If a Hermitian background admits a trace competitor, then in the doubled background the same competitor lies in the half-sized background. This gives the trace envelope positive mass and the model condition needed for equations in its ordinary type.
Theorem 20.3.35.5CC5BF99E84F Choose $\Omega _0$ and $v$ as in proposition 20.3.24 and set $\Omega =2\Omega _0$. Then $T=\mathcal T_\Omega (u;Y)$ belongs to $\mathrm{PSH}(Y,\Omega )$ and satisfies all the divisorial inequalities in (20.3.25). Moreover,
\[ P_\Omega [T]_{\mathcal I}=T,\qquad P_\Omega [T]=T, \qquad P_{\Omega /2}(T)\not\equiv -\infty . \]
Writing $r=\dim Y$, its extended measure satisfies
\[ 0<\int _ Y\mathrm{MA}^{\mathrm{ext}}_\Omega (T)<+\infty , \qquad \mathrm{MA}^{\mathrm{ext}}_\Omega (T) \leq \mathbf1_{\{ T=0\} }\Omega ^ r. \]
Proof.
Since $\theta |_ Y+\epsilon _ m\omega |_ Y\leq \Omega $ for all sufficiently large $m$, the restricted approximants belong to $\mathrm{PSH}(Y,\Omega )$. Theorem 20.3.27, applied with $\delta _ m=0$, gives $T$, its defining inequalities and both fixed-point identities.
Normalize $v$ so that $\sup _ Yv=0$. Since $v\in \mathrm{PSH}(Y,\Omega /2)$ is a trace competitor, it lies below $T$ and is a competitor for $P_{\Omega /2}(T)$. The contact estimate for envelope fixed points, proposition 16.28.7, applied on $Y$, gives the bound for the measure and its finiteness, and lemma 16.28.69 gives strict positivity.
$\square$
Corollary 20.3.36.C5512844A274 Let $\Omega $ and $T$ be as in theorem 20.3.35, 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 $\int _ Y f\, \mathrm{d}V>0$. For every $\lambda >0$ there is a unique $a\in \mathrm{PSH}(Y,\Omega )$ with $a\sim T$ and
\[ \mathrm{MA}^{\mathrm{ext}}_\Omega (a)=\mathrm{e}^{\lambda a}f\, \mathrm{d}V. \]
There are also a constant $c>0$ and $a\in \mathrm{PSH}(Y,\Omega )$ with $a\sim T$ and $\mathrm{MA}^{\mathrm{ext}}_\Omega (a)=cf\, \mathrm{d}V$, and the constant $c$ is unique.
Proof.
By theorem 20.3.35, $T$ satisfies both parts of the model condition (16.28.67) for the Hermitian background $\Omega $ on $Y$, so theorem 15.12.1 applies.
$\square$
On a curve, lemma 5.38.5 shows that the trace inequalities are equivalent to domination by the Green representative $w$ in the ordinary sense. Thus
\[ \mathcal T_{A\eta }(u;Y)=P_{A\eta }[w]=P_{A\eta }[w]_{\mathcal I} \]
for every $A>0$ with $A\geq D_ u(Y)$. The last equality, and the fact that the envelope has exactly the prescribed Lelong numbers, follow from lemma 20.3.20, also when the residual mass $A-D_ u(Y)$ is zero.
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