Proposition 8.13.12.FBE7E9A6BE5E Let $A<B$ and $f$ be smooth near $X\times [A,B]$, with $\partial _ t f\geq \kappa >0$ there, and let $\nu $ be a finite positive Radon measure. Suppose bounded admissible $u,v$ have values in $[A,B]$ and satisfy
\[ H_ m(u)=e^{f(x,u)+h_1}\nu ,\qquad H_ m(v)=e^{f(x,v)+h_2}\nu , \]
where $h_1,h_2$ are bounded Borel functions. Then
\[ \| u-v\| _\infty \leq \kappa ^{-1}\| h_1-h_2\| _{L^\infty (\nu )}. \]
Proof.
Put $K=\| h_1-h_2\| _{L^\infty (\nu )}$ and fix $s>K/\kappa $. On $E_ s=\{ v+s<u\} $, the actual values $u,v$ lie in $[A,B]$, so $f(x,u)-f(x,v)\geq \kappa s$. The equations and the essential bound on $h_2-h_1$ therefore give, as restricted measures,
\[ \mathbf1_{E_ s}H_ m(v+s) =\mathbf1_{E_ s}H_ m(v) \leq e^{K-\kappa s}\mathbf1_{E_ s}H_ m(u). \]
The constant is smaller than one. Applying 8.13.10 to $v+s,u$ gives $u\leq v+s$ pointwise. Let $s\downarrow K/\kappa $ and exchange $u,v$ to obtain the norm bound. The exponent is evaluated only at $u,v$; the shifted comparison potential need not take its values in $[A,B]$.
$\square$
Pome
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