[proofplan]
The proof applies the [Hörmander--Kohn $\bar\partial$-solvability theorem](/theorems/3493) in weighted $L^2$ degree $1 \le q \le n$. The explicit weight hypothesis in the theorem statement gives the smooth weight and the uniform positive complex Hessian lower bound required by the estimate, while pseudoconvexity is the domain hypothesis consumed by the solvability theorem. Applying the theorem to an arbitrary $\bar\partial$-closed weighted $L^2$ representative produces a weighted $L^2$ primitive in the previous degree, so every Hörmander-admissible positive-degree cohomology class is zero.
[/proofplan]
[step:Record the curvature hypothesis for the given weight]
Throughout the proof, $\mathcal{L}^{2n}$ denotes Lebesgue measure on $\mathbb{C}^n$ under the real-linear identification $\mathbb{C}^n \cong \mathbb{R}^{2n}$. Let $\varphi \in C^\infty(\Omega;\mathbb{R})$ denote the weight fixed in the theorem statement. The theorem statement assumes the following uniform curvature lower bound: there exists a constant $\lambda_\varphi > 0$ such that, for every $z \in \Omega$ and every $\xi = (\xi_1,\dots,\xi_n) \in \mathbb{C}^n$,
\begin{align*}
\sum_{j,k=1}^n \frac{\partial^2 \varphi}{\partial z_j \partial \overline{z}_k}(z)\,\xi_j \overline{\xi_k}
\ge \lambda_\varphi |\xi|^2.
\end{align*}
[guided]
We first fix the measure convention: $\mathcal{L}^{2n}$ is Lebesgue measure on $\mathbb{C}^n$ after identifying $\mathbb{C}^n$ with $\mathbb{R}^{2n}$. The proof must use the weight named in the theorem statement, not replace it by a new weight. Thus let $\varphi \in C^\infty(\Omega;\mathbb{R})$ be the given weight. The theorem statement explicitly assumes the following quantitative curvature condition: there is a constant $\lambda_\varphi > 0$ such that the complex Hessian of $\varphi$ satisfies
\begin{align*}
\sum_{j,k=1}^n \frac{\partial^2 \varphi}{\partial z_j \partial \overline{z}_k}(z)\,\xi_j \overline{\xi_k}
\ge \lambda_\varphi |\xi|^2
\end{align*}
for every point $z \in \Omega$ and every vector $\xi \in \mathbb{C}^n$. This is exactly the positivity condition that will be consumed when we invoke the [Hörmander--Kohn $\bar\partial$-solvability theorem](/theorems/3493).
[/guided]
[/step]
[step:Apply the higher-degree Hörmander existence theorem to a closed form]
Fix an integer $q$ with $1 \le q \le n$. Let
\begin{align*}
f \in L^2_{(0,q)}(\Omega,e^{-\varphi})
\end{align*}
be a $\bar\partial$-closed form satisfying the weighted integrability hypothesis of the Hörmander--Kohn theorem, namely that $f$ is a measurable $(0,q)$-form on $\Omega$ whose coefficient functions satisfy
\begin{align*}
\int_\Omega |f(z)|^2 e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)<\infty,
\end{align*}
with the coefficient norm taken with respect to the standard Hermitian metric on $(0,q)$-forms, and
\begin{align*}
\bar\partial f = 0
\end{align*}
in the [distributional](/page/Distribution) sense.
We use the [Hörmander--Kohn $\bar\partial$-solvability theorem](/theorems/3493) in the following weighted degree-$q$ form: if $D \subset \mathbb{C}^n$ is pseudoconvex, $\psi \in C^\infty(D;\mathbb{R})$ has complex Hessian bounded below by a positive constant $\lambda_\psi$, and $g \in L^2_{(0,q)}(D,e^{-\psi})$ is $\bar\partial$-closed for $1 \le q \le n$, then there exists $v \in L^2_{(0,q-1)}(D,e^{-\psi})$ such that $\bar\partial v=g$ in the distributional sense. Its hypotheses are satisfied with $D=\Omega$, $\psi=\varphi$, and $g=f$: $\Omega$ is pseudoconvex by the theorem statement, the weight $\varphi$ is smooth and satisfies the uniform lower bound recorded above, the integer $q$ lies in $1 \le q \le n$, and $f$ is $\bar\partial$-closed with the finite weighted $L^2$ norm displayed above. Hence there exists
\begin{align*}
u \in L^2_{(0,q-1)}(\Omega,e^{-\varphi})
\end{align*}
such that
\begin{align*}
\bar\partial u = f
\end{align*}
in the distributional sense.
[guided]
The point of choosing the weight in the previous step is that it puts us exactly in the setting of Hörmander's solvability theorem. We now fix $q$ with $1 \le q \le n$ and take a closed weighted $L^2$ form
\begin{align*}
f \in L^2_{(0,q)}(\Omega,e^{-\varphi}).
\end{align*}
This means that $f$ is a measurable $(0,q)$-form whose coefficients satisfy
\begin{align*}
\int_\Omega |f(z)|^2 e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z) < \infty,
\end{align*}
with the coefficient norm taken with respect to the standard Hermitian metric on forms. The closedness assumption is
\begin{align*}
\bar\partial f = 0
\end{align*}
as a distribution.
We apply the [Hörmander--Kohn $\bar\partial$-solvability theorem](/theorems/3493). In the weighted degree-$q$ form needed here, it says that if $D \subset \mathbb{C}^n$ is pseudoconvex, $\psi \in C^\infty(D;\mathbb{R})$ has complex Hessian bounded below by a positive constant $\lambda_\psi$, and $g \in L^2_{(0,q)}(D,e^{-\psi})$ is $\bar\partial$-closed for $1 \le q \le n$, then there exists $v \in L^2_{(0,q-1)}(D,e^{-\psi})$ satisfying $\bar\partial v=g$ in the distributional sense. We verify these hypotheses with $D=\Omega$, $\psi=\varphi$, and $g=f$. The domain hypothesis is exactly the pseudoconvexity assumed in the theorem statement; in the strictly pseudoconvex special case, it follows from the existence of a smooth strictly plurisubharmonic defining function. The weight hypothesis holds because the theorem statement assumes the uniform curvature lower bound recorded in the previous step. The degree hypothesis holds because $1 \le q \le n$. The form hypothesis is exactly the assumption imposed on $f$, including the finite weighted $L^2$ norm displayed above and distributional closedness.
Therefore Hörmander's theorem produces a form
\begin{align*}
u \in L^2_{(0,q-1)}(\Omega,e^{-\varphi})
\end{align*}
such that
\begin{align*}
\bar\partial u = f
\end{align*}
in the sense of distributions. This is the desired primitive of $f$ in the previous degree.
[/guided]
[/step]
[step:Conclude that every Hörmander-admissible positive-degree cohomology class vanishes]
Define $\operatorname{Dom}_{q,\varphi}(\bar\partial)$ to be the set of forms $a$ in the [weighted $L^2$ space](/page/Weighted%20L2%20Space) $L^2_{(0,q)}(\Omega,e^{-\varphi})$ whose [distributional](/page/Distribution) derivative $\bar\partial a$ belongs to $L^2_{(0,q+1)}(\Omega,e^{-\varphi})$. Define
\begin{align*}
Z^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)
&:= \{a \in \operatorname{Dom}_{q,\varphi}(\bar\partial) : \bar\partial a = 0 \text{ and } a \text{ satisfies the weighted hypotheses of the higher-degree Hörmander theorem}\},\\
B^{0,q}_{(2),\varphi}(\Omega)
&:= \{\bar\partial b : b \in \operatorname{Dom}_{q-1,\varphi}(\bar\partial)\}.
\end{align*}
The Hörmander-admissible weighted $L^2$ [Dolbeault cohomology](/page/Dolbeault%20Cohomology) group is the quotient [vector space](/page/Vector%20Space)
\begin{align*}
H^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)
:= Z^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)/B^{0,q}_{(2),\varphi}(\Omega),
\end{align*}
so two representatives define the same class exactly when their difference lies in $B^{0,q}_{(2),\varphi}(\Omega)$.
Let $[f] \in H^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)$ be a class with $1 \le q \le n$, represented by $f \in Z^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)$.
By the previous step, there exists such a form $u \in L^2_{(0,q-1)}(\Omega,e^{-\varphi})$ with $\bar\partial u = f$. Since $f \in L^2_{(0,q)}(\Omega,e^{-\varphi})$, this equality also shows that $u \in \operatorname{Dom}_{q-1,\varphi}(\bar\partial)$, so $f \in B^{0,q}_{(2),\varphi}(\Omega)$. Hence $f$ represents the zero class:
\begin{align*}
[f] = 0.
\end{align*}
Since $[f]$ was arbitrary, we obtain
\begin{align*}
H^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)=0
\end{align*}
for every $1 \le q \le n$.
[guided]
We spell out the quotient notation. Let $\operatorname{Dom}_{q,\varphi}(\bar\partial)$ be the set of forms $a$ in the [weighted $L^2$ space](/page/Weighted%20L2%20Space) $L^2_{(0,q)}(\Omega,e^{-\varphi})$ whose [distributional](/page/Distribution) derivative $\bar\partial a$ belongs to $L^2_{(0,q+1)}(\Omega,e^{-\varphi})$. The closed representatives are
\begin{align*}
Z^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)
:= \{a \in \operatorname{Dom}_{q,\varphi}(\bar\partial) : \bar\partial a = 0 \text{ and } a \text{ satisfies the weighted hypotheses of the higher-degree Hörmander theorem}\},
\end{align*}
and the exact representatives are
\begin{align*}
B^{0,q}_{(2),\varphi}(\Omega)
:= \{\bar\partial b : b \in \operatorname{Dom}_{q-1,\varphi}(\bar\partial)\}.
\end{align*}
The Hörmander-admissible weighted $L^2$ [Dolbeault cohomology](/page/Dolbeault%20Cohomology) group is the quotient
\begin{align*}
H^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)
:= Z^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)/B^{0,q}_{(2),\varphi}(\Omega).
\end{align*}
Thus a class $[f]$ is zero precisely when its representative $f$ lies in $B^{0,q}_{(2),\varphi}(\Omega)$, equivalently when there is a form
\begin{align*}
u \in \operatorname{Dom}_{q-1,\varphi}(\bar\partial) \subset L^2_{(0,q-1)}(\Omega,e^{-\varphi})
\end{align*}
such that
\begin{align*}
f = \bar\partial u.
\end{align*}
The previous step proves exactly this statement for every $\bar\partial$-closed representative satisfying the weighted hypotheses of the Hörmander theorem. Therefore every representative of every Hörmander-admissible positive-degree cohomology class is exact. Hence every such class is zero, and so
\begin{align*}
H^{0,q}_{(2),\varphi,\mathrm{Hor}}(\Omega)=0
\end{align*}
for all $1 \le q \le n$.
[/guided]
[/step]
