PomeCorollary 8.13.13.5460C4009B01 Under the hypotheses of 8.8.3, every bounded admissible solution with values in $[A,B]$ is its smooth solution. In particular, for every $a>0$ and smooth real $H$, the equation $H_ m(u)=e^{au+H}\omega ^ n$ has exactly one bounded admissible solution, and that solution is smooth and strictly admissible.
Corollary 8.13.135460C4009B01
Proof. For a continuous admissible potential, smooth decreasing admissible approximations converge uniformly on compact $X$ by Dini's theorem. Thus 8.11.12, 8.7.1 define the same Hessian measure on the continuous class. The smooth solution of 8.8.3 therefore solves the bounded equation as well. Apply 8.13.12 with $\nu =\omega ^ n$ and $h_1=h_2=0$ to identify any other bounded solution in $[A,B]$ with it. For the final assertion, smooth existence is already 8.6.1, and 8.13.11 with $\nu =e^ H\omega ^ n$ gives uniqueness among all bounded solutions. $\square$
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