Theorem 15.31.15.5F0343E9A338 Suppose $Q=Q_\mu $ is finite and continuous. Then $b_ H(\nu )=c$, attained by the bounded continuous $u$ of proposition 15.31.13. For every $\lambda >0$, the continuous potential
\[ U_\lambda =\lambda ^{-1}\log c+w_\lambda \circ \pi _ Y \]
satisfies
\[ \mathrm{MA}_ H(U_\lambda )=\mathrm e^{\lambda U_\lambda }\nu , \qquad \int _ X\mathrm{MA}_ H(U_\lambda )=NV_0. \]
It dominates every $p\in \mathrm{PSH}(X,H)$ with $\mathrm{MA}^{\mathrm{ext}}_ H(p)\geq \mathrm e^{\lambda p}\nu $, and it is the unique bounded solution. For $v_\lambda =U_\lambda -\sup _ XU_\lambda $ and $c_\lambda =\mathrm e^{\lambda \sup _ XU_\lambda }$,
\[ \begin{gathered} \mathrm{MA}_ H(v_\lambda )=c_\lambda \mathrm e^{\lambda v_\lambda }\nu , \qquad c_\lambda =c\, \mathrm e^{\lambda \sup _ Yw_\lambda },\\ \| v_\lambda -u\| _\infty =O(\lambda ),\qquad \lambda ^{-1}\log (c_\lambda /c)\longrightarrow \sup _ YQ-\int _ YQ\, \mathrm d\mu . \end{gathered} \]
All the constructed potentials belong to $\mathcal E_{\mathrm{env}}(X,H,0)$. The same coefficient $c$ is attained if the defining supremum for $b_ H(\nu )$ is restricted to bounded potentials or to this envelope class.
Proof.
Lemma 15.31.4 gives non-polarity, and theorem 15.31.8 constructs $w_\lambda $. Substituting it in (15.31.14) proves the exponential equation and its finite mass. Supremum normalization gives the displayed $c_\lambda $, and the uniform curve estimate gives both limits. For any bounded $H$-plurisubharmonic $p$ and $t\geq 1$, the constant $a_ t=t\inf _ Xp$ is an $H$-plurisubharmonic competitor below $tp$. Hence $P_ H(tp)$ is genuine for every $t$, which proves the envelope-class assertion.
Greatestness. Write $s=U_\lambda $. If $b$ is a bounded exponential subsolution, then on $\{ s<b-\epsilon \} $,
\[ \mathrm{MA}_ H(s)\leq \mathrm e^{-\lambda \epsilon }\mathrm{MA}_ H(b-\epsilon ). \]
The bounded strict domination theorem, theorem 14.19.10, gives $s\geq b-\epsilon $. Letting $\epsilon \downarrow 0$ proves $s\geq b$, and applying the argument in both directions proves bounded uniqueness.
For an arbitrary exponential subsolution $r$, fix $A>0$ and put $p=s-A$, $W=\max (r,p)$. These maxima are $H$-plurisubharmonic by lemma 5.24.1, and $W$ is bounded. Choose $j$ with $-j<\inf _ Xp$ and set $r_ j=\max (r,-j)$; then $W=\max (r_ j,p)$. Plurifine locality on $\{ r_ j>p\} $ and lemma 5.28.3, applied to $p\leq W$ on their contact set, give
\[ \mathrm{MA}_ H(W)\geq \mathbf1_{\{ r_ j>p\} }\mathrm{MA}_ H(r_ j) +\mathbf1_{\{ r_ j\leq p\} }\mathrm{MA}_ H(p). \]
On the first set $r_ j=r>-j$, so strict-superlevel locality identifies the measure with $\mathrm{MA}^{\mathrm{ext}}_ H(r)$. On the second, $W=p$ and $\mathrm{MA}_ H(p)=\mathrm e^{\lambda s}\nu \geq \mathrm e^{\lambda p}\nu $. Thus $W$ is a bounded exponential subsolution, and $r\leq W\leq s$. This uses bounded products before every restriction and applies also when the measure of $r$ has infinite mass.
The maximal coefficient. Suppose $\mathrm{MA}^{\mathrm{ext}}_ H(r)\geq a\nu $ with $a>0$, and put $B=\sup _ Xr$. For every $\lambda >0$ the potential $r+\lambda ^{-1}\log a-B$ is an exponential subsolution, since its exponential is at most $a$. Greatestness gives
\[ r+\lambda ^{-1}\log a-B \leq \lambda ^{-1}\log c+w_\lambda \circ \pi _ Y. \]
Evaluate at a fixed finite point of $r$, multiply by $\lambda $, and let $\lambda \downarrow 0$. The uniform barriers for $w_\lambda $ give $a\leq c$. The equation for $u$ attains $c$ in each stated class.
$\square$
Pome
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