[proofplan]
We prove the estimate on an arbitrary [open set](/page/Open%20Set) $\Omega\subset\mathbb C^n$; the pseudoconvexity hypothesis in the original statement is not used because compact support eliminates all boundary terms. The compactly supported weighted Kohn-Morrey identity writes the graph norm of $v$ as a sum of a non-negative first-derivative term and a curvature term involving the Hermitian matrix $(\varphi_{j\bar k})$. The lower bound $i\partial\bar\partial\varphi \ge c\omega$ says that this Hermitian matrix is bounded below by $cI$, and applying this bound to each of the $q$ antiholomorphic slots gives the factor $q$.
[/proofplan]
[step:Expand the form and introduce the slot coefficients]
Let $\mathcal{I}_q$ denote the set of strictly increasing multi-indices $J=(j_1,\dots,j_q)$ with $1\le j_1<\cdots<j_q\le n$. Write
\begin{align*}
v=\sum_{J\in\mathcal{I}_q}v_J\,d\bar z_J,
\end{align*}
where each coefficient is a smooth compactly supported function
\begin{align*}
v_J:\Omega\to\mathbb{C}.
\end{align*}
For each $K=(k_1,\dots,k_{q-1})\in\mathcal{I}_{q-1}$ and each $1\le j\le n$, define $v_{jK}:\Omega\to\mathbb{C}$ by the identity
\begin{align*}
v_{jK}\,d\bar z_j\wedge d\bar z_K
\end{align*}
as the corresponding signed coefficient of $v$; equivalently, $v_{jK}=0$ if $j\in K$, and if the increasing rearrangement of $(j,K)$ is $J\in\mathcal{I}_q$, then $v_{jK}$ is $v_J$ multiplied by the sign of that rearrangement. For each $K\in\mathcal{I}_{q-1}$, define the coefficient vector
\begin{align*}
a_K:\Omega&\to\mathbb{C}^n,\\
z&\mapsto (v_{1K}(z),\dots,v_{nK}(z)).
\end{align*}
[/step]
[step:Apply the weighted Kohn-Morrey identity]
Since $v\in C_c^\infty(\Omega;\Lambda^{0,q})$, there is a compact set $K_v\subset\Omega$ such that $\operatorname{supp}v\subset K_v$. The form $\bar\partial v$ is again smooth and compactly supported, hence belongs to $L^2_{0,q+1}(\Omega,e^{-\varphi})$. Moreover, the formal weighted adjoint
\begin{align*}
\bar\partial_\varphi^*v
=
-\sum_{j=1}^n
\left(\frac{\partial}{\partial z_j}-\frac{\partial\varphi}{\partial z_j}\right)\iota_{\partial/\partial\bar z_j}v
\end{align*}
is a smooth compactly supported $(0,q-1)$-form, where $\iota_{\partial/\partial\bar z_j}$ denotes contraction by the coordinate vector field $\partial/\partial\bar z_j$. [Integration by parts](/theorems/2098) against every test form in $C_c^\infty(\Omega;\Lambda^{0,q-1})$ therefore shows that $v\in\operatorname{Dom}(\bar\partial_\varphi^*)$ and that the displayed formula is the Hilbert-space adjoint on $v$. The hypotheses needed for the compactly supported weighted Kohn-Morrey computation are thus satisfied: $\varphi\in C^2(\Omega)$ supplies the second derivatives $\varphi_{j\bar k}$, and compact support eliminates boundary terms.
The compactly supported weighted Kohn-Morrey identity in degree $q$, obtained by expanding $\bar\partial v$ and $\bar\partial_\varphi^*v$ in coefficients, integrating by parts inside $K_v$, and using the commutator
\begin{align*}
\left[\frac{\partial}{\partial \bar z_k},\frac{\partial}{\partial z_j}-\frac{\partial\varphi}{\partial z_j}\right]
=
-\varphi_{j\bar k},
\end{align*}
gives
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
&=
\sum_{J\in\mathcal{I}_q}\sum_{j=1}^n
\int_\Omega
\left|\frac{\partial v_J}{\partial \bar z_j}(z)\right|^2
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)\\
&\quad+
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
Here
\begin{align*}
\varphi_{j\bar k}:\Omega&\to\mathbb{C},\\
z&\mapsto \frac{\partial^2\varphi}{\partial z_j\,\partial\bar z_k}(z).
\end{align*}
The first term on the right-hand side is non-negative because it is a sum of integrals of pointwise squared absolute values against the positive measure $e^{-\varphi}\,d\mathcal{L}^{2n}$. Therefore
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
\ge
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
[guided]
The point of the Kohn-Morrey computation is that it converts the two analytic quantities $\|\bar\partial v\|_{e^{-\varphi}}^2$ and $\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2$ into a pointwise coercive expression. First we check that the Hilbert-space adjoint is legitimate on this test form. Since $v\in C_c^\infty(\Omega;\Lambda^{0,q})$, there is a compact set $K_v\subset\Omega$ with $\operatorname{supp}v\subset K_v$, and the coefficient formula
\begin{align*}
\bar\partial_\varphi^*v
=
-\sum_{j=1}^n
\left(\frac{\partial}{\partial z_j}-\frac{\partial\varphi}{\partial z_j}\right)\iota_{\partial/\partial\bar z_j}v
\end{align*}
defines a smooth compactly supported $(0,q-1)$-form. [Integration by parts](/theorems/210) against compactly supported smooth $(0,q-1)$-forms shows that this formal adjoint equals the Hilbert-space adjoint on $v$, so $v\in\operatorname{Dom}(\bar\partial_\varphi^*)$.
The only regularity required for the local identity is $\varphi\in C^2(\Omega)$, because the computation differentiates the first derivatives $\partial\varphi/\partial z_j$ once more. Compact support is the condition that removes every boundary term; no pseudoconvexity of $\Omega$ is involved in this compactly supported version. Expanding the two operators in coefficients, integrating by parts inside $K_v$, and using
\begin{align*}
\left[\frac{\partial}{\partial \bar z_k},\frac{\partial}{\partial z_j}-\frac{\partial\varphi}{\partial z_j}\right]
=
-\varphi_{j\bar k}
\end{align*}
gives the identity
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
&=
\sum_{J\in\mathcal{I}_q}\sum_{j=1}^n
\int_\Omega
\left|\frac{\partial v_J}{\partial \bar z_j}(z)\right|^2
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)\\
&\quad+
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
The coefficient $\varphi_{j\bar k}$ is the function
\begin{align*}
\varphi_{j\bar k}:\Omega&\to\mathbb{C},\\
z&\mapsto \frac{\partial^2\varphi}{\partial z_j\,\partial\bar z_k}(z).
\end{align*}
The first sum is non-negative because every integrand is of the form $|w(z)|^2e^{-\varphi(z)}$ and $e^{-\varphi(z)}>0$. Dropping this non-negative term gives the lower bound
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
\ge
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
This is the only place where the local Kohn-Morrey computation is used.
[/guided]
[/step]
[step:Use the curvature lower bound on each antiholomorphic slot]
The hypothesis $i\partial\bar\partial\varphi\ge c\omega$ means 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 \varphi_{j\bar k}(z)\,\xi_j\overline{\xi_k}
\ge
c\sum_{j=1}^n|\xi_j|^2.
\end{align*}
Apply this pointwise with $\xi=a_K(z)$ for each $K\in\mathcal{I}_{q-1}$. Then
\begin{align*}
\sum_{j,k=1}^n
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
\ge
c\sum_{j=1}^n |v_{jK}(z)|^2.
\end{align*}
Multiplying by $e^{-\varphi(z)}$ and integrating over $\Omega$ with respect to $\mathcal{L}^{2n}$ gives
\begin{align*}
\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
\ge
c\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
Summing over $K\in\mathcal{I}_{q-1}$ yields
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
\ge
c\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
[guided]
The curvature assumption $i\partial\bar\partial\varphi\ge c\omega$ is a pointwise Hermitian matrix inequality. Written in coordinates, it says that the Hermitian matrix $(\varphi_{j\bar k}(z))_{j,k=1}^n$ dominates $cI_n$ at every point $z\in\Omega$. Thus for every vector $\xi=(\xi_1,\dots,\xi_n)\in\mathbb C^n$,
\begin{align*}
\sum_{j,k=1}^n \varphi_{j\bar k}(z)\,\xi_j\overline{\xi_k}
\ge
c\sum_{j=1}^n |\xi_j|^2.
\end{align*}
For a fixed $K\in\mathcal I_{q-1}$, the coefficient vector $a_K:\Omega\to\mathbb C^n$ was defined by $a_K(z)=(v_{1K}(z),\dots,v_{nK}(z))$. Applying the Hermitian lower bound with $\xi=a_K(z)$ gives
\begin{align*}
\sum_{j,k=1}^n
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
\ge
c\sum_{j=1}^n |v_{jK}(z)|^2.
\end{align*}
The weight $e^{-\varphi(z)}$ is positive for every $z\in\Omega$, so multiplying by it preserves the inequality. Integrating over $\Omega$ with respect to $\mathcal L^{2n}$ gives
\begin{align*}
\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
\ge
c\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
Finally, summing this inequality over all $K\in\mathcal I_{q-1}$ preserves the inequality and yields
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j,k=1}^n
\int_\Omega
\varphi_{j\bar k}(z)\,v_{jK}(z)\overline{v_{kK}(z)}
e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
\ge
c\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z).
\end{align*}
[/guided]
[/step]
[step:Count the coefficient multiplicity and recover the factor $q$]
For each fixed $J=(j_1,\dots,j_q)\in\mathcal{I}_q$, the coefficient $v_J$ appears exactly $q$ times among the signed coefficients $v_{jK}$: namely, for each $r\in\{1,\dots,q\}$, take $j=j_r$ and $K=J\setminus\{j_r\}$ in increasing order. The sign does not affect the square absolute value. Therefore
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n |v_{jK}(z)|^2
=
q\sum_{J\in\mathcal{I}_q}|v_J(z)|^2
\end{align*}
for every $z\in\Omega$. Integrating this pointwise identity gives
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
=
q\sum_{J\in\mathcal{I}_q}
\int_\Omega |v_J(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
=
q\|v\|_{e^{-\varphi}}^2.
\end{align*}
Combining this identity with the curvature estimate from the previous step and the Kohn-Morrey lower bound gives
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
\ge
qc\,\|v\|_{e^{-\varphi}}^2.
\end{align*}
This is the desired Hörmander estimate for the compactly supported smooth $(0,q)$-form $v$.
[guided]
The remaining task is to translate the slot sum back into the original norm of $v$. Fix $J=(j_1,\dots,j_q)\in\mathcal I_q$. The coefficient $v_J$ contributes to $v_{jK}$ precisely when $j$ is one of the entries of $J$ and $K$ is obtained from $J$ by deleting that entry. There are exactly $q$ such choices, one for each $r\in\{1,\dots,q\}$ with $j=j_r$ and $K=J\setminus\{j_r\}$ in increasing order. The signs introduced by rearranging $d\bar z_j\wedge d\bar z_K$ do not matter after taking absolute values. Hence, for every $z\in\Omega$,
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n |v_{jK}(z)|^2
=
q\sum_{J\in\mathcal{I}_q}|v_J(z)|^2.
\end{align*}
Multiplying this pointwise identity by the positive weight $e^{-\varphi(z)}$ and integrating over $\Omega$ with respect to $\mathcal L^{2n}$ gives
\begin{align*}
\sum_{K\in\mathcal{I}_{q-1}}\sum_{j=1}^n
\int_\Omega |v_{jK}(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
=
q\sum_{J\in\mathcal{I}_q}
\int_\Omega |v_J(z)|^2e^{-\varphi(z)}\,d\mathcal{L}^{2n}(z)
=
q\|v\|_{e^{-\varphi}}^2.
\end{align*}
Combining this identity with the curvature lower bound and the Kohn-Morrey lower bound gives
\begin{align*}
\|\bar\partial v\|_{e^{-\varphi}}^2
+
\|\bar\partial_\varphi^*v\|_{e^{-\varphi}}^2
\ge
qc\,\|v\|_{e^{-\varphi}}^2.
\end{align*}
This is exactly the asserted estimate.
[/guided]
[/step]
