We call the potentials $w_\lambda $ of theorem 11.44.2 the centered potentials. Choose a critical potential $\psi $ for $f$ with $\sup _ X\psi =0$, so that $\mathrm{MA}_\theta (\psi )=cf\, \mathrm{d}V$. The integral normalizations prevent the centered potentials from drifting upward by constants.
Lemma 11.44.15.FA5BBA96CCC3 There is $K<\infty $, independent of $0<\lambda \leq 1$, such that
CB3DF4196532
11.44.16
\begin{equation} \psi \leq w_\lambda ,\qquad 0\leq k_\lambda :=\sup _ Xw_\lambda \leq K,\qquad \psi -K\leq v_\lambda :=w_\lambda -k_\lambda \leq V_\theta . \end{equation}
Moreover $\int _ Xw_\lambda \, \mathrm{d}\nu \leq 0$ for every $\nu \in \mathcal P_ f$. The family $w_\lambda $ is relatively compact for locally uniform convergence on $\Omega $; every such cluster also converges in $L^1(X)$ to a potential of minimal type.
Proof.
The centered potential satisfies $\mathrm{MA}_\theta (w_\lambda )=c\mathrm{e}^{\lambda w_\lambda }f\, \mathrm{d}V$, while $\psi \leq 0$ gives $\mathrm{MA}_\theta (\psi )\geq c\mathrm{e}^{\lambda \psi }f\, \mathrm{d}V$. Part (2) of corollary 15.13.3 therefore gives $\psi \leq w_\lambda $, hence $k_\lambda \geq 0$. For each fixed $\lambda $, the density $\mathrm{e}^{\lambda w_\lambda }f$ lies in $L^ p$, and its critical scalar is $c$. Therefore
\[ N(a\mathrm{e}^{\lambda w_\lambda /n})=A. \]
The moment estimate above gives $aw_\lambda \in L^\ell $, so proposition 11.44.9, with $z=w_\lambda $ and $t=\lambda $, gives the asserted integral inequalities. Fix one $\nu \in \mathcal P_ f$. Then
\[ k_\lambda \leq -\int _ Xv_\lambda \, \mathrm{d}\nu \leq C_ q A^{-1}\| a v_\lambda \| _\ell \leq K \]
by (11.44.13), with $K$ independent of $\lambda $. The normalization $\sup _ Xv_\lambda =0$ gives $v_\lambda \leq V_\theta $, and subtracting $k_\lambda $ from the lower bound $\psi \leq w_\lambda $ proves (11.44.16).
The potentials $v_\lambda =u_\lambda -\sup _ Xu_\lambda $ form the normalized exponential family of proposition 15.17.3, which gives subsequences converging locally uniformly on $\Omega $ and in $L^1(X)$ to potentials of minimal type. Extracting a convergent subsequence of $k_\lambda \in [0,K]$ gives the same compactness for $w_\lambda =v_\lambda +k_\lambda $. The bound $|w_\lambda |\leq |\psi |+K$ also gives $L^1$ convergence along every subsequence converging locally uniformly on $\Omega $, since $X\setminus \Omega $ has zero volume.
$\square$
In particular,
\[ \mathrm{e}^{\lambda \sup _ Xu_\lambda } =c\mathrm{e}^{\lambda k_\lambda }\longrightarrow c. \]
Moreover, for $0<\lambda \leq 1$ the densities of the equations satisfy $\| c\mathrm{e}^{\lambda w_\lambda }f\| _ p\leq c\mathrm{e}^ K\| f\| _ p$. It matters that this bound includes the scalar factor: an $L^ p$ bound on the data alone, after an arbitrary rescaling, does not bound the normalized potentials. For example, on $\mathbb P^1$, with $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|z|^2)$ and $0<\epsilon <1$, the function
\[ \psi _\epsilon =\log \frac{|z|^2+\epsilon ^2}{1+|z|^2} \]
is smooth and $\omega _{\mathrm{FS}}$-plurisubharmonic, with supremum zero and infimum $\log \epsilon ^2$. Write $\mathrm{MA}_{\omega _{\mathrm{FS}}}(\psi _\epsilon )=g_\epsilon \omega _{\mathrm{FS}}$, and take $L^ p$ norms with respect to $\omega _{\mathrm{FS}}$ in this example. The data $g_\epsilon /\| g_\epsilon \| _ p$ have norm one, and $\psi _\epsilon $ solves the equation with these data and scalar $\| g_\epsilon \| _ p$; but the densities $g_\epsilon $ of the equations themselves are not bounded in $L^ p$, and the potentials are not uniformly bounded.
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