Lemma 7.5.7880C30054932

Proof. (1) Let $B=B(a,R)$, fix $K$, and choose $R'<R$ with $K\subset B'=B(a,R')$. Let $M\ge 1$ be a common bound for $|u|$ on $B'$, $u\in \mathcal U$. For $u_1,\ldots ,u_ q\in \mathcal U$ put $v_ l=(u_ l+M)/(2M)$ and $\phi =|z-a|^2/R'^2$; these functions take values in $[0,1]$ on $B'$, and so does

which is therefore a competitor for $\operatorname {Cap}_{B'}$. Expand $(\mathrm{dd}^{\mathrm{c}}F)^ n$ by additivity, positive homogeneity and symmetry (proposition 7.2.2, theorem 7.3.7). All terms are positive measures, and those in which each $\mathrm{dd}^{\mathrm{c}}v_ l$ occurs once and $\mathrm{dd}^{\mathrm{c}}\phi $ occurs $n-q$ times add up to $c_{n,q}\, \mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge (\mathrm{dd}^{\mathrm{c}}\phi )^{n-q}$, where $c_{n,q}=n^{-n}\, n!\, (n-q)^{n-q}/(n-q)!>0$ (with $0^0=1$) is the corresponding multinomial coefficient. Hence $\mathrm{dd}^{\mathrm{c}}v_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}v_ q\wedge (\mathrm{dd}^{\mathrm{c}}\phi )^{n-q}\le c_{n,q}^{-1}(\mathrm{dd}^{\mathrm{c}}F)^ n$. Since $\mathrm{dd}^{\mathrm{c}}\phi =\beta /(\pi R'^2)$ and $\mathrm{dd}^{\mathrm{c}}u_ l=2M\mathrm{dd}^{\mathrm{c}}v_ l$, it follows that $\int _ E\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge \beta ^{n-q}\le C\operatorname {Cap}_{B'}(E)$ for Borel $E\subset B'$. The bound for the coefficient measures follows from theorem 5.3.7, and proposition 7.4.3(4), applied to $B'\Subset B$ and $K$, replaces $\operatorname {Cap}_{B'}$ by $\operatorname {Cap}_ B$ on subsets of $K$.

(2) The weak limit $S$ is a positive current (theorem 5.3.7). Let $\sigma _ j$ be either the trace measure or a coefficient measure of $S_ j$, and $\sigma $ the corresponding measure of $S$; then $\int \psi \, \mathrm d\sigma _ j\to \int \psi \, \mathrm d\sigma $ for every test function $\psi $, since this integral is the value of $S_ j$ on a test form. Fix a continuous function $\chi $ with compact support in $B$ and a compact set $K\subset B$ with $\operatorname{Supp}\chi \subset \operatorname{Int}K$, and let $C$ be the constant of (1) for $K$; in particular $|\sigma _ j|(K)\le C\operatorname {Cap}_ B(K)$, which is finite by proposition 7.4.3(1). A similar bound holds for $\sigma $ on open subsets $U\subset \operatorname{Int}K$: for the trace measure, which is the vague limit of the trace measures of $S_ j$ (section 5.3), proposition 2.6.2 gives $\sigma (U)\le \varliminf _ j\sigma _ j(U)\le C\operatorname {Cap}_ B(U)$, and the total variations of the coefficient measures of the positive current $S$ are bounded by a constant multiple of its trace measure.

Since the total variations $|\sigma _ j|(K)$ are bounded, the convergence $\int \psi \, \mathrm d\sigma _ j\to \int \psi \, \mathrm d\sigma $ extends to continuous functions $\psi $ with support in $\operatorname{Int}K$: approximating $\psi $ uniformly by test functions supported in a fixed compact subset of $\operatorname{Int}K$ (lemma 2.5.1(5)) changes the integrals by at most $\sup |\psi -\psi _\varepsilon |$ times a bound for $|\sigma _ j|(K)$ and $|\sigma |(K)$.

Consider first the fixed weight $f$, and let $\eta >0$. Since $f$ is quasi-continuous, there is an open set $G\subset B$ with $\operatorname {Cap}_ B(G)<\eta $ such that $f$ is continuous on $K\setminus G$; let $g$ be a continuous extension of $f|_{K\setminus G}$ to $\mathbb C^ n$ with $\sup |g|\le \sup _ K|f|$ (lemma 7.5.3). Then

and similarly for $\sigma $, using the open set $G\cap \operatorname{Int}K$. Since $\chi g$ is continuous with support in $\operatorname{Int}K$, we have $\int \chi g\, \mathrm{d}\sigma _ j\to \int \chi g\, \mathrm{d}\sigma $. Hence $\varlimsup _ j|\int \chi f\, \mathrm{d}\sigma _ j-\int \chi f\, \mathrm{d}\sigma |\le 2C'\eta $ for every $\eta >0$, which proves $f\sigma _ j\to f\sigma $.

For varying weights, let $M_ K$ bound $|f_ j|$ and $|f|$ on $K$, and let $\delta >0$. By (1),

As $j\to \infty $, the second term tends to zero by convergence in capacity; then we let $\delta \downarrow 0$. Together with the fixed-weight case, this proves $f_ j\sigma _ j\to f\sigma $, and hence $f_ jS_ j\to fS$. $\square$