Corollary 14.19.46E9194EDCB1FF

Proof. Admissibility and $\Gamma \subseteq \Gamma _1$ give $\operatorname {tr}_\omega \mathrm{dd}^{\mathrm{c}}u\geq -C$. By theorem 11.4.2, there is a smooth function $\rho _0>0$ with $\mathrm{dd}^{\mathrm{c}}(\rho _0\omega ^{n-1})=0$, so that $\int _ X\mathrm{dd}^{\mathrm{c}}\varphi \wedge \rho _0\omega ^{n-1}=0$ for every smooth $\varphi $. Integrating the product rule $\mathrm{dd}^{\mathrm{c}}(u^2)=2u\mathrm{dd}^{\mathrm{c}}u+2\mathrm du\wedge \mathrm d^{\mathrm c}u$ against $\rho _0\omega ^{n-1}$, and using $0\leq u\leq C$ from proposition 14.19.44, we obtain a uniform bound for $\int _ X|\mathrm du|_\omega ^2\omega ^ n$, because $\rho _0$ is bounded above and below. Rellich's theorem therefore gives precompactness in $L^1$.

Given a sequence of such solutions, the Arzelà–Ascoli theorem gives a subsequence whose right-hand sides converge uniformly, since they are equicontinuous and bounded by $H_0$. A further subsequence of the solutions is Cauchy in $L^1$, and theorem 14.19.45 makes it uniformly Cauchy. This proves precompactness in $C^0(X)$. A common modulus of continuity follows from a finite uniform $\varepsilon /3$-net, since each of its finitely many members is uniformly continuous. $\square$