The endpoint estimate can fail for the family of all tests while the maximizing tests for a family of densities remain uniformly controlled. For smooth densities close in $C^{2,\alpha }$ to the normalized volume form $\omega ^ n/\int _ X\omega ^ n$ of the background, a fixed-point argument keeps the solution forms close to $\omega $ and their Gauduchon factors close to that of $\omega $. This gives uniform bounds for both the determinant density of the maximizing current and the density of the supporting probability.
In this subsection we assume $n\geq 3$ and fix $0<\alpha <1$. All Hölder norms are defined using a fixed finite atlas and fixed smooth connections. Put
\[ H=\frac{\omega ^ n}{\mathrm{d}V},\qquad V=\int _ X\omega ^ n,\qquad F_*=\log (H/V). \]
Thus $\mathrm{e}^{F_*}\mathrm{d}V=\omega ^ n/V$ is a probability measure. By theorem 11.4.2, there is a unique smooth positive function $\chi _0$, the normalized Gauduchon factor of $\omega $, such that
\[ \mathrm{dd}^{\mathrm{c}}(\chi _0\omega ^{n-1})=0,\qquad \int _ X\chi _0\omega ^ n=1. \]
Write $m=\min _ X\chi _0$, $a=\max _ X\chi _0$ and $C_0=\| \chi _0\| _{C^{2,\alpha }}$.
We first specify the constants that determine the size of the neighborhood. For a smooth positive $(1,1)$-form $\Omega $, set
\[ L_\omega v=\frac{n\mathrm{dd}^{\mathrm{c}}v\wedge \omega ^{n-1}}{\omega ^ n}, \qquad A_\Omega \eta =\frac{\mathrm{dd}^{\mathrm{c}}(\eta \Omega ^{n-1})}{\mathrm{d}V}. \]
The compatibility argument of proposition 11.23.2, proposition 14.4.1 shows that
\[ \mathcal L(v,c)=L_\omega v-c, \qquad \int _ Xv\, \mathrm{d}V=0, \]
is an isomorphism from the space of pairs $(v,c)\in C^{4,\alpha }\times \mathbb R$ with this normalization onto $C^{2,\alpha }$. Let $K\geq 1$ bound its inverse for the norm $\| v\| _{C^{4,\alpha }}+|c|$. The formal adjoint of $A_\omega $ with respect to $\mathrm{d}V$ is $(H/n)L_\omega $, whose kernel consists of the constants and whose index is zero. Consequently $A_\omega $ has kernel $\mathbb R\chi _0$ and range
\[ Z_0=\left\{ g\in C^{0,\alpha }:\int _ Xg\, \mathrm{d}V=0\right\} . \]
Its restriction to
\[ E_\chi =\left\{ \eta \in C^{2,\alpha }:\int _ X\eta \omega ^ n=0\right\} \]
has a bounded inverse $\mathcal R:Z_0\to E_\chi $. Fix $G\geq 1$ bounding this inverse.
We normalize the Hölder norms of forms and of endomorphism fields so that they dominate pointwise operator norms and are submultiplicative under composition of endomorphisms; the subscript $\mathrm{alg}$ below refers to this norm on endomorphism fields. Choose $D\geq 1$ so that
\[ \| \mathrm{dd}^{\mathrm{c}}v\| _{C^{2,\alpha }}\leq D\| v\| _{C^{4,\alpha }},\qquad \| \omega ^{-1}\mathrm{dd}^{\mathrm{c}}v\| _{C^{2,\alpha },\mathrm{alg}} \leq D\| v\| _{C^{4,\alpha }}. \]
Let $C_{\mathrm{tr}}\geq 1$ be a bound for the trace map on endomorphism fields, and put $Q=2C_{\mathrm{tr}}D^2$. Expanding $\Omega ^{n-1}-\omega ^{n-1}$ gives a constant $C_ A\geq 1$ such that
\[ \| A_\Omega -A_\omega \| _{C^{2,\alpha }\to C^{0,\alpha }} \leq C_ A\| \Omega -\omega \| _{C^{2,\alpha }} \quad \text{if }\| \Omega -\omega \| _{C^{2,\alpha }}\leq 1. \]
Only two derivatives of $\Omega $ enter this polynomial estimate. Define
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12.9.5
\begin{equation} r=\min \left\{ \frac1{2D},\frac1{4KQ}, \frac{m}{4GC_ ADC_0}\right\} , \qquad \varepsilon =\frac r{2K}. \end{equation}
These constants depend only on $\omega $, $\mathrm{d}V$, $\alpha $ and the chosen norms.
Lemma 12.9.6.9F57BA26805A Let $F$ be smooth with $\| F-F_*\| _{C^{2,\alpha }}\leq \varepsilon $. The solution form $\Omega _ F=\omega +\mathrm{dd}^{\mathrm{c}}u_ F$ of $\mathrm{MA}_\omega (u_ F)=\mathrm{e}^{B_0(F)+F}\mathrm{d}V$ and its Gauduchon factor $\chi _ F$, normalized as below, satisfy
\begin{gather*} \tfrac 12\omega \leq \Omega _ F\leq \tfrac 32\omega , \qquad \frac m2\leq \chi _ F\leq \frac{3a}2,\\ \mathrm{dd}^{\mathrm{c}}(\chi _ F\Omega _ F^{n-1})=0, \qquad \int _ X\chi _ F\omega ^ n=1. \end{gather*}
The potential can be normalized by $\int _ Xu_ F\, \mathrm{d}V=0$ with $\| u_ F\| _{C^{4,\alpha }}+|B_0(F)-\log V|\leq r$.
Proof.
For $\| v\| _{C^{4,\alpha }}\leq 1/(2D)$ put
\[ N(v)=\log \det (I+\omega ^{-1}\mathrm{dd}^{\mathrm{c}}v)-L_\omega v. \]
The logarithm series starts at degree two. If the endomorphism norms of $\omega ^{-1}\mathrm{dd}^{\mathrm{c}}v$ and $\omega ^{-1}\mathrm{dd}^{\mathrm{c}}w$ are at most $\rho \leq 1/2$, telescoping their powers bounds the difference series by $C_{\mathrm{tr}}\rho /(1-\rho )$ times the endomorphism difference. Hence
\begin{align*} \| N(v)\| _{C^{2,\alpha }}& \leq Q\| v\| _{C^{4,\alpha }}^2,\\ \| N(v)-N(w)\| _{C^{2,\alpha }} & \leq Q(\| v\| _{C^{4,\alpha }}+\| w\| _{C^{4,\alpha }}) \| v-w\| _{C^{4,\alpha }}. \end{align*}
On the closed radius-$r$ ball of normalized pairs $(v,c)$, the map
\[ (v,c)\longmapsto \mathcal L^{-1}(F-F_*-N(v)) \]
has image norm at most $K\varepsilon +KQr^2\leq 3r/4$ and Lipschitz constant at most $2KQr\leq 1/2$. Its fixed point $(u,c)$ satisfies
\[ \log \frac{(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n}{\omega ^ n}-c=F-F_*. \]
The estimate $Dr\leq 1/2$ gives the form comparison. The scalar is $b=\log V+c$. Differentiating the local equation gives a uniformly elliptic equation for each first derivative of $u$, with coefficients initially in $C^{2,\alpha }$. Interior Schauder estimates improve the regularity one order at a time, so $u$ is smooth. Uniqueness in theorem 14.2.1 identifies it with $u_ F$ and gives $b=B_0(F)$. No uniform higher-derivative bounds on $F$ enter the estimates above.
Put $\Delta A=A_{\Omega _ F}-A_\omega $. Stokes' theorem makes its image lie in $Z_0$. On the complete space $E_\chi $, the affine map
\[ \eta \longmapsto -\mathcal R\Delta A(\chi _0+\eta ) \]
has Lipschitz constant $\kappa =GC_ ADr\leq m/(4C_0)\leq 1/4$. Its fixed point obeys
\[ \| \eta \| _{C^{2,\alpha }} \leq \frac{\kappa C_0}{1-\kappa } \leq 2\kappa C_0\leq m/2. \]
Since $A_\omega \chi _0=0$ and $A_\omega \mathcal R$ is the identity on $Z_0$, the function $\chi _ F=\chi _0+\eta $ satisfies $A_{\Omega _ F}\chi _ F=0$, and it has the stated normalization and bounds. Elliptic regularity makes it smooth. Every Gauduchon factor of $\Omega _ F$ with the same normalization satisfies the same affine fixed-point equation, so this factor is the one given by theorem 11.4.2.
$\square$
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
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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 $\int _ X\mathrm{d}V=1$.
$\square$
Corollary 12.9.10.7877D1AB98D3 The ball of lemma 12.9.6 contains smooth injectively parametrized families, of arbitrarily large finite dimension, of data $F$ normalized by $\int _ X\mathrm{e}^ F\mathrm{d}V=1$. It also contains a smooth path of such data whose maximizing currents are pairwise distinct.
Proof.
For a small smooth function $f$, put
\[ F(f)=F_*+f-\log \int _ X\mathrm{e}^{F_*+f}\, \mathrm{d}V. \]
If $F(f)=F(g)$, then $f-g$ is constant, so the map $f\mapsto F(f)$ is injective on functions of zero $\mathrm{d}V$-mean. By its continuity in $C^{2,\alpha }$, it maps a sufficiently small neighborhood of zero in any fixed finite-dimensional space of such smooth functions into the ball, and these spaces can have arbitrary dimension.
For the path, choose a smooth real $\psi $ of zero $\mathrm{d}V$-mean whose complex Hessian at some point $x_0$ has, in an $\omega $-unitary frame, a matrix $Q_0$ that is not a multiple of the identity. A cutoff of $|z_1|^2-|z_2|^2$ in local coordinates, minus its mean, is such a function. For small real $t$ put
\[ \Omega _ t=\omega +t\mathrm{dd}^{\mathrm{c}}\psi ,\qquad V_ t=\int _ X\Omega _ t^ n,\qquad F_ t=\log \frac{\Omega _ t^ n}{V_ t\, \mathrm{d}V}. \]
These normalized data tend to $F_*$ in $C^{2,\alpha }$ as $t\to 0$, and by uniqueness in theorem 14.2.1 their solution forms are the forms $\Omega _ t$. If $F_ s=F_ t$, then $\log (\Omega _ s^ n/\Omega _ t^ n)$ is the constant $\log (V_ s/V_ t)$. The signs of the complex Hessian of $(s-t)\psi $ at a maximum and at a minimum force this constant to be zero. Integrating the derivative of $\log \det $ along the segment between $\Omega _ t$ and $\Omega _ s$ then gives a uniformly elliptic equation for $(s-t)\psi $ without zeroth-order term, and the maximum principle forces $(s-t)\psi $ to be constant. Since $\psi $ is not constant, $s=t$.
If the maximizing currents for $F_ s$ and $F_ t$ agree, (12.9.9) makes $\Omega _ s^{n-1}$ and $\Omega _ t^{n-1}$ proportional at $x_0$. Positive cofactor matrices determine positive forms up to scale, so $I+sQ_0$ and $I+tQ_0$ are proportional. Since $I$ and $Q_0$ are linearly independent, comparing coefficients gives $s=t$.
$\square$
The small $C^{2,\alpha }$ neighborhood gives a condition directly on the prescribed density. The next estimate instead assumes uniform bounds for the solution forms and covers a larger smooth family without derivative bounds on its densities. The question for arbitrary bounded families of continuous densities remains open when $v_-(\omega )=0$. The concentrating sequences of proposition 12.18.2 consist of near-maximizers, so these estimates do not apply to them.
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