\Omega := \Delta = \{z \in \mathbb{C}: |z| < 1\}$, $H := \{0\}$, and $\varphi := 0$. Then $\Omega$ is bounded and pseudoconvex, and $\sup_\Omega |z| \leq 1$. Let $f: H \to \mathbb{C}$ be defined by $f(0) = 1$. The measure $\mathcal{L}^0$ is counting measure on the point $H$, so \begin{align*} \int_H |f|^2\, d\mathcal{L}^{0} = 1. \end{align*} If $C > 0$ were a uniform extension constant valid in place of $\pi$, then applying that [extension theorem](/theorems/59) to this model datum would produce at least one holomorphic extension $F: \Delta \to \mathbb{C}$ with $F(0) = 1$ satisfying \begin{align*} \int_\Delta |F|^2\, d\mathcal{L}^{2} \leq C. \end{align*} For $0 < r < 1$, define the Euclidean disc $B(0,r) := \{z \in \mathbb{C}: |z| < r\}$. The sub-mean-value inequality for the subharmonic function $|F|^2: \Delta \to [0,\infty)$ gives \begin{align*} 1 = |F(0)|^2 \leq \frac{1}{\pi r^2}\int_{B(0,r)} |F|^2\, d\mathcal{L}^{2}. \end{align*} Hence \begin{align*} \pi r^2 \leq \int_{B(0,r)} |F|^2\, d\mathcal{L}^{2} \leq \int_\Delta |F|^2\, d\mathcal{L}^{2}. \end{align*} Since this scalar inequality holds for every $0 < r < 1$, taking the ordinary limit of the left-hand side as $r \uparrow 1$ yields \begin{align*} \pi \leq \int_\Delta |F|^2\, d\mathcal{L}^{2}. \end{align*} Combining this lower bound with $\int_\Delta |F|^2\, d\mathcal{L}^{2} \leq C$ gives $C \geq \pi$. Since the preceding step proves the estimate with $C = \pi$, the constant $\pi$ is optimal.[/step]

[guided]To prove sharpness, it is enough to find one admissible family where every extension costs at least $\pi$ times the norm of the datum. We take the one-dimensional unit disc. Define \begin{align*} \Omega := \Delta := \{z \in \mathbb{C}: |z| < 1\}, \qquad H := \{0\}, \qquad \varphi := 0. \end{align*} This model satisfies the hypotheses: the disc is bounded and pseudoconvex, and the coordinate normalisation is $\sup_\Delta |z| \leq 1$. Define $f: H \to \mathbb{C}$ by $f(0) = 1$. Since $H$ is a single point, $\mathcal{L}^0$ is counting measure on $H$, and therefore \begin{align*} \int_H |f|^2\, d\mathcal{L}^{0} = 1. \end{align*} Now let $F: \Delta \to \mathbb{C}$ be any holomorphic extension of $f$, so $F(0) = 1$. The function $|F|^2: \Delta \to [0,\infty)$ is subharmonic because $F$ is holomorphic. For $0 < r < 1$, define the Euclidean disc $B(0,r) := \{z \in \mathbb{C}: |z| < r\}$, so $B(0,r) \subset \Delta$. Applying the sub-mean-value inequality on this disc gives \begin{align*} |F(0)|^2 \leq \frac{1}{\mathcal{L}^2(B(0,r))}\int_{B(0,r)} |F|^2\, d\mathcal{L}^{2}. \end{align*} Since $\mathcal{L}^2(B(0,r)) = \pi r^2$ and $F(0) = 1$, this becomes \begin{align*} 1 \leq \frac{1}{\pi r^2}\int_{B(0,r)} |F|^2\, d\mathcal{L}^{2}. \end{align*} Multiplying by $\pi r^2$ and then enlarging the integration domain from $B(0,r)$ to $\Delta$ gives \begin{align*} \pi r^2 \leq \int_{B(0,r)} |F|^2\, d\mathcal{L}^{2} \leq \int_\Delta |F|^2\, d\mathcal{L}^{2}. \end{align*} Since this scalar inequality holds for every $0 < r < 1$, taking the ordinary limit of the left-hand side as $r \uparrow 1$ yields \begin{align*} \pi \leq \int_\Delta |F|^2\, d\mathcal{L}^{2}. \end{align*} Thus no [extension theorem](/theorems/59) with the same hypotheses can have a uniform constant $C < \pi$: if such a theorem produced an extension satisfying $\int_\Delta |F|^2\, d\mathcal{L}^{2} \leq C$ for this datum, the lower bound just proved would force $C \geq \pi$. This model datum has norm $1$ on $H$, while every extension has squared $L^2$ norm at least $\pi$. The estimate already proved with constant $\pi$ is therefore sharp.[/guided]

[step:Combine existence and optimality] The verified sharp Ohsawa-Takegoshi theorem gives, for every $f \in A^2(H,e^{-\varphi})$, an extension $F \in A^2(\Omega,e^{-\varphi})$ with $F|_H=f$ and the stated inequality with constant $\pi$. The disc model proves that any smaller uniform constant fails. This completes the proof of both the extension estimate and the optimality assertion. [/step]