Lower bounds $\langle T_ i^ n\rangle \geq f_ i\nu $ by densities with respect to a common measure $\nu $ pass to mixed products, as they do for bounded potentials (theorem 7.12.1): we apply the bounded inequality to the truncated potentials and restrict to the set where no truncation is active. A pluripolar set charged by $\nu $ causes no difficulty, because there the densities $f_ i$ vanish $\nu $-almost everywhere.
Proposition 13.5.8.A93A2F4C95C2 Let $T_1,\ldots ,T_ n$ be closed positive $(1,1)$-currents on a compact Kähler manifold $X$ of dimension $n\geq 1$, not necessarily distinct. Let $\nu $ be a positive Radon measure on $X$, and let $f_1,\ldots ,f_ n$ be nonnegative Borel functions on $X$ such that
\[ \langle T_ i^ n\rangle \geq f_ i\nu \qquad (1\leq i\leq n). \]
Then
\[ \langle T_1\wedge \cdots \wedge T_ n\rangle \geq \left(\prod _{i=1}^ n f_ i^{1/n}\right)\nu . \]
All the products are finite by theorem 13.5.4; in particular each $f_ i$ is $\nu $-integrable, and so is $\prod _ if_ i^{1/n}$.
Proof.
For $n=1$ the conclusion is the hypothesis, so let $n\geq 2$. Work on a relatively compact coordinate domain on which $T_ i=\mathrm{dd}^{\mathrm{c}}a_ i$ with plurisubharmonic potentials $a_ i$ bounded above, and put
\[ a_{i,j}=\max \{ a_ i,-j\} ,\qquad G_ j=\bigcap _{i=1}^ n\{ a_ i>-j\} . \]
Since $G_ j$ is contained in the level-$j$ set $\{ a_ i>-j\} $ of the tuple $(a_ i,\ldots ,a_ i)$, lemma 13.2.2 shows that the bounded measure $(\mathrm{dd}^{\mathrm{c}}a_{i,j})^ n$ agrees with $\langle T_ i^ n\rangle $ on $G_ j$. By positivity, therefore, on the whole coordinate domain
\[ (\mathrm{dd}^{\mathrm{c}}a_{i,j})^ n\geq \mathbf1_{G_ j}(\mathrm{dd}^{\mathrm{c}}a_{i,j})^ n\geq \mathbf1_{G_ j}f_ i\nu \qquad (1\leq i\leq n). \]
Now theorem 7.12.1, applied to the densities $\mathbf1_{G_ j}f_ i$, gives $\mathrm{dd}^{\mathrm{c}}a_{1,j}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}a_{n,j}\geq \mathbf1_{G_ j}\prod _ if_ i^{1/n}\nu $. Since $G_ j$ is the level-$j$ set of the tuple $(a_1,\ldots ,a_ n)$, the left side agrees on $G_ j$ with the non-pluripolar product of the $T_ i$, by lemma 13.2.2. Restricting to $G_ j$ and using positivity (lemma 13.2.1), we obtain
\[ \langle T_1\wedge \cdots \wedge T_ n\rangle \geq \mathbf1_{G_ j}\prod _ i f_ i^{1/n}\nu . \]
The complement of $\bigcup _ jG_ j$ is the union $P$ of the pole sets of the $a_ i$. Every $\langle T_ i^ n\rangle $ gives zero mass to the whole pluripolar set $P$, not only to the pole set of $a_ i$, so the inequality $\langle T_ i^ n\rangle \geq f_ i\nu $ forces $f_ i=0$ $\nu $-almost everywhere on $P$, for every $i$. Hence $\mathbf1_{G_ j}\prod _ if_ i^{1/n}$ increases to $\prod _ if_ i^{1/n}$ $\nu $-almost everywhere as $j\to \infty $, and monotone convergence on every Borel subset of the coordinate domain removes the indicator. Since finitely many such coordinate domains cover $X$, and an inequality of measures that holds on each set of a finite open cover holds on $X$ (decompose a Borel set into Borel pieces, each contained in one set of the cover), the asserted inequality follows.
Finally, $\int _ Xf_ i\, \mathrm d\nu \leq \int _ X\langle T_ i^ n\rangle <\infty $. To integrate the product, we use the inequality between weighted geometric and arithmetic means: for $y_1,\ldots ,y_ m\in [0,\infty ]$ and weights $\lambda _ i>0$ with $\sum _ i\lambda _ i=1$,
6B41F6267BFA
13.5.9
\begin{equation} \prod _ iy_ i^{\lambda _ i}\leq \sum _ i\lambda _ iy_ i . \end{equation}
If all $y_ i$ are positive and finite, this is lemma 2.2.4 for the exponential function on the probability space $\{ 1,\ldots ,m\} $ with masses $\lambda _ i$, on which the integrand takes the values $\log y_ i$; otherwise the left side vanishes or the right side is infinite. With $\lambda _ i=1/n$, it gives $\int _ X\prod _ if_ i^{1/n}\, \mathrm d\nu \leq \frac1n\sum _ i\int _ Xf_ i\, \mathrm d\nu <\infty $.
$\square$
Corollary 13.5.10.7FDDF6925B80 Let $T,S$ be closed positive $(1,1)$-currents on a compact Kähler manifold $X$ of dimension $n\geq 1$. Let $\nu $ be a positive Radon measure on $X$, and let $f,g$ be nonnegative Borel functions on $X$ such that
\[ \langle T^ n\rangle \geq f\nu ,\qquad \langle S^ n\rangle \geq g\nu . \]
Then for every integer $1\leq k\leq n-1$,
\[ \langle T^ k\wedge S^{n-k}\rangle \geq f^{k/n}g^{(n-k)/n}\nu . \]
The endpoint cases $k=0$ and $k=n$ are the hypotheses; for $n=1$ there are no other cases.
Proof.
For $t=0$ and $t=1$ this is the hypothesis. Otherwise, multilinearity (proposition 13.2.6) expands the left side as $\sum _{k=0}^ n\binom nk(1-t)^ kt^{n-k}\langle T^ k\wedge S^{n-k}\rangle $. Bounding the terms with $1\leq k\leq n-1$ by corollary 13.5.10 and the terms with $k=0,n$ by the hypotheses, we obtain
\[ \left\langle \big ((1-t)T+tS\big )^ n\right\rangle \geq \sum _{k=0}^ n\binom nk(1-t)^ kt^{n-k}f^{k/n}g^{(n-k)/n}\, \nu =\big ((1-t)f^{1/n}+tg^{1/n}\big )^ n\nu \]
by the binomial formula. The measure on the right is finite, because $f$ and $g$ are $\nu $-integrable by the hypotheses and each density $f^{k/n}g^{(n-k)/n}$ is at most $\frac knf+\frac{n-k}ng\leq f+g$: for $k=0$ and $k=n$ this is an equality, and for $1\leq k\leq n-1$ it is (13.5.9).
$\square$
Pome
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