Corollary 17.5.26.DAF36530C8C1 Let $p>1$, let $0\leq f\in L^ p(X,\omega ^ n)$ and $0\leq a<1$, and let $u\in \mathcal E(X,\theta ,\phi )$ satisfy $\sup _ Xu=0$ and
\[ \mu _ u\leq f\omega ^ n+a\mu _\phi . \]
Then $\phi -C\leq u\leq \phi $, where $C$ depends only on $X,\theta ,\omega ,p,m,\| f\| _{L^ p}$ and $(1-a)^{-1}$. In particular, if $\int _ X f\omega ^ n=m$, then the equation $\mu _ u=f\omega ^ n$ has a solution $u\in \mathcal E(X,\theta ,\phi )$ with $\sup _ Xu=0$, and this solution has bounded difference from $\phi $.
Proof.
Since $\sup _ Xu=0$, lemma 16.22.2 gives $u\leq \phi $; recall also that $\sup _ X\phi =0$, by definition 16.21.1. Let $q=p/(p-1)$ be the conjugate exponent. By corollary 10.3.2, the $L^ q(X,\omega ^ n)$ norms of $u$ and $\phi $ are at most a constant $M_ q$ depending only on $X,\theta ,\omega ,q$. Hölder's inequality gives
\[ I=\int _ X(\phi -u)f\omega ^ n \leq \| f\| _{L^ p}\| \phi -u\| _{L^ q} \leq 2M_ q\| f\| _{L^ p}. \]
By corollary 16.27.10 with $r=2$, the measure $\nu =f\omega ^ n$ satisfies the capacity bound in (17.5.21) with $\delta =1$ and $A=B_2\| f\| _{L^ p}/m^2$, where $B_2$ depends only on $X,\theta ,\omega ,p$. (If $f=0$ almost everywhere, take $A=1$ instead.) With $\delta =1$, the constant (17.5.22) equals $4+2^ nAI/(1-a)^2$. Inserting the bounds for $A$ and $I$, and recalling that the lower bound $\phi -C\leq u$ survives an increase of $C$, we see that theorem 17.5.20 gives the assertion with
\[ C=4+\frac{2^{n+1}B_2M_ q\| f\| _{L^ p}^2}{m^2(1-a)^2}, \]
which has the stated dependence.
Finally, the measure $f\omega ^ n$ is non-pluripolar, as noted after definition 17.5.2. If its mass is $m$, theorem 17.5.15 gives a solution $u\in \mathcal E(X,\theta ,\phi )$ of $\mu _ u=f\omega ^ n$ with $\sup _ Xu=0$, and the bound just proved, with $a=0$, shows that $u$ has bounded difference from $\phi $.
$\square$
Pome
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