16.28.42 Local regularity and strict weights5323E094C77D

Proof. In the pluripolar case, choose $b\in \mathcal L_1$ with $b=-\infty $ on $E$ by lemma 16.28.12, and a finite upper bound $C$ for $b$ on $B$. The functions

are competitors for $h_{E,B}$, and they tend to zero almost everywhere as $\epsilon \downarrow 0$. Hence $h_{E,B}^*\geq 0$ almost everywhere, and then everywhere by lemma 5.24.3. Since $h_{E,B}^*\leq 0$, it vanishes.

Now suppose that $E$ is non-pluripolar. Proposition 16.28.10 gives a finite upper bound $M\geq 1$ for $V_ E^*$ on $B$. For each competitor $v$ for $V_ E$, the function $\max (-1,v/M-1)$ is a competitor for $h_{E,B}$. Taking suprema and regularizing gives

As $V_ E^*\geq 0$, the equality $h_{E,B}^*(a)=-1$ forces $V_ E^*(a)=0$.

For the converse, choose $r<R_0<R$ and $k>0$ with $k\log (R_0/r)>1$. The plurisubharmonic function

belongs to $\mathcal L_ k$ and equals $-\infty $ at $a$. For a competitor $u$ for $h_{E,B}$, define $W=\max (u+1,L)$ on $B$ and $W=L$ outside $B$. On the collar $R_0<\lVert z-a\rVert <R$ we have $L>1\geq u+1$, so the two definitions agree there, and $W$ is plurisubharmonic by lemma 5.24.2. On $E$ one has $u=-1$ and $L\leq 0$, so $W=0$; on the ball $B(a,r)$ one has $L<0\leq u+1$, so $W=u+1$. Hence $W/k$ is a competitor for $V_ E$, with $k$ independent of $u$. Taking the supremum over $u$ and regularizing at $a$ gives

Thus $V_ E^*(a)=0$ forces $h_{E,B}^*(a)=-1$. $\square$

Proof. Put $M_ r=\sup _{K_ r}q$. Raw contact, monotonicity of envelopes with respect to the compact set, and (16.28.17) for $K_ r$ give

Since $q$ is upper semicontinuous and $a\in K_ r$, we have $M_ r\downarrow q(a)$ as $r\downarrow 0$, which gives $F_{\delta ,K,q}(a)=q(a)$.

Every compact subset $E\subseteq K$ inherits raw contact: on $E$, the raw envelope for $E$ is at least the raw envelope for $K$, which equals $q$, and at most the obstacle $q$. For fixed $s>0$, apply the same bounds with $r<s$ and $K_ s$ in place of $K$, and let $r\downarrow 0$. This proves local $(\delta ,q)$-regularity. By theorem 16.28.14, raw contact also holds for every larger parameter, and applying what has been proved to each of them gives the last assertion. $\square$

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 $. $\square$

Proof. The complex Hessian of $\Psi $ is semipositive. If its top power did not vanish at $a$, it would be positive definite at $a$, and by continuity $\mathrm{dd}^{\mathrm{c}}\Psi \geq c\omega _0$ on a neighborhood of $a$ for some $c>0$. Theorem 16.28.46 would then give regular contact at $a$ for every $\delta \geq \tau $, a contradiction. $\square$