Proposition 6.7 (003S) · ScalingStacks

003S

Proposition 6.7. Assume $\left\lVert\phi\right\rVert_{C^{0}}$ has a uniform bound independent of $t$ . Then after shrinking $U$ by a small amount independent of $t$ , we have

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The convex function $\bar{\phi}$ has a Lipschitz bound $|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.$
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There is an upper bound $\phi-\bar{\phi}\leq\frac{C}{|\log|t||^{1/2}}$ .
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On each logarithmic dyadic scale $U_{a}=\{a_{i}\leq\log|z_{i}|\leq 2a_{i},\forall i\}\subset U$ , the $L^{1}$ -integral

$\int_{U_{a}}|\phi-\bar{\phi}|\prod\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{i}}\leq\frac{C}{|\log|t||^{1/2}}.$
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There is an improved Skoda inequality with uniform constants $\alpha,C$ independent of $t$ :

$\int_{U}e^{-\alpha|\log|t||^{1/2}(\phi-\bar{\phi})}d\mu_{t}\leq C.$

003T

Proof. (Sketch)

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Bounded convex functions automatically have Lipschitz bound on slightly shrinked convex domains.
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The second item follows from a slightly tricky application of mean value inequality for subharmonic functions, cf. [52, section 4.3].
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The third item is because the function $\phi-\bar{\phi}$ has mean value zero, so an upper bound implies an $L^{1}$ -bound, cf. [52, section 4.3].
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One first apply the basic Skoda estimate Thm. 4.2 to the function $\phi$ on each logarithmic dyadic scale, where $\bar{\phi}$ is almost constant by the Lipschitz bound. Then we sum over all the logarithmic dyadic scales (cf. [52, section 4.6]).

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