Proposition 5.27.31026A64B7A5A2

Proof. Translate $a$ to the origin and write $z=(\zeta ,z')\in \mathbb C\times \mathbb C^ m$, $m=n-1$. It suffices to construct measures of mass one and then multiply the potentials by $c^{1/n}$. For $\delta >0$, put $H_\delta (z')=\log (|z'|^2+\delta ^2)$. With $s=|z'|^2$, the complex Hessian of $H_\delta $ has the eigenvalue $(s+\delta ^2)^{-1}$ with multiplicity $m-1$ in the directions orthogonal to $z'$ and the eigenvalue $\delta ^2(s+\delta ^2)^{-2}$ in the radial direction. Taking the determinant and integrating in polar coordinates gives

where $\lambda _{2m}$ is Lebesgue measure. These probability measures tend to $\delta _0$ as $\delta \downarrow 0$, since their mass outside each ball about the origin tends to zero.

Choose $R>\max (1,\sup _\Omega |\zeta |)$ and a ball $P\subset \mathbb C^ m$ containing the projection of $\Omega $ to the $z'$ variables. Set

Apply lemma 5.27.29 on $\mathbb C\times \mathbb C^ m$ to these data, and let $u_ j=h_ j-\eta _ j$, where $h_ j$ is the resulting function. The convolution satisfies $0\leq \chi _{\eta _ j}(t)-\max (t,0)\leq \eta _ j$, and $q+\max (T,0)=\max (q,v_ j-A_ j)$. Both branches of this maximum are negative on $\Omega $, so

Each $u_ j$ is smooth and bounded on $\Omega $, since the second branch $v_ j-A_ j$ is bounded below on the bounded set $P$. The logarithm has finite $L^ p(\Omega )$ norm for every finite $p$, by polar integration on a fixed polydisk containing $\Omega $. Hence $u_ j\to 0$ in all these norms.

Let $B'\Subset B\Subset \Omega $ be concentric balls, and let $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Since $|u_ j|\leq \varepsilon _ j|\log (|\zeta |^2/R^2)|+\eta _ j$, the weighted Chern–Levine–Nirenberg estimate applied to $\log (|\zeta |^2/R^2)$, together with the range estimate (5.27.3) for the constant term, gives

where $C_ B$ depends on the fixed balls and the logarithm but not on $j$ or $w$. Restricting the integral to $B'\cap \{ |u_ j|>t\} $, dividing by $t>0$ and taking the supremum over $w$ shows that the capacity of $B'\cap \{ |u_ j|>t\} $ relative to $B$ tends to zero.

In (5.27.30), the factor $\varepsilon _ j(\mathrm{dd}^{\mathrm{c}}v_ j)^ m$ equals $(\mathrm{dd}^{\mathrm{c}}H_{\delta _ j})^ m$. For $z'\in P$ and $|t|\leq \eta _ j$, the radii satisfy

Extend $f\in C_ c(\Omega )$ by zero to $\mathbb C^ n$. Outside the projection of $\operatorname {supp}f$ to the $z'$ variables, both $f(r_ t\mathrm e^{\mathrm i\vartheta },z')$ and $f(0,z')$ vanish. On this projection, the uniform continuity of $f$ and the bound on the radii make their difference tend to zero uniformly. Since the weights $w_{\eta _ j}(t)\, \mathrm dt$ and $\mathrm d\vartheta /(2\pi )$ and the measures $(\mathrm{dd}^{\mathrm{c}}H_{\delta _ j})^ m$ all have total mass one, it follows that

The second integral tends to $f(0)$, which proves the asserted vague limit. $\square$