Theorem 6.4.988735CB0C076

Proof. We regularize $p$ on relatively compact pseudoconvex subsets of $\Omega $, apply theorem 6.4.7 there, and pass to a weak limit. Since the weight $p+q$ is bounded above on compact sets, the discussion preceding theorem 6.4.7 shows that $\alpha \in L^2_{\mathrm{loc},(0,1)}(\Omega )$, so that $\bar\partial \alpha $ is defined. By corollary 6.3.5 there are pseudoconvex open sets $\Omega _1\subset \Omega _2\subset \cdots $ with $\Omega _ j\Subset \Omega $ and $\bigcup _ j\Omega _ j=\Omega $. Let $\rho _\varepsilon $ be the radial mollifier of section 2.5, and choose $\varepsilon _ j\downarrow 0$ with $\varepsilon _ j<\operatorname {dist}(\overline\Omega _ j,\mathbb C^ n\setminus \Omega )$. By proposition 5.9.2(2), $p_ j=p*\rho _{\varepsilon _ j}$ is defined, smooth and plurisubharmonic on the open neighborhood $\{ z\in \Omega :\operatorname {dist}(z,\mathbb C^ n\setminus \Omega )>\varepsilon _ j\} $ of $\overline\Omega _ j$, so its complex Hessian is positive semidefinite there, and $p_ j\downarrow p$ pointwise as $j\to \infty $. Put $\varphi _ j=p_ j+q$. Then $\varphi _ j$ is smooth with complex Hessian matrix $H_{\varphi _ j}\ge A\ge 0$ near $\overline\Omega _ j$, so it is plurisubharmonic there by proposition 5.9.2(1), and $\varphi _ j\ge p+q$. Since the dual energy decreases when the matrix increases,

By theorem 6.4.7 there is $u_ j$ on $\Omega _ j$ with $\bar\partial u_ j=\alpha $ and $\int _{\Omega _ j}|u_ j|^2\mathrm e^{-\varphi _ j}\, \mathrm dV\le M$.

Let $K\subset \Omega $ be compact. Then $K\subset \Omega _ k$ for some $k$, and for $j\ge k$ we have $\varphi _ j\le \varphi _ k$ on $K$, because $p_ j$ decreases. Hence $\int _ K|u_ j|^2\, \mathrm dV\le \mathrm e^{\max _ K\varphi _ k}M$ for $j\ge k$, so the sequence $(u_ j)$ is eventually bounded in $L^2(K)$ for every compact set $K$. Let $(K_ m)$ be a compact exhaustion of $\Omega $ (section 2.2). By proposition 6.2.2 and a diagonal argument, we find a subsequence that converges weakly in $L^2(K_ m)$ for every $m$. The weak limits are compatible, because restriction from $K_{m+1}$ to $K_ m$ is a bounded linear map, and they define a function $u\in L^2_{\mathrm{loc}}(\Omega )$; since every compact subset of $\Omega $ lies in some $K_ m$, the subsequence converges weakly to $u$ in $L^2(K)$ for every compact $K\subset \Omega $. Then $u_ j\to u$ in the sense of distributions, and $\bar\partial u=\alpha $.

Now let $K\subset \Omega $ be compact, and let $k$ be so large that $K\subset \Omega _ k$. On $L^2(K)$ the norm with weight $\mathrm e^{-\varphi _ k}$ is equivalent to the unweighted norm, so the subsequence also converges weakly for it, and the lower semicontinuity in proposition 6.2.2 gives

This holds for all large $k$, and $\mathrm e^{-\varphi _ k}$ increases to $\mathrm e^{-p-q}$ as $k\to \infty $; so the monotone convergence theorem gives $\int _ K|u|^2\mathrm e^{-p-q}\, \mathrm dV\le M$, and exhausting $\Omega $ by compact sets proves the estimate. The last assertion follows from the pointwise inequality $|\alpha |_{A^{-1}}^2\le \lambda ^{-1}|\alpha |^2$. $\square$