Theorem 14.19.19.EA32B8891E2B There are constants $C_ p,c_ p>0$, depending only on $(X,\omega ,\mathrm{d}V,p)$, such that, whenever $u\in \mathrm{PSH}(X,\omega )\cap L^\infty (X)$, $\sup _ Xu=0$, and
\[ \omega _ u^ n=F\, \mathrm{d}V,\qquad F\in L^ p(X,\mathrm{d}V),\quad F\geq 0, \]
we have
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14.19.20
\begin{equation} \| u\| _\infty \leq C_ p(1+\| F\| _ p), \qquad \| F\| _ p\geq c_ p. \end{equation}
Proof.
By corollary 14.19.11, $F$ is not almost everywhere zero. Apply lemma 14.19.16 with $v=0$, $H=1$, $\varepsilon =1/2$, and $A=C\| F\| _ p>0$, where $C$ is the constant of corollary 5.29.5. Fix $D=\delta _{1/2}/2$, and put $M=-\inf _ Xu$. Since $U_ D=\{ u<-M+D\} $, the resulting inequality reads
\[ \operatorname {Cap}(\{ u<-M+D\} )\geq c/\| F\| _ p. \]
If $M>D+1$, (5.29.4) bounds this capacity above by $C/(M-D)$, and the first assertion follows; the case $M\leq D+1$ is immediate. Bounding the same capacity above by $\operatorname {Cap}(X)$ gives the second assertion.
$\square$
Pome
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