Theorem 14.19.32B9E6E7C4706B

Proof. The density $\mathrm e^{F(x,u)}$ is bounded and strictly positive, so corollary 14.19.29 makes $u$ continuous. Choose smooth strictly $\omega $-plurisubharmonic $v_ j\downarrow u$ by theorem 5.32.1. They have a common bound $M$ and converge uniformly by Dini's theorem. Define

By theorem 14.7.2, there is a smooth strictly $\omega $-plurisubharmonic $a_ j$ with

Put $\delta _ j=\| G(x,v_ j)-G(x,u)\| _{C^0}\to 0$. The given $u$ solves the proper equation with coefficient $G(x,u)$. The weighted measures of $a_ j-\delta _ j$ and $a_ j+\delta _ j$ bracket its coefficient, so corollary 14.19.12 gives

Hence $a_ j\to u$ uniformly. By lemma 14.19.31, $C^{-1}\omega \leq \omega _{a_ j}\leq C\omega $ uniformly. In particular $\Delta _ ga_ j$ is uniformly bounded. Fixed-operator elliptic estimates give uniform $W^{2,q}$ bounds for every finite $q$, so $u\in C^{1,\alpha }(X)$ for every $0<\alpha <1$.

We explain the uniformly elliptic passage to higher regularity, using the concave extension from corollary 14.5.8. On a relatively compact coordinate chart, let $P$ send a real Hessian to its complex Hermitian part, with the coordinate normalization of $\mathrm{dd}^{\mathrm{c}}$. All matrices $g(x)+P(D^2a_ j)$ lie in a fixed compact interval $\mathcal K=\{ \lambda I\leq A\leq \Lambda I\} $ of positive matrices. Define

The tangent inequality for the concave logarithmic determinant shows that this is the original operator whenever $g(x)+P(N)\in \mathcal K$. It is concave and uniformly elliptic on real symmetric matrices: for $N'\geq 0$, $P(N')\geq 0$ and its trace is a fixed positive multiple of $\operatorname {tr}N'$, while $A^{\pm 1}$ have uniform bounds. Its dependence on $x$ is uniformly Lipschitz on this chart. Thus the $a_ j$ are classical, and hence viscosity, solutions of

Both sides converge uniformly except for the Hessian variable. Stability of viscosity solutions gives $\mathcal F(x,D^2u)=u+G(x,u)$ in the viscosity sense. The right-hand side is Hölder because $u\in C^{1,\alpha }$. The local Evans–Krylov estimate therefore gives $u\in C^{2,\beta }$ for some $\beta >0$. The metric bounds pass to the limit, so the original equation is now uniformly elliptic with $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Differentiating it and applying Schauder estimates repeatedly proves $u\in C^\infty (X)$. $\square$