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 $b_ h(f)=0$.
$\square$
Pome
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