PomeLemma 8.2.9.365519F85C5A If smooth uniformly bounded admissible $u_ j\downarrow u$ and $v_ j\downarrow v$ are given, there are smooth uniformly bounded admissible $w_ j\downarrow \max \{ u,v\} $. For any chosen $\epsilon _ j>0$ decreasing to zero, they can be chosen to satisfy $w_ j=u_ j$ on $\{ u_ j-v_ j>\epsilon _ j\} $ and $w_ j=v_ j$ on $\{ v_ j-u_ j>\epsilon _ j\} $. For smooth functions $f,g$ on an open subset of $X$ with $\mathrm{dd}^{\mathrm{c}}f,\mathrm{dd}^{\mathrm{c}}g\in \overline{\Gamma _ m(\omega )}$, the regularized maximum $M_\epsilon (f,g)$ constructed below satisfies the same unshifted cone condition.
Lemma 8.2.9365519F85C5A
Proof. Choose a nonnegative even smooth probability density $\zeta $ supported in $[-1,1]$, and set
\[ r_\epsilon (t)=\int |t-\epsilon s|\zeta (s)\, \mathrm ds, \qquad M_\epsilon (x,y)=\tfrac 12(x+y+r_\epsilon (x-y)). \]
The function $r_\epsilon $ is smooth and convex, $|r_\epsilon '|\leq 1$, and $r_\epsilon (t)=|t|$ for $|t|\geq \epsilon $. Jensen's inequality and the triangle inequality give
\[ \max \{ x,y\} \leq M_\epsilon (x,y) \leq \max \{ x,y\} +\epsilon /2. \]
Its partial derivatives in $x,y$ are nonnegative. For fixed $t$, $r_\epsilon (t)$ is an even convex function of $\epsilon $, since $\zeta $ is even. It is therefore nondecreasing for $\epsilon \geq 0$. Consequently $w_ j=M_{\epsilon _ j}(u_ j,v_ j)$ decreases pointwise to $\max \{ u,v\} $ and has the asserted equalities and bounds.
Put $h_ j=u_ j-v_ j$ and $a_ j=(1+r_{\epsilon _ j}'(h_ j))/2$. The chain rule gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}w_ j =a_ j(\omega +\mathrm{dd}^{\mathrm{c}}u_ j)+(1-a_ j)(\omega +\mathrm{dd}^{\mathrm{c}}v_ j) +\tfrac 12r_{\epsilon _ j}''(h_ j) \mathrm dh_ j\wedge \mathrm d^{\mathrm c}h_ j. \]
Here $0\leq a_ j\leq 1$, and the last form is semipositive. Convexity of the cone and its stability under semipositive addition prove admissibility. For smooth local functions satisfying the unshifted cone condition, omit $\omega $ from this chain rule; the same argument proves preservation of that condition. The pointwise maximum is itself admissible: an upper test for it at a point is an upper test for a potential attaining the maximum there. Its canonical representative is thus the indicated pointwise maximum. $\square$
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