Corollary 4.8 (00RN) · ScalingStacks

Verified tagged author-source HTML · 1912.02360v1 · cited publication edition alignment unverified.

Corollary 4.8. (Local $L^{1}$ -oscillation bound) Over one log scale inside $U_{w,\delta}^{s}$ ,

\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)|}|\phi-\bar{\phi}|d\mu_{s}\leq Cs^{-1/2}.

00RP

Proof. Recall the local oscillation of $\bar{\phi}$ in one log scale is $O(s^{-1})$ . Since the local sup of $\phi$ differs from the local average of $\phi$ by $O(s^{-1/2})$ , the local $L^{1}$ -oscillation is likewise bounded by $O(s^{-1/2})$ . ∎