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Corollary 18.28.17CE1417F951E3

Corollary 18.28.17.CE1417F951E3 Let $f\colon K\rightarrow \mathbb {R}$ be a bounded function, and $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$. Then

A99869D43853 18.28.18
\begin{equation} P_{\theta ,K}[\varphi ](f)=P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \end{equation}

Proof. Take any $\eta \in \mathrm{PSH}(X,\theta )$ with $\eta \preceq \varphi $ and $\eta |_ K\leq f$ quasi-everywhere, then

\[ \eta \leq P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[V_{\theta }](f). \]

Therefore,

\[ \eta \leq P_{\theta ,X}[\varphi ]\left( P_{\theta ,K}[V_{\theta }](f) \right). \]

So we conclude the $\leq $ direction in (18.28.18).

For the reverse direction, take $\eta \in \mathrm{PSH}(X,\theta )_{>0}$ such that

\[ \eta \preceq \varphi ,\quad \eta \leq P_{\theta ,K}[V_{\theta }](f) \quad \textup{quasi-everywhere}. \]

Then $\eta |_ K\leq f$ quasi-everywhere by proposition 18.28.6. Hence

\[ \eta \leq P_{\theta ,K}[\varphi ](f). \]

By corollary 18.28.13, the $\geq $ direction in (18.28.18) follows. $\square$

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