Theorem 13.15.23.18CDC482A9DD Let $\theta $ be a smooth real $(1,1)$-form, and for nonpositive $u\in \mathrm{PSH}(X,\theta )$ put $\mu _{\theta ,u}=\mathrm{MA}^{\mathrm{ext}}_\theta (u)$. The mass bounds, capacity estimates and capacity-continuity statements for damped measures in this chapter, including proposition 13.15.9, hold for the measures $\mu _{\theta ,u}$ in place of $\mu _ u$, with constants depending on $\theta $ and $\omega $. In particular, for $p>0$ and $q>r_ n$,
\[ \int _ X\mathrm{e}^{pu}\, \mathrm{d}\mu _{\theta ,u} \leq C(1+p^{-1})^{r_ n},\qquad \int _ X(1-u)^{-q}\, \mathrm{d}\mu _{\theta ,u}\leq C_ q. \]
The undamped tail criterion of theorem 13.15.13 and the integrability criterion of corollary 13.15.14 also hold with $\mu _{\theta ,u}$ in place of $\mu _ u$.
Proof.
Choose $C_0\geq 1$ such that $\theta \leq \Omega :=C_0\omega $. Then $u\in \mathrm{PSH}(X,\Omega )$, and theorem 13.14.3 gives
\[ 0\leq \mu _{\theta ,u}\leq \mathrm{MA}^{\mathrm{ext}}_\Omega (u). \]
All the mass, tail and capacity bounds therefore follow from the estimates already proved for $\Omega $. The metric comparison in section 5.29 gives, for every Borel set $E\subset X$,
\[ \operatorname {Cap}_\omega (E) \leq \operatorname {Cap}_\Omega (E) \leq C_0^ n\operatorname {Cap}_\omega (E). \]
For continuity, let nonpositive $u_ j,u\in \mathrm{PSH}(X,\theta )$ converge in capacity, and use (13.15.10) with the $\theta $-product. The truncations are bounded and $\Omega $-plurisubharmonic. The weighted capacity continuity of section 5.27, section 5.29 applies term by term to the local expansion of their signed $\theta $-products, with the same bounded quasi-continuous coefficients. Since the cutoff vanishes at the truncation level, the result is exactly the restricted positive measure. The uniform tails obtained from the domination by $\mathrm{MA}^{\mathrm{ext}}_\Omega $ remove the cutoff. Domination also gives the sublevel estimate, so the preceding proofs of the undamped tail and integrability criteria apply without change.
$\square$
Pome
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