Open Problems in Pluripotential Theory — Question 23

Definitions and mathematical statement

Let $n\geq 1$ be an integer and let $\Omega\subset{\mathbb{C}}^n$ be a bounded domain with smooth, strictly pseudoconvex boundary. Strict pseudoconvexity means that a smooth local defining function has positive Levi form on every nonzero complex tangent vector at the boundary. A function is plurisubharmonic if it is upper semicontinuous and its restriction to every complex line is subharmonic wherever defined.

Let $u$ be plurisubharmonic on $\Omega$, and assume \[ u\in\mathcal C^{1,\alpha}(\Omega),\qquad \alpha>1-\frac{2}{n}. \] In the usual range $0<\alpha\leq 1$, the Hölder condition means that the first derivatives are locally Hölder continuous with exponent $\alpha$. The source does not explicitly specify the admissible range of $\alpha$, or whether its Hölder assumption requires a finite global norm on $\Omega$.

Suppose that there exist $f\in\mathcal C^\infty(\Omega)$ and a constant $\varepsilon>0$ such that \[ (dd^c u)^n=f\,dV\quad\text{on }\Omega, \qquad f(z)\geq\varepsilon\quad\text{for every }z\in\Omega, \] where $dV$ is Euclidean volume measure. Since $u$ is locally bounded and plurisubharmonic, the left side is interpreted as its complex Monge--Ampère measure. The normalization of $d^c$ is not specified in the supplied source and must be used consistently with the density $f$.

Question [Blocki-Dinew]. Under these assumptions, must \[ u\in\mathcal C^\infty(\Omega)? \] The conclusion concerns interior smoothness. No boundary values, extension to $\overline\Omega$, or smoothness up to the boundary are specified.

Short English statement

Must a plurisubharmonic solution of the complex Monge--Ampère equation on a smooth bounded strictly pseudoconvex domain be smooth in the interior if its density is smooth and uniformly positive and its first derivatives have Hölder exponent $\alpha>1-2/n$? The source leaves the precise Hölder-space convention unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800892$, source problem code catalogue entry, entitled Open Problems in Pluripotential Theory --- Question 23. The record lists Dinew, Guedj, and Zeriahi as proposers, with proposed year 2015; the question itself carries the attribution [Blocki-Dinew].
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 23. Supplied source URL: https://arxiv.org/abs/1511.00705. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are source-record claims, not independent access or verification performed for this draft.
• The registry labels this problem open. This is supplied status information, not an independently verified assessment of the literature.