Lemma 16.5.26.8C85369857E4 Let $\varphi ,\psi ,\gamma \in \mathcal{E}(X,\theta ;\phi )$. Assume that $\gamma \geq \varphi \lor \psi $. Then
\[ \int _ X (\gamma -\varphi )\, \theta _{\psi }^ n\leq 2\int _ X (\gamma -\varphi )\, \theta _{\varphi }^ n+2\int _ X (\gamma -\psi )\, \theta _{\psi }^ n. \]
Proof.
Observe that
6786C4A2C71B
16.5.27
\begin{equation} \{ \gamma >\varphi +2t\} \subseteq \{ \gamma >\psi +t \} \cup \{ \psi >\varphi +t\} \, . \end{equation}
So
\[ \begin{aligned} \int _ X (\gamma -\varphi )\, \theta _{\psi }^ n =& 2\int _0^{\infty } \int _{\{ \gamma >\varphi +2t\} }\, \theta _{\psi }^ n\, \mathrm{d}t \\ \leq & 2\int _0^{\infty } \int _{\{ \gamma >\psi +t\} }\, \theta _{\psi }^ n\, \mathrm{d}t+ 2\int _0^{\infty } \int _{\{ \psi >\varphi +t\} }\, \theta _{\psi }^ n\, \mathrm{d}t\\ \leq & 2\int _ X (\gamma -\psi ) \, \theta _{\psi }^ n+2\int _0^{\infty } \int _{\{ \psi >\varphi +t\} }\, \theta _{\varphi }^ n\, \mathrm{d}t\\ \leq & 2\int _ X (\gamma -\psi ) \, \theta _{\psi }^ n+2\int _0^{\infty } \int _{\{ \gamma >\varphi +t\} }\, \theta _{\varphi }^ n\, \mathrm{d}t\\ =& 2\int _ X (\gamma -\psi ) \, \theta _{\psi }^ n+2\int _ X (\gamma -\varphi ) \, \theta _{\varphi }^ n, \end{aligned} \]
where the second line follows from (16.5.27), while the third line follows from proposition 16.5.5.
$\square$
Pome
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