Proposition 11.38.1.170451FB65CB Let $\theta =\omega $ be a Hermitian form, and let $\varphi =0$. There are real constants $A$ and, for every $q>1$, $a_ q$ such that every Borel probability measure $\nu $ satisfies
\[ a_ q+\frac1q\operatorname {Ent}_{\mathrm{d}V}(\nu ) \leq H_0(\nu ) \leq A+n\operatorname {Ent}_{\mathrm{d}V}(\nu ). \]
The inequalities include the value $+\infty $. Thus $H_0(\nu )<\infty $ if and only if $\nu $ has finite entropy, and a family of probability measures has uniformly bounded values of $H_0$ if and only if its entropies are uniformly bounded.
Proof.
In dimension one, $\omega $ is a Kähler form, and proposition 11.37.3, which rests on (11.37.2), gives $H_0=\log \int _ X\omega +\operatorname {Ent}_{\mathrm{d}V}$, so the assertions follow. Suppose that $n\geq 2$.
For $G\in C(X)$, apply the lower bound for $\| F\| _ q$ in theorem 14.19.19 to a bounded solution with density $F=\mathrm{e}^{B_0(G)+G}$. It gives a constant $c_ q>0$, independent of $G$, such that
\[ B_0(G)\geq \log c_ q-\frac1q\log \int _ X\mathrm{e}^{qG}\, \mathrm{d}V. \]
For the upper bound, fix a smooth Gauduchon metric $\gamma $, rescaled so that $\int _ X\omega \wedge \gamma ^{n-1}=1$, and write
\[ k_\gamma =\left(\frac{\gamma ^ n}{\mathrm{d}V}\right)^{(n-1)/n}, \qquad m=\min _ X k_\gamma >0. \]
For smooth $G$, the trace bound (11.5.6) gives
\[ \mathrm{e}^{B_0(G)/n}\int _ X\mathrm{e}^{G/n}k_\gamma \, \mathrm{d}V\leq 1. \]
Uniform approximation and continuity of $B_0$ extend this inequality to continuous $G$. Consequently
\[ B_0(G)\leq -n\log m-n\log \int _ X\mathrm{e}^{G/n}\, \mathrm{d}V. \]
Taking the defining suprema in the two scalar bounds and applying lemma 11.37.1 with $F=qG$ and $F=G/n$, respectively, proves both inequalities for every probability measure, including infinite values, with $a_ q=\log c_ q$ and $A=-n\log m$. The finite-domain and bounded-family assertions follow by taking $q=2$.
$\square$
Pome
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