Theorem 12.9.7.6A30AAE398CD For every smooth $F$ in the ball of lemma 12.9.6, let $T_ F$ be the maximizing current of theorem 12.7.3 and $\nu _ F$ the supporting probability of $B_0$ at $F$; both are unique. With
\[ H_{\max }=\max _ XH,\qquad C_\nu =\frac{3^{n+1}}4aH_{\max },\qquad C_ k=\frac{3^ n}2aH_{\max }^{(n-1)/n}, \]
their densities satisfy
\[ \nu _ F=q_ F\, \mathrm{d}V,\qquad \| q_ F\| _{L^\infty }\leq C_\nu ,\qquad \| k_{T_ F}\| _{L^\infty }\leq C_ k. \]
In particular, for $p=n/(n-1)$,
\[ \| q_ F\| _{L^ p}\leq C_\nu ^{1/n},\qquad \| k_{T_ F}\| _{L^ p}\leq C_ k. \]
The constants do not involve $v_-(\omega )$; in particular, these bounds hold also when $v_-(\omega )=0$.
Proof.
Set $\Omega =\Omega _ F$, $\chi =\chi _ F$ and
\[ Z=\int _ X\chi \omega \wedge \Omega ^{n-1}. \]
The form comparison and $\int _ X\chi \omega ^ n=1$ give
8926A66CEBEF
12.9.8
\begin{equation} 2^{-(n-1)}\leq Z\leq (3/2)^{n-1}. \end{equation}
Since $\mathrm{dd}^{\mathrm{c}}(\chi \Omega ^{n-1})=0$, Stokes' theorem also gives $Z=\int _ X\chi \Omega ^ n$. The form $T=(\chi /Z)\Omega ^{n-1}$ is normalized and is a constant multiple of the $(n-1)$st power of the Gauduchon metric conformal to $\Omega $. By theorem 12.7.3, it is the unique maximizing current, so $T=T_ F$. The equality case in the proof of that theorem gives $\nu _ F=\Omega \wedge T_ F$, and hence
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12.9.9
\begin{equation} \begin{aligned} T_ F& =\frac\chi Z\Omega ^{n-1},\\ \nu _ F& =\Omega \wedge T_ F=\frac\chi Z\Omega ^ n,\\ q_ F& =\frac\chi Z\frac{\Omega ^ n}{\mathrm{d}V},\\ k_{T_ F}& =\frac\chi Z \left(\frac{\Omega ^ n}{\mathrm{d}V}\right)^{(n-1)/n}. \end{aligned} \end{equation}
The last identity uses the positive root $\gamma =(\chi /Z)^{1/(n-1)}\Omega $, for which $T_ F=\gamma ^{n-1}$. Uniqueness of the supporting probability follows from theorem 12.8.2.
Now $\chi /Z\leq (3a/2)2^{n-1}$ and $\Omega ^ n/\mathrm{d}V\leq (3/2)^ nH_{\max }$, and multiplying gives exactly $q_ F\leq C_\nu $ and $k_{T_ F}\leq C_ k$. Since $\int _ Xq_ F\, \mathrm{d}V=1$, we have $\int _ Xq_ F^ p\, \mathrm{d}V\leq C_\nu ^{p-1}$, which gives the sharper $L^ p$ estimate for $q_ F$. The $L^ p$ estimate for $k_{T_ F}$ follows from $\int _ X\mathrm{d}V=1$.
$\square$
Pome
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