Pome

Lemma 15.14.1869D3138741D

Lemma 15.14.1.869D3138741D Fix $p>1$ and $L>0$. There is $m>0$ such that, whenever

\[ \rho \leq \eta \leq 0,\quad \eta \in \mathrm{PSH}(X,\theta ), \qquad g\geq 0,\quad \| g\| _{L^ p(\mathrm{d}V)}\leq L, \]

there exists $v\in \mathrm{PSH}(X,\theta )$ with

\[ \eta \leq v\leq \eta +1,\qquad \mathrm{MA}_\theta (v)\geq m g\, \mathrm{d}V. \]

The constant $m$ depends only on the fixed geometry and on $\rho ,p,L$.

Proof. Choose $1<q<p$. By Hölder's inequality and the finiteness of all polynomial moments of the fixed quasi-plurisubharmonic function $\rho $, the functions $h=g(2-\rho )^{2n}$ have uniformly bounded $L^ q$ norms. Lemma 15.3.1, applied in the Hermitian background $\omega $, gives $s\in \mathrm{PSH}(X,\omega )$ with $-1\leq s\leq 0$ and

\[ \omega _ s^ n\geq m h\, \mathrm{d}V \]

for a fixed $m>0$. Put $u_1=\rho +s$. Then $\theta _{u_1}\geq \omega _ s$ and $u_1\leq \rho \leq \eta $. On $\Omega $ define

\[ t=1+\eta -u_1,\qquad v=\eta +t^{-1}, \]

so that $1\leq t\leq 2-\rho $. Lemma 5.28.13, applied on relatively compact subsets of $\Omega $ with $a=\eta $, $h=u_1$ and $\chi (x)=(1-x)^{-1}$, gives

\[ \theta _ v=(1-t^{-2})\theta _\eta +t^{-2}\theta _{u_1} +2t^{-3}\, \mathrm{d}(\eta -u_1)\wedge \mathrm d^{\mathrm c}(\eta -u_1) \geq t^{-2}\theta _{u_1}. \]

Since $\eta \leq v\leq \eta +1$, lemma 5.24.7 extends $v$ across $Z$, and the bounds extend by lemma 5.24.3. On $\Omega $, positivity of the local products gives

\[ \mathrm{MA}_\theta (v)\geq t^{-2n}\theta _{u_1}^ n \geq (2-\rho )^{-2n}\omega _ s^ n \geq m g\, \mathrm{d}V. \]

Neither side charges $Z$, so the inequality holds on $X$. $\square$

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