Corollary 7.3.88BD7803A393B

Proof. Since weak convergence is a local property (lemma 7.3.6), it suffices to prove the convergence on every relatively compact open set $W\Subset \Omega $. On $W$ all the functions $v_{i,j}$ and $v_ i$ are bounded, because $v_ i\le v_{i,j}\le v_{i,1}$. Adding a constant $c_ i$ to each $v_{i,j}$ and to $v_ i$ replaces $P$ by the polynomial $P(x_1-c_1,\ldots ,x_ m-c_ m)$; so we may assume that all these functions are nonnegative on $W$. Since the weighted products depend linearly on the weight, it suffices to treat a monomial. A constant weight is covered by the unweighted assertion of the theorem. A monomial of degree $N\ge 1$ is a product $a_1\cdots a_ N$, where each $a_ l$ is one of the $v_ i$, repetitions being allowed; we write $a_{l,j}$ for the corresponding function $v_{i,j}$. For real numbers $x_1,\ldots ,x_ N$ we have the polarization identity

To prove it, expand $(\sum _{l\in I}x_ l)^ N=\sum _\alpha \frac{N!}{\alpha !}x^\alpha $, the sum running over the multi-indices $\alpha \in \mathbb N^ N$ with $|\alpha |=N$ and $\alpha _ l=0$ for $l\notin I$. Fix such an $\alpha $, and let $J=\{ l:\alpha _ l>0\} $ have $k$ elements. The monomial $x^\alpha $ occurs exactly in the terms with $I\supset J$, so its total coefficient on the right-hand side is

This vanishes unless $k=N$, in which case $\alpha =(1,\ldots ,1)$ and the coefficient is $N!$.

Each sum $s=\sum _{l\in I}a_ l$ is a bounded nonnegative plurisubharmonic function on $W$, and so is $s^ N$. For $N=1$ there is nothing to prove; for $N\ge 2$ and smooth $s$ this follows from

in which both terms are positive $(1,1)$-forms, the first because $s\ge 0$ and $s$ has a positive semidefinite complex Hessian, and the second by (7.3.1); hence $s^ N$ is plurisubharmonic, by both directions of proposition 5.9.2. In general, the regularizations $s*\rho _\varepsilon $ are smooth, plurisubharmonic and nonnegative, and they decrease to $s$ (proposition 5.9.2); since $t\mapsto t^ N$ is increasing on $[0,\infty )$, the plurisubharmonic functions $(s*\rho _\varepsilon )^ N$ decrease to $s^ N$, which is therefore plurisubharmonic by theorem 5.10.1. Likewise the sums $s_ j=\sum _{l\in I}a_{l,j}$ are nonnegative and decrease to $s$, so the $s_ j^ N$ are bounded plurisubharmonic functions on $W$ decreasing to $s^ N$. The weighted assertion of the theorem therefore applies with the weights $s_ j^ N$, and by (7.3.9), with $x_ l=a_{l,j}$, it applies to the monomial weights $a_{1,j}\cdots a_{N,j}$.

For the second assertion, let $a$ and $b$ be bounded plurisubharmonic functions on a neighborhood $U$ of $\overline B$ which are equal on a collar $V$ of $\partial B$. Let $d>0$ be smaller than the distances from $\partial B$ to $\mathbb C^ n\setminus V$ and from $\overline B$ to $\mathbb C^ n\setminus U$, and put $V'=\{ z:\operatorname {dist}(z,\partial B)<d/2\} $. For $\varepsilon =\varepsilon _ k\downarrow 0$ with $\varepsilon _ k<d/2$, the regularizations $a_\varepsilon =a*\rho _\varepsilon $ and $b_\varepsilon =b*\rho _\varepsilon $ are smooth and plurisubharmonic on a neighborhood of $\overline B$ and decrease to $a$ and $b$ (proposition 5.9.2); moreover $a_\varepsilon =b_\varepsilon $ on the collar $V'$, because $a=b$ on the ball $B(z,\varepsilon )\subset V$ for $z\in V'$. By lemma 7.3.3, applied to the smooth function $a_\varepsilon $ and the bounded plurisubharmonic function $b_\varepsilon $,

Choose $\chi \in C^\infty _ c(B)$ equal to $1$ on $B\setminus V'$. On $V'$ the measures $a_\varepsilon \mathrm{dd}^{\mathrm{c}}b_\varepsilon \wedge R$ and $b_\varepsilon \mathrm{dd}^{\mathrm{c}}a_\varepsilon \wedge R$ coincide, by locality, so the identity is equivalent to $\int \chi a_\varepsilon \, \mathrm{dd}^{\mathrm{c}}b_\varepsilon \wedge R=\int \chi b_\varepsilon \, \mathrm{dd}^{\mathrm{c}}a_\varepsilon \wedge R$. By the weighted assertion of the theorem with one factor and the fixed base current $R$, this passes to the limit: $\int \chi a\, \mathrm{dd}^{\mathrm{c}}b\wedge R=\int \chi b\, \mathrm{dd}^{\mathrm{c}}a\wedge R$. Since $a=b$ on $V$, the measures $a\, \mathrm{dd}^{\mathrm{c}}b\wedge R$ and $b\, \mathrm{dd}^{\mathrm{c}}a\wedge R$ also coincide on $V'$, and adding the integrals against $1-\chi $ over $B$ gives the identity for $a$ and $b$. $\square$