[proofplan]
This result is a synthesis corollary of three already-established theorems rather than a new proof of the Levi problem or Hörmander's estimate from first principles. We first invoke the previously proved [Levi-Oka exhaustion theorem](/page/Levi-Oka%20Exhaustion%20Theorem), in the precise form that pseudoconvex domains in $\mathbb{C}^n$ admit continuous plurisubharmonic exhaustions. We then invoke the previously proved [Hörmander weighted $L^2$ existence theorem for $\bar{\partial}$](/page/Hormander%20Weighted%20L2%20Existence%20Theorem), and the curvature lower bound $i\partial\bar{\partial}\varphi \ge c\omega$ gives the inverse curvature estimate with constant $1/c$ on $(0,1)$-forms. Finally, the already-proved [Cartan-Thullen formulation of the Levi problem](/page/Cartan-Thullen%20Theorem) identifies pseudoconvex domains, equivalently domains admitting plurisubharmonic exhaustions, with domains of holomorphy.
[/proofplan]
[step:Invoke the established Levi-Oka exhaustion theorem]
We use the previously established [Levi-Oka exhaustion theorem](/page/Levi-Oka%20Exhaustion%20Theorem) in the following exhaustion form: if $D \subset \mathbb{C}^n$ is a domain, then $D$ is pseudoconvex if and only if there exists a continuous plurisubharmonic map $\rho_D: D \to \mathbb{R}$ such that, for every $A \in \mathbb{R}$, the sublevel set $\{z \in D : \rho_D(z) < A\}$ is relatively compact in $D$. The hypotheses of this prior theorem apply with $D=\Omega$, because $\Omega \subset \mathbb{C}^n$ is a domain and is pseudoconvex by hypothesis. Hence there exists a continuous plurisubharmonic function $\rho: \Omega \to \mathbb{R}$ such that, for every $A \in \mathbb{R}$,
\begin{align*}
\{z \in \Omega : \rho(z) < A\}
\end{align*}
is relatively compact in $\Omega$. This proves the exhaustion assertion as a consequence of the cited prerequisite.
[/step]
[step:Apply the established Hörmander estimate to solve $\bar{\partial}u=f$]
Let $\varphi: \Omega \to \mathbb{R}$ be the given $C^2$ function satisfying $i\partial\bar{\partial}\varphi \ge c\omega$ for some constant $c>0$. Let $f \in L^2_{(0,1)}(\Omega,e^{-\varphi})$ be the given $\bar{\partial}$-closed $(0,1)$-form. We now use the previously established [Hörmander weighted $L^2$ existence theorem for $\bar{\partial}$](/page/Hormander%20Weighted%20L2%20Existence%20Theorem). Its hypotheses are satisfied: $\Omega$ is pseudoconvex, $\varphi$ is plurisubharmonic because its complex Hessian is bounded below by the positive form $c\omega$, and $f$ is $\bar{\partial}$-closed with finite weighted norm. Therefore there exists a function $u \in L^2(\Omega,e^{-\varphi})$ satisfying $\bar{\partial}u=f$ in the distributional sense and
\begin{align*}
\int_\Omega |u|^2 e^{-\varphi}\,d\mathcal L^{2n}
&\le
\int_\Omega \langle (i\partial\bar{\partial}\varphi)^{-1}f,f\rangle_\omega e^{-\varphi}\,d\mathcal L^{2n}.
\end{align*}
The curvature inequality $i\partial\bar{\partial}\varphi \ge c\omega$ means that the Hermitian form $i\partial\bar{\partial}\varphi$ dominates $c\omega$ pointwise. For each point $z \in \Omega$, let $T_z^{0,1*}\Omega$ denote the complex [vector space](/page/Vector%20Space) of $(0,1)$-covectors at $z$, equipped with the Hermitian inner product induced by the Euclidean Kähler metric $\omega$. Define
\begin{align*}
A_z: T_z^{0,1*}\Omega &\to T_z^{0,1*}\Omega
\end{align*}
to be the positive Hermitian endomorphism representing the Hermitian form $i\partial\bar{\partial}\varphi$ on $(0,1)$-covectors with respect to this $\omega$-inner product. The lower bound says that every eigenvalue of $A_z$ is at least $c$, so every eigenvalue of $A_z^{-1}$ is at most $1/c$. Hence
\begin{align*}
\langle (i\partial\bar{\partial}\varphi)^{-1}f,f\rangle_\omega
\le \frac{1}{c}|f|_\omega^2
\end{align*}
for $\mathcal L^{2n}$-almost every point of $\Omega$. Substituting this pointwise bound into Hörmander's estimate gives
\begin{align*}
\int_\Omega |u|^2 e^{-\varphi}\,d\mathcal L^{2n}
\le
\frac{1}{c}
\int_\Omega |f|_\omega^2 e^{-\varphi}\,d\mathcal L^{2n}.
\end{align*}
[guided]
We apply the previously established [Hörmander weighted $L^2$ existence theorem for $\bar{\partial}$](/page/Hormander%20Weighted%20L2%20Existence%20Theorem). This is a prerequisite theorem, not the statement currently being proved. It requires a pseudoconvex domain, a plurisubharmonic $C^2$ weight, and a $\bar{\partial}$-closed form with finite weighted $L^2$ norm. These are exactly the hypotheses in the present situation, except that we should explicitly check plurisubharmonicity of the weight. The inequality $i\partial\bar{\partial}\varphi \ge c\omega$ with $c>0$ says that the complex Hessian of $\varphi$ is positive semidefinite, indeed uniformly positive, so $\varphi$ is plurisubharmonic.
Hörmander's theorem therefore gives a function $u \in L^2(\Omega,e^{-\varphi})$ such that $\bar{\partial}u=f$ in the distributional sense and
\begin{align*}
\int_\Omega |u|^2 e^{-\varphi}\,d\mathcal L^{2n}
&\le
\int_\Omega \langle (i\partial\bar{\partial}\varphi)^{-1}f,f\rangle_\omega e^{-\varphi}\,d\mathcal L^{2n}.
\end{align*}
The remaining point is to compute the constant. For each point $z \in \Omega$, let $T_z^{0,1*}\Omega$ denote the complex [vector space](/page/Vector%20Space) of $(0,1)$-covectors at $z$, with Hermitian inner product induced by the Euclidean Kähler metric $\omega$. Define
\begin{align*}
A_z: T_z^{0,1*}\Omega &\to T_z^{0,1*}\Omega
\end{align*}
to be the positive Hermitian endomorphism representing the Hermitian form $i\partial\bar{\partial}\varphi$ on $(0,1)$-covectors with respect to this $\omega$-inner product. The inequality $i\partial\bar{\partial}\varphi \ge c\omega$ says that every eigenvalue of $A_z$ is at least $c$. Taking inverses reverses the order of positive Hermitian endomorphisms, so every eigenvalue of $A_z^{-1}$ is at most $1/c$. Therefore, for the coefficient vector of the $(0,1)$-form $f$,
\begin{align*}
\langle (i\partial\bar{\partial}\varphi)^{-1}f,f\rangle_\omega
\le \frac{1}{c}|f|_\omega^2.
\end{align*}
Multiplying this pointwise inequality by the non-negative weight $e^{-\varphi}$ and integrating with respect to $\mathcal L^{2n}$ gives
\begin{align*}
\int_\Omega |u|^2 e^{-\varphi}\,d\mathcal L^{2n}
\le
\frac{1}{c}
\int_\Omega |f|_\omega^2 e^{-\varphi}\,d\mathcal L^{2n}.
\end{align*}
This is precisely the asserted estimate.
[/guided]
[/step]
[step:Invoke the established Cartan-Thullen theorem to obtain that $\Omega$ is a domain of holomorphy]
We use the previously established [Cartan-Thullen formulation of the Levi problem](/page/Cartan-Thullen%20Theorem): for a domain $D \subset \mathbb{C}^n$, the following are equivalent: $D$ is pseudoconvex, $D$ admits a plurisubharmonic exhaustion, and $D$ is a domain of holomorphy. This prior theorem applies with $D=\Omega$, because $\Omega \subset \mathbb{C}^n$ is a domain and is pseudoconvex by hypothesis; moreover, the first step produced a plurisubharmonic exhaustion of $\Omega$. Therefore $\Omega$ is a domain of holomorphy.
[guided]
The final implication is an application of the previously established [Cartan-Thullen formulation of the Levi problem](/page/Cartan-Thullen%20Theorem). We are not reproving that equivalence here; this step records the corollary obtained from it. The theorem states that, for domains in $\mathbb{C}^n$, the following conditions are equivalent: pseudoconvexity, existence of a plurisubharmonic exhaustion, holomorphic convexity, and being a domain of holomorphy. We verify the hypotheses for the cited theorem: $\Omega \subset \mathbb{C}^n$ is a domain by the theorem statement, $\Omega$ is pseudoconvex by hypothesis, and the first step constructs a plurisubharmonic exhaustion $\rho: \Omega \to \mathbb{R}$. Therefore the Cartan-Thullen equivalence applies to $\Omega$ and gives that $\Omega$ is a domain of holomorphy.
[/guided]
[/step]
