Theorem 16.28.4609FE648CAD9A

Proof. Set $B=B(a,R)$ and $E=K_ r$. For a competitor $u$ for $h_{E,B}$, the function

is plurisubharmonic on $B$, because the assumption on $\mathrm{dd}^{\mathrm{c}}\Psi $ makes $\Psi -c\lVert z-a\rVert ^2$ plurisubharmonic there. Since $u\leq 0$, we have $H_ u\leq \Psi $ where $\lVert z-a\rVert \geq r$. Define $W=\max (\Psi ,H_ u)$ on $B$ and $W=\Psi $ outside $B$. The two definitions agree near $\partial B$, so $W$ is plurisubharmonic on $\mathbb C^ n$, and it belongs to $\mathcal L_\tau $ because it equals $\Psi $ outside $B$. On $E$, the equality $u=-1$ gives $H_ u\leq \Psi $, and on $K\setminus E$ the distance bound gives the same inequality. Thus $W=q$ on $K$, so $W$ is a competitor for $F_{\delta ,K,q}$ for each $\delta \geq \tau $. Since $W\geq H_ u$ near $a$, taking the supremum over $u$, regularizing, and using the continuity of $\Psi $, we obtain (16.28.47).

The function $\Psi $ itself is a competitor, so raw contact holds for every $\delta \geq \tau $. If regular contact holds for one such $\delta $, the left-hand side of the estimate vanishes, so $h_{K_ r,B(a,R)}^*(a)=-1$ for all $0<r<R$. By lemma 16.28.43, every $K_ r$ with $r<R$ is then non-pluripolar and satisfies $V_{K_ r}^*(a)=0$; for $r\geq R$ this follows by monotonicity. Conversely, local pluriregularity and lemma 16.28.45 give regular contact at $a$ for every $\delta \geq \tau $. $\square$