ScalingStacks

Proof. (cf. proof of Prop. 3.1 in [2]) Consider the local potentials $\phi=\varphi_{m}$ on various coordinate charts in section 3.1, both of the toric type and of the boundary type. The charts can be chosen so that the Lebesgue measures thereof are uniformly equivalent to $d\mu_{s}$ up to a scaling factor. We have $|\phi-\varphi|\leq C$ uniformly on charts. Suppose a coordinate ball $B(p,3R)$ is contained in (the universal cover of) the local chart. Since $\phi$ is psh and $\phi-C\leq 0$, for $z\in B(p,R)$,

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

hence

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

To deduce the global version of the Harnack type inequality we need a transitivity property, namely we can connect the chart containing the maximum point of $\varphi$ to any of the toric charts in $X_{s}^{toric}$ via a chain of $O(1)$ number of charts, such that $\inf_{B(p,R)}|\varphi|$ on charts increase by only $O(1)$ in each step. This last fact is because we can choose the chains of successive charts $B(p_{i},5R_{i})$ such that the measure of the overlap occupies a nontrivial portion of the previous chart:

$$|B(p_{i},R_{i})\cap B(p_{i+1},R_{i+1})|\gtrsim\frac{1}{10}|B(p_{i},R_{i})|,$$

which would force

$$\begin{split}\inf_{B(p_{i+1},R_{i+1})}|\varphi|\leq&\inf_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\\ \lesssim&\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p_{i},R_{i})}|\varphi|\lesssim 1+\inf_{B(p_{i},R_{i})}|\varphi|.\end{split}$$

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