Corollary 12.9.3.893A41D78AF5 For every $R>0$ and $1<q<n/(n-1)$ there is a constant $C_{R,q}$ with the following property. If $F\in C(X)$ and $\| F\| _\infty \leq R$, every supporting probability $\nu \in \mathcal A_0(F)$ of theorem 12.8.2 has a density $a$ satisfying
\[ \nu =a\, \mathrm{d}V,\qquad \| a\| _{L^ q(\mathrm{d}V)}\leq C_{R,q}. \]
If $v_-(\omega )>0$, this holds also for $q=n/(n-1)$.
Proof.
Recall that $M(F)=\exp (-B_0(F)/n)$, and represent $\nu $ by a maximizing current $T\in \mathscr M(F)$, using theorem 12.8.2. Since $\Gamma _ F(T)=\mathrm{e}^{F/n}\Gamma _0(T)$ and $B_0$ is $1$-Lipschitz,
\[ \nu =\frac{\mathrm{e}^{F/n}k_ T}{M(F)}\, \mathrm{d}V, \qquad M(F)\geq \mathrm{e}^{-R/n}M(0)>0. \]
Theorem 12.9.1 therefore gives the subcritical bound with constant $\mathrm{e}^{2R/n}M(0)^{-1}C_ q$. When $v_-(\omega )>0$, the same argument with the endpoint constant $v_-(\omega )^{-1/n}$ of theorem 12.9.2 gives the case $q=n/(n-1)$.
$\square$
Pome
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