PomeEADDEAF9FAD9Corollary 7.8.4.EADDEAF9FAD9 Let $u,v$ be locally bounded plurisubharmonic functions on $\Omega $, and let $f,g\ge 0$ be continuous functions on $\Omega $ such that $(\mathrm{dd}^{\mathrm{c}}u)^ n\ge f\eta ^ n$ and $(\mathrm{dd}^{\mathrm{c}}v)^ n\ge g\eta ^ n$. Then, for each integer $1\le k\le n-1$,
For $k=0$ and $k=n$ the same inequality is the hypothesis.
Proof. Let $u_\varepsilon =u*\rho _\varepsilon $ and $v_\varepsilon =v*\rho _\varepsilon $, with the same mollifiers for both functions. By the last assertion of proposition 7.8.3, applied to $u$ with the density $f$ and to $v$ with the density $g$,
For real $t$ and $s$, (7.2.4), applied to the smooth function $tu_\varepsilon +sv_\varepsilon $, gives $(t\mathrm{dd}^{\mathrm{c}}u_\varepsilon +s\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ n=\det \bigl (t\mathcal H(u_\varepsilon )+s\mathcal H(v_\varepsilon )\bigr )\eta ^ n$ for the exterior powers of smooth forms. Since $2$-forms commute, the left side can be expanded by the binomial formula. Comparing the coefficients of $\binom nkt^ ks^{n-k}$, and recalling that Bedford–Taylor products of smooth functions are exterior products of smooth forms (proposition 7.2.2(1)), we find
where the inequality is the second assertion of lemma 7.8.1, applied to the matrices $\mathcal H(u_\varepsilon )$ and $\mathcal H(v_\varepsilon )$, which are positive semidefinite by proposition 5.9.2. As $\varepsilon \downarrow 0$, the density on the right converges locally uniformly to $f^{k/n}g^{(n-k)/n}$, while the left side converges weakly to $(\mathrm{dd}^{\mathrm{c}}u)^ k\wedge (\mathrm{dd}^{\mathrm{c}}v)^{n-k}$ by theorem 7.3.7, since $u_\varepsilon $ and $v_\varepsilon $ are smooth plurisubharmonic functions decreasing to $u$ and $v$ (proposition 5.9.2). $\square$
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