Singular solutions to the complex Monge-Ampère equation II

Continuing our work in [Math. Ann. 392 (2025), 1503-1518], we construct new singular solutions to the complex Monge-Ampère equation with constant right-hand side in the Bedford-Taylor sense. Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$. For $n\ge 2$, these solutions include: (i) a bounded solution on ${\mathbb{D}}^n$ that is discontinuous almost everywhere and nowhere classically differentiable; (ii) a continuous exhaustion of ${\mathbb{C}}^n$ that does not belong to $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2,1}({\mathbb{C}}^n)$; (iii) a continuous solution on $\overline{{\mathbb{D}}^n}$ whose Laplacian has a singular part. We construct these solutions from singular harmonic functions using partial Legendre transforms and holomorphic pullbacks.