Proposition 15.17.3.D11439C1393F For the family $(v_\lambda ,c_\lambda )$ of lemma 15.16.5, the normalized potentials are relatively compact for uniform convergence of differences on $\Omega $. As $\lambda \downarrow 0$, every cluster limit $v$ is normalized, has minimal singularities, and satisfies
\[ \mathrm{MA}_\theta (v)=cf\, \mathrm{d}V, \]
where $c>0$ is the unique constant of theorem 15.16.7. Moreover,
\[ c_\lambda \longrightarrow c, \qquad c_\lambda \mathrm{e}^{\lambda v_\lambda }f \longrightarrow cf\quad \text{in }L^ p(\mathrm{d}V). \]
Every sequence of parameters in $(0,1]$ has a subsequence for which the potentials converge in $C^\beta $ on each compact subset of $\Omega $, for all
\[ 0<\beta <\frac{2(p-1)}{np+3(p-1)}. \]
Proof.
By lemma 15.16.5, all $v_\lambda $ lie in a common reference interval, and their densities are bounded in $L^ p$ by $C\| f\| _ p$. We may therefore apply corollary 15.15.3 with $\eta =V_\theta $. Given any sequence $\lambda _ j\downarrow 0$, we pass to a subsequence with $v_{\lambda _ j}\to v$ uniformly in differences on $\Omega $ and in $L^1(X)$, and with $c_{\lambda _ j}\to c_*\in [m,C]$. The limit satisfies the same reference bounds and is normalized. Off the pole set,
\[ 0\leq 1-\mathrm{e}^{\lambda v_\lambda } \leq 1-\mathrm{e}^{\lambda (V_\theta -B)}\longrightarrow 0 \]
as $\lambda \downarrow 0$. Multiplying by $f$ and using dominated convergence, we obtain $\mathrm{e}^{\lambda v_\lambda }f\to f$ in $L^ p$ for the whole family. The compactness corollary therefore identifies the limit equation as $\mathrm{MA}_\theta (v)=c_*f\, \mathrm{d}V$, and the uniqueness of the constant gives $c_*=c$. Since every subsequential limit of $c_\lambda $ as $\lambda \downarrow 0$ equals $c$, we conclude that $c_\lambda \to c$, and the stated convergence of the densities follows.
Finally, theorem 15.17.1 gives uniform local $C^\alpha $ bounds for every $\alpha $ below its endpoint, and the uniform limit inherits these bounds by pointwise convergence. For $0<\beta <\alpha $, uniform convergence and the elementary interpolation inequality on smaller coordinate balls give
\[ [v_{\lambda _ j}-v]_{C^\beta } \leq C\| v_{\lambda _ j}-v\| _\infty ^{1-\beta /\alpha } [v_{\lambda _ j}-v]_{C^\alpha }^{\beta /\alpha } \longrightarrow 0, \]
where the Hölder bounds are taken on slightly larger compact neighborhoods. The uniformly convergent subsequence given by compactness thus converges in $C^\beta $ on every compact subset of $\Omega $ and for every exponent below the endpoint; no further extraction depending on the exponent is needed.
$\square$
Pome
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