ScalingStacks

Proof. On either a log scale in the toric region, or a boundary type chart, we have by the local $L^{1}$-oscillation estimate and the mean value inequality that

$$|\sup_{loc}\varphi_{m}-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}\varphi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\psi_{m}-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}\psi_{m}|\leq Cs^{-1/2}.$$

Notice also the local averages of $\varphi_{m}$ and $\psi_{m}$ differ by $O(s^{-1/2})$, so

$$|\sup_{loc}\varphi_{m}-\sup_{loc}\psi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\varphi-\sup_{loc}\psi|\leq Cs^{-1/2}.$$

Combined with (28),

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{loc}e^{-\alpha\sqrt{s}(\varphi-\psi)}d\mu_{s}\leq C.$$

The summation argument as in the global Skoda estimate proves the claim. ∎