Corollary 15.18.50`AD8A02A6088D`

Corollary 15.18.50.AD8A02A6088D Let $0\leq f\in L^ p(X,\omega ^ n)$, $p>1$, be smooth and strictly positive on $\Omega $. Every minimal-singularity solution of $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )^ n=cf\, \omega ^ n$, $c>0$, is smooth and strictly $\theta $-plurisubharmonic on $\Omega $.

Proof. The analytic complement of $\Omega $ has zero mass for both measures, so it suffices to know the equation in the bounded-potential sense on $\Omega $. Apply theorem 15.17.1 with density $cf$ to obtain local Hölder continuity. All hypotheses of theorem 15.18.1 are now satisfied. $\square$

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@misc{pome-tag-AD8A02A6088D,
  author = {{Pome contributors}},
  title = {Corollary 15.18.50},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/AD8A02A6088D/}
}

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Corollary 15.18.50AD8A02A6088D

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Corollary 15.18.50`AD8A02A6088D`