Proposition 16.27.1204DE884FD6DB

Proof. All capacities involved are inner regular (proposition 16.25.6), so it suffices to treat compact sets $K$: the inequalities for a Borel set $E$ follow by applying them to the compact subsets of $E$ and taking suprema. If $K$ is pluripolar, all capacities of $K$ vanish. Otherwise fix a smooth function $g\geq 0$ with zero set $K$, and apply lemma 16.27.5 with this $g$ to $(\theta ,\phi )$ and to $(\kappa ,0)$; here $0=V_\kappa $ is a model potential for $\kappa $ of mass $\Omega $, by proposition 16.21.2(4). Denote the resulting envelopes by $S_\phi $ and $S_0$ and their suprema by $L_\phi $ and $L_0$. Every competitor for $P_\theta (\phi +jg)$ is $\kappa $-plurisubharmonic, because $\theta \leq \kappa $, and it is at most $\phi +jg\leq jg$ quasi-everywhere, so $P_\theta (\phi +jg)\leq P_\kappa (jg)$ for every $j$; hence $S_\phi \leq S_0$ and $L_\phi \leq L_0$. The two estimates of lemma 16.27.5 give

This proves (16.27.14) and, for $\theta =\kappa $, the first inequality in (16.27.13).

For the upper bound let $\theta =\kappa $, and choose $b>1$ with $b^ n(\Omega -m)<\Omega $. By theorem 16.22.4, applied to $\phi \preceq 0$, where the mass of $0$ is $\int _ X\kappa ^ n=\Omega $, the envelope $\eta =E_ b(\phi ,0)=P_\kappa (b\phi )$ is a potential, and it satisfies $b^{-1}\eta \leq \phi $; in particular $\eta =-\infty $ at the poles of $\phi $. Write $c=1-b^{-1}>0$. For fixed $j$ and $\delta >0$ consider

a convex combination of $\kappa $-plurisubharmonic functions minus a constant. The function $P_\kappa (jg)$ is upper semicontinuous and vanishes on $K$, because $0\leq P_\kappa (jg)\leq jg$. Hence $P_\kappa (jg)<\delta /c$ on a neighborhood $U$ of $K$, and there $v_ j<b^{-1}\eta \leq \phi $. Outside the poles of $\phi $, the difference $v_ j-\phi $ is bounded above by $cj\sup _ Xg$, and $g$ has a positive minimum on $X\setminus U$. Therefore $v_ j\leq \phi +lg$ for a sufficiently large integer $l$, so $v_ j\leq P_\kappa (\phi +lg)\leq S_\phi $. Letting $\delta \downarrow 0$ and then $j\to \infty $, and using that the increasing limit of $P_\kappa (jg)$ equals $S_0$ almost everywhere, we obtain

almost everywhere, hence everywhere.

Choose a Borel set $G$ of positive volume on which $b^{-1}\eta \geq -D$ for a constant $D$; such a set exists because $\eta $ is finite almost everywhere. By proposition 10.3.1, there is a constant $B$, independent of $K$, with $\int _ X|S_0-L_0|\, \kappa ^ n\leq B$. Hence $S_0\geq L_0-B/\int _ G\kappa ^ n$ at some point $x\in G$, for otherwise $\int _ G(L_0-S_0)\, \kappa ^ n>B$. At this point the last display gives $L_\phi \geq S_\phi (x)\geq cL_0-D'$ with $D'=cB/\int _ G\kappa ^ n+D$, which does not depend on $K$. If $L_0\geq \max \{ 1,2D'/c\} $, then $L_\phi \geq cL_0/2$, and lemma 16.27.5 gives

the last inequality is the lower bound (16.27.6) for the construction with reference potential $0$, which gives $\Omega /L_0^ n\leq \operatorname {Cap}_\kappa (K)$. In the remaining range $L_0<\max \{ 1,2D'/c\} $, the same lower bound shows that $\operatorname {Cap}_\kappa (K)$ is bounded below by a positive constant independent of $K$, and we use $\operatorname {Cap}_{\kappa ,\phi }(K)\leq m$, enlarging $A$ if necessary. The constant $A$ in the upper bound depends on $\phi $. $\square$