Corollary 4.7 (00RL) · ScalingStacks

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Corollary 4.7. (Local potential upper bound) On $U^{s}_{w,\delta}$ , then $\phi-\bar{\phi}\leq Cs^{-1/2}.$

Proof. The psh property of $\phi$ implies subharmonicity. By the Harnack inequality in Prop. 4.1 the average $L^{1}$ -integral is bounded, and by Prop. 4.4 there is a Lipschitz bound on the local average function $\bar{\phi}$ . ∎