The uniqueness just proved identifies the limit, as $\lambda \downarrow 0$, of the solutions of the exponential equations $\mathrm{MA}^{\mathrm{ext}}_\omega (u)=\mathrm{e}^{\lambda u}f\, \mathrm{d}V$. To control their normalization, we use the probability measure obtained from the maximizing form and the density. Normalized potentials have bounded moments with respect to this measure, and Jensen's inequality converts this bound into an estimate for the constants and a common lower barrier for the potentials.
Proposition 12.10.20.D804C5144A2D Assume the hypotheses of theorem 12.10.17, and let $u_0$ be its unique threshold solution with $\sup _ Xu_0=0$. For real $\lambda >0$, let $u_\lambda $ be the greatest solution of the exponential equation (15.35.1), with $\omega $ in place of $h$, given by corollary 15.35.3, and put
\[ v_\lambda =u_\lambda -\sup _ Xu_\lambda ,\qquad c_\lambda =\mathrm{e}^{\lambda \sup _ Xu_\lambda }. \]
There is a constant $C$ independent of $\lambda >0$ such that
\[ b\leq c_\lambda \leq b\, \mathrm{e}^{C\lambda },\qquad u_0-C\leq v_\lambda \leq 0. \]
Moreover $v_\lambda \to u_0$ in $L^1(X)$ as $\lambda \downarrow 0$.
Proof.
Write $p=f^{1/n}$ and introduce the probability measure
\[ \mathrm{d}\nu =b^{1/n}p k_ T\, \mathrm{d}V. \]
It has mass one because $T$ maximizes. Its density belongs to $L^ n$, since $p\in L^ n$ and the smooth semipositive form $T$ has bounded determinant density $k_ T$. By lemma 5.26.2, $\omega $-plurisubharmonic functions with supremum zero have uniformly bounded moments of every finite order. Hölder's inequality with the exponents $n$ and $n/(n-1)$, which is finite because $n\geq 2$, therefore gives
\[ \int _ X(-v_\lambda )\, \mathrm{d}\nu \leq C. \]
Since $\mathrm{MA}^{\mathrm{ext}}_\omega (v_\lambda )=c_\lambda \mathrm{e}^{\lambda v_\lambda }f\, \mathrm{d}V$, the trace inequality of lemma 13.14.12, applied with the positive regularizations of $T$ and passed to the limit as in the preceding proof, gives $(\omega +\mathrm{dd}^{\mathrm{c}}v_\lambda )\wedge T\geq (c_\lambda /b)^{1/n}\mathrm{e}^{\lambda v_\lambda /n}\, \nu $. Integrating, we obtain
\[ 1\geq (c_\lambda /b)^{1/n} \int _ X\mathrm{e}^{\lambda v_\lambda /n}\, \mathrm{d}\nu \geq (c_\lambda /b)^{1/n}\mathrm{e}^{-C\lambda /n}, \]
where the last step is Jensen's inequality. This proves the upper bound for $c_\lambda $. The lower bound and the comparison
\[ v_\lambda \geq u_0-\lambda ^{-1}\log (c_\lambda /b) \]
follow from corollary 15.35.3 applied with $v=u_0$, and together with the upper bound they give the common lower barrier $u_0-C$.
Take any sequence $\lambda _ j\downarrow 0$. By theorem 5.25.2, a subsequence of $(v_{\lambda _ j})$ converges in $L^1$ to some $v\in \mathrm{PSH}(X,\omega )$ with supremum zero, and proposition 15.28.6 makes $v$ a threshold subsolution. Uniqueness in theorem 12.10.17 therefore gives $v=u_0$. Since every sequence tending to zero has a subsequence with this same $L^1$ limit, the whole family converges.
$\square$
Under the additional envelope condition of the next corollary, the lower barrier upgrades this to convergence in capacity. If the maximizing form is strictly positive everywhere, corollary 12.10.13 already gives uniform convergence.
Corollary 12.10.21.D179EF5E5E25 Under the hypotheses of proposition 12.10.20, suppose in addition that the obstacle envelope $P_{\delta \omega }(u_0)$ of section 5.40 is not identically $-\infty $ for some real $0<\delta <1$. Then $v_\lambda \to u_0$ in capacity as $\lambda \downarrow 0$. In particular this holds if the threshold solution $u_0$ is bounded.
Proof.
Fix a sequence $\lambda _ j\downarrow 0$. Proposition 12.10.20 gives $u_0-C\leq v_{\lambda _ j}\leq 0$ and $v_{\lambda _ j}\to u_0$ in $L^1$. For all $j$ with $\lambda _ j\leq 1$,
\[ \mathrm{MA}^{\mathrm{ext}}_\omega (v_{\lambda _ j}) =c_{\lambda _ j}\mathrm{e}^{\lambda _ jv_{\lambda _ j}}f\, \mathrm{d}V \leq b\mathrm{e}^{C}f\, \mathrm{d}V. \]
The bounded functions $|\mathrm{e}^{v_{\lambda _ j}}-\mathrm{e}^{u_0}|$ tend to zero in $\mathrm{d}V$-measure. Since $f\in L^1$, their integrals against $f\, \mathrm{d}V$ tend to zero, so the error term in (15.9.2) tends to zero. We apply proposition 15.9.3 with $\theta =\omega $ and $\varphi =u_0$; its additional hypothesis that the $L^1$ limit $u$ satisfies $u\preceq \varphi $, in the notation of definition 16.7.1, holds trivially, since $u=u_0$. This proves convergence in capacity along every sequence, and hence for the family. Finally, if $u_0$ is bounded, a constant below it is a competitor for $P_{\delta \omega }(u_0)$, so this envelope is not identically $-\infty $.
$\square$
In general, the condition on $P_{\delta \omega }(u_0)$ is an additional hypothesis. The next subsection shows that it holds when the Monge–Ampère mass of $\omega $ is invariant, or when the singularity of $u_0$ satisfies a quantitative bound.
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