PomeD00B685B1791Proposition 5.29.13.D00B685B1791 Let $(X,\omega )$ be compact Hermitian. Fix $0<\alpha \leq 1$ and $N>0$. There are $a,C>0$ such that every $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$ with $\| u\| _{C^{0,\alpha }}\leq N$ satisfies
Consequently, for each $s>0$ there is a uniform $C_ s$ with $\omega _ u^ n(E)\leq C_ s\operatorname {Cap}_\omega (E)^ s$.
Proof. Fix nested coordinate balls $B'\Subset B$ and a smooth quadratic $Q$ with $\mathrm{dd}^{\mathrm{c}}Q\geq \omega $ on $B$. The function $p=u+Q$ is plurisubharmonic with a common Hölder bound, and $\omega _ u^ n\leq (\mathrm{dd}^{\mathrm{c}}p)^ n$ there. Regard $B$ as a ball in the affine chart $\mathbb C^ n\subset \mathbb P^ n$ and restrict this local measure to $B'$, extending it by zero. By lemma 5.29.10 on $(\mathbb P^ n,\omega _{\mathrm{FS}})$, it has a uniform exponential tail on every normalized global Fubini–Study potential. Its total mass is uniformly bounded by local Chern–Levine–Nirenberg estimates.
For a non-pluripolar compact $K\subset B'$, choose a normalized potential $v$ with $\sup _ Kv\leq \log T(K)+1$. For small $T(K)$ the tail estimate gives
The common mass bound handles the remaining $T(K)$, and pluripolar sets have zero local product mass. A bounded local potential for $\omega _{\mathrm{FS}}$ turns every global test $0\leq v\leq 1$ into a plurisubharmonic function of a fixed bounded range on $B$. Hence
where the last inequality is the gluing construction of lemma 5.29.2 on the original manifold. These inequalities have the required direction after taking negative inverse powers. Inner regularity of the finite local product extends the bound from compact to Borel subsets of $B'$. Summing over a fixed finite cover proves the first assertion. The second follows by bounding $x^{-s}\exp (-ax^{-1/n})$ on $0<x\leq \operatorname {Cap}_\omega (X)$. $\square$
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