Lemma 15.36.2.91B8E02B5759 There is a $\kappa $-plurisubharmonic function $w_{\lambda ,\alpha }$ on $Y$ such that
\[ U_{\lambda ,\alpha }=\lambda ^{-1}\log b+w_{\lambda ,\alpha },\qquad w_{\lambda ,\alpha }\geq u_\alpha , \qquad \mathrm{MA}^{\mathrm{ext}}_\kappa (w_{\lambda ,\alpha }) =\mathrm e^{\lambda w_{\lambda ,\alpha }}q_\alpha \kappa ^ k. \]
The last measure has mass one. On the affine chart centered at $o$, set $s=\log |z|^2$, $\rho (s)=\log (1+\mathrm e^ s)$ and $m(s)=\rho '(s)$. The function $w=w_{\lambda ,\alpha }$ is radial and satisfies
C1A1056DFED6
15.36.3
\begin{equation} \begin{aligned} Q_{\lambda ,\alpha }(s) & =\int _{-\infty }^ s\mathrm e^{\lambda w(r)}q_\alpha (r)\, \mathrm d(m(r)^ k),\\ w'(s)& =Q_{\lambda ,\alpha }(s)^{1/k}-m(s), \qquad Q_{\lambda ,\alpha }(-\infty )=0, \quad Q_{\lambda ,\alpha }(+\infty )=1. \end{aligned} \end{equation}
Proof.
Choose a smooth nondecreasing concave function $\chi $ on $[0,+\infty )$, equal to the identity on $[0,1]$ and constant on $[2,+\infty )$, and put $q_ j=j\chi (q_\alpha /j)$. Concavity gives $q_ j\uparrow q_\alpha $ and
\[ \min (q_\alpha ,j)\leq q_ j\leq \min (q_\alpha ,2j). \]
The functions $q_ j$ are smooth and positive on $Y$: near $o$ they are constant, and elsewhere the density is smooth. Let $w_ j$ be the continuous solution of $\mathrm{MA}_\kappa (w_ j)=\mathrm e^{\lambda w_ j}q_ j\kappa ^ k$ from theorem 14.19.3. It is the greatest subsolution by theorem 15.35.2 and bounded uniqueness. Since $u_\alpha \leq 0$ and $\mathrm{MA}^{\mathrm{ext}}_\kappa (u_\alpha )=q_\alpha \kappa ^ k$, it is a subsolution for every $q_ j$, whence $u_\alpha \leq w_ j$. Density comparison gives $w_ j\downarrow w$.
The pullback $W_ j=\lambda ^{-1}\log b+w_ j$ is a bounded solution on $X$, by the product expansion in theorem 15.30.3. For the smooth density $q_ j\mu $ it is therefore also the greatest subsolution. Every subsolution for $q_\alpha \mu $ is a subsolution for $q_ j\mu $, so $U_{\lambda ,\alpha }\leq W_ j$.
Monotone convergence in capacity and theorem 15.34.2 on $Y$ identify the equation for $w\geq u_\alpha $. Moreover, Stokes gives $\int _ Y\mathrm e^{\lambda w_ j}q_ j\kappa ^ k=1$. The integrands are dominated by $\mathrm e^{\lambda \sup _ Yw_1}q_\alpha $, so dominated convergence gives the same identity for $w$. Its pullback is a product solution, hence lies below the greatest solution on $X$. Together with $U_{\lambda ,\alpha }\leq \lim _ jW_ j$, this proves separation and all measure assertions.
Uniqueness makes each $w_ j$ invariant under the unitary action on the affine coordinates $z\in \mathbb C^ k$, and the limit has the same invariance. A radial $\kappa $-plurisubharmonic function has a convex nondecreasing profile $\rho +w$. Symmetric convolution of this convex profile on compact $s$ intervals, decreasing as the scale decreases, and bounded-product continuity give its radial measure $\mathrm d((\rho +w)'^ k)$, as in lemma 15.30.1. The bound $w\geq u_\alpha $, whose growth is $o(|s|)$ at $-\infty $, forces the initial derivative to vanish. The positive density makes the flux continuous and positive at finite $s$, yielding (15.36.3). Its final value is the mass one already proved. The profile is consequently smooth on finite $s$ intervals.
$\square$
Pome
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