PomeAEDE3F4F53ECLemma 5.42.8.AEDE3F4F53EC Let $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$, $0<\alpha \leq 1$, and let $\mu $ be a positive measure with $\mu \leq A\omega _ u^ n$. For bounded $\varphi \in \mathrm{PSH}(X,\omega )$ with $-M\leq \varphi \leq 0$, the Chern averages satisfy
for small $t>0$. The constant is uniform under fixed bounds for $A,M$ and $\| u\| _{C^{0,\alpha }}$.
Proof. First we check a common lower curvature bound for the averages. The base part of the classical Chern curvature estimate stated before lemma 5.42.4 gives
Its constants are fixed under the stated bounded range. In the local kernel form used in lemma 5.42.2,
with uniformly bounded first derivatives and compact support in its rescaled second variable. Differentiating with respect to $t$ at fixed $x,y$ and changing variables shows that
Thus $|t\partial _ tR_ t\varphi |\leq CM$, so $\lambda $ is uniformly bounded and the displayed curvature estimate supplies one fixed lower bound $\mathrm{dd}^{\mathrm{c}}R_ t\varphi \geq -C\omega $. Both $R_ t\varphi $ and $\varphi $ also lie in $[-M,0]$.
Apply corollary 5.29.9 to this pair, whose common lower bound has just been established. By lemma 5.42.2, its smooth-volume $L^1$ error is at most $Ct^2$. Domination by $A\omega _ u^ n$ therefore gives
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