Corollary 7.8.8.142C8110B540 Let $u_1,\ldots ,u_ n$ be locally bounded plurisubharmonic functions on an open set $\Omega \subset \mathbb C^ n$, $n\geq 1$, and let $f_1,\ldots ,f_ n\geq 0$ be continuous functions on $\Omega $ such that
\[ (\mathrm{dd}^{\mathrm{c}}u_ i)^ n\geq f_ i\eta ^ n\quad (1\leq i\leq n), \qquad \eta =\mathrm{dd}^{\mathrm{c}}|z|^2. \]
Then
\[ \mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n \geq \left(\prod _{i=1}^ n f_ i^{1/n}\right)\eta ^ n. \]
Proof.
For $n=1$ this is the hypothesis, so let $n\geq 2$. Put $u_{i,\varepsilon }=u_ i*\rho _\varepsilon $, with the same radial mollifiers for all $i$. By the last assertion of proposition 7.8.3, applied to each $u_ i$ with the density $f_ i$,
\[ \det \mathcal H(u_{i,\varepsilon })\geq (f_ i^{1/n}*\rho _\varepsilon )^ n\qquad (1\leq i\leq n). \]
By (7.2.4), applied to the smooth function $\sum _ it_ iu_{i,\varepsilon }$, the exterior powers of smooth forms satisfy
\[ \Bigl (\mathrm{dd}^{\mathrm{c}}\sum _{i=1}^ nt_ iu_{i,\varepsilon }\Bigr )^ n=\det \Bigl (\sum _{i=1}^ nt_ i\mathcal H(u_{i,\varepsilon })\Bigr )\eta ^ n\qquad (t_1,\ldots ,t_ n\in \mathbb R). \]
Both sides are polynomials in $t_1,\ldots ,t_ n$. Since $2$-forms commute, the coefficient of $t_1\cdots t_ n$ on the left is $n!\, \mathrm{dd}^{\mathrm{c}}u_{1,\varepsilon }\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{n,\varepsilon }$, and on the right it is $n!\, D(\mathcal H(u_{1,\varepsilon }),\ldots ,\mathcal H(u_{n,\varepsilon }))\eta ^ n$. Since the Bedford–Taylor product of smooth functions is the exterior product of the forms $\mathrm{dd}^{\mathrm{c}}u_{i,\varepsilon }$ (proposition 7.2.2(1)), we obtain
\[ \mathrm{dd}^{\mathrm{c}}u_{1,\varepsilon }\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_{n,\varepsilon } =D\bigl (\mathcal H(u_{1,\varepsilon }),\ldots ,\mathcal H(u_{n,\varepsilon })\bigr )\eta ^ n \geq \prod _{i=1}^ n(f_ i^{1/n}*\rho _\varepsilon )\, \eta ^ n, \]
where the inequality is lemma 7.8.6, applied to the complex Hessians of the smooth plurisubharmonic functions $u_{i,\varepsilon }$, which are positive semidefinite (proposition 5.9.2). As $\varepsilon \downarrow 0$, the density on the right converges locally uniformly to $\prod _ if_ i^{1/n}$, and the left side converges weakly to $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$ by theorem 7.3.7, since $u_{i,\varepsilon }\downarrow u_ i$ for each $i$ (proposition 5.9.2).
$\square$
Pome
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