Lemma 8.6.18B310CC2C1A1

Proof. At a maximum of $w$, the admissible form $\omega +\mathrm{dd}^{\mathrm{c}}w$ is at most $\omega $, hence $e^{aw+H}\leq 1$ there. At a minimum it is at least $\omega $, and the reverse inequality holds. These give the bounds. For two solutions, a positive maximum of their difference would imply $H_ m(w_1)\leq H_ m(w_2)$ there, contradicting the exponential factor. This proves uniqueness.

First take $a=1$ and use the path $H_ m(w_ t)=e^{w_ t+tH}\omega ^ n$, starting with $w_0=0$. The trace estimate of 8.4.6 applies with prescribed potential $v=0$, fixed smooth comparison $0$, and $h(x,v)=tH(x)$; its constants are uniform for $t\in [0,1]$. 8.3.3 then gives a uniform gradient and complex-Hessian bound. 8.5.1 gives uniform $W^{2,p}$ and hence $C^{1,\alpha }$ bounds, for any fixed $\alpha <1$. Consequently $r_ t=e^{(w_ t+tH)/m}$ has a uniform Hölder bound.

The extension of 8.5.3 is uniformly elliptic and concave on all real Hessians, with coefficient oscillation bounded by $C|x-y|(1+\| R\| )$. The analytic estimate recalled in section 8.5 therefore gives uniform $C^{2,\beta }$ bounds on a finite cover. The original smooth Hessian operator is uniformly elliptic near the attained compact cone set. Differentiating and applying Schauder estimates gives bounds for every higher derivative.

The linearization of $\log (H_ m(w)/\omega ^ n)-w-tH$ is $L_ w-1$, where

Its principal coefficients are positive definite. The maximum principle gives zero kernel; smooth elliptic Fredholm theory gives surjectivity and Schauder inversion. The implicit-function theorem proves openness. The uniform estimates prove closedness, so the path reaches $t=1$. For general $a$, use the metric $a\omega $ and potential $\widehat w=aw$. Both sides of its coefficient-one equation acquire the same factor $a^ n$, so it is exactly the stated equation after division by that factor. $\square$