Theorem 20.3.15.AFC6C3B6368C Let $\mathrm{d}V$ be a smooth positive volume form on $Y$, and let $0\leq f\in L^ p(\mathrm{d}V)$, $p>1$, with $\int _ Yf\, \mathrm{d}V>0$. There is a solution $v$ in the atomic trace type of
\[ \mathrm{MA}^{\mathrm{ext}}_{\theta |_ Y}(v)=c f\, \mathrm{d}V, \qquad c>0, \]
if and only if $M>d$. In that case
\[ c=\frac{M-d}{\int _ Yf\, \mathrm{d}V}, \]
and $v$ is unique up to an additive constant. If $M=d$, there is instead a potential in this type with extended measure zero, unique up to an additive constant.
Proof.
Assume that $M>d$, take the value of $c$ given above, and solve
\[ \mathrm{dd}^{\mathrm{c}}s=d\, \eta +c f\, \mathrm{d}V-\theta |_ Y. \]
The right-hand side has integral zero, so the elliptic $L^ p$ theory gives a solution $s\in W^{2,p}(Y)$, unique up to constants. In real dimension two, the Sobolev embedding for $p>1$ makes $s$ continuous and bounded. The potential $v=w+s$ has the atomic trace type and satisfies
\[ \theta |_ Y+\mathrm{dd}^{\mathrm{c}}v=D_ u+c f\, \mathrm{d}V\geq 0. \]
Its extended measure is $c f\, \mathrm{d}V$, since this absolutely continuous measure does not charge the polar set of poles, while the atoms lie in it. For $M=d$, the same construction with $c=0$ has a smooth right-hand side and gives the asserted potential with zero extended measure.
To prove necessity and uniqueness, let $a$ be any $\theta |_ Y$-plurisubharmonic potential in the ordinary type of $w$, and put $\mu =\theta |_ Y+\mathrm{dd}^{\mathrm{c}}a$. By corollary 5.27.15, ordinary equivalence with $w$ makes the polar part of $\mu $ equal to $D_ u$. Consequently
\[ \mathrm{MA}^{\mathrm{ext}}_{\theta |_ Y}(a)=\mu -D_ u, \qquad \int _ Y\mathrm{MA}^{\mathrm{ext}}_{\theta |_ Y}(a)=M-d. \]
Integrating the equation therefore forces the stated value of $c$, so a positive $c$ requires $M>d$. If two potentials solve the equation with the same density and constant, their curvature currents both equal $D_ u+c f\, \mathrm{d}V$; their difference has zero $\mathrm{dd}^{\mathrm{c}}$ and is therefore constant on the compact connected curve. The same argument proves uniqueness in the case of zero extended measure.
$\square$
Pome
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