Pome

Lemma 18.28.655C6401E18EB3

Lemma 18.28.65.5C6401E18EB3 Let $v\in C^0(K)$, $\varphi \in \mathrm{PSH}(X,\theta )$, and $\eta \in \mathrm{PSH}(X,\theta )$. Assume that $\eta \preceq \varphi $. Then

2DBAE7A20C99 18.28.66
\begin{equation} \sup _ K (\eta -v) = \sup _{\{ \eta \neq -\infty \} } \left(\eta - P'_{\theta ,K}[\varphi ](v)\right)=\sup _{\left\{ P'_{\theta ,K}[\varphi ](v)\neq -\infty \right\} } \left(\eta - P'_{\theta ,K}[\varphi ](v)\right). \end{equation}

Proof. We prove the first equality first. We write $S=\{ \eta =-\infty \} $.

By definition, $P'_{\theta ,K}[\varphi ](v)|_ K\leq v$, so

\[ \left.\left(\eta -P'_{\theta ,K}[\varphi ](v)\right)\right|_{K\setminus S}\geq \eta |_{K\setminus S}-v|_{K\setminus S}\, . \]

This implies that

\[ \sup _ K (\eta -v) \leq \sup _{X\setminus S} \left(\eta - P'_{\theta ,K}[\varphi ](v)\right). \]

Conversely, observe that $\sup _ K(\eta -v)>-\infty $ as $K$ is non-pluripolar. It follows that $\eta -\sup _ K (\eta -v)$ is a candidate in the definition of $P'_{\theta ,K}[\varphi ](v)$. Hence,

3504DA07F50F 18.28.67
\begin{equation} \eta -\sup _ K (\eta -v)\leq P'_{\theta ,K}[\varphi ](v). \end{equation}

This implies that

\[ \sup _ K (\eta -v) \geq \sup _{X\setminus S} \left(\eta - P'_{\theta ,K}[\varphi ](v)\right), \]

finishing the proof of the first identity in (18.28.66).

In order to prove the latter identity, observe that

\[ \left\{ P'_{\theta ,K}[\varphi ](v) =-\infty \right\} \subseteq S \]

as a consequence of (18.28.67). On $\{ P'_{\theta ,K}[\varphi ](v)\neq -\infty \} \cap S$, we have $\eta -P'_{\theta ,K}[\varphi ](v)=-\infty $. Hence the last equality of (18.28.66) also follows. $\square$

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