Corollary 11.38.5A13D502ACAAC

Proof. For $G\in C(X)$, let $u\sim \varphi $ solve the equation with density $c\mathrm{e}^ G$, where $c=\mathrm{e}^{B_\varphi (G)}$. The lower bound $b_ q>0$ of lemma 15.12.2 for the $L^ q$ norm of the density gives $c\| \mathrm{e}^ G\| _ q\geq b_ q$, hence

For the upper bound, choose a Hermitian form $\Omega \geq \theta $, and let $B_{\Omega ,0}$ and $H_{\Omega ,0}$ denote its normalization scalar and dual functional for the zero reference. Let $v$ be a bounded solution for $G$ in this background; being bounded, it has a smaller-background envelope $P_{\delta \Omega }(v)\geq \inf _ Xv$ for every $0<\delta <1$. Moreover $u\in \mathrm{PSH}(X,\Omega )$, $u\preceq v$, and positivity gives

If $c>\mathrm{e}^{B_{\Omega ,0}(G)}$, the singular strict domination principle, applied in the background $\Omega $ to $v$ and $u+t$ for every real $t$, would give $v\geq u+t$ for all $t$, which is impossible. Hence $B_\varphi \leq B_{\Omega ,0}$, and therefore $H_\varphi \leq H_{\Omega ,0}$; proposition 11.38.1 now gives the upper bound.

Taking the defining supremum in the lower scalar estimate and applying lemma 11.37.1 with $F=qG$ gives the lower bound for every probability measure, including infinite values, with $a_ q=\log b_ q$. The finite-domain and bounded-family assertions follow by taking $q=2$. Finally, lemma 15.12.2 is uniform in the reference for fixed $\theta $ and $\varepsilon $, and the upper bound uses only $\Omega $, which proves the last assertion. $\square$