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Lemma 7.10.1C09E42A43E93

Lemma 7.10.1.C09E42A43E93 Let $D\subset \mathbb C^ n$ be a bounded open set, let $u,v$ be locally bounded plurisubharmonic functions on $D$, and let $s\in \mathbb R$ and $t>0$. If $\{ u<v-s\} \Subset D$, then

\[ t^ n\operatorname {Cap}_ D(\{ u<v-s-t\} ) \le \int _{\{ u<v-s\} }(\mathrm{dd}^{\mathrm{c}}u)^ n. \]

Proof. Let $w\in \mathrm{PSH}(D)$ with $-1\le w\le 0$. Then

\[ \{ u<v-s-t\} \subset \{ u<v-s+tw\} \subset \{ u<v-s\} , \]

so the middle set is relatively compact in $D$. Hence every $x\in \partial D$ has a neighborhood on whose intersection with $D$ we have $u\ge v-s+tw$, and in particular $\varliminf _{D\ni z\to x}\bigl (u(z)-(v(z)-s+tw(z))\bigr )\ge 0$. Expanding $(\mathrm{dd}^{\mathrm{c}}v+t\mathrm{dd}^{\mathrm{c}}w)^ n$ by additivity and symmetry of the products (proposition 7.2.2, theorem 7.3.7), and using that all the mixed products of the locally bounded plurisubharmonic functions $v$ and $w$ are positive, we get $\bigl (\mathrm{dd}^{\mathrm{c}}(v-s+tw)\bigr )^ n\ge t^ n(\mathrm{dd}^{\mathrm{c}}w)^ n$. The comparison principle theorem 7.3.11, applied to $u$ and $v-s+tw$, therefore gives

\[ \begin{aligned} t^ n\int _{\{ u<v-s-t\} }(\mathrm{dd}^{\mathrm{c}}w)^ n & \le \int _{\{ u<v-s+tw\} }\bigl (\mathrm{dd}^{\mathrm{c}}(v-s+tw)\bigr )^ n\\ & \le \int _{\{ u<v-s+tw\} }(\mathrm{dd}^{\mathrm{c}}u)^ n \le \int _{\{ u<v-s\} }(\mathrm{dd}^{\mathrm{c}}u)^ n. \end{aligned} \]

As $w$ ranges over these functions, $w+1$ ranges over the competitors for $\operatorname {Cap}_ D$ (definition 7.4.2). Taking the supremum over $w$ proves the lemma. $\square$

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