PomeTheorem 14.19.2 (Kołodziej–Nguyen).C8D798C2D2F0 There are $c>0$ and $u\in \mathrm{PSH}(X,\omega )\cap C(X)$ such that
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^ n=c f\, \mathrm{d}V. \]
The potential can be normalized by $\sup _ Xu=0$.
14.19.1 Continuous solvability49F0FF1AED46
There are two natural ways to normalize the prescribed density: by an unknown positive constant, or by an exponential of the potential.
The constant $c$ is uniquely determined by $f$, by proposition 14.19.34. The normalized potential, however, need not be unique, even for a smooth density; see theorem 14.26.1.
Theorem 14.19.3 (Nguyen).CFA579B45DF9 For every $\lambda >0$, there is a unique continuous $\omega $-plurisubharmonic function $u_\lambda $ satisfying
C01B1AA0FBCA
14.19.4
\begin{equation} (\omega +\mathrm{dd}^{\mathrm{c}}u_\lambda )^ n=\mathrm{e}^{\lambda u_\lambda }f\, \mathrm{d}V. \end{equation}
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