Corollary 5.29.9.F701C1999BDC Let $(X,\omega )$ be compact Hermitian, and let $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$, $0<\alpha \leq 1$. If uniformly bounded quasi-plurisubharmonic functions $a,b$ have a fixed common smooth lower curvature bound, then
\[ \int _ X|a-b|\, \omega _ u^ n \leq C\| a-b\| _{L^1(\mathrm dV)}^{\alpha /(\alpha +2n)}. \]
The constant depends on the bounds for $a,b$, their lower curvature bound and the $C^{0,\alpha }$ norm of $u$.
Proof.
On each of finitely many coordinate balls, choose a fixed smooth strictly plurisubharmonic quadratic $Q$ such that $a+Q,b+Q,u+Q$ are plurisubharmonic and $\mathrm{dd}^{\mathrm{c}}Q\geq \omega $. There $0\leq \omega _ u\leq \mathrm{dd}^{\mathrm{c}}(u+Q)$, so positivity of bounded products gives $\omega _ u^ n\leq (\mathrm{dd}^{\mathrm{c}}(u+Q))^ n$. Apply lemma 5.29.7 on smaller balls covering $X$. The quadratic cancels in the difference, and the local volumes are equivalent to $\mathrm dV$. Summing proves the assertion.
$\square$
Pome
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