Theorem 8.8.23B8FDE2FF95D

Proof. Choose smooth admissible $v_ j\to \varphi $ uniformly, by 8.6.2. Their values lie in a fixed compact interval on which $f$ is smooth. Put $h(x,t)=-t+f(x,t)$ and solve

The preceding stability proposition shows that $u_ j$ converges uniformly to a continuous admissible $u$ satisfying

8.7.9, applied in both directions, gives $u=\varphi $.

The zero-order bounds for $u_ j$ are uniform. Fix $M$ bounding them and choose $B>\| \varphi \| _\infty +1$. Extend $h$ smoothly away from a neighborhood of the range of $\varphi $ to $X\times [-B-1,B+1]$; this does not change the equations for large $j$. The tolerance $\varepsilon $ in 8.4.6 depends on these bounds and on $h$, without derivatives of the smooth comparison function. Choose a fixed smooth $\widetilde\varphi $ with $\| \widetilde\varphi -\varphi \| _\infty <\varepsilon /2$ and $\| \widetilde\varphi \| _\infty \leq B$. For large $j$, $\| v_ j-\widetilde\varphi \| _\infty <\varepsilon $. 8.4.7 then gives uniform gradient and complex-Hessian bounds for $u_ j$, independent of derivatives of $v_ j$. 8.5.1 yields $\varphi \in W^{2,p}$ for every finite $p$, and $\varphi \in C^{1,\alpha }$ for every $\alpha <1$.

8.5.4 makes $\varphi $ a continuous viscosity solution of the concave uniformly elliptic extension, with right-hand side $e^{f(x,\varphi )/m}$. This right-hand side is Hölder. The analytic estimate in section 8.5 therefore gives $\varphi \in C^{2,\beta }$, for some $\beta >0$. The operator agrees with the original Hessian root on the attained compact cone set; the now classical equation thus has smooth dependence on $x,\varphi $ and its complex Hessian, and is uniformly elliptic there. Difference quotients and interior Schauder estimates give $C^{3,\beta }$; iteration proves smoothness. The estimate applies directly to the viscosity limit. $\square$