Theorem 12.10.141D17749328B

Proof. For a smooth strictly positive $p$, we have $D_ p=\mathfrak D_{n\log p}$ and $\mathrm{e}^{-B_0(n\log p)/n}=\rho _\omega (p)$, so theorem 12.7.3, applied to $F=n\log p$, proves (12.10.2). By Hölder's inequality and theorem 12.9.1, for every real $r>n$ there is a constant $C_ r$ such that, for every $q\in L^ r$,

Next, approximate a bounded nonnegative $p$ in $L^{2n}$ by smooth strictly positive functions. This estimate and (15.32.8) give uniform convergence on $\mathscr G_\omega $ of the corresponding functionals $D$ and convergence of the root norms, so the supremum formula holds for bounded $p$. For such $p$, the functional $D_ p$ is also upper semicontinuous and concave, by lemma 12.7.2 and uniform convergence. For general $p\in L^ n$, we use the truncations $\min (p,K)$ at positive integer levels $K$. Monotone convergence and the definition of $b_\omega $ through truncations give

This step only interchanges two suprema and gives no attainment.

Now suppose $p\in \mathcal D_\omega ^0$. The supremum formula applied to the tail gives

uniformly in $T$, by proposition 15.32.9. Hence $D_ p$ is finite and is a uniform limit of upper semicontinuous concave functionals, and compactness of $\mathscr G_\omega $ gives a maximizer. Finally, fix $1<q<n/(n-1)$. By Hölder's inequality and theorem 12.9.1, for every measurable set $E$,

Choosing first $K$ large and then the volume of $E$ small proves uniform absolute continuity. Since the total masses are bounded by $\rho _\omega (p)$, the densities are uniformly integrable. $\square$