Lemma 10.31.53.F816A9EF0F9D Let $u\in \mathcal E_\varphi $ with $u\leq \varphi $, and let $\chi :\mathbb R_{\leq 0}\to \mathbb R_{\leq 0}$ be increasing and concave, with $\chi (0)=0$. Suppose that $\chi $ is $C^{1,1}$ on compact intervals and that, for a finite constant $L\geq 1$,
\[ 1\leq \chi '\leq L. \]
Then the envelope
\[ p=P_\theta (\varphi +\chi (u-\varphi )) \]
belongs to $\mathcal E_\varphi $, and
\[ \mu _ p\leq \mathbf1_{\{ p=\varphi +\chi (u-\varphi )\} } \chi '(u-\varphi )^ n\mu _ u. \]
Here the obstacle and the contact set are defined off the pluripolar set where $u-\varphi $ is undefined, which all these measures ignore.
Proof.
For $t\leq 0$, the bounds on $\chi '$ give $Lt\leq \chi (t)\leq t$. Thus
\[ Q_ L(u,\varphi )\leq p\leq u\leq \varphi . \]
The lower potential belongs to $\mathcal E_\varphi $ by lemma 16.28.63, so $p$ is not identically $-\infty $ and belongs to $\mathcal E_\varphi $ by upward stability. The obstacle is finite and quasi-continuous off the poles, so the contact theorem theorem 15.8.6 applies.
Let $\tau =\chi ^{-1}$. It is convex, with $L^{-1}\leq \tau '\leq 1$. We justify the transformation
\[ w=\varphi +\tau (p-\varphi ) \]
without assuming that $\varphi $ is locally bounded on some open set. On a coordinate ball choose a smooth function $q$ with $\theta +\mathrm{dd}^{\mathrm{c}}q\geq 0$. The functions
\[ p_ m=\max (p,q-m),\qquad \varphi _ m=\max (\varphi ,q-m) \]
are locally bounded $\theta $-plurisubharmonic functions on the ball, and $p_ m\leq \varphi _ m$. Lemma 5.28.13 applies to $w_ m=\varphi _ m+\tau (p_ m-\varphi _ m)$ and gives
\[ \theta _{w_ m}\geq 0,\qquad \mu _{w_ m}\geq \tau '(p_ m-\varphi _ m)^ n\mu _{p_ m}. \]
The function $(s,t)\mapsto t+\tau (s-t)$ on $\{ s\leq t\} $ is nondecreasing in each variable. Consequently $w_ m$ decreases as $m$ increases. Since $p\leq p_ m\leq w_ m\leq \varphi _ m$, the limit is not identically $-\infty $ on the ball. Off the poles it equals $w$. The local limits therefore agree on overlaps after regularization and define a global $\theta $-plurisubharmonic function.
On $D_ m=\{ p>q-m\} $, neither $p$ nor $\varphi $ is truncated. Thus $w_ m=w$ there, and plurifine locality transfers the product inequality to $D_ m$. These sets exhaust the complement of the poles of $p$ in the ball. Since all products ignore the discarded pluripolar set,
\[ \mu _ w\geq \tau '(p-\varphi )^ n\mu _ p. \]
The obstacle bound $p\leq \varphi +\chi (u-\varphi )$ gives $w\leq u$, with equality on the contact set of $p$. Ordered contact, the identity $\tau '(p-\varphi )=1/\chi '(u-\varphi )$ on that set, and contact support of $\mu _ p$ prove the assertion.
$\square$
Pome
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