ScalingStacks

Proof. Let the global maximum of $u$ be achieved at $q_{0}\in E_{J}^{0}$, and denote the local potential of $u$ as $u_{\beta}$. Without loss of generality $u_{\beta}\leq 0$. We have $u_{\beta}(q_{0})\geq-C$ since $|u-u_{\beta}|\leq C$. Applying the mean value inequality around $q_{0}$, we find that the local average function $\bar{u}_{\beta}$ produced in Lemma 2.2 satisfies $\sup\bar{u}_{\beta}\geq-C$ for another uniform constant $C$. By the convexity of $\bar{u}_{\beta}$ its sup is almost achieved at the boundary of the chart, which is contained in a union of less deep strata $E_{J^{\prime}}^{0}$ with $J^{\prime}\subsetneq J$. Thus we can find a point $q^{\prime}$ with $u(q^{\prime})\geq-C$ that belongs to a less deep stratum; an induction shows that there is some $i\in I$, such that $\sup_{E_{i}^{0}}u\geq-C$.

For the $L^{1}$-bound we recall the following Harnack inequality argument. Suppose a coordinate ball $B(q,3R)$ is contained in a local chart in a small neighbourhood of $E_{i}^{0}$. Applying the mean value inequality to the local psh function associated to $u$, we see for $y\in B(q,R)$ that

$$u(y)\leq C+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(y,2R)}u\lesssim 1+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(q,R)}u.$$

hence the Harnack inequality

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(q,R)}|u|\lesssim 1+\inf_{B(q,R)}(-u).$$

Applying this to a chain of balls connecting any two points in $E_{i}^{0}$ gives the $L^{1}$-bound $\int_{E_{i}^{0}}u\omega_{\mathcal{X}}|_{X_{t}}^{n}\geq-C^{\prime}$; the bound is uniform because the number of balls involved in the chain can be controlled independent of $t$. ∎