Theorem 15.21.125FC2412C22E9

Proof. Set $R=1/8$ and choose $M$ so that $P_0=P+M$ satisfies $0\leq P_0\leq L$ on $B_ R\Subset U\times W$ for some finite $L$. Choose $C>0$ with

and put $Q(Z)=C(|Z|^2-R^2/4)$ for $Z=(z,w)$. Let $\chi $ be the convolution of $|t|$ with an even nonnegative smooth probability kernel supported in $(-1,1)$. Then $\chi $ is smooth and convex, $|\chi '|\leq 1$, $\chi '(0)=0$, and $\chi (t)=|t|$ for $|t|\geq 1$. The function

is bounded and plurisubharmonic, because $m$ is convex and nondecreasing in each variable. The collar inequalities imply

Indeed, $Q\leq -2\leq P_0-2$ on the first ball and $Q>L+2\geq P_0+2$ on the collar. Define

The collar identity makes $u$ identically zero near $\partial B_ R$ and gives a global bounded potential with $\theta +\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}v$ on the ball and $\theta +\mathrm{dd}^{\mathrm{c}}u=\theta $ outside. Thus $u$ is admissible and has minimal ordinary singularities.

Outside the pluripolar set $(S\times W)\cap B_ R$, set $\lambda =(1+\chi '(P_0-Q))/2$. Differentiating the regularized maximum gives

Both $H_ P$ and $CI$ are positive definite there; the final term is positive semidefinite. Hence the classical density is positive. Bounded products charge no pluripolar sets, so there is no further measure on the exceptional set. Their mass on compact $X$ is finite by proposition 5.28.2. This proves $\mathrm{MA}_\theta (u)=F\theta ^2$ for an almost-everywhere positive $F\in L^1$. On $\Omega =B_{R/4}$, (15.21.9) gives the representative $F=C^{-2}$. There $u=P+M-Q$ is discontinuous at $(0,w_0)$ for $0<|w_0|<R/4$.

We prove failure of every global $L^ p$ directly in the transition region. For large $k$, set

Then $w_ k>0$, $P_0(Z_ k)=Q(Z_ k)=m_ k$, and (15.21.13) gives $|Z_ k|^2<9R^2/16$. Near the isolated pole $a_ k$,

where $H_ k$ is smooth and harmonic. Consequently $b$ and $P_0$ are continuous near $Z_ k$. There is a product neighborhood $V_ k\Subset B_ R$ of $Z_ k$, with $|w|$ bounded below by a positive number, on which $1/4\leq \lambda \leq 3/4$.

At smooth points of $V_ k$, positivity and the two-dimensional determinant expansion give

where $c_ k>0$ may depend on $k$ and $C$. Here $P_{z\bar z}=b^{-1}+|w|^2b_{z\bar z}$. With $\zeta =z-a_ k$, shrink the first factor of $V_ k$ so that

for some $c'_ k>0$. Since $F=C^{-2}\det H_ v$, this lower bound implies $F\notin L^ p$ whenever $p(1-\alpha _ k)\geq 1$: the $w$-disk has positive area and the radial integral in $z$ contains

For each $p>1$ such a $k$ exists because $\alpha _ k\downarrow 0$. $\square$