2.5. Envelopes [019B]

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2.5. Envelopes

Let $\theta$ be a form as in §2.4.

Proposition 2.13.

[BFJ11, Theorem 7.9]. If $(\varphi_{\alpha})_{\alpha\in A}$ is a family of $\theta$ -psh functions that is uniformly bounded above, then the usc upper envelope $(\sup_{\alpha}\varphi_{\alpha})^{*}$ is also $\theta$ -psh.

Recall that the usc regularization $u^{*}$ of a function $u:X\to[-\infty,+\infty[$ is the smallest usc function such that $u^{*}\geq u$ .

Definition 2.14.

Let $f:X\to[-\infty,+\infty[$ be any function. We define its $\theta$ -psh envelope $P_{\theta}(f)$ as follows. If there does not exist any $\varphi\in\PSH(X,\theta)$ such that $\varphi\leq f$ on $X$ then we set $P_{\theta}(f)\equiv-\infty$ . Otherwise, we define $P_{\theta}(f)$ as the usc upper envelope of the set of all $\theta$ -psh functions $\varphi$ such that $\varphi\leq f$ on $X$ , i.e. we set

P_{\theta}(f):=\left(\sup\left\{\varphi\mid\varphi\in\PSH(X,\omega),\,\varphi\leq f\right\}\right)^{*}.

Thanks to Proposition 2.13 $P_{\theta}(f)$ is either $-\infty$ or belongs to $\PSH(X,\theta)$ . If $f$ is usc, then clearly $P_{\theta}(f)\leq f$ on $X$ , and $P_{\theta}(f)$ is then the largest $\theta$ -psh function with this property.

Proposition 2.15.

[BFJ11, Proposition 8.1]

(i)

$P_{\theta}$ is non-decreasing: $f\leq g\Rightarrow P_{\theta}(f)\leq P_{\theta}(g)$ .
(ii)

$P_{\theta}(f)$ is concave in both arguments:

$P_{t\theta+(1-t)\theta^{\prime}}\left(tf+(1-t)g\right)\geq tP_{\theta}(f)+(1-t)P_{\theta^{\prime}}(g)$

for $0\leq t\leq 1$ .
(iii)

For each $c\in\mathbf{R}$ we have $P_{\theta}(f+c)=P_{\theta}(f)+c$ .
(iv)

$P_{\theta}$ is $1$ -Lipschitz continuous, i.e. $\sup_{X}|P_{\theta}(f)-P_{\theta}(g)|\leq\sup_{X}|f-g|$ .
(v)

Given a bounded function $f$ and a convergent sequence $\theta_{m}\to\theta$ in $N^{1}(\mathcal{X}/S)$ we have $P_{\theta_{m}}(f)\to P_{\theta}(f)$ uniformly on $X$ .