PomeD716F01D638BLemma 15.3.1.D716F01D638B Fix $p>1$ and $L>0$. There is $b>0$ such that, for every $f\geq 0$ with $\| f\| _{L^ p(\mathrm{d}V)}\leq L$, one can find $s\in \mathrm{PSH}(X,\theta )$ satisfying
The constant $b$ depends only on the fixed geometry and on $\rho ,a,p,L$.
Proof. Choose $1<q<p$. Since a fixed quasi-plurisubharmonic function belongs to every finite $L^ r$ space, Hölder's inequality gives a uniform $L^ q$ bound for
By theorem 14.19.2 there are $c>0$ and a continuous $\omega $-plurisubharmonic $v$, normalized by $\sup _ Xv=-1$, such that
The construction in that theorem gives a uniform upper bound for $c$, because in its Gauduchon trace estimate the root integral of $1+g$ is bounded below by the root integral of $1$. The lower-density estimate of theorem 14.19.19 gives $c\| 1+g\| _ q\geq c_ q>0$, so $c$ is also bounded below. The potential estimate of the same theorem then gives $-C\leq v\leq -1$ with a uniform $C$.
Put $h=\rho +av\leq -1$. Then $\theta _ h\geq a\omega _ v$. On $(-\infty ,-1]$ define
and set $s=\chi (h)$, with value $-1$ at the poles of $h$. To justify both the positivity of $\theta +\mathrm{dd}^{\mathrm{c}}s$ and the computation of its Monge–Ampère measure, we first work with the bounded functions $h_ k=\max (h,-k)$, $k\geq 1$. The chain rule for bounded potentials and $\theta \geq 0$ give
These bounded potentials decrease to $s$, so $s\in \mathrm{PSH}(X,\theta )$ and $-1\leq s\leq 0$. On $\{ h>-k\} $, locality and positivity of mixed products give
Letting $k\to \infty $, we obtain this inequality on the set where $h$ is finite; its complement is pluripolar and has zero volume. Since $|h|\leq (-\rho )+aC\leq (1+aC)(-\rho )$, the assertion follows with any common lower bound for $a^ nc(1+aC)^{-2n}$. $\square$
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