ScalingStacks

Proof. We induct on the depth of the strata. For $p=0$ this follows from Prop. 2.4. So let us assume the bound is achieved for depth $<p$. For a given chart, we consider the local psh function $u_{\beta}$ associated to $u$ and produce the convex average function $\bar{u}_{\beta}$ as in Lemma 2.2. Since a definite neighbourhood of the boundary of the chart lies inside less deep strata, we know that near the boundary $|\bar{u}_{\beta}|\leq C$ by the induction hypothesis and the convexity condition. Using convexity again in the interior of the chart we see $|\bar{u}_{\beta}|\leq C$ in the whole chart.

Within any log scale, by construction the local average $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}(u_{\beta}-\bar{u}_{\beta})=0.$ But $u_{\beta}\leq C$ by $u\leq 0$, hence

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}|u_{\beta}-\bar{u}_{\beta}|\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}(u_{\beta}-\bar{u}_{\beta})_{+}\leq C.$$

Using $|u-u_{\beta}|\leq C$ we conclude the local $L^{1}$-estimate on $u$. ∎