[proofplan] This is the unweighted consequence of the Hörmander-Kohn $L^2$ method. One proves a weighted estimate using a strictly plurisubharmonic exhaustion, applies the Hilbert-space closed range lemma to obtain a weak solution, and then removes the weight on bounded exhaustion subdomains. Exhaustion and weak compactness pass the solution to the original pseudoconvex domain. [/proofplan] [step:Establish the basic $L^2$ estimate] On a smoothly bounded strictly pseudoconvex exhaustion subdomain $\Omega_j\Subset\Omega$, choose a smooth strictly plurisubharmonic weight $\varphi_j$. The Kohn-Morrey-Hörmander identity gives a constant $c_j>0$ such that, for test $(0,q)$-forms $v$ in the domains of $\bar\partial$ and $\bar\partial^*$, \begin{align*} \|\bar\partial v\|_{\varphi_j}^2+\|\bar\partial^*_{\varphi_j}v\|_{\varphi_j}^2\ge c_j\|v\|_{\varphi_j}^2. \end{align*} The positivity comes exactly from pseudoconvexity and strict plurisubharmonicity of the weight. [/step] [step:Apply the Hilbert-space existence lemma] The estimate implies that the range of $\bar\partial$ is closed in the weighted $L^2$ complex and that the functional \begin{align*} \bar\partial^*_{\varphi_j}v\longmapsto \langle f,v\rangle_{\varphi_j} \end{align*} is bounded. By the Riesz representation theorem, there exists $u_j\in L^2_{0,q-1}(\Omega_j)$ such that $\bar\partial u_j=f|_{\Omega_j}$ in the weak sense and \begin{align*} \|u_j\|_{\varphi_j}\le C_j\|f\|_{\varphi_j}. \end{align*} [/step] [step:Pass from exhaustion subdomains to $\Omega$] Choose the exhaustion and weights so that the estimates are uniform on compact subsets and compatible with the given $L^2$ norm. The sequence $u_j$ is bounded on each compact subset of $\Omega$ in $L^2$. By weak compactness and a diagonal argument, a subsequence converges weakly on compact subsets to a form $u\in L^2_{0,q-1,\mathrm{loc}}(\Omega)$. Closedness of $\bar\partial$ in the distributional sense gives $\bar\partial u=f$. [/step] [step:Recover the global estimate] For bounded pseudoconvex $\Omega$, the exhaustion and weight can be chosen so that the constants are controlled by the geometry of $\Omega$. Lower semicontinuity of the $L^2$ norm under weak convergence gives \begin{align*} \|u\|_{L^2(\Omega)}\le C_\Omega\|f\|_{L^2(\Omega)}. \end{align*} This proves the asserted solvability and estimate. [/step]