Lemma 5.27.34.0950BB9BC118 Let $\Omega \subset \mathbb C^ n$ be a domain, let $K$ and $L$ be compact subsets of $\Omega $ with $K$ contained in the interior $L^\circ $ of $L$, and put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. Let $b$ be a locally bounded plurisubharmonic function on $\Omega $, and let $T$ be a smooth positive closed $(r,r)$-form, $1\leq r<n$. Then
\[ \int _ K(\mathrm{dd}^{\mathrm{c}}b)^{n-r}\wedge T \leq C_{K,L,n}(1+\| b\| _{L^\infty (L)})^{n-r} \int _ L T\wedge \beta ^{n-r}. \]
Proof.
Put $M=1+\| b\| _{L^\infty (L)}$. Cover $K$ by finitely many balls whose larger concentric balls have closures in $L^\circ $. Replacing $b$ by $b+M$, which has the same Hessian and takes values in $[0,2M]$ on $L$, we may assume $0\leq b\leq 2M$ there. For nested cutoff functions $\psi $, integration by parts against closed forms gives
\[ \int \psi \mathrm{dd}^{\mathrm{c}}b\wedge W =\int b\mathrm{dd}^{\mathrm{c}}\psi \wedge W \leq CM \int _{\operatorname {supp}\psi }\beta \wedge W, \]
where $W$ is $T$ multiplied by the remaining positive closed factors. Apply this successively to the $n-r$ Hessians of $b$. For bounded $b$, the integrations by parts are justified by local decreasing regularization and theorem 5.27.1, with one common bound on the range. Summing over the finite cover gives the asserted constant.
$\square$
Pome
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