Theorem 17.17.8062DEC0E6527

Proof. By lemma 5.41.1, for each $x$ the function $t\mapsto u_ t(x)-ct$ is either finite and convex or identically $-\infty $. It is bounded above by zero, so it is decreasing. Since $\sup _ X(u_ t-ct)=0$, theorem 5.25.2, corollary 5.25.3 give

For $\tau <c$, we have $\widehat u_ c\leq \widehat u_\tau \leq u_1-\tau $, so proposition 5.41.3 gives $\widehat u_\tau \in \mathrm{PSH}(X,\theta )$. Also

while the bound $\widehat u_\tau \leq t(c-\tau )$ with $t\to \infty $ shows $\widehat u_\tau \equiv -\infty $ for $\tau >c$. Concavity and monotonicity in $\tau $ hold because $\widehat u_\tau $ is an infimum of affine functions of $\tau $ with negative slopes $-t$.

Fix $\tau <c$ and write $g=\widehat u_\tau $. The inequality $g\leq u_ M-M\tau $, valid for every $M>0$, gives

For $C>0$ put $q_ C=P_\theta (g+C,0)$. Since $g\leq 0$, the function $g$ is a competitor, so $q_ C\not\equiv -\infty $, and

The affine path $q_ C-Ct/M$ is admissible for the endpoints $\varphi $ and $u_ M-M\tau $ on $[0,M]$. Adding $-t\tau $ gives a bijection between the competitor families for the endpoints $\varphi ,u_ M$ and $\varphi ,u_ M-M\tau $, by proposition 17.16.2, so the Perron segment for the latter endpoints is $u_ t-t\tau $. For $0<s\leq 1$ this yields

Taking the infimum over $M>0$ gives $q_ C-Cs\leq g$, and letting $s\downarrow 0$ gives $q_ C\leq g$. Thus $q_ C=g$ for every $C>0$, and letting $C\to \infty $ in the definition of $P_\theta [g]$ proves $P_\theta [g]=g$.

Let $\psi =(\sup _{\tau <c}\widehat u_\tau )^*$; since $\widehat u_\tau =P_\theta [\widehat u_\tau ]\leq \varphi $, we have $\psi \leq \varphi $. Because $u_ t-ct$ is decreasing in $t$, the limit of $u_ t(x)$ as $t\downarrow 0$ exists at every $x$, and an almost everywhere convergent sequence given by the $L^1$ convergence $u_ t\to \varphi $ identifies it with $\varphi $ almost everywhere. At these points, the endpoint identity in lemma 5.41.5 gives

Therefore $\psi =\varphi $ by lemma 5.24.3.

As $\tau \uparrow c$, the transforms decrease to $\widehat u_ c$, since the infima over $t>0$ and over $\tau <c$ commute; this is the convention for the value at the cutoff. Finally, lemma 5.41.5 recovers $u_ t$ from its inverse transforms, also at points where $t\mapsto u_ t(x)$ is identically $-\infty $, and lemma 5.41.8 recovers each test curve from its forward transform. Hence the correspondence is a bijection. $\square$