Example 15.21.4DF2D43B90C47

Example 15.21.4.DF2D43B90C47 For every $n\geq 1$, the trace criterion applies to a bounded continuous solution whose global density belongs to $L^1$ but to no $L^ p$ with $p>1$, and which is not Hölder continuous at the points of $X\setminus \Omega $.

Let $E=\mathbb C/(\mathbb Z+\mathrm{i}\mathbb Z)$, $X=E^ n$, and

Take $\mathrm{d}V=\theta ^ n$ and $\rho =0$. Choose a nonnegative smooth cutoff $\chi $ supported in a coordinate disk $|z|<r_0<1$ of $E$ and equal to one near zero. For $\beta >0$, define

and extend by zero. On the smaller punctured disk put $t=|z|^2$ and $L=-\log t$. Differentiation gives

The first expression is integrable and the second is the radial flux. Integration by parts on punctured disks, using also $\psi \to 0$, shows that the distributional Hessian has no atom at zero. On the cutoff annulus its negative part is bounded. Hence, for a sufficiently small fixed $\eta >0$, $u=\eta \psi (z_1)$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \theta /2$.

Only one coordinate contributes to $\mathrm{dd}^{\mathrm{c}}u$. Locally approximate the bounded subharmonic potential in that coordinate from above and use theorem 5.27.1; the smooth diagonal determinant identity passes to the limit and yields

Give $f$ any positive value on the null set $D$. It is smooth and positive on $\Omega =X\setminus D$, where $u$ is smooth and has smooth traces on all relatively compact coordinate balls. Near $D$ its singular term is comparable to $|z_1|^{-2}(-\log |z_1|^2)^{-\beta -2}$. Its radial $L^1$ integral converges, whereas

Also $(-\log |z|^2)^{-\beta }/|z|^\alpha \to +\infty $ for every $\alpha >0$, so $u$ is not Hölder at $D$. Finally, boundedness gives minimal ordinary singularities, since every global $\theta $-plurisubharmonic competitor is bounded above.