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For a smooth strictly positive probability density, the normalization scalar is an infimum of Gauduchon quotients by theorem 12.2.3. We extend this dual formula to the threshold of an arbitrary integrable density. A uniform bound for all Gauduchon tests in a single $L^ r$ norm allows the passage from smooth to bounded densities, and monotone truncation then reaches all integrable densities, including those with zero threshold. The formula is most naturally written for the $n$th root of the density.
Recall that a Hermitian form $\gamma $ is Gauduchon if $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$ (definition 11.4.1); in dimension one every Hermitian form is Gauduchon. For a Gauduchon form $\gamma $ and a nonnegative measurable function $p$, put
\[ L_{h,\gamma }(p)= \frac{\displaystyle \int _ Xp(\mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n}{\displaystyle \int _ Xh\wedge \gamma ^{n-1}} =\int _ Xp\, a_\gamma \, \mathrm{d}V, \qquad a_\gamma =\frac{(\gamma ^ n/\mathrm{d}V)^{(n-1)/n}}{\int _ Xh\wedge \gamma ^{n-1}}. \]
Since the coefficient $a_\gamma $ is smooth and strictly positive, each individual test $L_{h,\gamma }$ is finite on $L^ n$; the supremum over $\gamma $ may be infinite. In dimension one the test is $\int _ Xp\, \mathrm{d}V/\int _ Xh$. For real or complex $p\in L^ n(\mathrm{d}V)$, define the root functional
\[ \rho _ h(p)=b_ h(|p|^ n)^{-1/n}\quad (p\ne 0),\qquad \rho _ h(0)=0, \]
with the convention $0^{-1/n}=+\infty $.
Theorem 15.29.1.346B39C3F0B6 For every real or complex $p\in L^ n(\mathrm{d}V)$,
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15.29.2
\begin{equation} \rho _ h(p)=\sup _{\gamma \text{ Gauduchon}}L_{h,\gamma }(|p|). \end{equation}
Equivalently, for every nonzero nonnegative $f\in L^1(\mathrm{d}V)$,
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15.29.3
\begin{equation} b_ h(f)^{1/n}= \inf _{\gamma \text{ Gauduchon}} \frac{\displaystyle \int _ Xh\wedge \gamma ^{n-1}}{\displaystyle \int _ X(f\, \mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n}. \end{equation}
Both sides of (15.29.2) may be $+\infty $, and both sides of (15.29.3) may vanish.
Proof.
It suffices to prove (15.29.2) for $p\geq 0$, since both sides depend only on $|p|$; the case $p=0$ is immediate.
Step 1: smooth strictly positive $p$. Apply theorem 12.2.3, with background $h$, to the probability measure $\mu =p^ n\mathrm{d}V/Z$, where $Z=\int _ Xp^ n\, \mathrm{d}V$. By homogeneity (lemma 15.27.1) its scalar is $c_ h(\mu )=Zc_ h(p^ n)$, while the denominator of each Gauduchon quotient is $Z^{-1/n}\int _ Xp(\mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n$. The factors of $Z$ cancel, and the dual formula becomes
\[ c_ h(p^ n)^{-1/n}=\sup _\gamma L_{h,\gamma }(p). \]
Since $b_ h(p^ n)=c_ h(p^ n)$ by proposition 15.28.7, this proves (15.29.2) for such $p$.
Step 2: a uniform bound for the tests. The lower-density estimate of theorem 14.19.19, with exponent two, gives a constant $m_ h>0$ such that
\[ c_ h(p^ n)\| p^ n\| _{L^2}\geq m_ h \]
for smooth strictly positive $p$. By Step 1, therefore,
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15.29.4
\begin{equation} L_{h,\gamma }(p)\leq m_ h^{-1/n}\| p\| _{L^{2n}} \quad \text{for every }\gamma . \end{equation}
Approximating nonnegative functions in $L^{2n}$ by smooth strictly positive ones extends this bound to all nonnegative $p\in L^{2n}$. Since $a_\gamma >0$, duality gives $\| a_\gamma \| _{L^{2n/(2n-1)}}\leq m_ h^{-1/n}$ for every $\gamma $. In particular, $p\mapsto \sup _\gamma L_{h,\gamma }(p)$ is Lipschitz on the cone of nonnegative functions in $L^{2n}$, with the same constant.
Step 3: bounded $p$. Let $p\geq 0$ be bounded and nonzero, and approximate it in $L^{2n}$ by smooth strictly positive functions $p_ j$. By Hölder's inequality, $p_ j^ n\to p^ n$ in $L^2$, so lemma 15.27.2 gives
\[ c_ h(p_ j^ n)\longrightarrow c_ h(p^ n)>0. \]
By the Lipschitz bound derived from (15.29.4),
\[ \sup _\gamma L_{h,\gamma }(p_ j) \longrightarrow \sup _\gamma L_{h,\gamma }(p). \]
Passing to the limit in the identity of Step 1 for $p_ j$, and using $b_ h(p^ n)=c_ h(p^ n)$ (proposition 15.28.7), we obtain (15.29.2) for bounded $p$.
Step 4: general $p$. For nonnegative $p\in L^ n$, put $p_ k=\min (p,k)$. Then $p_ k^ n=\min (p^ n,k^ n)$ are truncations of $p^ n$, so the definition of $b_ h$ and Step 3 give
\[ \rho _ h(p)=\lim _{k\to \infty }c_ h(p_ k^ n)^{-1/n} =\sup _ k\sup _\gamma L_{h,\gamma }(p_ k). \]
For each fixed $\gamma $, monotone convergence gives $\sup _ kL_{h,\gamma }(p_ k)=L_{h,\gamma }(p)$. Interchanging the two suprema proves (15.29.2). Finally, every denominator in (15.29.3) is positive; taking reciprocals in (15.29.2) with $p=f^{1/n}$ proves (15.29.3), including the case $b_ h(f)=0$.
$\square$
For a nonsmooth density we do not claim that the infimum in (15.29.3) is attained by a Gauduchon form; attainment by Gauduchon currents is studied in section 12.10.
For $n\geq 2$, the coefficient of the test has norm
\[ \| a_\gamma \| _{L^{n/(n-1)}} =\frac{(\int _ X\gamma ^ n)^{(n-1)/n}}{\int _ Xh\wedge \gamma ^{n-1}} \]
in the exponent dual to $n$. By (12.15.5) in theorem 12.15.4, positive lower mass is therefore equivalent to a uniform bound for these norms, that is, for all tests on $L^ n$ simultaneously. Without positive lower mass, Step 2 of the proof still bounds the coefficients uniformly in the weaker norm of $L^{2n/(2n-1)}$, on every compact Hermitian manifold. This suffices to pass from smooth to bounded densities, and monotone truncation then treats all integrable densities.
Proposition 15.29.5.429BB0FF98A9 Let $f$ be a nonzero nonnegative function in $L^1(\mathrm{d}V)$ and let $p\in L^ n(\mathrm{d}V)$ be real or complex. If $h_1,h_2$ are Hermitian forms on $X$ and $m h_1\leq h_2\leq M h_1$ for positive constants $m,M$, then
\[ m^ n b_{h_1}(f)\leq b_{h_2}(f)\leq M^ n b_{h_1}(f), \qquad M^{-1}\rho _{h_1}(p)\leq \rho _{h_2}(p)\leq m^{-1}\rho _{h_1}(p). \]
If $h_2-h_1=\mathrm{dd}^{\mathrm{c}}a$ for a smooth real function $a$, then $b_{h_1}(f)=b_{h_2}(f)$ and $\rho _{h_1}(p)=\rho _{h_2}(p)$.
Proof.
For every Gauduchon form $\gamma $, the numerators in (15.29.3) satisfy
\[ m\int _ Xh_1\wedge \gamma ^{n-1}\leq \int _ Xh_2\wedge \gamma ^{n-1} \leq M\int _ Xh_1\wedge \gamma ^{n-1}, \]
while the denominators do not depend on the background. Taking the infimum over $\gamma $ and then $n$th powers proves the bounds for the thresholds; taking reciprocal $n$th roots gives the bounds for $\rho $, with the stated conventions at zero and infinity. If $h_2-h_1=\mathrm{dd}^{\mathrm{c}}a$, the numerators agree by Stokes' theorem, since $\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$.
$\square$
Any two Hermitian metrics on the compact manifold $X$ are comparable. Hence the positivity of $b_ h(f)$, and the set of real or complex $p\in L^ n$ with $\rho _ h(p)<+\infty $, do not depend on the choice of $h$. The dependence on the reference volume is equally simple. If $\mathrm{d}V$ is replaced by $\mathrm{d}V'=\sigma \, \mathrm{d}V$ with a smooth function $\sigma >0$, the measure $f\, \mathrm{d}V$ has density $f'=f/\sigma $ with respect to $\mathrm{d}V'$, and theorem 15.28.4 shows that the threshold of $f'$ relative to $\mathrm{d}V'$ equals $b_ h(f)$. Correspondingly, the map $p\mapsto \sigma ^{-1/n}p$ from $L^ n(\mathrm{d}V)$ to $L^ n(\mathrm{d}V')$ preserves both the $L^ n$ norm and the root functional.
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