PomeLemma 16.9.4.2490CFA511EC Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then
\[ d_ S(\varphi ,\psi )\leq d_ S(\varphi ,\varphi \lor \psi )+d_ S(\psi ,\varphi \lor \psi ) \leq C_ n d_ S(\varphi ,\psi ), \]
where $C_ n=3(n+1)2^{n+2}$.
Lemma 16.9.42490CFA511EC
Proof. We claim that
FE54EE184B7A
16.9.5
\begin{equation} \ell ^{\varphi }\lor \ell ^{\psi }=\ell ^{\varphi \lor \psi }. \end{equation}
Recall that $\lor $ is defined in definition 17.10.85. Note that this assertion implies our desired inequality by 17.10.90.
In proving this assertion, we may assume that $\varphi ,\psi \leq 0$ since
\[ \ell ^{\varphi +C}=\ell ^{\varphi },\quad \ell ^{\psi +C}=\ell ^{\psi },\quad \ell ^{(\varphi +C)\lor (\psi +C)}=\ell ^{\varphi \lor \psi } \]
for any $C\in \mathbb {R}$.
In fact, it is clear that
\[ \ell ^{\varphi }\leq \ell ^{\varphi \lor \psi },\quad \ell ^{\psi }\leq \ell ^{\varphi \lor \psi }, \]
so the $\leq $ direction in (16.9.5) holds.
Conversely, if $\ell '\in \mathcal{R}^1(X,\theta )$ and $\ell '\geq \ell ^{\varphi }\lor \ell ^{\psi }$, then for each $t\geq 0$,
\[ \ell '_ t\geq \Bigl ((V_{\theta }-t)\lor \varphi \Bigr ) \lor \Bigl ((V_{\theta }-t)\lor \psi \Bigr )=(V_{\theta }-t)\lor (\varphi \lor \psi ). \]
See (17.10.96) for the first inequality. Therefore, thanks to the definition of the Perron envelopes (cf. (17.10.21)), we have
\[ \ell '_ s\geq \ell ^{\varphi \lor \psi ,t}_ s \]
for any $0\leq s\leq t$. It follows from (17.10.96) that $\ell '_ s\geq \ell ^{\varphi \lor \psi }_ s$ for any $s\geq 0$. $\square$
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