Open Problems in Pluripotential Theory — Question 16

Definitions and mathematical statement

Let $n\geq 1$ and let $\Omega\subset{\mathbb{C}}^n$ be a bounded pseudoconvex domain. Let $\mu$ be a positive Borel measure on $\Omega$ with $\mu(\Omega)=1$. A function is plurisubharmonic (psh) if it is upper semicontinuous and its restriction to each complex line is subharmonic wherever defined. Use the convention $dd^c=(i/\pi)\partial\bar\partial$; for locally bounded psh functions, $(dd^c u)^n$ denotes the complex Monge--Ampère measure in the Bedford--Taylor sense.

For a precise global interpretation of Hölder continuity, require a continuous extension to $\overline\Omega$ and constants $C<\infty$ and $0<\alpha\leq 1$ such that \[ |u(z)-u(w)|\leq C\lVert z-w\rVert^\alpha \qquad(z,w\in\overline\Omega). \] The boundary condition means that this extension vanishes on $\partial\Omega$. The supplied source says Hölder continuous in $\Omega$, without explicitly specifying whether the estimate is global or merely local; the global interpretation used here therefore requires confirmation.

Question 16 (attributed to Zeriahi). Suppose there exists a Hölder continuous psh function $u$ with zero boundary values such that \[ \mu\leq(dd^c u)^n\quad\text{in }\Omega, \] where domination means $\mu(E)\leq(dd^c u)^n(E)$ for every Borel set $E\subset\Omega$. Must there exist a Hölder continuous psh function $\varphi$, also extending continuously with zero values on $\partial\Omega$, such that \[ (dd^c\varphi)^n=\mu\quad\text{in }\Omega? \] The Hölder exponent and constant for $\varphi$ are existential and need not equal those for $u$; the source prescribes no relation between them. No boundary smoothness or stronger geometric assumption on $\Omega$ is supplied.

Short English statement

Given a probability measure on a bounded pseudoconvex domain, does a Hölder continuous plurisubharmonic subsolution with zero boundary values guarantee a Hölder continuous plurisubharmonic solution of the complex Monge--Ampère equation with the same boundary values? The precise scope of Hölder continuity requires confirmation. The supplied registry labels this question open; that status has not been independently verified.

Sources

• Local UnsolvedMath registry, permanent ID $800885$, source problem code catalogue entry, title: Open Problems in Pluripotential Theory --- Question 16. The supplied record attributes the collection to Dinew, Guedj, and Zeriahi (2015), and the individual question to Zeriahi.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 16. Supplied source URL: https://arxiv.org/abs/1511.00705. The provenance reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access or source checking.