ScalingStacks

Proof. Since we know the local $L^{1}$-oscillation estimate holds in every local region, in a log scale in $U^{s}_{w}$, not necessarily in the shrinked region $U^{s}_{w,\delta}$,

$$\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}|\varphi_{m}-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\varphi_{m}|d\mu_{s}\leq Cs^{-1/2}.$$

Since $\bar{\phi}_{m,w}$ is convex, a local $L^{1}$-bound implies a local $L^{\infty}$-bound in a slightly shrinked region, so in the log scale,

$$|\bar{\phi}_{m,w}-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\varphi_{m}|\leq Cs^{-1/2}.$$

Likewise for $\bar{\phi}_{m^{\prime},w^{\prime}}$. By definition the local potentials differ by

$$\varphi_{m}-\varphi_{m^{\prime}}=\langle m^{\prime}-m,\text{Log}_{s}(z)\rangle$$

Notice that for a given point $P$ on $\partial\Delta_{\lambda}^{\vee}$, the log scales on $U^{s}_{w}$ and $U^{s}_{w^{\prime}}$ around $P$ have a nontrivial percentage of overlapping measure. Thus

$$\begin{split}|\bar{\phi}_{m,w}-\bar{\phi}_{m^{\prime},w^{\prime}}+(m-m^{\prime})|&\lesssim s^{-1/2}+|\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\varphi_{m}-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}+\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{overlap}|\varphi_{m}-\varphi_{m^{\prime}}+(m-m^{\prime})|\\ &\lesssim s^{-1/2}.\end{split}$$

∎