Under the vanishing-tail condition, we can also perturb the density by continuous multiplicative factors $\mathrm{e}^ F$. The logarithm of the threshold is then a concave function of $F$, and its supporting probabilities are given by the maximizing currents, just as for the continuous densities in section 12.8.
Let $f\in L^1(\mathrm{d}V)$ be nonzero and nonnegative, and assume $p=f^{1/n}\in \mathcal D_\omega ^0$. For a continuous real $F$, set
\[ B_ f(F)=\log b_\omega (\mathrm{e}^ Ff) =-n\log \rho _\omega (p \mathrm{e}^{F/n}),\qquad M_ f(F)=\rho _\omega (p \mathrm{e}^{F/n}). \]
Both quantities are finite and $M_ f(F)>0$, since $\mathcal D_\omega ^0$ is an order ideal. Let $\mathscr M_ f(F)$ be the set of currents in $\mathscr G_\omega $ maximizing $D_{p \mathrm{e}^{F/n}}$, which is nonempty by theorem 12.10.1, and put
\[ \nu _{T,F}=\frac{p \mathrm{e}^{F/n} k_ T}{M_ f(F)}\, \mathrm{d}V \quad (T\in \mathscr M_ f(F)). \]
These are probability measures. Let $\mathcal A_ f(F)$, the set of supporting probabilities of $B_ f$ at $F$, consist of all Borel probability measures $\nu $ satisfying
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12.10.8
\begin{equation} B_ f(G)\leq B_ f(F)-\int _ X(G-F)\, \mathrm{d}\nu \qquad (G\in C(X,\mathbb R)). \end{equation}
Proposition 12.10.9.CCE6CFE41CBF The functional $B_ f$ is concave, decreasing and $1$-Lipschitz for the uniform norm, and $B_ f(F+a)=B_ f(F)-a$ for constants $a$. Moreover,
\[ \mathcal A_ f(F)=\{ \nu _{T,F}:T\in \mathscr M_ f(F)\} . \]
This set is nonempty, convex and weakly compact, and its measures are absolutely continuous. For each $R>0$, the densities of all these measures, for all $F$ with $\| F\| _\infty \leq R$, are uniformly integrable. For $H\in C(X,\mathbb R)$, the one-sided directional derivatives of $B_ f$ at $F$ are
\[ (B_ f)'_+(F;H)=-\max _{\nu \in \mathcal A_ f(F)}\int _ XH\, \mathrm{d}\nu , \qquad (B_ f)'_-(F;H)=-\min _{\nu \in \mathcal A_ f(F)}\int _ XH\, \mathrm{d}\nu . \]
Proof.
For each $T$, Hölder's inequality makes $F\mapsto \int _ Xp \mathrm{e}^{F/n}k_ T\, \mathrm{d}V$ logarithmically convex, and taking suprema preserves the inequality defining logarithmic convexity. Thus $M_ f$ is logarithmically convex and $B_ f=-n\log M_ f$ is concave. Monotonicity and homogeneity of $\rho _\omega $ give the translation rule and, by comparison with constant translates, the Lipschitz estimate.
For fixed $F$, the weight $p\mathrm{e}^{F/n}$ belongs to $\mathcal D_\omega ^0$, so the densities $p\mathrm{e}^{F/n}k_ T$, $T\in \mathscr G_\omega $, are uniformly integrable by theorem 12.10.1. If $T_ j\to T$ and $D_{p\mathrm{e}^{F/n}}(T_ j)\to M_ f(F)$, proposition 12.10.6 makes $T$ a maximizer and gives convergence of the weighted measures. For $T_1,T_2\in \mathscr G_\omega $ and real $0\leq t\leq 1$, matrix determinant concavity gives
\[ p \mathrm{e}^{F/n}k_{tT_1+(1-t)T_2}\, \mathrm{d}V \geq t p \mathrm{e}^{F/n}k_{T_1}\, \mathrm{d}V +(1-t)p \mathrm{e}^{F/n}k_{T_2}\, \mathrm{d}V. \]
For two maximizing currents, the current on the left also maximizes, and equal total masses turn this inequality into an equality. The set $\mathscr N_ f(F)=\{ \nu _{T,F}:T\in \mathscr M_ f(F)\} $ is therefore nonempty, compact and convex.
A maximizing $T$ gives
\[ M_ f(G)\geq M_ f(F)\int _ X\mathrm{e}^{(G-F)/n}\, \mathrm{d}\nu _{T,F}. \]
Taking $-n$ times the logarithm and applying Jensen's inequality proves (12.10.8), so $\mathscr N_ f(F)\subset \mathcal A_ f(F)$. For the reverse inclusion, a Taylor expansion of the exponential gives
\[ D_{p \mathrm{e}^{(F+tH)/n}}(T) =D_{p \mathrm{e}^{F/n}}(T)+\frac tn\int _ X H p \mathrm{e}^{F/n} k_ T\, \mathrm{d}V+O(t^2), \]
where the error is uniform in $T$ since $D_{p \mathrm{e}^{F/n}}(T)\leq M_ f(F)$. Maximizers at $F+tH$ are near-maximizers at $F$, by the multiplicative bounds $\mathrm{e}^{-t\| H\| _\infty /n}\leq M_ f(F+tH)/M_ f(F)\leq \mathrm{e}^{t\| H\| _\infty /n}$. The convergence of their weighted measures proved above and this uniform expansion are exactly the two inputs of the one-sided differentiation argument in the proof of theorem 12.8.2, which gives
\[ (B_ f)'_+(F;H)=-\max _{\nu \in \mathscr N_ f(F)}\int _ XH\, \mathrm{d}\nu . \]
The separation argument of that proof then identifies all supporting probabilities with $\mathscr N_ f(F)$, and replacing $H$ by $-H$ gives the left derivative.
Finally, $M_ f(F)\geq \mathrm{e}^{-R/n}\rho _\omega (p)$ when $\| F\| _\infty \leq R$, so the densities of the probabilities $\nu _{T,F}$ are at most $\mathrm{e}^{2R/n}\rho _\omega (p)^{-1}p k_ T$. The uniform integrability of the densities $pk_ T$ in theorem 12.10.1 proves the remaining assertions.
$\square$
When $f=1$, we have $B_ f=B_0$, and the proposition reduces to theorem 12.8.2. For general $f$, the functional $B_ f$ is defined through the subsolution threshold and does not presuppose that the corresponding equation has a solution. If $v_-(\omega )>0$, every nonzero nonnegative $f\in L^1$ satisfies $f^{1/n}\in \mathcal D_\omega ^0$ by proposition 15.32.9, and corollary 15.37.32 identifies $b_\omega (f)$ as the unique constant for which the equation $\mathrm{MA}^{\mathrm{ext}}_\omega (v)=b\, f\, \mathrm{d}V$ has a solution $v$ in the envelope class $\mathcal E_{\mathrm{env}}(X,\omega ,0)$ of that corollary. Without positive lower mass, the uniform tail condition is an explicit sufficient hypothesis for this compact dual description.
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