Pome

Theorem 15.3.396A341EAF1D4

Theorem 15.3.3.96A341EAF1D4 Let $0\leq f\in L^ p(\mathrm{d}V)$, $p>1$, be nonzero, and let $\lambda >0$. There is a unique bounded $v\in \mathrm{PSH}(X,\theta )$ such that

\[ \mathrm{MA}_\theta (v)=\mathrm{e}^{\lambda v}f\, \mathrm{d}V. \]

For fixed $\lambda $, these solutions are uniformly bounded when $\| f\| _ p$ is bounded above and $\int _ X f^{1/n}\, \mathrm{d}V$ is bounded away from zero.

Proof. For $0<\varepsilon \leq 1$, set $\theta _\varepsilon =\theta +\varepsilon \omega $. By the Hermitian existence theorem theorem 14.19.3, there is a continuous solution $v_\varepsilon $ of

\[ \mathrm{MA}_{\theta _\varepsilon }(v_\varepsilon ) =\mathrm{e}^{\lambda v_\varepsilon }f\, \mathrm{d}V. \]

Take $s,b$ from lemma 15.3.1, and put $w=s+\lambda ^{-1}\log b$. Positivity of bounded products gives

\[ \mathrm{MA}_{\theta _\varepsilon }(w) \geq \mathrm{MA}_\theta (s)\geq bf\, \mathrm{d}V \geq \mathrm{e}^{\lambda w}f\, \mathrm{d}V, \]

and bounded exponential comparison (corollary 14.19.12) yields $w\leq v_\varepsilon $. If $0<\varepsilon '<\varepsilon $, then $v_{\varepsilon '}$ is also $\theta _\varepsilon $-plurisubharmonic and

\[ \mathrm{MA}_{\theta _\varepsilon }(v_{\varepsilon '}) \geq \mathrm{e}^{\lambda v_{\varepsilon '}}f\, \mathrm{d}V, \]

so the same comparison gives $v_{\varepsilon '}\leq v_\varepsilon $. Hence $v_\varepsilon $ decreases to a bounded $\theta $-plurisubharmonic function $v$ with $w\leq v\leq v_1$.

To pass to the limit in the equation, we use the fixed Hermitian background $\Omega =\theta +\omega $ and the expansion

\[ (\theta _\varepsilon +\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ n =\sum _{k=0}^ n\binom nk (\Omega +\mathrm{dd}^{\mathrm{c}}v_\varepsilon )^ k \wedge \bigl (-(1-\varepsilon )\omega \bigr )^{n-k}. \]

Continuity of bounded products along monotone sequences applies to every term, and the smooth coefficients converge. Dominated convergence applies to the right-hand sides. This proves the equation. Uniqueness follows from bounded domination in the semipositive big background (theorem 16.28.56, proposition 16.28.55) by the usual exponential comparison argument.

For the uniform assertion, note first that the lower barrier $w$ is uniform by lemma 15.3.1. The upper barrier $v_1$ solves an equation in the fixed Hermitian background $\Omega $. The proof of lemma 14.19.26 bounds its supremum using only an upper bound for the density norm and a positive lower bound for the root integral: these two bounds control, respectively, the Hölder term and the denominator in (14.19.27), and they survive the smooth approximation used there. Thus $v_1$, and hence $v$, has a uniform upper bound. $\square$

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