ScalingStacks

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 $O(s^{-1/2})$. ∎